Observational Signatures and Effects of Magnetic Fields in Astrophysical Systems

Home , Abell 2163, Abell 2256, Abell 400, Abell 665, Galactic ridge, Lucie Green

Observational signatures and eﬀects of magnetic ﬁelds in astrophysical systems

Alvina Yee Lian On

Thesis submitted for the degree of Doctor of Philosophy of University College London

Mullard Space Science Laboratory Department of Space and Climate Physics University College London

May 2020 2

Declaration of originality

I, Alvina Yee Lian On, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. In Chapters3 and4, I collaborated with Jennifer Chan on the polarised radiative transfer calculations, where I focused on the galaxy cluster scales, and Jennifer on the cosmological aspect. David Barnes kindly provided the galaxy cluster data obtained from his GCMHD+ simulations for my analysis in Chapter 4. In Chapter5, I collaborated with Ellis Owen on the ionisation and heating effects of cosmic rays in molecular clouds, where I focused on the effects caused by the presence of magnetic fields, and Ellis on the microscopic interactions between cosmic ray particles and the medium. Some parts of this thesis were published/submitted in the following articles: (i) Alvina Y. L. On, Jennifer Y. H. Chan, Kinwah Wu, Curtis J. Saxton and Lidia van Driel-Gesztelyi, “Polarised radiative transfer, rotation measure fluctuations and large-scale magnetic fields”, December 2019, MNRAS 490 (2): 1697–1713

(ii) Jennifer Y. H. Chan, Kinwah Wu, Alvina Y. L. On, David J. Barnes, Jason D. McEwen and Thomas D. Kitching, “Covariant polarized radiative transfer on cosmological scales for investigating large-scale magnetic ﬁeld structures”, April 2019, MNRAS 484 (2): 1427–1455

(iii) David J. Barnes, Alvina Y. L. On, Kinwah Wu and Daisuke Kawata, “SPMHD simulations of structure formation”, May 2018, MNRAS 476 (3): 2890– 2904

(iv) Ellis R. Owen, Alvina Y. L. On, Shih-Ping Lai and Kinwah Wu, “Interac- tions of energetic cosmic rays in molecular clouds: astrophysical implications and Declaration of originality 3 observational signatures”, submitted March 2020 to MNRAS

Signed: Date: 4

Abstract

Magnetic fields are ubiquitous, permeating across all scales from interstellar space to voids. Their origins and evolution still remain as open questions. On galactic scales and beyond, Faraday rotation measure (RM) at radio wavelengths is commonly used to diagnose magnetic fields, and its spatial correlation gives the characteristic length scales of the field variation. This work shows how the RM is derived from the polarised radiative transfer equations under restrictive conditions and assesses the merit of RM fluctuations (RMF) for large-scale magnetic field diagnostics. The interpretation of RMF analyses is ambiguous for an ill-defined characteristic density, such as lognormal-distributed and fractal-like structures. The RMF approach also falls short under radiative absorption, emission, Faraday mixing and the contribution of non-thermal electrons. Notably, correlations along the line-of-sight and across the sky plane are generally dissimilar, therefore the context of RMF must be clarified when inferring from observations. Magnetic fields can also imprint observational signatures in the radio synchrotron emission, whose total intensity reveals the field strength and polarisation traces the field orientation. A point-by-point comparison between the X-ray and radio emissions of a simulated galaxy cluster follows a linear best-fit slope of almost unity, indicating that the magnetic field scales with density locally. On smaller scales, magnetic fields may have an important role during the formation and evolution of molecular clouds. The effects of magnetic fields on the ionisation and heating rates of cosmic rays in IC 5146 are quantified, assuming that the fields are traceable via optical and near-infrared starlight polarisations. While the ionisation rate is fairly constant across the cloud, cosmic-ray heating is capable of raising temperatures by order of 1 K in a Galactic environment, or even higher in actively star-forming regions. This may lead to an increase in the Jeans mass and consequently affect the onset of star formation. 5

Impact statement

The origins and evolution of cosmic magnetism is still an astrophysical puzzle. This research has shifted our current understanding of cosmic magnetic fields. Notably, this work has shown that the conventional diagnostic of large-scale magnetic fields is not universally applicable. As such, it has raised awareness within the scientific community of the validity of assumptions which have been previously overlooked in literature. This consequently motivates the work to develop a more robust theoretical method of radiation transport as a cosmic magnetic field probe. These studies are crucial as a means to probe the magnetic fields in galaxy clusters and beyond, which are particularly timely with the advent of the world’s largest radio telescope, the Square Kilometre Array.

As part of this research, the conventions of polarisation and magnetic field have been clarified. These definitions were previously ambiguous across multi-disciplinary literature, including plasma physics and astrophysics. Furthermore, this research has contributed to our knowledge and understanding of magnetic fields in the intracluster medium and molecular clouds. The work presented in this thesis has been presented at local, national, and international conferences. The research works have also been published and submitted to leading, international, peer-reviewed academic journals. The cumulative academic reach1 of the three publications, from February 2018, are 9 citations, 486 reads, and 232 downloads, as of May 2020. Additionally, this research has established a network of local and international collaborations, across the UK, Taiwan (ROC), USA, Canada, Israel, France, and Hungary.

Beyond academia, this research has stimulated broader public interest and en- gagement through posters, talks, and social media. From the extensive public en- gagement activities undertaken during the duration of this research, it is clear that

1Using the metrics from the NASA’s Astrophysics Data System Impact statement 6 astronomy is a subject that fascinates people of all ages. Parts of this research have been widely communicated through Facebook, VoxCharta, and the UCL-MSSL As- trophysics Blog, where posts on topics including “The invisible world of magnetic ﬁelds” and scientiﬁc experiences in various conferences, have been well-received. Moreover, the opportunity to tutor GCSE and A-Level students, as well as discuss with visitors to the laboratory about research student life, has allowed this work to be disseminated in informal and non-specialist settings. This has ensured that the results of this research are accessible to wider audiences, including school children and prospective BSc/MSc/PhD students. In some cases, this has inspired younger students to pursue higher education in physics and astronomy. Finally, there has been fruitful knowledge and research transfer between astrophysics and industry, by consulting start-ups in renewable energy and technology on big data analytics. 7

Acknowledgements

First of all, I would like to express my deepest gratitude towards Prof Kinwah Wu and Prof Lidia van Driel-Gesztelyi for their unwavering support and encourage- ment throughout my PhD journey. I am incredibly grateful to be working with two supervisors who inspired me to become a better scientist through their passion, knowledge, insight and wisdom. Special thanks to Dr Curtis Saxton for the helpful science discussions, and for patiently teaching me IDL and various statistical techniques. I am appreciative of Dr Wim van Driel for being a wonderful mentor in radio astronomy. I would also like to thank Prof Daisuke Kawata, Prof Mark Cropper and Dr Ignacio Ferreras for their technical help, scientiﬁc discussions and bits of life anecdotes. I thank Prof Lucie Green and Prof Chung-Ming Ko for exam- ining this thesis and for their helpful suggestions. Very special thanks to Mr Kwan Keng Kwek for introducing me to the wonders of physics at Pengiran Jaya Negara Pengiran Haji Abu Bakar Secondary School. I am lucky to have met many amazing people who have assisted me along the way and contributed immensely to my research experience at MSSL. Many thanks to Dr David Barnes, Dr Idunn Jacobsen, Dr Jennifer Chan, Dr Ellis Owen, Dr Kirthika Mohan, Dr Luke Pratley and Brian Yu for their help, discussion and companionship in achieving our PhD degrees together. Special thanks to Dr Jennifer Chan for being my long-time collaborator and good friend; and to Dr Ellis Owen for the fruitful collaboration and his generosity in sharing ideas, techniques and methods in science. I am happy to have had the opportunity to work with and alongside other past and present members of the MSSL Astrophysics group, especially Dr Ziri Younsi, Dr Awat Rahimi, Dr Stefano Pasetto, Dr Robert Grand and Dr Jason Hunt. I would also like to particularly thank Dr Curtis Saxton, Dr Ziri Younsi, Dr Idunn Jacobsen, Dr Ellis Owen, Dr Martin Spinrath, Dr Jennifer Chan, Jane Yap and SiNan Long for proofreading my chapters. Acknowledgements 8

I would like to thank the hospitality of the Institute of Astronomy, National Tsing Hua University (NTHU), ROC (Taiwan), where I spent substantial time as a visitor and wrote parts of this thesis. Special thanks to Prof Albert Kong for kindly hosting and supporting my visits under the Ministry of Science and Technology of the ROC (Taiwan) grant 105-2119-M-007-028-MY3. I have also been supported by the Center for Informatics and Computation in Astronomy (CICA) at NTHU through a grant from the Ministry of Education of the ROC (Taiwan). I am grateful for the opportunity to learn from Prof Albert Kong, Prof Tomotsugu Goto, Dr Martin Spinrath, Prof Shih-Ping Lai, Dr Kuo-Chuan Pan and Dr Hsiang-Yi Yang during my visits. I enjoy the astro-ph sessions and Tomo-san’s group meetings, especially the science discussions and banter with Dr Tetsuya Hashimoto, Dr Seong- Jin Kim, Ting-Wen Wang, Daryl Santos, Chien-Chang Ho, Ting-Yi Lu, Yu-Yang Hsiao and Chih-Teng Ling. Special thanks to Chun-Hao Lee, Sheng-Jun Lin, SiNan Long, Anand Hegde, Travis Thieme, Dr Chris Lau, Jane Yap, Crimson Chan, Dr Shang-Yu Chien, Shu-Wei Yeh, Bin Wang, Hao-Yuan Duan, Hui-Hsuan Chung, Dr Kate Pattle, Dr Yi-Han Wu, Li-Wen Liao and En-Tzu Lin for their friendship, science discussions, fun chats and badminton/yoga sessions. Most of all, to my parents, my sister Emily, and my brother Kevin, thank you for the unconditional love and support in all my pursuits. Lastly, I am grateful for the ﬁnancial support from the Ministry of Education, Brunei Darussalam, without which the opportunity to pursue higher education in the UK, and hence this thesis, would not have been possible. I also thank the Royal Astronomical Society and the UCL Graduate Research Conference Fund for their ﬁnancial support towards conferences and research visits. This thesis has made use of NASA’s Astrophysics Data System. 9

Contents

Page

Declaration of originality2

Abstract4

Impact statement5

Acknowledgements7

List of ﬁgures 13

List of tables 15

1 Magnetic fields in the Universe 17 1.1 Astrophysical magnetic fields ...... 21 1.1.1 Stars and sub-stellar systems ...... 21 1.1.2 Galaxies, groups, interstellar and intergalactic media . . . . . 25 1.1.3 Galaxy clusters and the intracluster medium ...... 25 1.1.4 Large-scale structures: filaments and voids ...... 26 1.2 Origins of large-scale magnetic fields ...... 28 1.2.1 Cosmic inflation ...... 28 1.2.2 Phase transitions in the early Universe ...... 29 1.2.3 Density perturbations before recombination ...... 30 1.2.4 Biermann battery mechanism ...... 31 1.2.5 The first stars and first galaxies ...... 33 1.2.6 Contribution from star-forming and starburst galaxies . . . . 34 1.2.7 Contribution from Active Galactic Nuclei ...... 35 1.3 Evolution of large-scale magnetic fields ...... 35 Contents 10

1.4 Summary ...... 38

2 Observational signatures and diagnostics of magnetic fields 40 2.1 Radio synchrotron radiation ...... 41 2.1.1 Radiation from a single electron ...... 41 2.1.2 Relativistic beaming and spectrum of emission from a single electron ...... 42 2.1.3 Spectrum of emission from a population of relativistic electrons with a power-law energy distribution ...... 43 2.1.4 Synchrotron self-absorption ...... 44 2.2 Polarisation and Faraday rotation ...... 45 2.2.1 Polarisation of a photon ...... 45 2.2.2 Polarisation of an electromagnetic wave ...... 45 2.2.3 Jones vectors and Mueller matrices ...... 45 2.2.4 Stokes parameters and the Poincarésphere ...... 46 2.2.5 Physics of Faraday rotation ...... 49 2.3 Polarised Universe observable in the radio wavelengths ...... 51 2.3.1 VLA and JVLA ...... 51 2.3.2 IRAM 30-m telescope ...... 51 2.3.3 ATCA and the Parkes Observatory ...... 52 2.3.4 VLBA ...... 53 2.3.5 GMRT ...... 53 2.3.6 GBT, Effelsberg 100-m telescope and Jodrell Bank Lovell telescope ...... 54 2.3.7 LOFAR and WSRT ...... 54 2.3.8 ALMA and MWA ...... 55 2.3.9 SKA and its precursors ...... 56 2.4 Other magnetic-field diagnostics ...... 56 2.4.1 Zeeman effect ...... 56 2.4.2 Goldreich-Kylafis effect ...... 59 2.4.3 Interstellar dust ...... 60 2.4.4 High-energy cosmic rays ...... 63 2.4.5 Astronomical transients ...... 65 Contents 11

2.5 Summary ...... 68

3 Rotation measure, fluctuations, and large-scale magnetic fields 70 3.1 Rotation measure ...... 72 3.1.1 In the context of polarised radiative transfer ...... 72 3.1.2 Derivation of rotation measure ...... 74 3.2 Rotation measure fluctuations ...... 75 3.2.1 Computing rotation measure in a discrete lattice ...... 75 3.2.2 Rotation measure fluctuations as a restrictive autoregression (AR) process ...... 76 3.2.3 Remarks on the fluctuations along the line-of-sight and across the sky plane ...... 79 3.3 Solution to the polarised radiative transfer equation ...... 81 3.4 Absorption, emission and Faraday mixing coefficients ...... 82 3.4.1 Thermal plasma ...... 82 3.4.2 Non-thermal plasma ...... 83 3.5 Numerical algorithms and verifications ...... 85 3.6 Summary ...... 90

4 Magnetism in galaxy clusters 92 4.1 Variance of rotation measure fluctuations ...... 94 4.1.1 Assessing the rotation measure fluctuation approach . . . . . 94 4.1.2 Subtleties in the interpretation of magnetic field properties from polarisation data ...... 108 4.2 Cosmological MHD simulations of cluster formation ...... 115 4.2.1 Numerical method ...... 116 4.2.2 Cluster simulations ...... 117 4.2.3 RM and X-ray correlations during the assembly of galaxy clusters ...... 117 4.2.4 Properties of cluster X-ray and radio emissions ...... 135 4.3 Summary ...... 141

5 Cosmic rays in magnetised molecular clouds 143 5.1 Cosmic rays in molecular clouds ...... 146 Contents 12

5.1.1 Molecular clouds ...... 146 5.1.2 Cooling and absorption of cosmic rays ...... 148 5.1.3 Cosmic ray propagation ...... 152 5.1.4 Effects of heating and ionisation ...... 160 5.2 Propagation of cosmic rays in magnetised media ...... 166 5.2.1 Empirical characterisation of cosmic-ray propagation . . . . . 166 5.2.2 Diffusion estimation ...... 168 5.2.3 Observing magnetic fields in molecular clouds ...... 170 5.3 Semi-quantitative analysis: model and results ...... 170 5.3.1 Molecular cloud model ...... 170 5.3.2 Illustrative case ...... 179 5.3.3 Additional remarks ...... 184 5.4 Summary ...... 187

6 Summary and outlook 189

Appendices 193 A Errata in the GCMHD+ analysis code output.f ...... 194 B Numerical scheme for solving the transport equation ...... 195 B.1 Primary protons ...... 195 B.2 Primary electrons ...... 196 B.3 Secondary electrons ...... 196 C Abundance ratios ...... 196 D Angular dispersion function and power spectrum ...... 199 E Spatial dependence of the empirical diﬀusion parameter in IC 5146 . 200

Bibliography 203

Nomenclature 272 13

List of Figures

1.1 The Parker spiral magnetic field ...... 23 1.2 Optical image of M51 overlaid by radio contours and magnetic field vectors ...... 26 1.3 Magnetic field strength from a cosmological simulation ...... 27

2.1 Schematic of the Poincar´esphere ...... 49 2.2 Schematic of the Faraday rotation eﬀect ...... 50 2.3 Images of the polarised intensities and vectors in M33 ...... 61

3.1 Radio synchrotron intensity and polarisation maps for a mock spherical galaxy cluster ...... 89

4.1 Synthetic RM maps of GdGb and UdUb* ...... 99 4.2 Numerical diﬀerences between the CDF of every single-phased model with the reference CDF of GdGb(z)...... 103 4.3 Cross-sections of Cd3 and Cd5 at lines-of-sight x, y and z ...... 106 4.4 Column densities of Cd3 and Cd5 at lines-of-sight x, y and z .... 106 4.5 Synthetic RM maps of Cd2Gb and Cd2Ub* ...... 107 4.6 Numerical diﬀerences between the CDF of every cloudy model with the reference CDF of GdGb(z)...... 113 4.7 Schematic of how higher-redshift structures can contaminate the observed power spectrum ...... 115 4.8 Properties of cluster C01 at various redshifts ...... 123 4.9 Properties of cluster C02 at various redshifts ...... 124 4.10 Properties of cluster C03 at various redshifts ...... 125 4.11 Properties of cluster C04 at various redshifts ...... 126 4.12 Properties of cluster C05 at various redshifts ...... 127 List of Figures 14

4.13 Properties of cluster C06 at various redshifts ...... 128 4.14 Properties of cluster C07 at various redshifts ...... 129 4.15 Properties of cluster C08 at various redshifts ...... 130 4.16 Properties of cluster C09 at various redshifts ...... 131 4.17 Properties of cluster C10 at various redshifts ...... 132 4.18 Properties of cluster HR01 at various redshifts ...... 133 4.19 Properties of cluster HR02 at various redshifts ...... 133

4.20 How the slope in theσ ¯R I relation varies with redshift ...... 134 − X 4.21 X-ray images, radio contours and their point-by-point comparison for a high resolution cluster ...... 137

5.1 Schematic of the different CR propagation regions for an idealised MC149 5.2 Distribution of primary CRs above a GeV propagating into the MC model ...... 173 5.3 Energy densities of primary CRs throughout the cloud under Galactic conditions ...... 174 5.4 Ionisation profiles due to HECR and LECR components ...... 175 5.5 Abundance ratios due to CR ionisation within the MC model . . . . 176 5.6 Specific CR total, Alfvénicand ionisation heating rates in an idealised MC model ...... 177 5.7 Total specific CR heating rates in the MC under increasing ambient CR energy densities ...... 178 5.8 Temperature profiles when CR heating is in equilibrium with gas cooling under Galactic conditions ...... 179 5.9 Dispersion functions in the Rc-, i’-, H-, and K-bands ...... 181 5.10 Equilibrium temperature as a result of only CR heating of a MC in different starburst environments ...... 186

1 Empirical diﬀusion parameters in each region A, B, C and D for each of the four bands ...... 202 15

List of Tables

3.1 Summary of transfer coeﬃcients and emission coeﬃcients through a simulated galaxy cluster with thermal and non-thermal electron contributions ...... 90

4.1 The standard deviations of RM for various models of single-phased, magnetised plasma ...... 101

4.2 The KS statistics D and p-value probabilities of various single-phased, magnetised models ...... 104

4.3 Summary of cloudy model properties ...... 105

4.4 The standard deviations of RM for various models of cloudy, magnetised plasma ...... 110

4.5 The KS statistics D and p-value probabilities of various cloudy, magnetised models ...... 111

4.6 Properties of cluster C01 at various redshifts ...... 118

4.7 Properties of cluster C02 at various redshifts ...... 118

4.8 Properties of cluster C03 at various redshifts ...... 119

4.9 Properties of cluster C04 at various redshifts ...... 119

4.10 Properties of cluster C05 at various redshifts ...... 119

4.11 Properties of cluster C06 at various redshifts ...... 120

4.12 Properties of cluster C07 at various redshifts ...... 120

4.13 Properties of cluster C08 at various redshifts ...... 120

4.14 Properties of cluster C09 at various redshifts ...... 121

4.15 Properties of cluster C10 at various redshifts ...... 121

4.16 Properties of cluster HR01 at various redshifts ...... 121

4.17 Properties of cluster HR02 at various redshifts ...... 122 List of Tables 16

5.1 Summary of CR properties and ﬁlamentary properties in the IC 5146 region ...... 183

1 Summary of rate coeﬃcients for principal formation/destruction reactions of chemical tracers in MCs ...... 198 2 Summary of the RA, Dec, and number of points in each analysis region of IC 5146 ...... 200 17

Chapter 1

Magnetic ﬁelds in the Universe

The Universe is magnetised. Magnetic fields are present in stars, sub-stellar objects (including the Earth, some planets and moons), galaxies (e.g. the Milky Way), and galaxy clusters and groups. They are present also in larger structures beyond galaxy cluster scales, permeating the inter-cluster filaments and cosmological voids (see e.g. Akahori et al. 2018, for a recent review). Magnetic fields are thought to be necessary for stars to form, by decreasing angular momentum of proto-stellar clouds through ambipolar diffusion (see e.g. Beck and Wielebinski 2013). The fields may also influence the dynamics of the interstellar medium (ISM) and mediate star formation by providing additional pressure against gravitational collapse for dense molecular clouds (see e.g. Mouschovias and Ciolek 1999; Beck and Wielebinski 2013). It has been suggested too that magnetic re-connection can heat up the ISM, while its field lines can divert the transport of cosmic rays (CRs) accelerated in supernova remnants (SNRs), thus affecting their density and distribution within the ISM (Fermi 1954). However, magnetic fields are not concrete objects with well- defined boundaries. This makes it challenging for them to be measured directly and their presence can only be inferred through indirect methods.

In 1896, Pieter Zeeman discovered the splitting of spectral lines of a sodium flame in the presence of magnetic fields (Zeeman 1897). This breakthrough was awarded the Nobel Prize in Physics 1902, shared between Pieter Zeeman and his mentor, Hendrik A. Lorentz, for the discovery of the Zeeman effect. More than a decade later, George Ellery Hale used the Zeeman effect to establish that sunspots are magnetic (Hale 1908). This is the first time a non-terrestrial magnetic field was detected. Hale also discovered that sunspots tend to exist in pairs of opposite 18 magnetic polarities, with one polarity leading the other in one hemisphere and vice versa in the other hemisphere. The polarities appear to switch every 11 years, ∼ therefore the period of a solar magnetic cycle is approximately 22 years. This is known as the Hale’s polarity law (Hale et al. 1919; Hale and Nicholson 1925). In the ensuing years, magnetic fields have been detected in the ISM (e.g. Hall 1949; Hiltner 1949), on different types of main-sequence stars (e.g. Babcock 1947; Robinson et al. 1980), white dwarfs (e.g. Kemp et al. 1970; Angel and Landstreet 1974), and neutron stars (e.g. Truemper et al. 1978).

Most planets in the Solar System, including the Earth, also behave like a giant magnet (see e.g. Gilbert and Wright 1600, for the discovery of the Earth’s magne- tosphere). The planetary magnetic fields are either powered by the hydro-magnetic dynamos of their metallic cores (see e.g. Elsasser 1939, for the Earth’s geodynamo theory), or convection of electrically-conducting fluids close to their surfaces (see e.g. Ness et al. 1989, for magnetic fields on Neptune). The strength and direction of the magnetic field around the planets are usually measured using spacecraft mag- netometers. However not all planets have detectable magnetic fields. The magnetic field of Venus is almost non-existent, in part due to its slow rotation and lack of stratified core (see e.g. Stanley 2014, for a review). Some of these planets, e.g. Jupiter and Saturn, also have moons with diverse magnetic properties (see e.g. Jia et al. 2010, for a review).

In 1932, Karl Jansky discovered radio waves emanating from the Milky Way (Jansky 1933), which was later confirmed to be polarised synchrotron radiation (Westerhout et al. 1962; Wielebinski et al. 1962), emitted by relativistic cosmic-ray electrons gyrating in the galactic magnetic fields. Jansky’s pioneering efforts laid the groundwork for the development of radio astronomy to study magnetic fields. Soon, the presence of magnetic fields in nearby galaxies was hinted by observations of starlight polarisation (e.g. M31, Ohman¨ 1942) and observations of radio polarisation (e.g. the Crab Nebula, Cygnus-A, and Centaurus-A, Mayer et al. 1957, 1962; Bracewell et al. 1962, respectively). Beginning the 1970s, several ground-based radio telescopes (e.g. Westerbork, Effelsberg, the Very Large Array (VLA)) became operational and led the way to greatly improve the understanding of magnetic fields in galaxies. Galactic magnetic fields are dynamically important, affecting the formation of spiral arms, as well as the general evolution of galaxies. In particular, the 19 formation of jets and radio lobes in active galaxies can only be comprehended with the existence of magnetic fields (see e.g. Beck and Wielebinski 2013, for a recent review).

Beyond galactic scales, the magnetic fields are relatively weaker and more difficult to be observed. Their large-scale properties are often inferred from the observational signatures in their radio synchrotron emission and Faraday rotation measure (RM) (e.g. Govoni and Feretti 2004; see e.g. Carilli and Taylor 2002; Vacca et al. 2018, for reviews). Galaxy clusters, galaxy groups and the intracluster medium (ICM) are known to be magnetised at several to tens of µG (e.g. Vacca et al. 2010; Bonafede et al. 2010; Feretti et al. 2012) through the detection of diffuse, Mpc-scale radio emission across the ICM (see e.g. Feretti et al. 2012; van Weeren et al. 2019, for reviews) and the Faraday RMs of bright polarised radio sources inside or behind the galaxy clusters (e.g. Clarke et al. 2001; Clarke 2004; Govoni et al. 2010; Bonafede et al. 2013). The cluster volume is a patchwork of magnetic fields, typically coherent on kpc length scales but disordered on the larger scales, which are likely attributed to cluster interactions and mergers (e.g. Clarke 2004; Stasyszyn and de los Rios 2019; Dom´ınguez-Fernándezet al. 2019). These dynamical processes wipe out the memory of the first magnetic fields. It is still uncertain whether the seed fields are primordial in origin or generated through astrophysical processes after the first stars and the first galaxies form.

The origin(s) of cosmic magnetism remains an open question. The other big science questions which are as important are: how are the magnetic fields maintained in present-day astrophysical systems, how can the large-scale fields be properly diagnosed, and how can their evolutionary history be inferred (see e.g. West et al. 2019). The dynamo amplification and evolution of large-scale magnetic fields with structure formation of the Universe requires the existence of a non-zero initial seed field. In galaxies, it is natural to consider supernova remnants (SNRs) as a source which distributes stellar magnetic fields over large ISM volumes (Rees 1987) and drives turbulent dynamo actions (Balsara et al. 2004). Suppose that a plerion (centre-filled) supernova remnant has a typical field strength of B 10−3 G and a ∼ length scale of r 1 pc. Given that turbulent diffusivity is D 103 pc km s−1, ∼ t ∼ it would take approximately 106 years for SNRs to magnetise a galaxy (Zweibel 2006). As the galaxy forms, the magnetic field can be amplified up to B µG rms ∼ 20 by the turbulent small-scale dynamo, which efficiently converts kinetic energy into magnetic energy (e.g. Schober et al. 2013; Brandenburg and Subramanian 2005; Balsara and Kim 2005). Since 106 years is much shorter than the Hubble time, ∼ SNRs are likely able to seed magnetic fields in galaxies.

On larger scales, however, it becomes increasingly difficult for SNRs to pollute the intergalactic medium (IGM) with magnetic fields. For a rich galaxy cluster with a typical size of about r 1 Mpc and a turbulent diffusivity of D 107 pc km ∼ t ∼ s−1, the diffusion timescale for stellar fields to spread beyond galaxies is estimated to be about 1011 years, which far exceeds the age of the Universe (see Zweibel 2006). Hence this cannot explain the presence of weak magnetic fields in clusters and larger-scale structures such as cosmic filaments and voids (e.g. Vernstrom et al. 2017). It is argued that cosmological filaments and voids may have been magnetised during cosmic inflation or by some exotic mechanisms in the early Universe. Alter- natively, they could have been magnetised by cosmic rays from isolated star-forming galaxies within the voids or void-bordering AGN (see e.g. Rees 1987; Widrow 2002; Kulsrud and Zweibel 2008; Kandus et al. 2011; Widrow et al. 2012; Katz et al. 2019, for overviews on the origins of cosmic magnetic fields). Nevertheless, the earliest evidence of cosmological magnetogenesis should be imprinted within the cosmic web.

Future magnetic field observations with the Square Kilometre Array (SKA) including its pathfinders, the Low Frequency Array (LOFAR), the Murchison Wide- field Array (MWA), the Expanded Very Large Array (EVLA), and its precursors, the Australian SKA Pathfinder (ASKAP) and MeerKAT, would be able to constrain fundamental physics on cosmic magnetism (see e.g. Gaensler et al. 2010; Beck 2015; Johnston-Hollitt et al. 2015). The dense all-sky maps of radio synchrotron polarisation and Faraday rotation measures would shed new light on the magnetic fields in the Milky Way, intervening galaxies and media, galaxy clusters and groups, cosmic filaments and voids. It is therefore important to properly characterise magnetic fields beyond the scale of galaxy clusters in both theoretical and observational astrophysics.

This thesis is organised as follows. Chapter1 presents an introduction to the magnetic fields in the Universe, with an overview of astrophysical magnetic fields, and the origins and evolution of large-scale magnetic fields. Chapter2 discusses the observational signatures and diagnostics of magnetic fields, with a focus on the 1.1. Astrophysical magnetic fields 21 physics of radio synchrotron radiation, polarisation and Faraday rotation. Faraday RM at radio wavelengths is commonly used to diagnose magnetic fields, and its spatial correlation gives the characteristic length scales of the field variation. Chapter3 shows how the RM is derived from the polarised radiative transfer equations under restrictive conditions, and presents the theoretical background and solutions to the polarised radiative transfer equations. Chapter4 assesses the merit of RM fluctuations (RMF) for large-scale magnetic field diagnostics, and presents point-by-point correlation analyses between the X-ray and radio emissions of galaxy clusters from the cosmological magneto-hydrodynamic (MHD) simulations. On smaller scales, magnetic fields may also be important during the formation and evolution of molecular clouds (MCs). The effects of magnetic fields on the ionisation and heating rates of cosmic rays in MCs are quantified and discussed in Chapter5, assuming that the fields are traceable via optical and near-infrared starlight polarisation. Fi- nally, Chapter6 summarises the main results of this thesis, along with some remarks on the extension of these works. In this chapter, Section 1.1 provides an overview of the current knowledge on astrophysical magnetic fields. Sections 1.2 and 1.3 present and discuss the origins and evolution of large-scale magnetic fields, respectively. Lastly, Section 1.4 summarises this chapter. Unless otherwise stated, this thesis uses c.g.s. Gaussian units.

1.1 Astrophysical magnetic ﬁelds

1.1.1 Stars and sub-stellar systems

Magnetic fields are present in stars. They play a role at nearly all stages of stellar evolution. Magnetic fields are thought to be essential for the onset of star formation, by providing additional pressure against gravitational collapse of dense cores and clumps within molecular clouds (Mouschovias and Ciolek 1999). The magnetic field lines can also guide cosmic ray transport as they are accelerated in SNRs, therefore affecting their density and distribution within the ISM (Fermi 1954). Observa- tionally, Zeeman splitting is often used to probe solar and stellar (sometimes also proto-stellar) magnetic fields (Landi Degl’Innocenti 2003). In essence, the magnetic field splits the atomic energy levels into more than one, which causes a spectral line to become a line multiplex, each with slightly different frequencies. The amount of frequency splitting is proportional to the magnetic field strength (see Section 2.4 for 1.1. Astrophysical magnetic fields 22 more details on the Zeeman effect).

In general, stellar magnetic field strengths can extend from several µG to TG (e.g. Donati and Landstreet 2009). For solar-like stars, the magnetic fields are thought to be generated in and at the bottom of the convective envelope (Parker 1993). Subsequently, the field is amplified by the turbulent, electrically-conducting plasma which acts like a dynamo (Parker 1955). The fields are carried from the convection zone base towards the surface via buoyancy. As the magnetic fields are transported, they are stretched and sheared by the fluid motion, which in turn leads to an increase in the magnetic energy. The stellar field topology varies greatly, from being mainly poloidal to mainly toroidal, and from having almost axisymmetric dipoles to complicated structures with no obvious symmetry (Donati and Landstreet 2009). The complex field topology is largely attributed to the physics and dynamics of the star. For e.g., in the solar case, the omega-effect, also known as differential rotation, converts poloidal fields to toroidal fields and amplifies them during the process. From the toroidal layer at the bottom of the convection zone, strong flux tubes become buoyant and rise to the surface. When the flux loop breaks through the photosphere, a pair of sunspots is created. On the other hand, the alpha-effect due to the Coriolis force transforms the toroidal fields back to poloidal, but in the opposite direction (see e.g. Bushby and Mason 2004, for a review on the solar dynamo).

After amplification, the solar magnetic field strength is typically 10 103G. ∼ − Solar magnetism is often inferred from the flaring and eruptive activities at the surface of the Sun. Sunspots are located where very strong magnetic lines of force break through the solar surface. They usually come in pairs with opposite polarities and appear dark because they are cooler than the rest of the solar surface. This is the result of intense magnetic fields in sunspots, hindering heat convection to the solar surface. Meanwhile, solar flares are sudden outbursts of energy which occur in the solar atmosphere. They are thought to be due to magnetic reconnection, a process in which oppositely-oriented magnetic field lines break and reconnect in a plasma, therefore converting magnetic energy into plasma kinetic and thermal energies (see e.g. Parnell 2000). Often, the plasma is being confined to coronal loops. Coronal loops connect opposite polarity magnetic field concentrations. Prominences are observed to be suspended above the solar surface and are believed to be supported 1.1. Astrophysical magnetic fields 23

Fig. 1.1: The yellow arrows depict the Parker spiral magnetic ﬁeld. As the Sun rotates, the solar wind creates a neutral current sheet (in blue) which is a thin layer between the opposing ﬁeld directions. As the magnetic and rotation axes are not identical, the sheet gets wound up into a “ballerina skirt”. Image courtesy of J. Jokipii (University of Arizona).

by and threaded by magnetic fields. Coronal loops and prominences exhibit the frozen-in condition (Alfvén 1942), where the plasma and magnetic fields appear to move together. This is because the solar plasma has a very high electric conductivity, such that the magnetic Reynolds number1 is very much larger than 1, indicating that magnetic advection dominates over magnetic diffusion. Furthermore, the magnetic field of the Sun extends into the heliosphere, being carried by the expanding solar corona which, due to its high temperature and consequent high conductivity, cannot be in a hydrostatic equilibrium (Parker 1958). The solar wind, along with the rotation of the Sun, results in a wavy spiral shape of the heliospheric current sheet, which is also known as the “Parker spiral” (see Figure 1.1).

Most planets and some moons in the Solar System also behave like a giant magnet (see e.g. Stevenson 1983, 2003; Jones 2011, for reviews). The Earth has a planetary dipole, whose magnetic ﬁeld lines travel from the magnetic North to South pole. Similarly, Mercury and Jupiter are also known to have dipolar magnetic ﬁelds (e.g. Christensen 2006). On the other hand, Saturn, Uranus and Neptune exhibit

1The magnetic Reynolds number in the intracluster medium is also large, typically of the order 107 (e.g. Ruzmaikin et al. 1989). On the other hand, across a star-forming molecular cloud, the magnetic Reynolds number can vary substantially from the order of unity or less (e.g. Li and Houde 2008) to the order 104 (e.g. Draine 2011). 1.1. Astrophysical magnetic fields 24 magnetic multipoles (e.g. Connerney 1993). The multipoles on Uranus and Neptune are rather irregular and much weaker than those found on Saturn. The reason for this may be that, unlike the Earth whose magnetic field is generated by the hydromagnetic dynamos of its molten core, the magnetic fields on Uranus and Neptune are likely to be generated closer to their surfaces where icy salts or water lie (see e.g. Stanley and Bloxham 2006). Intriguingly, Mars does not have a dynamo-generated field but is found to have a magnetised surface, which indicates that it must have been magnetised before (Acuna et al. 1999). Venus is the only planet that appears to be non-magnetised as it lacks the magnetic material and currents (see e.g. Stanley 2014, for a review). Some of these planets, e.g. Jupiter and Saturn, have moons with various magnetic properties, which were possibly created during meteorite impacts (see e.g. Jia et al. 2010, for a review).

As stars in the main sequence approach the final stage of evolution, they become either white dwarfs, neutron stars or black holes. Low mass stars end up as white dwarfs, typically with r 109 cm and B 105 107 G (see e.g. Chanmugam ∼ ∼ − 1992). High mass stars, on the other hand, evolve into neutron stars, typically with r 106 cm and B 1011 1013 G. These degenerate stars have strong magnetic ∼ ∼ − field strengths because as stars collapse, their surface area and volume reduce, thus increasing their field strength, following the conservation of magnetic flux approximated by Br2 (Woltjer 1964). Observationally, the magnetic fields of neutron stars are inferred from either the changes in their pulsar radio emission (e.g. Nagase et al. 1991), or cyclotron resonant scattering in their X-ray spectra (e.g. Santangelo et al. 1999; Staubert et al. 2019).

Neutron stars are also born spinning very rapidly due to the conservation of angular momentum. A special characteristic of spinning neutron stars is the release of “pulses” of radio synchrotron and X-ray radiation. This can be explained by the acceleration of charged particles near the magnetic poles, which may not be aligned with the stellar rotation axis. More specifically, an observer sees beams of radiation whenever the magnetic pole is aligned to the line-of-sight. Hence the pulses are periodic and neutron stars with such behaviour are also known as pulsars. The strongest magnetic field in nature exceeds 1012 G, and is found on a type of neutron star known as the magnetar. Magnetars are believed to be the sources of high-energy electromagnetic (EM) emission in the form of X-rays and gamma 1.1. Astrophysical magnetic fields 25 rays (Thompson and Duncan 1993). They are speculated to have formed during a supernova when a star collapses into a mass of neutrons with exceptionally high gravity and a very strong magnetic field.

1.1.2 Galaxies, groups, interstellar and intergalactic media

Magnetic fields permeate the interstellar medium (ISM). They influence the ISM dynamics such as the gas flow in spiral arms, disks and galaxy halos. They contribute to the pressure within the ISM against gravity and are essential for the onset of star formation. Through magnetic reconnection, they may heat up the ISM and distribute energy within the medium through magnetohydrodynamic (MHD) turbulence. They also divert cosmic rays and affect their distribution in the ISM (Beck 2009a). According to numerical simulations (see e.g. Govoni and Feretti 2004; Ryu et al. 2008) and observational results (see e.g. Carilli and Taylor 2002; Clarke 2004), the typical strength of galactic magnetic fields is of the order of 10−6 G. Radio emissions reveal that the magnitude of the total magnetic field in the Milky Way is about 6 µG (see e.g. Beck 2001), similar to the field strength of neighbouring radio- faint galaxies such as M31 and M33. The first observation of a coherent magnetic field traversing the Magellanic Bridge has been made by measuring the Faraday rotation towards 167 polarised extragalactic radio sources with the Australia Tele- scope Compact Array (ATCA) (Kaczmarek et al. 2017). Gas-rich galaxies such as M51 (see Figure 1.2) have magnetic field strengths of about 20 30 µG in their − spiral arms (e.g. Beck 2016). The magnetic fields follow along the arms and are strongest at the inner edges of the arms and in between them (see e.g. Beck 2015). Starburst galaxies have the strongest galactic magnetic fields, about 50 100 µG − (e.g. Chy˙zyand Beck 2004), which could be due to their exceptionally high rate of star formation. For example, the star formation of M82 is ten times the rate of the Milky Way.

1.1.3 Galaxy clusters and the intracluster medium

The detection of diffuse radio sources in galaxy clusters on Mpc-scales indicates that magnetic fields are likely to be present within the galaxy cluster, permeating throughout the intracluster and intergalactic media (see e.g. Ferrari et al. 2008; Feretti et al. 2012, for reviews). Clarke(2004) observed that the RM of Abell clusters are of the order 100 200 rad m−2, hence implying that the magnetic ∼ − 1.1. Astrophysical magnetic fields 26

Fig. 1.2: An optical image of M51 from the Hubble Space Telescope overlaid by total radio intensity contours from the VLA and the Eﬀelsberg telescope at 6-cm wavelength. The magnetic ﬁeld vectors of polarised emission are also shown. Image courtesy of A. Fletcher & R. Beck (MPIfR Bonn).

fields are of the order of 5 10 µG. Meanwhile, Bonafede et al.(2010) derived the ∼ − magnetic field strengths and radial profiles of 7 sources in the Coma cluster through numerical simulations that best reproduce the observations. They found that the magnetic field is best represented by a Kolmogorov power spectrum and the central field strength to lie between 3.9 µG and 5.4 µG. Their simulated RMs suggest that, in general, the standard deviation of RM, σR, and the mean of RM, , decrease hRi outwards from the cluster centre.

1.1.4 Large-scale structures: ﬁlaments and voids

According to the Lambda Cold Dark Matter (ΛCDM) model, large-scale structures (LSS) are formed from the hierarchical clustering of objects (e.g. Widrow et al. 2012). The early Universe sees the manifestation of gravitational instabilities, which causes linear density perturbations. Small-scale objects are created ﬁrst, then they collapse and merge into objects of ever-increasing size. This process is thought to 1.1. Astrophysical magnetic ﬁelds 27

Fig. 1.3: Projected (mass-weighted) magnetic ﬁeld strength at z = 0 across a (50 Mpc)3 cosmological simulation (Vazza et al. 2014). The colours represent the ﬁeld strength between 10 nG 0.1 µG. ∼ −

create a cosmic web of structure, in which large number of galaxies reside in filaments, forming a network that is separated by voids and dominated by clusters at the intersections (see e.g. Bond et al. 1996). Filaments consist of millions of gravitationally-bound galaxies, which form massive, thread-like structures (see Fig- ure 1.3) with a typical length of a few hundred Mpc (Kronberg 2010). They form boundaries between large voids in the Universe and are believed to be permeated by weak magnetic fields. Understanding the nature of large-scale magnetic fields in filaments and voids is challenging as observations outside of clusters are limited. Numerical simulations (e.g. Ryu et al. 1998; Ryu et al. 2008; Akahori and Ryu 2010; Vazza et al. 2014), radio observations (of e.g. nearby superclusters, Xu et al. 2006), and a combination of both (e.g. Brown et al. 2017) indicate that the field strengths are of the order of nG and the root-mean-square (rms) of the RM is predicted to ∼ be 1 rad m−2. Recent observational evidence of Mpc-scale diffuse radio emission ∼ from cosmic filaments (Giovannini et al. 2010, 2015) constrains the magnetic field strength to be B < 1 µG(Govoni et al. 2019). 1.2. Origins of large-scale magnetic fields 28

1.2 Origins of large-scale magnetic ﬁelds

It is still unclear whether the first magnetic fields were generated via some exotic mechanisms in the early Universe or through some astrophysical processes after the recombination era (see e.g. Widrow 2002; Kulsrud and Zweibel 2008; Subramanian 2016, for reviews). The processes that may have generated the first seed magnetic fields are as follows.

1.2.1 Cosmic inﬂation

Magnetic fields may have first appeared in the very early Universe. When the Universe was merely a tiny fraction of a second old, it underwent a brief period of accelerated expansion known as inflation (e.g. Guth 1981; Sato 1981). It is widely accepted that the initial density perturbations generated at the time of inflation could give rise to form large-scale structures. Therefore, it is natural to question whether the large-scale magnetic fields observed today could also be seeded from the quantum fluctuations in the electromagnetic fields during inflation (Turner and Widrow 1988). The inflationary expansion is also capable of stretching the wave modes exponentially to generate coherent magnetic fields on the very large scales of galaxies and beyond. Furthermore, the rapid expansion dilutes the pre-existing charge densities, which substantially decreases the conductivity of the Universe, and therefore generates primeval magnetic fields from a zero seed field (Turner and Widrow 1988; see e.g. Subramanian 2019, for a recent review).

An open issue with this theory lies upon the fact that electromagnetism is conformally coupled to gravity in a flat Universe. An expanding Universe in the Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) framework has its energy density proportional to a−4, where a is the cosmological scale factor. The inflationary theory creates an initial magnetic field strength of B 10−50 G on co-moving length scales ∼ of order 10 kpc, which is too low to be of astrophysical or cosmological interest (Widrow et al. 2012). In order to grow seed fields super adiabatically during inflation such that they become physically signficant, conformal invariance of the Maxwell’s equations and/or conformal flatness of the Friedmann spacetimes must be broken. Turner and Widrow(1988) suggested that the addition of terms to the Lagrangian breaks conformal invariance, and in turn, constrains the magnetic field to act as a minimally-coupled scalar field. Alternatively, the EM field can be coupled to other 1.2. Origins of large-scale magnetic fields 29

fields in string-inspired cosmologies, such as the inflaton, as pointed out by Ratra (1992); or the dilaton, which is discussed in Gasperini and Veneziano(1994). Other possible models of inflation are reviewed and discussed by Martin and Yokoyama (2008) and Subramanian(2010). In essence, all of these models involve the breaking of invariance in various “open” cosmologies, so that inflation is able to seed magnetic fields up to present- −15 −9 day strength of 10 G . B . 10 G in large-scale structures (see e.g. Beck et al. 2013a). It is plausible for magnetic fields with an inflationary origin to penetrate into and magnetise cosmological voids as well as seed the dynamos of other gravitationally bound systems. For inflationary magnetogenesis to be feasible, new physics beyond the Standard Model would be necessary to break the conformal and gauge invariances. In view of this, void magnetic fields provide an interesting window into understanding the theories of inflation and the science that extends beyond the Standard Model.

1.2.2 Phase transitions in the early Universe

Primordial magnetic fields could also be generated during various phase transitions in the early Universe. It may be possible that a tiny fraction of the free energy released during electroweak or quantum-chromo-dynamics (QCD) transitions can be transformed into large-scale magnetic fields (see e.g. Kandus et al. 2011; Durrer and Neronov 2013; Subramanian 2016, for reviews). The electroweak transition occurs when the electromagnetic and weak forces become distinguishable: from the time in which the photon, W and Z bosons are identical and massless, to the point when the W and Z bosons acquire mass, but the photon remains massless (Coleman and Wein- berg 1973). In the case of this being a first-order phase transition, bubble nucleation can take place, creating non-equilibrium conditions well-suited for baryogenesis and leptogenesis, thus leading to magnetic field generation (Quashnock et al. 1989; see e.g. Widrow et al. 2012, for a review). The idea is such that the bubbles grow and percolate, driving turbulence within the charged plasma, which therefore induces currents and subsequently amplifies magnetic fields through a dynamo process (see e.g. Brandenburg and Subramanian 2005, for a review). Magnetogenesis is also possible during the QCD transition that takes place when the temperature of the Universe cools below T 150 MeV (see e.g. Kahni- ∼ ashvili et al. 2013; Bazavov et al. 2014). The Universe changes from a quark-gluon 1.2. Origins of large-scale magnetic fields 30 plasma to a hadronic phase under the strong nuclear force. Here the bubbles collide and the electric fields behind their shock fronts generate current flow, and in turn, create small-scale magnetic fields (see e.g. Subramanian 2016, for a review). Phe- nomenologically, the co-moving magnetic field strength on a 1 Mpc-scale generated during the QCD transition is estimated to be about B 1 10−29 G, whilst during ∼ × the electroweak transition is approximately B 1 10−23 G (see e.g. Widrow et al. ∼ × 2012, for a review). Hence, strong fields can indeed arise from post-inflationary phase transitions, but their correlation length scales are limited by the Hubble radius in the early Universe. The coherence length of the primordial magnetic fields generated by phase transitions is expected to be of the order of the largest bubble size formed, which is 0.01 pc for the electroweak transition and 10 pc for the ∼ ∼ QCD transition (see e.g. Banerjee and Jedamzik 2004). These lengths are too short, as compared to the kpc scales observed in galaxies and galaxy clusters today. Unless some dynamical mechanisms occur, such as the inverse cascade of magnetic energy, to generate helicity which can grow the scales non-linearly (see e.g. Field and Carroll 2000; Banerjee and Jedamzik 2004; Tevzadze et al. 2012).

1.2.3 Density perturbations before recombination

Prior to the epoch of recombination, the particle-photon interactions can trigger tiny fluctuations in the cosmological density field, which eventually creates a seed magnetic field (Ichiki et al. 2006; Maeda et al. 2011). The notion is that photons, electrons and protons are efficiently scattered and tightly coupled. The anisotropic pressure of photons and their preferential scattering off electrons over protons lead to differences in the rotational velocities of the charged particles, yielding an electric current and hence magnetic field. Although the expansion of the Universe quickly damps away the first-order linear perturbation modes, the second-order couplings between photons and electrons are strong enough to magnetise the early Universe. Since density perturbations of photons and electric currents are linked, the magnetic fields produced via this coupling process must be correlated with temperature fluctuations at recombination. The generation of magnetic fields takes into account three main factors: the baryon-photon slip, vorticity difference and anisotropic pressure. By adopting the standard cosmological model (i.e. ΛCDM), the density perturbations of these parameters are numerically solved to derive the power spectrum of the magnetic field 1.2. Origins of large-scale magnetic fields 31

(Widrow 2002; Ichiki et al. 2006). It is shown that magnetic fields of 10−18.1 G can be produced on a 1 Mpc-scale and they can be even stronger on smaller scales of 10 kpc, at 10−14.1 G(Ichiki and Takahashi 2006; Ichiki et al. 2006). These fields then decay adiabatically with the expanding Universe to present-day strengths of about B 10−22.8 G at 1 Mpc-scale and 10−18.8 G at 10 kpc-scale, which are capable of ∼ ∼ seeding galactic magnetic fields and playing a role in early star formation processes (Takahashi et al. 2005). When recombination ends, so does the generation of magnetic fields. This is because Compton scattering is no longer viable as the majority of the electrons have combined into hydrogen atoms. As a result, the magnetic field produced this way is essentially a “fingerprint” of primordial density perturbations. The detection of such magnetic fields may be able to shed some light on the physics of the early Universe.

1.2.4 Biermann battery mechanism

The equation of motion for electrons in a ﬂuid can be expressed as

dv v B n m e = n e E + e × P n m g + F , (1.1) e e dt − e c − ∇ e − e e e,i

where ne is the electron number density, me is the electron mass, Pe (= neκBTe) is 2 the electron pressure , e is the electron charge, c is the speed of light, and Fe,i is the electron-ion frictional force (see e.g. Kulsrud and Zweibel 2008). The inertial term neme(dve/dt) and the gravitational term nemeg are negligible since me is small. The force term Fe,i is also negligible because it is related to the current term j, which approaches zero on large scales, since j 1/L. The resulting equation becomes ∝

v B P E = e × ∇ e . (1.2) − c − nee

2Electron pressure can become signiﬁcant when the number density of electrons is large and/or the electron temperature is high. The electron gyroradius is therefore assumed to be much smaller than the length scale of the system. This would result in a small battery term which is often neglected in ideal MHD. 1.2. Origins of large-scale magnetic ﬁelds 32

Taking the curl of equation 1.2 and substituting it into Faraday’s law, ∂B/∂t = c( E) , gives − ∇ × ∂B v B P = c e × + ∇ e (1.3) ∂t ∇ × c nee c Pe = (ve B) + ∇ ∇ × × ∇ × nee ∂B c ( n P ) v B e e = ( e ) + ∇ 2× ∇ . (1.4) ∂t ∇ × × nee

Assuming that the plasma is partially ionised, the ionisation fraction χ can be deﬁned as n n χ = e = e , (1.5) np + nH nB where np is the ionised hydrogen (i.e. proton) number density, nH is the neutral hydrogen number density and nB is the baryonic number density (see e.g. Kulsrud and Zweibel 2008). Here χ is treated as a constant in space.

Then, assuming thermal equilibrium within the plasma, i.e. Te = Tgas, the ratios between the electron pressure Pe = neκBTe and the total gas pressure P =

(np + nH + ne) κBTgas, can be written as

P n e = e (1.6) P np + nH + ne n = e (1.7) nB + ne n = e (1.8) nB (1 + χ) P χ e = . (1.9) P 1 + χ

The total plasma density ρ can be expressed as ρ = mpnp + mene + mHnH, where mp is the proton mass and mH is the neutral hydrogen mass. Since the electron mass is small, i.e. m m , and the plasma can be considered as locally p e charge neutral, i.e. ne = np, the plasma density is therefore

Mn ρ m n + m n = e , (1.10) ≈ p p H H χ where M is the average mass per particle (see e.g. Xu et al. 2008). By substituting equations 1.9 and 1.10, equation 1.4 becomes

∂B Mc ρ P = (v B) + ∇ × ∇ . (1.11) ∂t ∇ × e × e (1 + χ) ρ2 1.2. Origins of large-scale magnetic ﬁelds 33

In the case of a hydrogen gas,

∂B m c ρ P = (v B) + p ∇ × ∇ . (1.12) ∂t ∇ × e × e(1 + χ) ρ2

The first term in equation 1.12 is the advection of the magnetic flux and shows that a small seed field can be amplified by shear flows and compression in the plasma. The second term is the Biermann(1950) battery mechanism. It arises due to pressure and density gradients, which give rise to vorticity and create a flow of electric current. This, in turn, generates a weak seed magnetic field. 1.2.5 The first stars and first galaxies

The first stars and first galaxies are plausible source candidates of seed magnetic fields. The seed field can be generated during the star formation process, as a result of disk accretion and proto-stellar gravitational collapse (see e.g. Machida et al. 2007), which can then be further enhanced by magneto-rotational instabilities (see e.g. Silk and Langer 2006; Widrow et al. 2012). The gas within the first star-forming halos is turbulent (see e.g. Abel et al. 2002; O’Shea and Norman 2007; Yoshida et al. 2008; Turk et al. 2009) and drives a small-scale dynamo, which exponentially increases weak seed fields, as shown in semi-analytical and numerical studies (see e.g. Schleicher et al. 2010; Sur et al. 2010). Assuming an initial field strength of 10−9 G, Sur et al.(2010) carried out simulations where the field was evolved ∼ with the central collapsing core during the first star formation. They found that the total magnetic field was amplified by six orders of magnitude, resulting in a 10−3 ∼ G field strength, which is strong enough to seed dynamos in galaxies and in the IGM. The stars can release material into the interstellar medium, via stellar winds and supernova explosions, which subsequently carry magnetic fields throughout the entire proto-galaxy (e.g. Michel and Yahil 1973; Chy˙zy et al. 2016). An ensemble of small-scale stellar magnetic fields subsequently produces a large- scale component which seeds a mean-field galactic dynamo (see e.g. Widrow 2002). Galactic magnetic fields typically comprise of nearly similar random and regular field components. The random, tangled fields are thought to be driven by the turbulent flow motions and active star formation, whilst the regular fields are made coherent by the large-scale dynamo and galactic rotation (see e.g. Beck 2016). Moreover, it is possible to transport the magnetic fields from the disk into the halo by means of a galactic, magnetised wind. This is supported by observations of similar scale heights 1.2. Origins of large-scale magnetic fields 34 of several spiral galaxies, which suggested that all of them host a galactic wind (Krause 2015), as also demonstrated by numerical simulations (see e.g. Brandenburg et al. 1993; Moss et al. 2010). It is also proposed that galactic winds may be able to carry the magnetic fields into the IGM and regions where larger structures reside. In order to explore the likelihood of galactic winds being a source of magnetic fields in the ICM, Don- nert et al.(2009) coupled the galaxies created in their cosmological MHD simulations (GADGET-2) with a semi-analytic model for magnetised galactic winds following Bertone et al.(2006). Their results show that the magnetic field amplifies as the cluster forms, well reproducing the observed magnetic field strength and structure in galaxy clusters at present time.

1.2.6 Contribution from star-forming and starburst galaxies

Most galaxies in the early Universe are believed to have experienced one or more periods of star formation. The galactic centres are usually sites of active star formation and µG magnetic fields (see e.g. Beck 2012, for a review). During episodes of star formation, the galaxies can efficiently drive magnetised metal-enriched hot gas, as well as cosmic rays, from their central regions (e.g. Klein 2011). The relativistic ejecta are often in the form of galactic bubbles, winds, and/or outflows, which can quickly transport metals and magnetic fields from the stellar nurseries to seed a substantial fraction of the IGM (e.g. Chy˙zyet al. 2016). A fraction of these galaxies are gas-rich and have an extraordinarily high rate of star formation, relative to their mean star formation rate in the long term or the star formation rate of other galaxies. These are known as starburst galaxies, and they can form new, massive stars, which strongly ionise the surrounding clouds and oftentimes exploding into supernovae, thus triggering a cascade of star formation throughout the region. The supernova remnants inject cosmic rays and magnetic seed fields into the IGM, where the latter gets amplified by the cosmic-ray driven dynamo (e.g. Hanasz et al. 2004, 2013; Siejkowski et al. 2018). The starburst activities often occur in abrupt, short-lived phases, as they end when the available gas reservoir is either fully consumed to form stars or blown out of the galaxy. Over the lifetime of each galaxy, and the combined action of many starburst events, it is possible to magnetise a significant volume of the surrounding IGM in the present day Universe (e.g. Kronberg 2016). Notably, dwarf galaxy 1.3. Evolution of large-scale magnetic fields 35 starburst outflows of magnetised gas is most effective in seeding the IGM at z & 7 (Kronberg et al. 1999).

1.2.7 Contribution from Active Galactic Nuclei

Daly and Loeb(1990) proposed Active Galactic Nuclei (AGN) as a possible origin of large-scale fields. A supermassive black hole at the centre of the active galaxy accretes mass to form a disk, which produces a strong seed magnetic field via the Biermann battery mechanism. These fields are amplified by the galactic dynamo and then ejected into the surrounding IGM by relativistic jets and winds of the AGN (see e.g. Kronberg et al. 2001). Assuming equipartition between magnetic and kinetic energies within a 1 Mpc3 volume, the required seed field strength for AGN contribution is estimated to be B 10 nG (Ryu et al. 2012). ∼ Further works have been carried out to explore the effects of a population of AGN jets and its volume filling fraction to magnetise the Universe. Using the cosmological AMR-MHD simulation of galaxy cluster formation (ENZO), Xu et al.(2010) found that a single, powerful radio jet at z = 3 is capable of magnetising the entire ICM to the observed µG levels, after undergoing cluster-wide turbulent amplification and diffusion. The timing of the AGN injection is crucial in this scenario: to obtain cluster-wide magnetic fields, the injection must occur at a sufficiently high redshift, otherwise the resulting fields are only confined within radio lobes and isolated bubbles. On the contrary, there appears to be no significant effect on the final magnetic energy and final magnetic field strength as the number of AGN injections increases. Xu et al.(2012) injected magnetic fields from 30 simulated AGN at z = 2.5 and found that the cluster, although filled to a larger volume, was still magnetised to µG levels. This is because the final magnetic energy and final magnetic field strength in the ICM are mainly determined by the amplification with the ICM turbulent energy (Xu et al. 2010). Hence, the magnetic growth saturates at µG levels, in spite of additional AGN injections spreading the magnetic fields wider to fill a larger volume of the cluster.

1.3 Evolution of large-scale magnetic ﬁelds

Various observations (see e.g. Govoni et al. 2001a,b; Carilli and Taylor 2002; Gov- oni and Feretti 2004; Govoni et al. 2004; Bonafede et al. 2010; Ryu et al. 2012) have indicated that galaxies and clusters are magnetised, with field strengths in the µG 1.3. Evolution of large-scale magnetic fields 36 range and ordered typically on scales of 1 10 kpc for galaxies and 0.1 1 ∼ − ∼ − Mpc for clusters. Some superclusters (e.g. Virgo, Hercules and Perseus-Pisces) also appear to be permeated by large-scale magnetic fields with an upper limit of several 10−7 G, assuming field reversal scales of hundreds of kpc (Xu et al. 2006). Cos- mological MHD simulations (see e.g. Brüggenet al. 2005; Ryu et al. 2008; Vazza et al. 2014) predict the magnetisation of cosmic filaments to reach strengths of a few nG, lower than the upper limit of 0.1 µG constrained by RM observations ∼ ∼ of background quasars3 (see e.g. Ryu et al. 1998; Xu et al. 2006; Ryu et al. 2012). There has also been evidence supporting the presence of magnetic fields in cosmic voids. The Large Area Telescope (LAT) on the Fermi Gamma-ray Space Telescope (Fermi-LAT) observations of distant TeV blazars4 place lower limits on the intergalactic magnetic field strength at B 3 10−16 G if assuming a constant ≥ × TeV flux emitted over long timescales (Neronov and Vovk 2010), while a more −18 robust limit of B & 10 G is obtained from restricting the active TeV emission to a shorter period (Dermer et al. 2011). Independent constraints on the properties of intergalactic magnetic fields are also possible with gamma-ray observations of GRBs, the first of which is GRB 190114C, limiting B > 10−19.5 G for a coherence length of l 1 Mpc (Wang et al. 2020b, see also Section 2.4 for more details). Whether ≤ or not this level can be detected by the Cherenkov Telescope Array (CTA) is still uncertain (Dzhatdoev et al. 2020). Seeding of these large-scale magnetic fields can either be attributed to astrophysical or primordial processes. Whilst the gas collapses during structure formation and galactic interactions, it experiences turbulence, compression and shear, all of which amplifies, stretches, twists and folds the seed field. Hence, it is not only crucial to trace the origin(s) of the large-scale fields, but also understand how they evolve with the structure formation of the Universe. The magnetic field evolution is determined by the induction equation: ∂B = c ( E) . (1.13) ∂t − × ∇ × In the context of an astrophysical plasma, it is expressed as ∂B m c ρ P = (v B) + η 2B + p ∇ × ∇ . (1.14) ∂t ∇ × × ∇ e(1 + χ) ρ2

3A quasar is a rapidly accreting supermassive black hole at the centre of a massive galaxy.

4A blazar is an AGN with a relativistic jet almost directed towards an observer. 1.3. Evolution of large-scale magnetic ﬁelds 37

The first term (v B) amplifies the magnetic fields due to the vorticity of ∇ × × plasma motions. It is of the form

(v B) = (v )B + (B )v B( v) , (1.15) ∇ × × − · ∇ · ∇ − ∇ · such that the term (v )B amplifies the fields by advection, (B )v by shearing, · ∇ · ∇ and B( v) by compression. The second term η 2B describes how the field ∇ · ∇ diffuses through the plasma, where η is the magnetic diffusivity. This term is usually neglected in astrophysical settings because the ICM and IGM are generally good conductors. Lastly, the third term is the battery term, which accounts for the seeding of magnetic fields from baroclinity. This means that the magnetic fields can be generated from the misalignment of pressure and density gradients, e.g. during cluster mergers and shocks. Neglecting the vorticity and battery terms, the time for the magnetic field to decay is

L2 t = , (1.16) d η where L is the characteristic length scale for the diﬀusion to occur. For clusters, where L 1 Mpc, the decay time is estimated to be many orders of magnitude larger ∼ than the Hubble time, implying that turbulent motions must play an important role in the system. In a co-moving frame, the magnetic ﬁelds evolve with the expanding Universe, following

∂B 1 a˙ 1 m c ρ P v B B p = [ ( )] 2 + ∇ ×2 ∇2 , (1.17) ∂t a ∇ × × − a BN tN e(1 + χ) a ρ where BN and tN are the normalisation constants for the magnetic field and time, respectively (e.g. Kulsrud et al. 1997; Chan and Wu 2014, private communication). The cosmological scale factor a quantifies the expansion of the Universe. The expansion rate of the Universe is characterised by the first Friedmann equation:

a˙ 2 Ω Ω Ω = H2 r,0 + m,0 + k,0 + Ω , (1.18) a 0 a4 a3 a2 Λ where Ωr is the energy density of radiation, Ωm is the energy density of matter, Ωk is the curvature, and ΩΛ is the energy density of dark energy. Observations of the cosmic microwave background (CMB) indicate that the Universe is flat so Ω 0. k ∼ The subscript “0” denotes present-day values of the cosmological parameters. The 1.4. Summary 38 scale factor is also related to the redshift by a = 1/(1 + z), and to the physical distance by arco, where rco is the co-moving distance. The term 2(˙a/a)B in equation 1.17 dilutes the field strength as the Universe − expands. In the case of a slow-moving, barotropic fluid, equation 1.17 simplifies to

∂B a˙ = 2 B , (1.19) ∂t − a which leads to ∂ a2B = 0 . (1.20) ∂t The above expression is simply flux conservation. Since the magnetic field strength B declines with a2 as the Universe expands, there has to be some mechanism to compensate for this field reduction. It is argued that there could be a galactic dynamo capable of amplifying the seed fields to the µG strengths observed in galaxies and galaxy clusters today (e.g. Moss et al. 1991; Gressel et al. 2008; see also Bran- denburg and Subramanian 2005; Brandenburg 2015, for reviews on galactic dynamo theory). To this end, the vorticity term has to dominate, through e.g. structure formation, cluster mergers and shocks. The seed fields must have a coherence length scale longer than 10 kpc and strengths ranging between 10−22 and 10−12 G (see e.g. Kandus et al. 2011, for a review on primordial magnetogenesis).

1.4 Summary

This chapter provides an overview of magnetic fields across all scales from interstellar space to cosmological voids. Magnetic fields are omnipresent and hence they must play an important role in shaping the Universe. They are thought to contribute significantly, amongst gravity, turbulence, and cosmic rays, to the evolution of the interstellar medium, and particularly to the formation of stars in molecular clouds (see Chapter5). Stellar, planetary, and lunar magnetic fields are likely to have arisen from the hydro-magnetic dynamos of their liquid metallic cores or convection of electrically-conducting fluids close to their surfaces. Their field structures are observed to be dynamically complex, and their field strengths can vary over many orders of magnitude, e.g. from 10 103 G in the Sun to 1015 G in magnetars. The − ∼ conventional methods to trace stellar and sub-stellar magnetic fields are starlight and dust polarisations, as well as Zeeman and Goldrich-Kylafis spectropolarimetry. Cyclotron resonant scattering features are also a direct observational evidence for 1.4. Summary 39 strongly-magnetised neutron stars. On galactic scales and beyond, it becomes more challenging to detect and study magnetic fields. The properties of magnetic fields in these large-scale systems are often inferred from observational signatures in their radio synchrotron emission and Faraday RMs. Notably, magnetism in galaxies, galaxy groups, and the intergalactic medium, have been constrained by radio observations to be of the order of several µG, and even stronger in starburst galaxies. There is also observational evidence for galaxy clusters and their intracluster medium to be magnetised at several to tens of µG, through the detection of diffuse, Mpc-scale radio emission across the ICM and the RMs of bright radio point sources at 100s rad m2. The cluster volume ∼ is a patchwork of magnetic fields which are typically coherent on kpc length scales but more disordered on the large scales, presumably due to cluster interactions and mergers. These dynamical processes smear out the memory of the first magnetic fields. The Biermann battery amplification and evolution of large-scale magnetic fields with structure formation of the Universe requires the presence of a non-zero seed field. It is still unclear whether the seed fields are primordial in origin or generated through astrophysical processes after the recombination era. The earliest evidence of cosmological magnetogenesis may be imprinted in the largest scale structures of the Universe, such as the cosmic filaments and voids, which are weakly magnetised at 10−16 nG levels. The advent of the all-sky SKA RM survey will provide ∼ − solid constraints on when and how the first magnetic fields are generated. It is therefore important to study the observational signatures that magnetic fields will leave in the radio wavebands (see Chapter2) and assess the large-scale magnetic field diagnostics in the RM fluctuations for galaxy cluster scales and beyond (see Chapters3 and4). 40

Chapter 2

Observational signatures and diagnostics of magnetic ﬁelds

Most of what is known about magnetic fields in galaxy clusters and larger scale structures comes from radio observations. The intracluster medium (ICM) is weakly magnetised and consists of relativistic electrons. The relativistic electrons gyrate around the magnetic field lines to produce synchrotron emission. Therefore radio synchrotron radiation is often used as a probe of large-scale magnetic fields. It is important to understand the physics of radio synchrotron radiation, especially its polarisation and effects under Faraday rotation, which are commonly used to characterise the magnetic fields. Observationally, many radio experiments have contributed immensely to the understanding of magnetic fields in astrophysical systems. Meanwhile, observations in the other wavebands, such as optical and near-infrared, or even using high-energy particles, e.g. cosmic rays, have proven to be useful in diagnosing magnetic fields on the smaller scales, e.g. in molecular clouds. This chapter therefore focuses on the observational signatures and diagnostics of magnetic fields.

This chapter is organised as follows. Section 2.1 presents an overview on the physics of radio synchrotron radiation, starting with the radiation from a single electron and a description of its relativistic beaming and spectral properties. The spectrum from a power-law distribution of electrons and the physics of synchrotron self-absorption are also discussed. Section 2.2 describes the polarisation of light and the physics of Faraday rotation. In particular, the polarisation of a photon and the polarisation of an electromagnetic wave are presented using the formalism of Jones vectors, Mueller matrices, and the Stokes parameters. Section 2.3 outlines 2.1. Radio synchrotron radiation 41 the polarised Universe observable in the radio wavelengths and how it improves the understanding of astrophysical magnetic fields. In Section 2.4, other magnetic field diagnostics, such as the Zeeman effect and high-energy cosmic rays, are discussed. Lastly, Section 2.5 summarises this chapter.

2.1 Radio synchrotron radiation

Charged particles gyrate around magnetic fields and radiate. In the non-relativistic case, the frequency of emission is the same as the frequency of gyration, which is known as cyclotron radiation. In the highly relativistic regime, the frequency of emission goes beyond the frequency of gyration and is subject to beaming effects. This is known as synchrotron radiation. The total intensity provides a direct estimate of the equipartition field strength, whereas the polarisation reveals the field orientation and degree of field ordering. For more details, see e.g. Rybicki and Lightman(1979) and Ulrich and Fletcher(2015). 2.1.1 Radiation from a single electron

When an electron travels in a magnetic ﬁeld B, it experiences a Lorentz force and accelerates, following dv e γm = (v B) , (2.1) e dt −c × where e is the electron charge, me is the electron mass, c is the speed of light, v − is the electron velocity, and γ = 1 v2/c2 1/2 is the Lorentz factor. It can be − shown that the electron velocity component parallel to the magnetic ﬁeld, vk, obeys

dvk = 0 , (2.2) dt implying that vk is invariant. The rate of change of electron velocity component

(i.e. electron acceleration) perpendicular to the magnetic ﬁeld, v⊥, is

dv⊥ e = (v⊥ B) . (2.3) dt −γmec ×

As v is constant and v is invariant, v⊥ is also constant. Therefore, the electron | | k | | gyrates in a uniform, helical motion along B at an angular frequency

e B ωB = | || | , (2.4) γmec and it accelerates in the direction perpendicular to B with a magnitude a⊥ = 2 ω v⊥ = ω v⊥ (as only e is relevant to the radiative processes, without losing B| | B 2.1. Radio synchrotron radiation 42 generality, the negative sign of the electron charge e is dropped hereafter). As a result, the total radiation emitted for a single electron is

2 P = r2 c β2 γ2 B2 , (2.5) 3 e ⊥

2 2 where β⊥ = v⊥/c, and re = e /mec is the classical radius of the electron. For an isotropic distribution of velocities, the total power is the mean over all angles at a given speed β,1 so equation 2.5 becomes

4 P = σ c β2 γ2 U , (2.6) 3 T B

2 2 where σT = 8πre /3 is the Thomson cross section and UB = B /8π is the magnetic energy density.

2.1.2 Relativistic beaming and spectrum of emission from a single electron

Due to relativistic boosting, synchrotron emission is strongly beamed in the direction of motion. The emission concentrates in narrow angles of the order 1/γ along the velocity direction. The beaming of radiation has a significant effect on the observed synchrotron spectrum. As the electron traverses around the magnetic field, its emission can only be seen when aligned with the observer’s line-of-sight. The observer sees a pulse of radiation whose period is much shorter than the gyration period by a factor of γ3. The synchrotron spectrum is fairly broad, extending to the order of its critical frequency

3 ω = γ3 ω sin α , (2.7) c 2 B which depends on the Lorentz factor and the magnetic ﬁeld strength perpendicular to the line-of-sight. The power of each synchrotron pulse is

√3 e3 B sin α ν P (ν) = 2 F , (2.8) mec νc where νc = ωc/2π. The function F is the Airy integral of the modiﬁed Bessel function K5/3(ξ) and can be written in the form of

ν ν Z ∞ F = K5/3(ξ) d(ξ) . (2.9) νc νc ν/νc

1 R 4π 2 Note that 0 dΩ sin θ = 2/3. 2.1. Radio synchrotron radiation 43

Hence, a single-electron power spectrum varies asymptotically, following

P (ν) ν1/3 for ν/ν 1 and (2.10) ∝ c P (ν) e−ν/νc for ν/ν 1 , (2.11) ∝ c with a peak emission occurring at νmax = 0.29 νc. 2.1.3 Spectrum of emission from a population of relativistic electrons with a power-law energy distribution

The synchrotron radiation from an ensemble of relativistic electrons with a power- law energy distribution generally follows a power-law spectrum

P (ν) ν−α , (2.12) ∝ characterised by a spectral index α. The relation between the spectral power-law index and the electron energy distribution power-law index can be derived as follows.

Consider the number density of relativistic electrons with energies between γ1 and γ2 to be n (γ) dγ = γ−p dγ , (2.13) re C where is a normalisation constant and p is the electron distribution index. The C total power radiated per unit volume by this distribution of electrons is therefore

Z γ2 Z γ2 −p ν −p Ptot(ν) = P (ν) γ dγ F γ dγ . (2.14) C γ1 ∝ γ1 νc Since ν γ2, equation 2.14 can be re-written as c ∝ Z x2 −(p−1)/2 (p−3)/2 Ptot(ν) ν F (x) x dx , (2.15) ∝ x1 where x ν/ν . Over a large range of energies, such that x 0 and x , the ≡ c 1 ≈ 2 ≈ ∞ integral term becomes a constant. Hence,

P (ν) ν−(p−1)/2 . (2.16) tot ∝ Comparing equations 2.12 and 2.16 gives p 1 α = − , (2.17) 2 where p 2.2 for freshly-accelerated electrons (by, e.g. supernova remnants and ≈ cosmic rays), whereas p 3.0 4.0 for aged electrons (e.g. radio jets in X-ray bina- ∼ − ries) (see e.g. Ginzburg and Syrovatskii 1968; Achterberg et al. 2001; Tsouros and Kylaﬁs 2017). 2.1. Radio synchrotron radiation 44

2.1.4 Synchrotron self-absorption

Not all of the synchrotron radiation emitted can reach the observer because of synchrotron self-absorption. At lower frequencies, more photons are likely to be scattered oﬀ and absorbed by electrons within the source. This results in an optically thick region, where the photon mean free path is much smaller than the characteristic size of the source. The average energy per particle E for a relativistic gas is

E = 3kBTeﬀ , (2.18)

2 where kB is the Boltzmann constant and Teﬀ is the eﬀective temperature of the synchrotron-emitting electrons.3 Therefore,

2 γmec Teﬀ = , (2.19) 3kB which, at a given γ, the maximum synchrotron emission occurs at ν γ2ν , so ≈ c 2 1/2 mec ν Teﬀ , (2.20) ≈ 3kB νc and since ν B, this becomes c ∝

T ν1/2B−1/2 . (2.21) eﬀ ∝

Recall that the surface brightness Sν of a blackbody is given by the Planck function 2hν3 1 Sν (Te) = , (2.22) c2 ehν/kBTe 1 − where Te is the blackbody temperature. At radio wavelengths, the low-frequency limit (i.e. hν k T ) applies, leading to the Rayleigh-Jeans law (1905): B e 2ν2k T S (T ) = B e , (2.23) ν e c2

(Rayleigh 1900, 1905a,b; Jeans 1905b,c,a). Lastly, substituting equation 2.21 into 2.23 gives S ν5/2B−1/2 . (2.24) ν ∝

2Although synchrotron emission is non-thermal, an eﬀective temperature depending on frequency can still be assigned to the electrons.

3 This leads to a speciﬁc heat ratio of cP/cV = 4/3. 2.2. Polarisation and Faraday rotation 45

This shows that, at low frequencies, severe self-absorption causes a turnover in the synchrotron spectrum. At low frequencies, the region is said to be optically thick when the synchrotron emission is proportional to ν5/2. At high frequencies, the synchrotron emission is proportional to ν−(p−1)/2 and the region is said to be optically thin (see e.g. Rybicki and Lightman 1979). Synchrotron self-absorption is a notable feature in the spectra of compact, highly energetic sources, for instance, the nuclei of active galaxies and quasars. Nonetheless, on larger scales (e.g. the IGM), synchrotron self-absorption is relatively low and can often be neglected.

2.2 Polarisation and Faraday rotation 2.2.1 Polarisation of a photon

The polarisation of a photon is described by the orientation of the axis along which its electric ﬁeld oscillates. A photon that is in a superposition of eigenstates can have linear, circular or elliptical polarisation. The quantum polarisation state vector for the photon is identical to the Jones vector, usually used to describe the polarisation of a classical wave. For more details, see e.g. Bohren and Huﬀman(2008). 2.2.2 Polarisation of an electromagnetic wave

Light is an electromagnetic wave. The electric field of light oscillates perpendicular to the direction of propagation. The orientation of the electric field determines the type of polarisation being observed. In the case of linear polarisation, the electric field travels in a single plane along the direction of propagation. When the two orthogonal linear components of the electric field has the same amplitude but are out of phase by π/2, the electric field rotates in a circle, resulting in circular polarisation. Depending upon the direction of rotation as seen by an observer, the polarisation is either deemed left- (anti-clockwise) or right-handed (clockwise). Circular polarisation is a special case of elliptical polarisation, where the electric field describes an ellipse. Elliptical polarisation arises from the superposition of two linear components which have different amplitudes and/or a phase difference that is not π/2. Light is considered unpolarised when the direction of its electric field does not have a preferential direction. 2.2.3 Jones vectors and Mueller matrices

The Jones vector is named after Robert Clark Jones (1941a; 1941; 1941b; 1942) and is only applicable to light that is already fully polarised. Light which is ran- 2.2. Polarisation and Faraday rotation 46 domly polarised, partially polarised, or incoherent, must be treated using Mueller (1943) calculus (see Collett 2005, for more details). Consider a monochromatic plane wave of light to be propagating in the z-direction, with angular frequency, ω, and wavevector, k. The corresponding Jones vectors are   1   (2.25) 0 for linear polarisation in the horizontal direction,

  0   (2.26) 1 for linear polarisation in the vertical direction,

  1 1   (2.27) √2 1 for linear polarisation in +45◦,

  1 1   (2.28) √2 1 − for linear polarisation in 45◦, −

  1 1   (2.29) √2 i − for right-hand circular polarisation,

  1 1   (2.30) √2 +i for left-hand circular polarisation.

2.2.4 Stokes parameters and the Poincar´esphere

Consider an electric vector   −iωt E1 −iωt E = E0 e =   e , (2.31) E2 2.2. Polarisation and Faraday rotation 47

iφ1 where E0 is a general complex vector in terms of amplitudes E1 = ε1e and

iφ2 E2 = ε2e . Taking only the real part of E, the electric ﬁeld in the xy plane is

E = ε cos(ωt φ ) , (2.32) x 1 − 1 E = ε cos(ωt φ ) , (2.33) y 2 − 2 which become

0 Ex = ε0 cos χ cos ωt , (2.34) 0 Ey = ε0 sin χ sin ωt , (2.35) after tilting the xy plane over an angle of ψ, where π/2 χ π/2. Rotating E0 − ≤ ≤ x 0 and Ey through ψ again result in

Ex = ε0(cos χ cos ψ cos ωt + sin χ sin ψ sin ωt) , (2.36)

E = ε (sin χ sin ψ cos ωt sin χ cos ψ sin ωt) . (2.37) y 0 −

Comparing the equations 2.32 gives

ε1 cos φ1 = ε0 cos χ cos ψ , (2.38)

ε1 sin φ1 = ε0 sin χ sin ψ , (2.39)

ε2 cos φ2 = ε0 cos χ sin ψ , (2.40)

ε sin φ = ε sin χ cos ψ . (2.41) 2 2 − 0

The above equations can be condensed into the four Stokes parameters of polarisation [ I, Q, U, V ]. I deﬁnes the total intensity, Q and U correspond to the linearly polarised intensities and V is the circularly polarised intensity. Any polarisation state can be uniquely represented on or within the Poincar´esphere. For monochromatic waves, the Stokes parameters are

I ε2 + ε2 = ε2 , (2.42) ≡ 1 2 0 Q ε2 ε2 = ε2 cos 2χ cos 2ψ , (2.43) ≡ 1 − 2 0 U 2ε ε cos(φ φ ) = ε2 cos 2χ sin 2ψ , (2.44) ≡ 1 2 1 − 2 0 V 2ε ε sin(φ φ ) = ε2 sin 2χ , (2.45) ≡ 1 2 1 − 2 0 2.2. Polarisation and Faraday rotation 48 where 2χ and 2ψ are the azimuthal and polar angles, respectively, as denoted in 2 Figure 2.1. ε0 is the intensity of the beam. Alternatively, the wave has

ε0 = √I, (2.46) V sin 2ψ = , (2.47) I U tan 2χ = (2.48) Q

(Rybicki and Lightman 1979). A fully polarised wave satisﬁes I2 = Q2 + U 2 + V 2, whilst a partially polarised wave follows I2 > Q2 + U 2 + V 2. Meanwhile, for other common polarisation states, the Stokes vectors are   1      1    (2.49)    0    0 for linear polarisation in the horizontal direction,

  1      1   −  (2.50)    0    0 for linear polarisation in the vertical direction,

  1      0    (2.51)    1    0 for linear polarisation in +45◦,

  1      0    (2.52)    1   −  0 for linear polarisation in 45◦, − 2.2. Polarisation and Faraday rotation 49

Fig. 2.1: Schematic of the Poincar´esphere, where P is a point on the sphere such that OP = (Collett 2005). | | 0

  1      0    (2.53)    0    1 for right-hand circular polarisation,

  1      0    (2.54)    0    1 − for left-hand circular polarisation, and

  1      0    (2.55)    0    0 for unpolarised light. Note that the Stokes parameters can convert into one another except between Q and V . The Stokes parameters are often used to represent Faraday rotation. 2.2.5 Physics of Faraday rotation

A linearly polarised wave can be decomposed into two circularly polarised states. The circularly polarised states interact diﬀerently in a magnetised medium and 2.2. Polarisation and Faraday rotation 50

Fig. 2.2: Schematic of the Faraday rotation eﬀect (Silva et al. 2012).

therefore propagate at diﬀerent speeds resulting in a phase shift between them. The recombination of the out-of-phase circularly polarised states into a linearly polarised wave causes a rotation in the polarisation plane. This phenomenon is known as Faraday rotation, named after Michael Faraday (1845 – see e.g. Knudsen 1976, for a review on Faraday rotation; see also Bhat 2002, for a review on the works of Faraday: the father of electromagnetism). The rotation measure (RM, denoted as ) is deﬁned as R

= (∆ϕ)λ−2 = (ϕ ϕ ) λ−2 , (2.56) R − 0 where ϕ is the measured polarisation angle, ϕ0 is the intrinsic polarisation angle at the source, and λ is the radiation wavelength (see Figure 2.2). ϕ0 is usually not known and any polarisation angle ϕ has an nπ ambiguity. In order to resolve these issues, a minimum of three diﬀerent reference wavelengths have to be used to correctly determine the slope of ϕ against λ2, and hence the RM. The RM for radiation traversing a magnetised plasma between an interval s0 and s is

Z s 0 0 0 ds ne,th(s ) Bk(s ) −2 (s) = 0.812 −3 radm (2.57) R s0 pc cm µG (Rybicki and Lightman 1979), if only thermal electrons are present. The thermal electron number density is ne,th and the magnetic field strength along the line- of-sight is Bk (see Chapter3 for derivation). The usual convention for the RM is positive for a magnetic field pointing towards the observer, and negative when pointing away from the observer. Therefore, the RM is sensitive to the regular component of the line-of-sight magnetic field. 2.3. Polarised Universe observable in the radio wavelengths 51

2.3 Polarised Universe observable in the radio wavelengths

As it is impractical to describe every radio telescope, this section provides an overview of several major telescopes that have been highly productive and made signiﬁcant impact in radio astronomy (see e.g. Trimble and Ceja 2010, for an extensive list).

2.3.1 VLA and JVLA

Sited on the plains of San Agustin, New Mexico, is the Very Large Array (VLA), which consists of twenty seven 25-m dishes in a Y-shaped re-configurable array. It has four array configurations: A (with a maximum separation of 36 km), B (10 km), C (3.6 km) and D (1 km). In every three to four months, the configuration is switched from one to another, allowing the VLA to cover a wide range of angular resolution (0.2 to 0.004 arcsec) and frequency (74 MHz to 50 GHz). Since its completion in 1980, the VLA has been ranked the most productive ground-based telescope according to its number of publications and citations. In addition, the VLA is also part of the National Radio Astronomy Observatory (NRAO) and has produced the largest radio survey to date, the NRAO VLA Sky Survey (NVSS), which covers the entire sky north of 40◦ declination. The VLA has been crucial to numerous im- − portant science discoveries, some of which include the imaging of magnetic filaments at the galactic centre (Yusef-Zadeh et al. 1984), the discovery of the first Einstein ring (Hewitt et al. 1988), and the revelation of internal structures within radio jets (Swain et al. 1998). To further enhance its scientific capabilities, the VLA has been upgraded and renamed in 2012 as the Jansky Very Large Array (JVLA). The JVLA is now in operation, achieving at least an order of magnitude improvement in all observational capabilities, compared to the VLA. Some of its science highlights include the imaging of merging cluster Abell 2256 to an unprecedented detail (Owen et al. 2014), and full polarisation observations of a sample of AGN with large RMs (Pasetto et al. 2016).

2.3.2 IRAM 30-m telescope

The 30-metre single dish radio telescope is located at the Spanish Pico de Veleta where atmospheric water content is minimal to allow for the observations of millimetre-wavelength signals from the cosmos. Operated by the Institute for Radio 2.3. Polarised Universe observable in the radio wavelengths 52

Astronomy in the Millimetre Range (IRAM), it is the second largest millimetre-wave telescope in the world after the Large Millimetre Telescope (LMT). The 30-m dish received its first light in 1984, and since then, it has explored the southern skies and the Milky Way centre at wavelengths between 0.9 and 3 mm. This wavelength range is ideal for studying spectral lines of interstellar molecules and tracing star formation within giant molecular clouds of the Milky Way out to nearby galaxies. One such example is the large-scale survey of line emission in the Orion molecular cloud (Bernéet al. 2014). In 2015, the second generation Neel-IRAM-KID-Array (NIKA2) has been installed at the telescope to enable simultaneous observations in the 1.15 mm and 2.0 mm bands of sub-mJy point sources, map the SZ effect in clusters out to z 2, as well as trace dust in nearby galaxies and star forming ∼ regions (e.g. Catalano et al. 2016).

2.3.3 ATCA and the Parkes Observatory

The Australia Telescope Compact Array (ATCA) and the Parkes Observatory are run by the Commonwealth Scientific and Industrial Research Organisation (CSIRO) as part of the Australia Telescope National Facility (ATNF). The ATCA and the Parkes observatory are frequently operated together with the Mopra telescope to form a very long baseline interferometer. The ATCA is located west of Narrabri, Australia and started its operations in 1987. It is an aperture synthesis telescope with an array of six 22-m movable dishes on a 6-km East-West baseline. The array can be used for observations between frequencies of 1.1 GHz and 105 GHz. Primarily scanning the southern skies, the ATCA has achieved many scientific highlights, including making the first detailed radio image of SN 1987A (Staveley-Smith et al. 1993) and, the first observation of a neutron star with an ultra-relativistic jet (Fender et al. 2004). At the outskirts of the New South Wales town of Parkes is a 210-feet parabolic dish known as the Parkes 64-m radio telescope. It is the second largest dish in the southern hemisphere and is also considered as the predecessor of multi-beam receivers, which are now available on many radio telescopes. After its completion in 1961, the dish has operated almost daily up to present day, due to regular upgrades such as the addition of smooth metal plates to improve the detection of radio waves. Over the decades, the Parkes observatory has found more than two-thirds of known pulsars, including the discovery of the the first known double pulsar system (Lyne 2.3. Polarised Universe observable in the radio wavelengths 53 et al. 2004). Not only that, a series of occultation observations made using the Parkes dish led to the historical discovery of the first quasar 3C273 (Hazard et al. 1963; Schmidt 1963; Greenstein and Schmidt 1964), as well as the discovery of the Magellanic Stream (Mathewson et al. 1974). Now the Parkes observatory is a vital component of the Parkes Pulsar Timing Array in the quest to detect gravitational waves (see e.g. Wang et al. 2015; Kerr et al. 2020).

2.3.4 VLBA

The Very Long Baseline Array (VLBA) is an interferometer consisting of ten 25-m parabolic dishes, stretching from Mauna Kea, Hawaii to the US Virgin Islands. Com- pleted in 1993, the VLBA spans approximately 8000 km, making it one of the long baseline interferometers in the world4. The VLBA typically observes at wavelengths of 3 mm to 28 cm, corresponding to a frequency range between 1.2 GHz and 96 GHz. Some of the key VLBA observations include: the structure of a circum-nuclear disk of maser emission (Miyoshi et al. 1995), the first direct and accurate measurement of the distance to a faraway galaxy NGC 4258 (Herrnstein et al. 1999), the recent discovery of a nearly naked supermassive black hole (Condon et al. 2017), as well as the first direct, accurate measure of the distance to a star-forming region on the far side of the Milky Way (Sanna et al. 2017). As the VLBA is also operated by the NRAO, its sensitivity can be improved five-fold by running it simultaneously with the Arecibo radio telescope, the GBT, the VLA and the Effelsberg radio telescope.

2.3.5 GMRT

The National Centre for Radio Astrophysics in India is home to the Giant Metrewave Radio Telescope (GMRT) since 1995. The GMRT is primarily designed to search for the highly redshifted 21 cm signal emitted by neutral hydrogen to determine the epoch of reionisation (see e.g. Ghosh et al. 2011). Pulsar astrophysics is also another key science goal of the GMRT (see e.g. Frail et al. 2016). To this end, the GMRT is made up of 30 fully steerable 45-m dishes, with a range of baselines up to 25 km, providing an angular resolution of 1 arcsecond at 21 cm. With its high sensitivity at a frequency range of 150 MHz to 1.5 GHz, the GMRT is often used to study a variety

4Other very long baseline interferometers (VLBI) include the European VLBI Network (EVN) and the Event Horizon Telescope (EHT). The EVN is a global network of 22 telescopes, including Eﬀelsberg and Jodrell Bank, whereas the EHT will be an Earth-sized interferometer, linking various radio telescopes across the world such as ALMA and IRAM. 2.3. Polarised Universe observable in the radio wavelengths 54 of celestial objects, for instance, the Sun, HII regions, nearby galaxies, pulsars and supernova remnants. The GMRT is also one of the SKA pathﬁnders.

2.3.6 GBT, Eﬀelsberg 100-m telescope and Jodrell Bank Lovell telescope

The Green Bank Telescope (GBT) is the world’s largest fully steerable radio telescope, followed by the Effelsberg 100-m telescope and the Lovell telescope at Jodrell Bank. Located at the US, the GBT has a 100-m diameter collecting area, grant- ing access to 85% of the entire celestial sphere, with observations at wavelengths of 3.0 m to 2.6 mm. Since 2001, the GBT has made several important discoveries, which include the detection of 3 new milli-second pulsars in M62 (Jacoby et al. 2002), the first observation of a helical magnetic field in the Orion molecular cloud (Robishaw and Heiles 2005), and the discovery of the most massive neutron star yet known (Demorest et al. 2010). Meanwhile, the Effelsberg 100-m dish near Bonn, Germany has been in operation since 1972. Its high sensitivity and frequency coverage between 0.3 and 96 GHz not only allow for the observations of clusters, star formation regions and jets, but also the ability for high-precision pulsar timing. One of the key discoveries made using the Effelsberg is the all-sky radio emission at 408 MHz (Haslam et al. 1982), which has now become the basis for the estimation of non-thermal foreground to the CMB. The third largest fully steerable radio telescope in the world belongs to Jodrell Bank, whose Lovell dish extends over 76 metres in diameter. Operating since the summer of 1957, it was originally designed to detect radio echoes from cosmic ray air showers. It was soon found that the echoes originated from ionised meteor trails, which then opened up a flurry of research on meteors, quasars, pulsars, masers and gravitational lenses. Some of the milestones achieved by the Lovell 76-m telescope include the discovery of the first pulsar in globular cluster, as well as the detection of the first gravitational lens (see e.g. Morison 2007, for a review). The telescope is now operated by the University of Manchester.

2.3.7 LOFAR and WSRT

Both the Low Frequency Array (LOFAR) and the Westerbork Synthesis Radio Tele- scope (WSRT) are operated by the Netherlands Institute for Radio Astronomy (ASTRON). Observations began in 2012 at LOFAR. Designed as a radio interfer- 2.3. Polarised Universe observable in the radio wavelengths 55 ometric array, LOFAR consists of about 20000 omni-directional antennas, which are concentrated in at least 48 stations distributed across Germany, the UK, Fin- land, France and Sweden. It is optimised for the low frequency range between 10 and 250 MHz, with sub-mJy sensitivity and sub-arcsecond spatial resolution. This makes it an excellent probe for cosmic magnetism of the nearby Universe, including one of its recent observations of the Milky Way magnetic field possessing a spaghetti-like structure (Jelićet al. 2015; Zaroubi et al. 2015). LOFAR is also a technology precursor for the SKA. Meanwhile, operations started in 1970 at the WSRT. Located at the north of Westerbork, it uses methods of aperture synthesis and Earth rotation synthesis to make observations, targeting frequencies between 120 MHz and 8092 line spectral resolution. WSRT is a linear array of fourteen 25-m dish-shaped antennas on a 2.7 km East-West line. Its equatorial mount makes it ad- vantageous for polarised emission research of the northern sky because the antennas do not change orientation during observation. Over years of operation, the WSRT has made many scientific advances, including the first time where polarised emission and its Faraday rotation have been observed to 10 µJy/beam rms for a large sample of galaxies (Heald et al. 2009). Often, the WSRT is operated simultaneously with other telescopes to perform (Very Long Baseline Interferometry) VLBI observations.

2.3.8 ALMA and MWA

The Atacama Large Millimetre/Submillimetre Array (ALMA) is by far the most expensive ground-based telescope. Sited at northern Chile, it consists of 66 high- precision antennas that can be moved over distances of 150 m to 16 km, allowing ALMA to zoom into the millimetre and sub-millimetre wavelengths. Observations started from the end of 2011, including the detailed imaging of star birth and planetary nebulae, comet studies and investigations of properties of starburst galaxies in the early Universe. Located at Western Australia, the Murchison Widefield Array (MWA) is the first large-N array made up of 128 phased tiles, each tile composed of 16 crossed dipoles arranged in a 4 by 4 square. Operating in the frequency range of 80 to 300 MHz, the MWA aims to detect neutral atomic hydrogen emission from the epoch of reionisation, to study the Sun and the Earth’s ionosphere, as well as to hunt for radio transient activities. The MWA started observations in 2013. A recent scientific highlight is the first-time, direct observation of field-aligned plasma ducts in the Earth’s ionosphere (Loi et al. 2015). The MWA is also one of the technology 2.4. Other magnetic-field diagnostics 56 and science pathfinders for the SKA.

2.3.9 SKA and its precursors

The Square Kilometre Array (SKA) is the upcoming new radio telescope that will address fundamental questions about the Universe. It will be made up of thousands of receptors connected across a total collecting area of a million square metres. Along with the MWA, the South-African MeerKAT and Australian ASKAP, this will make it the world’s largest and most sensitive radio telescope ever built. It is predicted to have 50 times the sensitivity and 10000 times the survey speed of current radio telescopes.5 It has recently been confirmed that the telescope will have a dual site, in which the low frequency aperture arrays for Phases I and II will be built in Australia, while the dishes and mid-frequency aperture arrays in South Africa. One of the five key science projects that the SKA will undertake is to understand the origin and evolution of cosmic magnetism (see e.g. Beck 2009b). The SKA will revolutionise the study of cosmic magnetism by an All-Sky RM Survey. This survey will measure the RMs for 108 polarised extragalactic sources across an entire hemisphere, with a 6000 resolution (Stepanov et al. 2008), thus providing a high density background of radio polarised sources for RM mapping towards nearby galaxies for the study of magnetic fields. The construction of the background RM grid and high-sensitivity polarisation maps will fully characterise the evolution of cosmological magnetic fields from redshifts z > 3 to today (Schilizzi 2007). This will enable the properties of magnetic fields in LSS to be determined and compared with theoretical predictions. The SKA will cover radio frequencies between 70 MHz and 10 GHz, in which the higher frequencies (1 10 GHz) are useful in tracing the − details of magnetic fields in galactic centres, disks and cluster relics (see e.g. Beck 2011). The lower frequencies will map weak magnetic fields in the outskirts and halos of the Milky Way, galaxies, and galaxy clusters.

2.4 Other magnetic-ﬁeld diagnostics

2.4.1 Zeeman eﬀect

The strength and orientation of the interstellar magnetic ﬁeld can be measured using the Zeeman eﬀect (Zeeman 1897) in H I, OH, and CN in extended gas; and in

5See SKA oﬃcial website for more details. 2.4. Other magnetic-ﬁeld diagnostics 57

OH, CH3OH, and H2O in masers (see e.g. Crutcher and Kemball 2019, for a recent review). The Zeeman effect is defined by the splitting of spectral lines of an atom in the presence of an external magnetic field. The amount of splitting depends on the strength and orientation of the external magnetic field, the spectroscopic selection rules and the properties of the atom. The selection rules govern the number of allowed transitions between atomic orbitals. For a weak magnetic field, the normal Zeeman effect occurs, in which a spectral line is split into a central π-component and two symmetric σ±-components. This is commonly referred to as the spin-0 state because the spin S is zero and the total angular momentum J is equivalent to the orbital angular momentum L due to LS- coupling. If the magnetic field is aligned along the line-of-sight, π disappears and only the pair of circularly polarised right-hand σ+ and left-hand σ− components are visible. The frequencies ν for the σ± components are

eB ν = ν0 , (2.58) ± 4πmec where ν0 is the line frequency in the absence of a magnetic field, e is the electron charge, me is the electron mass, c is the speed of light, and B is the magnetic field strength (see e.g. Crutcher and Kemball 2019, for a recent review). When the magnetic field is normal to the line-of-sight, all three components can be seen, for example in the hydrogen atom, with π being linearly polarised to Bk and σ± linearly polarised to B⊥. This is attributed to the selection rules: ∆mS = 0, i.e. no change in electron spin quantum number; ∆mL = 0, i.e. no frequency shift for the π-component; and ∆m = 1, i.e. symmetric, but opposite shifts in the L ± orbital angular momentum for σ±-components. When the spin of the initial and/or final states is non-zero, the anomalous Zeeman effect can take place (Knutson and Burikham 2003). Unlike the normal Zeeman effect, the splitting of energy levels in the anomalous case are reliant on all contributions of ∆mJ, ∆mL, and ∆mS. When the upper and lower states experience different amounts of splitting, this leads to more than three distinct transitions, and therefore gives rise to multiple spectral lines. The splitting of the sodium doublet is a good example, whereby the selection rule ∆m = 0, 1 can either generate four or six lines, depending on the choice J ± of transition. When the external magnetic field becomes stronger, the Zeeman splitting dominates over the spin-orbit coupling, giving rise to the Paschen-Back 2.4. Other magnetic-field diagnostics 58 effect. In essence, this occurs when L decouples from S, causing each of them to precess almost independently about the magnetic field. Since S 0, the amount of ≈ splitting is nearly uniform under the selection rules of ∆m = 0, 1 and ∆m = 0. L ± S Subsequently, only three spectral lines are observed, similar to the normal Zeeman effect, but each of the lines is actually made up of a very closely-spaced doublet (Knutson and Burikham 2003). The Paschen-Back effect applies to all atoms in the strong magnetic field limit.

The amount of Zeeman splitting depends on the magnetic field strength, and the polarisation of the Zeeman line components depends on the magnetic field orientation with respect to the line-of-sight. The Sun has magnetic field strengths between a few G and several kG, which are strong enough for Zeeman splitting to be observed. These observations are used by the Helioseismic and Magnetic Imager (HMI ) instrument on the Solar Dynamics Observatory (SDO) to produce magnetograms. Magnetograms show how magnetic fields on the Sun changes, depending on the line-of-sight magnetic field strength and its associated polarity. Regions hosting the strongest magnetic fields of several kG are identified as sunspots (Hale 1908). Sunspots with their fields directed outwards (north polarity) are convention- ally shown in white colour, whereas those with their fields directed inwards (south polarity) are shown in black.

Apart from studying solar magnetic fields, the Zeeman effect is also particularly useful in testing star formation paradigms. Zeeman observations in the molecular lines of cyanogen (CN) can directly measure magnetic field strengths and morphologies in dense molecular clouds. The first observations of the CN Zeeman effect were carried out with the IRAM telescope towards the Orion Molecular Cloud and reported a magnetic field strength of B = 0.36 0.08 mG (Crutcher et al. 1999). k − ± Following this, the first interferometric CN Zeeman detections and observations of dense molecular cores were carried out using the CARMA telescope and found an upper limit of B 4 mG (Crutcher 2014). Zeeman splitting used to be limited to k ∼ observations within the Milky Way but it is now able to probe magnetic fields in some extragalactic environments with the help of high sensitivity radio instruments. The detections of extragalactic Zeeman splitting are few and far between, including an absorption line in HI towards the Perseus cluster (Kazes and Baan 1991; Sarma et al. 2005) and in OH megamasers (OHMs) within 15 starburst galaxies (Robishaw 2.4. Other magnetic-field diagnostics 59 et al. 2008; McBride and Heiles 2013). There is also the potential of measuring magnetic fields in other galaxies using dual-polarisation VLBI observations of OHMs, such as that successfully demonstrated in Arp 220, whose magnetic field is measured to be in the pc scale with strengths B 1 5 mG (McBride et al. 2014). ∼ − The typical strengths of interstellar magnetic fields lie within 10 µG B ≤ ≤ 1 mG, which are estimated using equation 2.58 to cause frequency splits between 28 Hz ∆ν 2.8 kHz. These frequency splits are orders of magnitude lower than ≤ ≤ the observed line widths, which are often Doppler broadened. Moreover, overlapping features due to complex emission from high-energy density systems such as OHMs can smear the Zeeman-splitted lines. In view of these limitations, additional tools, such as Stokes polarisation, are needed, together with Zeeman splitting to assess the roles of magnetic fields in the Sun, stars, and stellar nurseries in the Milky Way and some nearby galaxies on smaller spatial scales.

2.4.2 Goldreich-Kylaﬁs eﬀect

When the Zeeman effect is very weak, it can be rather challenging to resolve the three Zeeman line components. The components have very low linear polarisations, as a result of symmetric, equally-likely transitions, i.e. ∆m = 1, 0, 1. This is not − always the case because the transitions ∆m can happen at different rates, leading to increased linear polarisation in the molecular spectral lines. This is known as the Goldreich-Kylafis (GK) effect (Goldreich and Kylafis 1981; Kylafis 1983). It occurs when an anisotropic source exists within the medium, such as velocity gradients lying parallel or normal to the magnetic field; or when an external anisotropic radiation is applied to excite the σ± and π line components differently, therefore causing unequal populations in magnetic sub-levels. In general, the quicker the radiative transitions, the brighter are the line intensities. Subject to which transitions are excited, the net linear polarisation can either be parallel or perpendicular to the magnetic field direction (Cortes and Crutcher 2006). This is one of the unfortunate issues in measuring the GK effect because the magnetic field orientation is ambiguous by π/2. Although the GK effect is unable to distinguish coherent from ordered fields, it is able to tell them apart from random fields. Moreover, the GK effect is prominent in observations of CO line transitions which is an excellent tracer of shocked gas due to its abundance and dipole moment (Gusdorf et al. 2014). As such, the GK effect is often used to probe magnetic 2.4. Other magnetic-field diagnostics 60

field morphologies in active star-forming regions, young stellar outflows, supernova remnants and molecular clouds. There has been a long-running debate whether magnetic fields affect star formation, owing in large part to the difficulty of observing stellar nurseries in the Milky Way from Earth. Stellar nurseries are defined as dense regions within molecular clouds that collapse under their own gravity to form stars. There are two competing models that seek to answer the fundamental question of what drives star formation. The first model proposed that molecular clouds are magnetised, such that the magnetic fields influence how the cloud assembles and break apart, hence controlling the speed and efficiency of star birth (see e.g. Crutcher 2014). In contrast to this, the second model suggests that magnetic fields are unimportant because turbulence dominates during molecular cloud formation and its dissipation is what causes core collapse to form stars (see e.g. Elmegreen et al. 2000). In order to address this issue, the observations of magnetic field alignment in six giant molecular clouds in the spiral arms of the Triangulum Galaxy M33 are made for the first time using the Submillimetre Array (SMA; Li and Henning 2011). M33 is chosen for its almost face-on view and close proximity to the Milky Way. Using CO line (GK effect) and synchrotron emissions, polarised light from these molecular clouds are analysed and used to detect the magnetic fields within them (see Figure 2.3). The fields within the clouds are discovered to be aligned with the spiral arms, suggesting that the galactic field of M33 anchors the clouds. Their results strongly indicate that magnetism is crucial in the formation of dense molecular clouds and stars. Another recent work carried out using the SMA unravels the magnetic field structure in the flattened envelope and jet in HH 211 (Lee et al. 2014). More interestingly, a first-time detection of a SiO line emission in the jet due to the GK effect is reported but can only be confirmed with further observations of higher sensitivity. It is worth pointing out that the GK effect is hard to detect, since its degree of polarisation is typically 2 4%, which is only slightly above the − instrumental polarisation. Observations of the GK effect is thus limited to nearby galaxies only.

2.4.3 Interstellar dust

Dust particles are solid, irregular-shaped grains, mainly composed of silicate, graphite and dirty ice. Although they merely constitute to 1% of the gaseous in- 2.4. Other magnetic-ﬁeld diagnostics 61

Fig. 2.3: These images show the polarisation intensities (grey contours) and vectors (red lines) within the six giant molecular clouds observed in the spiral arms of M33 (Li and Henning 2011). The polarisation properties are derived from the CO line (GK eﬀect) and radio synchrotron emissions using the SMA. The thick grey lines indicate the tangents of the local optical arms, whereas the ellipses correspond to the beam sizes.

terstellar medium permeating the Milky Way, dust grains play significant roles in astrophysics. The formation of molecules, stars and planetary systems are thought to be attributed to dust grains. In view of this, the molecular production of H2 is believed to be catalysed by dust grains whose surfaces act as primary sites for colliding hydrogen atoms in their surrounding gas to combine. The birth of stars and proto-planetary systems is understood to be formed by the gravitational collapse of a rotating cloud of dust and gas. In addition to that, the physical interactions between dust and starlight store important information on the interstellar magnetic field structure, grain physical properties and chemical composition, as well as their alignment mechanisms. These parameters of interest can be extracted through studies of: (i) extinction, which is the obscuring of light from distant stars; (ii) red- dening, which is the dimming of blue light over red; (iii) scattering of starlight, by which the reflection nebulae and diffuse Galactic light are generated; (iv) polarising 2.4. Other magnetic-field diagnostics 62 starlight, which is due to elongated dust alignment, causing the preferential extinction of one linear polarisation over another; (v) absorption ( 30 50%) of optical ∼ − photons, thus heating up the interstellar medium; and (vi) emission of the excess energy gained in the sub-millimetre (sub-mm) wavelengths, most of which is in the form of thermal infra-red radiation.

The polarisation of starlight by dust not only shows that dust grains are pro- late in nature, but also places observational constraints on interstellar magnetic fields. Optical starlight polarisation is the result of selective absorption by dust grains whose magnetic moments tend to align themselves with the interstellar magnetic field. The preferential grain alignment was first discovered by Hiltner(1949) and Hall(1949) during their study of starlight polarisation attenuated by interstellar dust, and later ascertained via the trace of magnetic fields in the Milky Way (Davis and Greenstein 1951), in M51 (see e.g. Scarrott et al. 1987), and also along the Galactic plane (see e.g. Fosalba et al. 2002). The physics of alignment is not yet well understood, although it is widely accepted that the rotating dust grains con- tinuously collide with the surrounding hydrogen atoms and interact with starlight photons, then precess and align themselves with the magnetic fields, either via paramagnetic dissipation of energy (Davis and Greenstein 1951) or radiative torque alignment (Draine and Weingartner 1996a; Hoang and Lazarian 2008). The latter alignment mechanism is more efficient and is supported by observations (Andersson 2015; Cashman and Clemens 2014). The dust particles preferentially absorb light that is polarised along their major axes, thus the net starlight becomes linearly polarised in the direction parallel to the magnetic field lines. At the same time, the hot dust grains cool down by emitting thermal infra-red photons. This net sub-mm emission is inherently polarised perpendicular to the incident optical photons and magnetic field lines.

Assuming equipartition between turbulent magnetic energy and turbulent kinetic energy, the dispersion of starlight polarisation orientations together with estimates of gas volume density and line-of-sight gas velocity dispersion can be used to determine the magnetic field strength across the sky plane. This is known as the Davis-Chandrasekhar-Fermi (Davis 1951; Chandrasekhar and Fermi 1953, hereafter DCF) method, and its developments (e.g. Hildebrand et al. 2009; Houde et al. 2009, 2011; Cho and Yoo 2016; Yoon and Cho 2019; Cho 2019) have been widely used to 2.4. Other magnetic-field diagnostics 63 infer magnetic field information, particularly in the ISM and molecular clouds, from dust polarisation data (e.g. Marchwinski et al. 2012; Cashman and Clemens 2014; Wang et al. 2020a). Sub-mm polarisation has also been used to trace the structure of magnetic fields, and in particular the direction of field lines on the sky plane, as seen in the Orion Molecular Cloud (Vaillancourt et al. 2008) and along the plane of the Milky Way (Planck Collaboration 2015). The latest Planck image reveals the presence of stronger polarised emission along the Galactic plane, threaded by regular magnetic fields lying predominantly parallel to it. Above and below the plane, the polarised emission is reduced by tangled local magnetic fields within gas and dust clouds. These works are essential in order to quantify and separate the Galactic foreground contributions from CMB polarisations, therefore improve the knowledge and understanding of the early Universe. Unfortunately optical and infrared observations have their limitations. The observed degree of polarisation is unable to distinguish between the intrinsic polarisation properties of dust and the magnetic field geometry due to the lack of understanding in grain alignment. Furthermore, anisotropic scattering of light in the interstellar medium can contaminate starlight polarisation. Bright sources are required as the degree of polarisation is very low, hence the field resolution is limited to the local Universe only.

2.4.4 High-energy cosmic rays

Cosmic rays (CRs) and magnetic fields are important in the dynamics of interstellar and intracluster media. The CRs, being composed of charged particles, interact with the magnetic fields as they propagate. In particular, the energies and arrival directions of CRs give some indication of their origins, as well as the strength and direction of the intervening magnetic fields. CRs are energetic particles travelling nearly at the speed of light. The primary CRs are usually made up of protons, and sometimes other heavier nuclei (see e.g. Grenier et al. 2015, for a recent review; Neronov et al. 2016). When such a particle interacts with the Earth’s atmosphere, it creates a shower of lower-energy secondary particles. While solar CRs have energies between 107 1010 eV, the rest have a wide range of energies, typically between − 109 1020 eV (e.g. Gaisser and Stanev 2006). For the lower end of energies up to − approximately 1018 eV, the CRs are believed to have originated from within the Milky Way (see e.g. Blasi 2013, for a review). 2.4. Other magnetic-field diagnostics 64

The most plausible source of these Galactic CRs is supernova remnants (SNRs) (e.g. Koyama et al. 1995; Ackermann et al. 2013). Their explosions are thought to drive magnetic fields that are generally amplified via a turbulent dynamo mechanism (e.g. Uchiyama et al. 2007; Inoue et al. 2009; Xu and Lazarian 2017). The strong magnetic fields are essential for an efficient diffusive shock acceleration of CRs. These magnetic fields subsequently trap and mirror the CR particles up to energies of around 1015 eV (e.g. Aharonian et al. 2006). Consequently, young galactic SNRs or Fermi bubbles may be able to further accelerate these particles up to 1018 eV (e.g. Ptuskin et al. 2010; Cheng et al. 2012; Chernyshov et al. 2017). However, these shock acceleration mechanisms would struggle to accelerate CRs to higher energies above 1018 eV. These high-energy CRs are likely to have an extragalactic origin. Arguably, from their anisotropic distribution, the high-energy CRs may be associated with nearby galaxies, such as radio galaxies and/or AGN (e.g. The Telescope Array Collaboration 2014, 2018; The Pierre Auger Collaboration 2018).

Nearby AGN jets may have injected CRs and magnetic fields into the ICM. Ob- servationally, the transport of CRs in clusters of galaxies can be probed in the radio, X-ray and gamma-ray wavelengths (e.g. Miniati et al. 2001; Pfrommer et al. 2008). Proton-proton collisions in the ICM produce secondary particles, including neutral pions which decay into gamma-rays and relativistic electrons. The relativistic electrons generate radio synchrotron emission when gyrating around the intracluster magnetic fields. They also emit X-rays by inverse Compton scattering of cosmic microwave background photons. The transport of CRs in galaxy clusters is therefore governed by the properties of magnetic fields and turbulence in the ICM.

Notably, the synchrotron and inverse Compton cooling time is of the order of 108 years, implying that the CR electrons must have been recently injected or re- accelerated in the galaxy cluster environment (e.g. Kang et al. 2012; Fujita et al. 2015; see e.g. van Weeren et al. 2019, for a review). On the other hand, the CR protons can be as old as their hot cluster – potentially being the source of radio halos observed today. These radio halos are thought to be linked to the dynamical state of the clusters (e.g. Dolag and Enßlin 2000; Donnert et al. 2010; Keshet and Loeb 2010). How cosmic rays propagate in diﬀerent clusters may provide a means to model the observed bi-modality among the cluster radio halos (e.g. Brunetti 2009). Moreover, cosmic rays can store a large amount of energy, which can be released 2.4. Other magnetic-ﬁeld diagnostics 65 at the cluster centres. They could therefore play an important role in stabilising a cluster cool core against cooling instabilities (e.g. Colafrancesco and Marchegiani 2008; Fujita and Ohira 2011; Fujita et al. 2013; Ruszkowski et al. 2017, see also Chapter4 for more details on galaxy clusters and radio halos).

On smaller scales, the propagation and interactions of CRs are also affected by magnetic fields in the interstellar medium (ISM). The ISM is a mixture of magnetic fields; hot, tenuous gases; and cold, dense molecular clouds (MCs). CRs tend to scatter randomly from the magnetohydrodynamic and turbulent plasma waves in the diffuse, interstellar regions (e.g. Kulsrud and Pearce 1969; Ko 1992; Lazarian and Beresnyak 2006; see e.g. Zweibel 2013, for a recent review). When CRs penetrate the dense cores and clumps of the MCs, they may be confined by the stronger, mG-level magnetic fields (e.g. Li et al. 2015b). These magnetic fields, which are amplified by the gravitational collapse of materials, enable focusing and entrapment of CRs in the MCs. As such, magnetic fields must play a significant role in CR transport and interactions in the ISM, as well as, mediating star formation alongside the forces of turbulence and gravity (see Chapter5 for cosmic ray interactions in magnetised molecular clouds).

2.4.5 Astronomical transients

Apart from radio galaxies and AGN, gamma-ray bursts (GRBs) have also been identified as candidate high-energy cosmic ray (CR) sources. Being the most luminous objects in the Universe after the Big Bang, GRBs are widely recognised as astrophysical powerhouses for high-energy CRs and magnetic fields. By definition, GRBs are intense, irregular, and short pulses of gamma-rays, which either last less than 2 seconds (short GRBs) or more than 2 seconds (long GRBs) (e.g. Klebesadel et al. 1973; Gendre et al. 2013; see e.g. Kumar and Zhang 2015, for a review). The bursts appear to arrive from random directions in the sky and are most certainly extragalactic with redshifts measured between z 0.16 8.2 (see e.g. Berger 2014, for ∼ − a review). Usually the GRBs are followed by an afterglow6 of radiation with lower energies and longer timescales, e.g. in the X-ray, optical, and radio wavelengths (see e.g. Dainotti and Del Vecchio 2017, for a recent review).

6The observed afterglow has been argued to be produced by synchrotron processes, in which magnetic ﬁelds would play a crucial role (e.g. Lloyd and Petrosian 2000). 2.4. Other magnetic-ﬁeld diagnostics 66

Since high-energy cosmic rays propagate through the large-scale magnetic fields, they consequently carry information about the strength and direction of the fields. The observed prompt emission is probably generated by a “fireball” of energetic particles accelerated within the relativistic, magnetised shocks during cataclysmic events, e.g. binary neutron star mergers (Narayan et al. 1992) or massive star collapse (Woosley 1993). The internal shocks or the reverse shock in GRBs are among several central engines in the Universe where the conditions for the acceleration of high-energy CRs can be satisfied following the Hillas (1984) criterion (e.g. Waxman 1995). Alternatively, it has also been proposed that protons can be accelerated to greater than 1020 eV7 within the inner engines of GRBs with magnetic fields larger than 1015 G (e.g. de Souza and Opher 2010).

Observations of pair echoes from high-redshift GRBs at z & 5 can also be used to probe intergalactic magnetic fields in the early Universe (Takahashi et al. 2011). The GRBs emit primary photons with GeV TeV energies. These photons interact − with the photons in the diffuse IGM, producing electron-positron pairs. The pairs of charged particles are then deflected by the intergalactic magnetic fields, before emitting secondary gamma rays via inverse Compton scattering with the CMB photons. The secondary gamma rays are known as a pair echo, and they experience a time delay relative to the primary gamma rays. This time delay depends on the properties of the intergalactic magnetic fields, and has been used to constrain the field strengths in several instances, e.g. B 10−16 10−15 G(Takahashi et al. 2011), ∼ − B 10−21 10−17 G(Veres et al. 2017), and B > 10−19.5 G(Wang et al. 2020b), for ∼ − a coherence length of l 1 Mpc. The next-generation gamma-ray telescopes such ≤ as the Cherenkov Telescope Array (CTA) may be able to observe pair echoes from luminous GRBs up to z . 10 (e.g. FernándezAlonso et al. 2018), and therefore tighten the constraints on the nature of intergalactic magnetic fields.

Aside from GRBs and their afterglows, fast radio bursts (FRBs) have also been identiﬁed as a potential powerful probe to the intergalactic magnetic ﬁelds (e.g. Akahori et al. 2016; Vazza et al. 2018). FRBs are millisecond, brief8 radio signals

7The highest energy CRs above 1019 eV are attenuated due to the Greisen-Zatsepin-Kuzmin (GZK – Greisen 1966; Zatsepin and Kuz’min 1966) eﬀect with the CMB photons. Hence, any observed CRs at energies above the GZK limit must originate from nearby sources.

8Several FRBs have been observed to repeat in their lifetimes, hinting at the possibility of two 2.4. Other magnetic-field diagnostics 67 emanating from distant galaxies (Lorimer et al. 2007). Their short duration, fre- quent occurrence, and extragalactic origin make them ideal for large-scale magnetic field diagnostics (e.g. Zheng et al. 2014). In particular, combining the estimates of the dispersion measure (DM) and rotation measure (RM) along the line-of-sight of the FRBs is proven to be useful in magnetic field inference (e.g. Akahori et al. 2016; Hackstein et al. 2019). The DM gives the free electron column density along the line-of-sight, whereas the RM is partly determined by the magnetic field strength along the line-of-sight. As such, both the DMs and RMs of FRBs can be used simultaneously to investigate the intergalactic magnetic fields, as shown below, following Akahori et al.(2016).

Consider a source of FRB at redshift z = zi. In the observer’s frame, it has a dispersion measure

Z zi ne(z) dl(z) −3 = CD dz pc cm (2.59) D 0 (1 + z) dz

(e.g. Deng and Zhang 2014) and a rotation measure

Z 0 ne(z)Bk(z) dl(z) −2 = CR dz 2 rad m (2.60) R zi (1 + z) dz

(e.g. Akahori and Ryu 2010), where ne(z) is the proper electron density in units −3 of cm , Bk(z) is the line-of-sight magnetic ﬁeld strength in µG, and dl(z) is the distance of radiation propagation along the line-of-sight in pc. The normalisation constants are CD = 1 and CR = 0.812 (Akahori et al. 2016). At z = 0, the magnetic ﬁeld strength along the line-of-sight is

−1 CD Bk = R = 12.3 R −2 3 D −3 nG. (2.61) CR 10 rad m 10 pc cm D In a cosmological context, equation 2.61 becomes CD 1 + z Bk = R h i (2.62) CR fD D 1 + z −1 = 12.3 h i R −2 3 D −3 nG fD 10 rad m 10 pc cm

(Akahori et al. 2016), where fD and 1 + z are the fraction of DM and weighted h i redshift by the warm-hot intergalactic medium (WHIM) in the cosmological ﬁla- ments, respectively. The RM and DM are cluster-subtracted measures for an FRB. diﬀerent classes of progenitors (see e.g. Cordes and Chatterjee 2019, for review). 2.5. Summary 68

Given a cosmological model, fD and 1 + z can be calculated for a known DM. h i In practice, the Bk estimate has its limitations in, e.g. the variance of observed DMs, and the foreground contamination due to the host galaxies or the Milky Way. However, these observational limits will improve with the advent of the SKA, which is expected to detect at least 104 FRBs sky−1 day−1 at z 6 in the 0.35 0.95 GHz & − band (Fialkov and Loeb 2017).

2.5 Summary

This chapter presents the observational signatures and diagnostics of magnetic fields from interstellar to cosmological scales. The intracluster medium is filled with relativistic electrons and weak magnetic fields. The relativistic electrons gyrate around the magnetic fields to produce synchrotron radiation. Therefore radio synchrotron emission and its polarisation are often used to diagnose large-scale magnetic fields. Firstly, this chapter outlines the physics of radio synchrotron emission from a single electron. A relativistic electron gyrates in a uniform, helical motion around a magnetic field line, emitting synchrotron radiation which is strongly beamed in the direction of motion. The spectral properties from a single electron and from a power-law distribution of electrons are also described (the latter is used in Chapters 4 and5).

Next, the physics of polarisation is presented, with an overview on its various representations by the Jones vectors, Mueller matrices and the Stokes parameters. The Stokes parameters of polarisation are defined on the Poincarésphere. They are often used in calculations of Faraday RMs and polarised radiative transfer (see also Chapter3). These theories alongside many radio experiments have contributed immensely to the understanding of magnetic fields in astrophysical systems. From single-dish radio telescopes (e.g. IRAM) to radio interferometers (e.g. LOFAR), numerous scientific advances have been achieved in mapping the radio sky to unprecedented detail.

On the other hand, observations in other wavelengths, such as optical and near- infrared, have proven to be useful in probing magnetic fields on the smaller scales, such as molecular clouds. The polarisation of interstellar dust is often used as a proto-stellar magnetic field diagnostic, especially when direct methods such as the Zeeman effect and the Goldreich-Kylafis effect are particularly difficult to perform 2.5. Summary 69 in dusty star-forming regions. Lastly, other large-scale magnetic field diagnostics, in the form of high-energy cosmic rays, gamma-ray bursts, and fast radio bursts, are discussed. Notably, the first GRB observational constraint on the strength of intergalactic magnetic field is B > 10−19.5 G for a coherence length of l 1 Mpc. ≤ 70

Chapter 3

Rotation measure, ﬂuctuations, and large-scale magnetic ﬁelds

This chapter presents results which have been published in whole or in part in: (i) Alvina Y. L. On, Jennifer Y. H. Chan, Kinwah Wu, Curtis J. Saxton and Lidia van Driel-Gesztelyi, “Polarised radiative transfer, rotation measure fluctuations and large-scale magnetic fields”, December 2019, MNRAS 490 (2): 1697–1713, doi:10.1093/mnras/stz2683. Published by Oxford University Press on behalf of the Royal Astronomical Society. All rights reserved. I am the lead author of this paper and wrote most of the manuscript. I formulated the research project and identified the majority of the scientific methods involved in the analysis. I also derived parts of the analytical formulation, built most of the simulated models, and computed the numerical RMF simulations. I also led the interpretation of the statistical results from the simulated RMF analyses. (ii) Jennifer Y. H. Chan, Kinwah Wu, Alvina Y. L. On, David J. Barnes, Jason D. McEwen and Thomas D. Kitching, “Covariant polarised radiative transfer on cosmological scales for investigating large-scale magnetic field structures”, April 2019, MNRAS 484 (2): 1427–1455, doi:10.1093/mnras/sty3498. Published by Oxford Uni- versity Press on behalf of the Royal Astronomical Society. All rights reserved. I am a co-author of this paper. My main role was to carry out an independent verification of the absorption, emission and Faraday coefficients, which are essential for the polarised radiative transfer calculations, the foundation of the key scientific theme of the paper. I also contributed to clarifying the conventions of polarisation and magnetic fields, which have been ambiguous in multi-disciplinary literature across 71 plasma physics and astrophysics, hence establishing a consistent convention for these expressions to be used in the cosmological polarised radiative transfer formulation derived by the lead author. I also performed some test calculations in the limit of galaxy cluster scales, conducted consistency and accuracy checks for the science, and assisted the lead author in some parts of the manuscript preparation.

This chapter presents the theoretical framework of polarised radiative transfer, fluctuations, and large-scale magnetic fields. While stellar magnetic fields have been extensively studied (see e.g. Wickramasinghe and Ferrario 2000; Wu and Wickra- masinghe 1990; Staubert et al. 2019), the determination of magnetic field properties in larger astrophysical systems is less direct. Faraday rotation measure (RM)1 has been identified as a diagnostic tool for magnetic fields in the Milky Way (see e.g. Simard-Normandin and Kronberg 1980; Han et al. 1999; Brown et al. 2003; Gaensler et al. 2004; Brown et al. 2007; Haverkorn et al. 2008; Oppermann et al. 2012; Han et al. 2015; Han 2017), nearby galaxies (e.g. Gaensler et al. 2005; Beck et al. 2020; Mao et al. 2010, 2017) and also some galaxy clusters (e.g. Carilli and Taylor 2002; Vogt and Enßlin 2003; Clarke 2004; Govoni and Feretti 2004; Brentjens and de Bruyn 2005; Bonafede et al. 2010; Kuchar and Enßlin 2011; Vazza et al. 2018).

There have also been studies using the Faraday rotation of distant polarised radio sources such as quasars (e.g. Kronberg et al. 2008; Xu and Han 2014b) and Fast Radio Bursts (FRBs) (e.g. Xu and Han 2014a; Zheng et al. 2014; Akahori et al. 2016; Ravi et al. 2016; Vazza et al. 2018; Hackstein et al. 2019), as a means to detect and probe cosmological magnetic fields. These fields permeating the cosmic web of filaments and voids are weak, and their properties are often inferred statistically (e.g. Akahori et al. 2014; Vernstrom et al. 2019), or indirectly constrained through the non-detection of GeV gamma-rays (e.g. Neronov and Vovk 2010; Taylor et al. 2011; Dermer et al. 2011; Takahashi et al. 2013).

Faraday Rotation Measure Fluctuation (RMF) analysis is proposed as a means to probe the structures of large-scale magnetic ﬁelds (e.g. Akahori and Ryu 2010; Beck et al. 2013b). RM and RMF analyses are essentially based on the theory of

1“Faraday depth” and “rotation measure” can only be used interchangeably in the case of a single point source along the line-of-sight. 3.1. Rotation measure 72 polarised radiative transfer under certain restricted conditions. It is therefore important to have a proper understanding of the information extracted from the analyses and under what conditions the analyses enable unambiguous interpretations. This chapter is organised as follows. In Section 3.1, the polarised radiative transfer equation is presented and shown to reduce to the standard RM equation under certain conditions. In Section 3.2, the RMF analysis is examined in the context of polarised radiative transfer. The mathematical and statistical properties of the analyses are also identified. In Section 3.3, the formal solution to the polarised radiative transfer equation is derived. Section 3.4 describes the absorption, emission and Faraday mixing coefficients in the cases of thermal and non-thermal plasma. In Section 3.5, the numerical algorithms and verifications of the polarised radiative transfer calculations are presented. Lastly, Section 3.6 summarises this chapter.

3.1 Rotation measure 3.1.1 In the context of polarised radiative transfer

The transport of polarised radiation along a distance s is governed by the diﬀerential equation         I κ q u v I I                 d  Q   q κ f g   Q   Q    =  −    +   , (3.1) ds   −        U   u f κ h   U   U     −      V v g h κ V − V where the coeﬃcients for absorption are κ , q , u and v, for emission , for Faraday rotation f, and for Faraday conversion g and h (see Sazonov 1969, for derivation; Jones and O’Dell 1977; Pacholczyk 1977). The Stokes parameters [ I, Q, U, V ] are observables, whose combination gives the total degree of polarisation, Πtot = p p Q2 + U 2 + V 2/I ( 1), the degree of linear polarisation Π = Q2 + U 2/I, ≤ lin the degree of circular polarisation Πcir = V/I, and the polarisation angle ϕ = (1/2) tan−1(U/Q) (see e.g. Rybicki and Lightman 1979). In the absence of absorption and emission, κ = q = u = v = 0, and [ I , Q , U , V ] = 0, respectively. This implies dI/ds = 0, and also       Q 0 f g Q    −    d        U  =  f 0 h   U  . (3.2) ds   −  −    V g h 0 V − 3.1. Rotation measure 73

In the highly relativistic limit, the emission from magnetised astrophysical plasma (e.g. synchrotron radiation and curvature radiation) are strongly linearly polarised. As such, the amount of circular polarisation can be considered as negligible. The linear polarisation properties are therefore determined by the inter-conversion between the two Stokes parameters of linear polarisation, Q and U. The inter- conversion between the linear and circular polarisations of the Stokes parameters are insignificant. Thus, it is sufficient to consider only the two linearly polarised Stokes parameters. This greatly simplifies the polarised radiative transfer calculations. In the absence of Faraday conversion, the polarised radiative transfer equation is simply       d Q 0 f Q   =     . (3.3) ds U − f 0 U − The Faraday rotation coefficient f, which is determined by the local number density (and energy distribution) of the free electrons and the line-of-sight component of the magnetic field, is therefore the only parameter. The transfer matrix does not have any diagonal elements, implying that the total linear polarisation is conserved along the propagation of radiation. Generally, thermal and non-thermal electrons co-exist in astrophysical environments. However, the inter-conversion between the two linearly polarised Stokes parameters is mostly determined by the thermal electrons, provided that the non- thermal relativistic electrons are relatively fewer in population. With only thermal electrons present, the Faraday rotation coefficient is given by

2 2 ωp cos θ ωB fth = 2 2 (3.4) c ωB ω ω − B

(Pacholczyk 1977), where ω = 2πν is the angular frequency of radiation, ωp = 2 1/2 (4πne,the /me) is the plasma frequency, ωB = (eB/mec) is the electron gyro- frequency, ne,th is the thermal electron number density, B is the magnetic ﬁeld strength and θ is the angle between the magnetic ﬁeld vector and the line-of-sight.

Here, c is the speed of light, e is the electron charge, and me is the electron mass. In the high-frequency limit (i.e. ω ω ), the Faraday rotation due to only thermal B electrons can be expressed as

3 1 e 2 fth = 2 4 ne,th Bk λ , (3.5) π mec 3.1. Rotation measure 74 where B = B cos θ is the magnetic ﬁeld along the line-of-sight and λ = 2πc/ω k | | is the wavelength of radiation. The corresponding expression for Faraday rotation due to only non-thermal electrons is

3 1 e 2 fnt = 2 4 ζ(p, γi) ne,nt Bk λ , (3.6) π mec where the factor (p 1)(p + 2) ln γi ζ(p, γi) = − 2 , (3.7) (p + 1) γi for p > 1, assuming an isotropic distribution of non-thermal electrons with a power- law energy spectrum of index p (Jones and O’Dell 1977). The number density of non-thermal electrons is ne,nt, and γi is their low-energy cut-oﬀ. In a plasma consisting of thermal electrons plus non-thermal electrons, the relative strength of their contributions to the Faraday rotation is therefore fnt ne,nt ζ(p, γi) , (3.8) fth ≈ ne,th provided that neither ne,nt nor ne,th correlates or anti-correlates signiﬁcantly with 2 Bk . 3.1.2 Derivation of rotation measure

From the restrictive polarised radiative transfer equation (3.3) which only has two Stokes linear polarisation components, it can easily be shown that the change in the linear polarisation angle along the line-of-sight is

dϕ 1 1 dU dQ f = Q U = . (3.9) ds 2 U 2 + Q2 ds − ds 2

With only thermal electrons present in a sufficiently weak magnetic field where ω ω, a direct integration of equation (3.9) with f = f yields B th 3 Z s 2πe 0 0 0 ϕ(s) = ϕ0 + 2 2 ds ne,th(s ) Bk(s ) . (3.10) me(c ω) s0 Rotation measure (RM) is defined as

= (∆ϕ)λ−2 = (ϕ ϕ ) λ−2 . (3.11) R − 0 2A similar relation was given in Jones and O’Dell(1977) for the relative contributions of relativistic and thermal electrons to the Faraday rotation. Their relation is expressed in terms of the spectral index α of the optically thin power-law synchrotron spectrum. The relation in (3.8) is expressed in terms of the power-law index p of the electron energy distribution, which is intrinsic to the magneto-ionic medium. Note that α = (p − 1)/2. 3.2. Rotation measure ﬂuctuations 75

The polarised radiative transfer equations (3.1), (3.2) and (3.3) are linear, thus the contributions to the Faraday rotation coeﬃcient by a collection of thermal and non-thermal electrons are additive. The RM for radiation traversing a magnetised plasma between an interval s0 and s is therefore

3 Z s e 0 0 0 0 (s) = 2 4 ds ne(s ) Θ(s ) Bk(s ) , (3.12) R 2πmec s0 where ne is the total electron number density, and Θ(s) = 1 Υ(s) [1 ζ(p, γi)] is − − s the weighting factor of ne contributing to the Faraday rotation eﬀect, accounting for both thermal and non-thermal electron populations, with Υ(s) as the local fraction of non-thermal electrons. If only thermal electrons are present, Υ(s) = 0 such that Θ(s) = 1, hence recovering the widely-used formula in RM analysis of magnetised astrophysical media (see e.g. Carilli and Taylor 2002):

Z s 0 0 0 ds ne,th(s ) Bk(s ) −2 (s) = 0.812 −3 radm . (3.13) R s0 pc cm µG 3.2 Rotation measure ﬂuctuations 3.2.1 Computing rotation measure in a discrete lattice

In practical calculations, polarised radiative transfer in an inhomogeneous medium often require sampling the medium into discrete segments in which the variations in physical properties are suﬃciently small. With this condition satisﬁed, the radiation propagation path length L can be divided into N intervals of length ∆s, i.e. L = PN i=1 ∆s(i). The integral in equation (3.12) can be approximated by summing contributions from all segments

N e3 X (s) = ∆s(i) n (i) Θ(i) B (i) , (3.14) R 2πm2c4 e k e i=1 where ne, Θ and Bk are evaluated at the centre of each interval, si. If the magnetic ﬁelds have uniform strengths and no preferential orientations, Bk will have a symmetric probability distribution: P (B ) = P ( B ). With ∆s > 0, n 0 and k − k e > Θ [ 0, 1 ], the symmetry in the probability distribution of B implies that the ∈ k expectation value of RM is

N e3 X = ∆s ne Θ Bk = 0 , (3.15) hRi 2πm2c4 h i i e i=1 where ... denotes the ensemble average of the variables. h i 3.2. Rotation measure ﬂuctuations 76

If ne, Θ and Bk are incoherent among the intervals ∆s, then ne, Θ and Bk are the only independent variables for computing the RM of a cell3 deﬁned by an interval. For a medium that does not evolve during the radiation’s propagation, ne,

Θ, Bk, and their products are exchangeable variables. Under the ergodic condition, the ensemble averages of independent and exchangeable variables can be replaced by the averages of over the path length, i.e. for a suﬃciently large N,

N 1 X X = X(s ) = X(s ) . (3.16) h i N j h j is j=1

Thus, e3 = 2 4 N ∆s ne Θ Bk s = 0 . (3.17) hRi 2πmec h i

Moreover, if there is no correlation between B, ne, and Θ,

e3 = 2 4 N ∆s s ne s Θ s Bk s = 0 . (3.18) hRi 2πmec h i h i h i h i 3.2.2 Rotation measure ﬂuctuations as a restrictive autoregression (AR) process

Note that an observable ˜ on the lattice grid k in an AR(1) (autoregression of Ok order one) process on a 1-D lattice is given by the recursive relation:

˜ = ρ ˜ − + ε Ok Ok 1 k = ρ ρ ˜ − + ε − + ε Ok 2 k 1 k

······ m−1 m X j = ρ ˜ − + ρ ε − (3.19) Ok m k j j=0

(see e.g. Box and Jenkins 1976; Anderson 1976; Grunwald et al. 1996), where ρ is a parameter, and εk is an iid (independent, identically distributed) variable with an expectation value E(ε ) = ε = 0 and a variance Var(ε ) = [σ(ε )]2. For a finite k h ki k k or semi-infinite lattice, which is truncated at j = 0, at which the observable ˜ is O0 well defined, equation (3.19) can be re-written as

k X ˜ = ρk ˜ + ρk−j ε . (3.20) Ok O0 j j=1

3The terms “cell” and “voxel” are used interchangeably in this thesis. 3.2. Rotation measure ﬂuctuations 77

For a polarised radiation’s propagation path consisting of N segments with approximately coherent Faraday rotation properties, the polarisation angle at the end of the k-th segment is given by

ϕk = ϕk−1 + ∆ϕk

= ϕk−2 + ∆ϕk−1 + ∆ϕk

······ k X = ϕ0 + ∆ϕj , (3.21) j=1

th where ∆ϕk is the rotation of the polarisation angle in the k segment, and the polarisation angle measured by the observer is simply ϕN . Comparing equations (3.21) and (3.20) reveals that the evolution of the polarisation angle along the radiation’s propagation is an AR(1) process with a constant parameter ρ = 1, provided that ∆ϕ = 0 and that Var(∆ϕ ) is well deﬁned and computable. An AR(1) process h ji j is a Markov process (see Anderson 1976), and an AR(1) process with ρ = 1 is also known as a simple random-walk.

The rotation measure across the propagation path of the radiation is = R N (ϕ ϕ )λ−2. Hence, from equation (3.17), N − 0 N X ∆ϕ = λ2 = 0 . (3.22) j hRis j=1 s

As the expectation value and the variance of (ϕ ϕ ) are N − 0

E ϕ ϕ = 0 ; (3.23) N − 0 Var ϕ ϕ = N σ2 , (3.24) N − 0 respectively, with σ2 = ∆ϕ 2 , the standard deviation of in the radiation’s j R propagation direction is therefore

2 1/2 −2 σR = √N ∆ϕj λ 3 e √N 2 2 2 2 1/2 = 2 4 ∆s ne Θ Bk 2πmec 3 e √N 2 2 2 2 1/2 = 2 4 ∆s ne Θ Bk s . (3.25) 2πmec Note that the rotation measure ﬂuctuation along a radiation propagation path consisting of coherent segments is proportional to the square root of the number of 3.2. Rotation measure ﬂuctuations 78

p the segments (√N = L/ ∆s ), a characteristic of a simple random-walk process, h is where the root mean square displacement is proportional to the square root of the number of steps. Here, the root mean square of properties within a step size is 2 2 2 2 1/2 ∆s ne Θ Bk s . In the speciﬁc condition that the interval segments have equal length, ∆s, and

(ne Θ) does not vary along the line-of-sight, equation (3.25) becomes

3 r e L 2 1/2 σR = 2 4 ∆s ne Θ Bk s . (3.26) 2πmec ∆s

A similar but more rigorous expression can be obtained if there is no correlation between electron number density and the magnetic ﬁelds. In this case, equation (3.25) becomes 3 r e L 2 1/2 σR = 2 4 ∆s ne Θ Bk s , (3.27) 2πmec ∆s with ne Θ denoting the mean value of (ne Θ). Additionally, with the presence of only thermal electrons, then Θ = 1 uniformly, and r e3 L σR = 2 4 ∆s ne,th Bkrms 2πmec ∆s r L ∆s n Bkrms = 0.812 e,th rad m−2 . (3.28) ∆s pc cm−3 µG

Most observational and numerical studies use either one of the expressions given in equations (3.27) or (3.28) in their RM fluctuation analyses. These include investigations of magnetic fields in galaxy clusters or in large-scale structures (e.g. Sokoloff et al. 1998; Blasi et al. 1999; Dolag et al. 2001; Govoni and Feretti 2004; Subramanian et al. 2006; Cho and Ryu 2009; Sur 2019). Note that the two expressions above are not always explicitly distinguished in studies of RM fluctuations.

The σR derivations from equation (3.25) to equations (3.26), (3.27) and (3.28) rely on subtly different assumptions regarding the electron density spatial distributions and their relation or correlation with the magnetic fields. For instance, it matters whether local quantities are multiplied before spatial averaging, or averaged separately then multiplied. Note also that, in reality, the condition of constant electron number density, or/and the condition of electron number density and magnetic field being uncorrelated, are generally not satisfied. One should therefore bear in mind which underlying assumptions have been used, and they should be stated explicitly when interpreting the magnetic field structures using the observed RM statistics. 3.2. Rotation measure fluctuations 79

Furthermore, while σR is observed on the sky plane, it is calculated over the radiation’s propagation path, with the application of a random walk model along the line-of-sight and invoking other explicit assumptions made above. 3.2.3 Remarks on the ﬂuctuations along the line-of-sight and across the sky plane

The statistics of observed polarisation across a sky plane depend on the spatial variations of the magnetic fields and the properties of the magneto-ionic plasma along the line-of-sight (specified by the ray propagation unit vector kˆ). On cosmological scales, the radiative transfer along a line-of-sight is also the radiative transfer from the past to the present. Therefore, the transfer of polarised radiation as it reaches the sky plane perpendicular to kˆ is affected by the cosmological evolution of both the intervening magnetic fields and the magneto-ionic plasma. More specifically, the spatial properties of the magnetic fields and the magneto- ionic plasma may not be independent from their temporal properties as the Universe evolves. There is also no guarantee that the fluctuations in the magnetic fields and the fluctuations in the plasma properties are statistically identical. In the context of polarised radiative transfer, it is the convolution between the magnetic field properties and the plasma properties that determines the observables on the sky plane. For an all-sky polarisation map, the common observables are, for e.g., the fluctuations in the degree of polarisation and the fluctuations in the orientation of the polarisation vectors. The statistical fluctuations of an observable x are usually determined by the variables observed on the sky plane. These fluctuations are not identical to the underlying fluctuations along the line-of-sight. Suppose there are two independent orthogonal components, where one is parallel ( ) to kˆ, and the k other perpendicular ( ) to kˆ. Clearly, the relation σ ⊥ = σ does not always ⊥ x, x, k hold. Furthermore, there are usually two correlation lengths, `k and `⊥, for each plasma quantity, e.g. the electron number density ne (which is a scalar) and the magnetic field B (which is a vector). This leads to a serious question: how valid is the correlation length derived from the polarisation statistics across the sky plane be treated as the correlation length scale over which the magnetic fields vary spatially along the line-of-sight or, alternatively, as the length scale over which the magnetic fields evolve over 3.2. Rotation measure fluctuations 80 cosmological time? Take the rotation of the polarisation angle as an example. This observable depends on the number density of the free electrons and the magnetic field strength along the line-of-sight Bk. The perpendicular component of the magnetic

field strength B⊥ is irrelevant here. Therefore, the fluctuations in the polarisation are the result of the fluctuations in the thermal electron number density ne,th and their energy distributions, as well as the fluctuations in the line-of-sight magnetic

field strength Bk. Moreover, the fluctuations of polarisation along a ray cannot be observed directly. For an all-sky observation, a polarisation map is typically generated on the celestial sphere, which represents the polarisation signatures at the end-points of a bundle of rays. Assuming these rays are independent, the variations seen in their polarisation signatures, such as the RM fluctuations, can be statistically uniform across the sky, even when the magneto-ionic plasma has no spatial structures at any cosmological epoch. The observed RM fluctuations among the rays actually reflect the convolution of the fluctuations in B and n along the line-of-sight. Hence, | k| e,th the RM fluctuations are not quite the signatures left by the spatial structures in a slice of space perpendicular to the ray propagation vector kˆ.

There is also another subtle, yet significant, issue when assessing the local variations in polarisation along a ray. Consider that the electron number density and its energy spectrum are uniform in space and do not change with time. Therefore the rotation of the polarisation angle only depends on B = B cos θ, where θ = | k| | | cos−1(kˆ Bˆ) and Bˆ = B/ B . Magnetic fields are vectors, so there are two aspects · | | in the magnetic field fluctuations: one related to the field strength (magnitude), and the other to the field orientation. The fluctuations in the field magnitude and the fluctuations in the field orientation are not necessarily due to the same physical process.

For instance, the spatial variations of the magnetic field orientation Bˆ can be a consequence of the presence of local magnetic sub-domains within the plasma. The variations in the magnetic field strength along the line-of-sight B are usually | k| caused by the changes in the global magnetic energy density. This particular effect is associated with the co-evolution of magnetic fields and structure formation of the Universe. The fluctuations in Bˆ and the fluctuations in B can therefore operate | | on different characteristic length scales and time scales. This causes ambiguity 3.3. Solution to the polarised radiative transfer equation 81 in the interpretation of polarisation fluctuations for cosmic magnetism. Thus, the assumption for σ|B| k or ⊥ = σBˆ k or ⊥ does not universally hold and has to be treated with caution in applications.

3.3 Solution to the polarised radiative transfer equation

In the absence of scattering, the radiative transfer equation is simply

dI ν = κ I + , (3.29) ds − ν ν ν where Iν is the specific intensity of radiation at frequency ν for a location s along the line-of-sight, and κν and ν are the specific absorption and emission coefficients respectively. The transfer equation can be expressed in terms of the optical depth,

τν, such that dIν = Iν + Sν , (3.30) dτν − where S /κ is the source function, and the diﬀerential optical depth is deﬁned ν ≡ ν ν as dτν = κν ds. Integration of equation 3.30 gives the formal solution

τν Z 0 −τν 0 0 −(τν −τν ) Iν(τν) = Iν(0) e + dτν Sν(τν) e , (3.31) 0 where Iν(0) is the initial speciﬁc intensity (e.g. Rybicki and Lightman 1979).

Consider the Stokes parameters of polarisation Iν = [ Iν,Qν,Uν,Vν ]. The subscript ν will henceforth be implicit for simplicity. In a homogeneous medium, the polarised radiative transfer equation is         I κ q u v I I                 d  Q   q κ f g   Q   Q    =  −    +   , ds   −        U   u f κ h   U   U     −      V v g h κ V − V which is the same as equation 3.1. In a local coordinate system where the magnetic

ﬁeld lies in the yz plane and the radiation propagates in the z-direction, U = 0, and u = g = 0. The polarised radiative transfer equation therefore becomes         I κ q 0 v I I                 d  Q   q κ f 0   Q   Q    =     +   . (3.32) ds   −        U   0 f κ h   U   0     −      V v 0 h κ V − V 3.4. Absorption, emission and Faraday mixing coeﬃcients 82

The solutions to it are −Ks I = IP + IH e , (3.33) in which the particular solution IP satisﬁes       κ q 0 v IP I              q κ f 0   QP   Q      =   , (3.34)        0 f κ h   UP   0   −      v 0 h κ V − P V and the general solution IH satisﬁes

I0 = qQ vV ; (3.35) H − H − H Q0 = qI fU ; (3.36) H − H − H U 0 = fQ hV ; (3.37) H H − H V 0 = vI + hU (3.38) H − H H

(e.g. Pacholczyk and Swihart 1975; Pacholczyk 1977; Jones and O’Dell 1977; Meg- gitt and Wickramasinghe 1982).

3.4 Absorption, emission and Faraday mixing coeﬃ- cients 3.4.1 Thermal plasma

The transfer coeﬃcients for a magnetised thermal plasma are

2 ω ωB cos θ f = p ; (3.39) th c(ω2 ω2 ) − B ω2ω2 sin2 θ h = p B ; (3.40) th 2cω(ω2 ω2 ) − B ω2(2ω4 + 2ω2ω2 3ω2ω2 sin2 θ + ω4 sin2 θ) κ = p B − B B ν ; (3.41) th 2cω2(ω2 ω2 )2 c − B ω2ω2 sin2 θ(3ω2 ω2 ) q = p B − B ν ; (3.42) th 2cω2(ω2 ω2 )2 c − B 2 2ωω ωB cos θ v = p ν (3.43) th c(ω2 ω2 )2 c − B (Sazonov 1969; Pacholczyk and Swihart 1975; Pacholczyk 1977; Meggitt and Wick- ramasinghe 1982; also consistent with Wickramasinghe and Meggitt 1985, after 2 correcting the sign of qth and omitting ω from the denominator of vth). Here 3.4. Absorption, emission and Faraday mixing coeﬃcients 83

p 2 ω = 2πν is the angular frequency of the radiation, ωp = 4πnee /me is the plasma frequency, ωB = (eB/mec) is the electron gyro-frequency, and θ is the angle between the magnetic ﬁeld vector and the direction of wave propagation. The electron-ion collisional frequency is given by

4e4n √2π ν = e ln Λ , (3.44) c 3/2 3√me(kBTe) where T is the electron temperature in thermal equilibrium. For ω ω , the e p Coulomb factor is

5 1 2 2 2 kBTe kBTe 5 Λ = 2 (Te 3.16 10 K) ; (3.45) 1.781 me e ω ≤ × 8πk T Λ = B e (T > 3.16 105 K) (3.46) 1.781hω e ×

(Lang 1974, after correcting for a typo on page 46, Table 2; Pacholczyk 1977). In a local thermal equilibrium, Kirchhoﬀ’s law applies, and hence

th I = κth Bω ; (3.47) th Q = qth Bω ; (3.48) th V = vth Bω , (3.49) where, in the limit of low frequencies, the Planck function can be approximated by the Rayleigh-Jeans intensity ω2k T B = B e (3.50) ω 2π2c2 3 (Kawabata 1964, after correcting for a misprint of 8π in the denominator of Bω; Meggitt and Wickramasinghe 1982, after correcting for a misprint of 4π3 in the th denominator of Bω). Here vth and V are positive, following the same convention in Chan et al.(2019).

3.4.2 Non-thermal plasma

For a power-law distribution of relativistic electrons, one can express

dn = [n γp]γ−p Θ(γ γ ) g(Ψ) dγ dΩ , (3.51) re γ − i Ψ where [n γp] is the normalisation constant4, Θ(γ γ ) is the step function, and g(Ψ) γ − i R is the pitch-angle distribution, normalised to g(Ψ) dΩΨ = 1 (Jones and O’Dell

4 p Note that [nγ γ ] is analogous to the normalisation constant C in Chapters2 and4. 3.4. Absorption, emission and Faraday mixing coeﬃcients 84

1977). As such, the number density of relativistic electrons is

− Z ∞ [n γp]γ(1 p) n = dγ [n γp]γ−p = γ i (p > 1) , (3.52) re γ p 1 γi − where γi is the low-energy cutoﬀ of electrons. The transfer coeﬃcients for the non-thermal plasma are

ω 2 − α + 2 d ln g(θ) f = f κ B⊥ (ln γ ) γ 2(α+1) cot θ 1 + ; (3.53) nt α ⊥ ω i i 2α + 3 d ln (sin θ) " α−1/2 # ω 3 − − 1 (ω /ω) h = h κ B⊥ γ (2α 1) − i ; (3.54) nt α ⊥ ω i α 1/2 − ω α+5/2 κ = κ κ B⊥ ; (3.55) nt α ⊥ ω ω α+5/2 q = q κ B⊥ ; (3.56) nt α ⊥ ω ω α+3 1 d ln g(θ) v = v κ B⊥ cot θ 1 + , (3.57) nt α ⊥ ω 2α + 3 d ln (sin θ)

2 p for α > 1/2, where ω γ ω , ω = ω sin θ, and κ⊥ = (2πr c/ω )[4πg(θ)][n γ ] i ≡ i B⊥ B⊥ B e B⊥ γ for g(θ) = 1/4π (assuming an isotropic electron distribution). The dimensionless functions are

α + 3/2 f = 2 ; (3.58) α α + 1

hα = 1 ; (3.59) 3α+1 α 25 α 5 κ = Γ + Γ + ; (3.60) α 4 2 12 2 12 α + 3/2 q = κ ; (3.61) α α + 13/6 α 3α+1/2 α + 2 3 α 7 α 5 v = α + Γ + Γ + , (3.62) α 2 α + 1 2 2 6 2 6 where the spectral index of radiation α is related to the power-law index of the energy distribution of electrons p via α = (p 1)/2. The emission coeﬃcients for − the Stokes I, Q, and V parameters are

ω α nt = I B⊥ ; (3.63) I α ⊥ ω ω α nt = Q B⊥ ; (3.64) Q α ⊥ ω ω α+1/2 1 d ln g(θ) nt = V B⊥ cot θ 1 + , (3.65) V α ⊥ ω 2α + 3 d ln (sin θ)

2 p 2 2 where ⊥ = (mec )(re/2πc) ωB⊥ [4πg(θ)][nγγ ], and re = e /mec is the classical 3.5. Numerical algorithms and veriﬁcations 85 radius of the electron. The corresponding dimensionless functions are

3α+1/2 α 11 α 1 I = Γ + Γ + ; (3.66) α 4(α + 1) 2 6 2 6 (α + 1) Q = I ; (3.67) α (α + 5/3) α 3α (α + 3/2) α 11 α 7 V = Γ + Γ + (3.68) α 2 (α + 1/2) 2 12 2 12

(Jones and O’Dell 1977; Chan et al. 2019).

3.5 Numerical algorithms and veriﬁcations

The polarised radiative transfer equation (3.32) is a set of four coupled first-order differential equations, which can be solved numerically by direct integration. The numerical scheme adopted in Chan et al.(2019) and On et al.(2019) for the calculations of polarised radio emission from galaxy clusters is described as follows. Consider the propagation of a ray in a 4-Stokes representation. The properties of the incident ray define the initial conditions of polarisation. Let the ray travel along a line-of-sight distance s discretised into Ncell number of cells. The size of each cell is therefore s/Ncell. For each cell, the change in the Stokes parameters of polarisation is calculated using the 4th-order Runge-Kutta (RK – Runge 1895; Kutta 1901) integrator.5 In the first cell, the Stokes parameters are initialised as

[ I0,Q0,U0,V0 ]. With this, the output Stokes parameters are computed by solving the polarised radiative transfer equation numerically. The output of one cell becomes the input of the next adjacent cell. The final transmitted ray seen on the sky plane is thus the net Stokes parameters of polarisation [ If ,Qf ,Uf ,Vf ]. There are subtle issues in the practical calculations, even though the numerical scheme for the solution of the polarised radiative transfer equation seems conceptu- ally straightforward and simple. These issues make the numerical computation for specific astrophysical applications challenging. The transfer coefficients are determined by the line-of-sight properties of the magneto-ionic plasma, where one of the important variables is the electron number densities. On local scales, the electron number density can either be thermal or non-thermal, and this assumption greatly

5This work uses the Fortran implementation of the 4th-order RK method in Press et al.(1992). For a comparison, similar results have also been obtained using the Dormand & Prince (DP – Dor- mand and Prince 1980) integrator and the Linear Algebra PACKage (LAPACK). The DP integrator is a 5th-order RK method and its Fortran subroutine in Hairer et al.(2008) is used here. 3.5. Numerical algorithms and veriﬁcations 86 reduces the computational demand in the polarised radiative transfer calculations. However, for calculations over large scales, the above assumption clearly breaks down.

Within the diffuse ICM, thermal and non-thermal electrons co-exist. This leads to a hybrid of thermal and non-thermal interactions and emissions. The transfer coefficients in Section 3.4 depend, not only on the electron number density, but also on their energy spectrum, as well as on the magnetic field properties. The co- existence of thermal and non-thermal electrons, with the added complexity from the magnetic field contributions, would lead to situations where the transfer coefficients can differ by several orders of magnitude. In such a case, the polarised radiative transfer equations become stiff. Stiffness occurs when the numerical solution being sought is varying slowly, but there are also nearby solutions varying quickly. The stiff system has proven problematic at times in this work.

In order to mitigate the stiffness, the following algorithm is implemented. The light travel distance is used as a scaling factor for the transfer coefficients, such that their values become as close to 1 as possible (see Chan et al. 2019, for more details). This formulation provides numerical stability for calculations with the integration solver. However, it makes the application of the RK family of integrators computationally expensive. This is because, to reduce stiffness, an integration step size is chosen for numerical stability rather than numerical accuracy. The computational cost can be offset by implementing an alternative solver in the cell where the stiffness is severe. A useful solver is the Linear Algebra Package (LAPACK – Anderson et al. 1999). The LAPACK routines diagonalise the transfer matrix and solve for the eigen- values and eigenvectors. The eigensolutions are then transformed back to obtain the Stokes parameters [ I, Q, U, V ]. This work adopts the LAPACK implementation for some of the polarised radiative transfer calculations.

The verification and validation of the Stokes ray-tracing computational algorithms are carried out for a number of test problems designed in an astrophysical context. In particular, the numerical accuracy in calculations of the variables along the ray is assessed. Their applications to the interpretation of polarisation effects in the radio wavelengths are also discussed. Here are two examples of the test problems constructed on galaxy cluster scales. 3.5. Numerical algorithms and verifications 87

The first test considers a mock spherical galaxy cluster6. The polarised radiative transfer equations are solved to compute the polarisation of radio synchrotron radiation from the cluster. The mock cluster is a spherical volume filled with magneto- ionic hydrogen plasma, with a virial radius of r 2.64 Mpc. The electron number vir ≈ density in the ICM decreases radially. The strength of the magnetic field throughout the ICM is uniform, and its field vectors have no preferential orientation. The magnetic field is made solenoidal through a divergence-free filter (see Chapter4 for more details). The normalised values of the thermal electron number density and the magnetic field strength are chosen to be of similar orders to those observed in galaxy clusters: n = 10−3 cm−3 and B = 1 µG. For the non-thermal h e,thi rms electrons, their energy density is set to be 1% of that in the thermal electrons.

This mock galaxy cluster is placed inside a simulation box, which is discretised into (512)3 voxels. The radiative transfer calculations are carried for each of the voxels. The initial conditions are defined for the incident rays with Stokes parameters [ I0,Q0,U0,V0 ] = [ 0, 0, 0, 0 ]. The radiation frequency is set at ν = 1.4 GHz, which is similar to the observing bandwidths of many radio telescopes (e.g. Arecibo, FAST, ATCA, and SKA and its precursors). The rays are traced cell by cell through the ICM, along the line-of-sight z. In every cell, the opacity calculator computes the thermal and non-thermal absorption, emission and Faraday mixing coefficients. The polarised radiative transfer equations are then solved directly using the 4th-order RK method or the DP method. In voxels with extreme stiffness, the LAPACK solver would be switched on.

The Stoke parameters [ If ,Qf ,Uf ,Vf ] of the line-of-sight rays emerging from the ﬁnal 512 512 cells are used to calculate the total Stokes intensity I , the × tot p 2 2 linearly polarised Stokes intensity Ilin = Q + U , and the circularly polarised Stokes intensity I = V , in the ﬁeld-of-view of the sky plane. The results are cir | | shown in Figure 3.1. It is within expectations for the total Stokes intensity to peak at the cluster centre and radially decrease outwards, as it follows the change in cluster density. Notably, the circularly polarised intensity is weaker compared to the total intensity and the linearly polarised intensity. This is because the relativistic electrons in the ICM produce synchrotron radiation that is mainly linearly polarised.

6Note that this cluster is not generated from the GCMHD+ simulation. 3.5. Numerical algorithms and veriﬁcations 88

Therefore, only a small amount of circular polarisation is being produced through Faraday conversion from the linear polarisation.

In the second test, a small galaxy cluster is used from the high-resolution, magnetic divergence-cleaned GCMHD+ simulations (see Barnes et al. 2018, for details of the cluster simulations). The cluster has a virial radius of r 1.44 Mpc and a vir ≈ 13 gas mass of m 10 M . The magnetic ﬁeld and the electron number density in gas ∼ the ICM are determined self-consistently throughout the simulations. The magnetic ﬁeld strength peaks at 2.5 µG near the cluster centre. The mean thermal electron ∼ number density is approximately 10−5 cm−3 (Barnes et al. 2018; see also Chapter 4 for more details on the X-ray and radio properties of galaxy clusters). The ratio of energy density of the non-thermal electrons to the energy density of the thermal electrons is set at 0.01.

Table 3.1 shows the transfer coefficients and emission coefficients from the thermal and non-thermal electron contributions in this galaxy cluster. The absorption and emission processes are contributed mainly by the non-thermal electrons. The non-thermal coefficients for Faraday rotation fnt and Faraday conversion hnt are sensitive to the low-energy cutoff of electrons γi. Generally, fnt decreases as γi in- 2 creases, following (ln γi)/γi (see equation 3.53). This implies that Faraday rotation weakens when electrons become relativistic. On the other hand, hnt increases from low to intermediate γi, but declines steeply once γi rises higher (see equation 3.54, and also e.g. Figure 2, Huang and Shcherbakov 2011). This indicates that, in the relativistic limit, Faraday conversion also becomes less important. Thus, as γ increases, the radiation mechanism converts from being mainly circularly-polarised cyclotron to mainly linearly-polarised synchrotron.

Notably, the minimum emissivity in V is negative for both cases of thermal and non-thermal electrons in this simulated galaxy cluster. This occurs when the magnetic ﬁeld vector and the wavevector are largely misaligned at an angle θ. More th speciﬁcally, V < 0 when vth < 0. This occurs when cos θ < 0 between π/2 < θ < nt 3π/2. Meanwhile, V < 0 when cot θ < 0 at π/2 < θ < π and 3π/2 < θ < 2π.

Hence, a negative V in this case may be interpreted as a fraction of the total emission being directed away from the observer. Furthermore, it is clear that Faraday rotation dominates over other radiative processes in this galaxy cluster. These results can be generalised: for electron number densities and magnetic ﬁelds typical of the ICM, 3.5. Numerical algorithms and veriﬁcations 89

Fig. 3.1: Radio synchrotron intensity and polarisation maps calculated along the line-of- sight z for a mock spherical galaxy cluster with r 2.64 Mpc at the present-day Universe. vir ≈ The panels from top to bottom show the total intensity Itot, the linearly polarised intensity p I = Q2 + U 2, and the circularly polarised intensity I = V . lin cir | | 3.6. Summary 90

Coeﬃcient Thermal Non-thermal Minimum Maximum Minimum Maximum f 4.026 10−22 5.172 10−22 3.746 10−31 4.830 10−31 − × × − × × h 5.652 10−53 1.180 10−30 5.501 10−35 7.621 10−59 × × − × − × κ 7.984 10−42 5.583 10−32 6.065 10−61 4.768 10−36 × × × × q 3.545 10−75 5.504 10−51 4.603 10−61 3.619 10−36 × × × × v 1.121 10−42 1.236 10−42 2.415 10−40 2.009 10−40 − × × − × × 1.481 10−53 1.623 10−44 3.374 10−61 7.353 10−41 I × × × × 1.790 10−87 8.334 10−62 2.382 10−61 5.191 10−41 Q × × × × 1.671 10−53 1.515 10−53 2.331 10−45 2.466 10−45 V − × × − × ×

Table 3.1: Summary of transfer coefficients and emission coefficients from thermal and non- thermal electron contributions through a high-resolution galaxy cluster from the GCMHD+ simulations (Barnes et al. 2012, 2018). The coefficients quantify the effects of Faraday rotation f; Faraday conversion h; absorption κ , q, and v; and emission I , Q , and V . 1 The absorption and Faraday mixing coefficients have units of cm− , while the emission 1 3 1 1 coefficients have units of erg s− cm− Hz− str− . The parameters for the non-thermal calculations are chosen at ν = 1.4 GHz, γi = 30, and α = 0.6.

Faraday rotation is signiﬁcant on the Mpc scales at a frequency of about 1.4 GHz. The Stokes intensity and polarisation maps obtained from this test are also presented in Chan et al.(2019).

3.6 Summary

This chapter discusses the theoretical framework of polarised radiative transfer, fluctuations, and large-scale magnetic fields. First of all, the polarised radiative transfer equation is presented and used to derive the standard rotation measure (RM) equation under several conditions. These conditions are: (i) the absence of emission and absorption processes, (ii) the significance of linear polarisation over circular polarisation, and (iii) the sole contribution of thermal electrons to the RM in an environment with a sufficiently weak magnetic field. The analyses of RM fluctuations are then examined in the context of polarised radiative transfer.

There are two expressions commonly used to quantify the RM fluctuations, but they are not always explicitly distinguished in many studies of RM fluctua- 3.6. Summary 91 tions, including those probing the magnetic fields in galaxy clusters and larger-scale structures. In this work, the two expressions are shown to be derived based on the theory of simple random-walk but with subtly different conditions. The conditions are generally specified by the spatial distributions of electron density and their relation with the magnetic fields. Usually, assumptions are made on having a constant electron number density and/or a non-correlation between the density and the magnetic field. However, these assumptions do not generally hold in real astrophysical environments. It also matters whether the local quantities are multiplied before spatial averaging, or averaged independently before multiplication. Notably, the fluctuations along the line-of-sight and across the sky plane are generally dissimilar. Therefore it is important to bear in mind which underlying assumptions have been made before using the statistics of RM observations to interpret the magnetic field properties. This chapter also presents the formal solution to the polarised radiative transfer (PRT) equation. The absorption, emission and Faraday mixing coefficients, in the cases of thermal and non-thermal plasma, are verified. The transfer and emission coefficients are essential for the PRT calculations. The PRT equations are solved using a numerical scheme based on ray-tracing along a number of discretised voxels. The solution to the PRT is calculated using either the 4th-order Runge-Kutta method or the Dormand & Prince method. In severe cases of stiffness, the LAPACK implementation is called instead. Lastly, the Stokes ray-tracing computational algorithms are verified and vali- dated for a number of astrophysical test problems. This work shows two examples: a mock spherical galaxy cluster, and a small galaxy cluster from the high-resolution GCMHD+ simulations. The first test shows that circular polarisation is relatively weak in the mock galaxy cluster. This may be attributed to the radio synchrotron radiation being mainly linearly polarised, so only a small amount of circular polarisation can be generated through Faraday conversion. Meanwhile, the second test shows that Faraday rotation mainly dominates over other radiative processes in the simulated galaxy cluster, and it can be a significant effect on Mpc scales. Hence, these works, on building a solid theoretical foundation and combining statistical methods for data analysis, are crucial for making a reliable interpretation of radio observations. 92

Chapter 4

Magnetism in galaxy clusters

This chapter presents results which have been published in whole or in part in: (i) Alvina Y. L. On, Jennifer Y. H. Chan, Kinwah Wu, Curtis J. Saxton and Lidia van Driel-Gesztelyi, “Polarised radiative transfer, rotation measure fluctuations and large-scale magnetic fields”, December 2019, MNRAS 490 (2): 1697–1713, doi:10.1093/mnras/stz2683. Published by Oxford University Press on behalf of the Royal Astronomical Society. All rights reserved. I am the lead author of this paper and wrote most of the manuscript. I formulated the research project and identified the majority of the scientific methods involved in the analysis. I also derived parts of the analytical formulation, built most of the simulated models, and computed the numerical RMF simulations. I also led the interpretation of the statistical results from the simulated RMF analyses. (ii) David J. Barnes, Alvina Y. L. On, Kinwah Wu and Daisuke Kawata, “SPMHD simulations of structure formation”, May 2018, MNRAS 476 (3): 2890–2904, doi:10.1093/mnras/sty400. Published by Oxford University Press on behalf of the Royal Astronomical Society. All rights reserved. I am a co-author of this paper. I carried out all of the radiative transfer calculations for the X-ray and radio emissions by galaxy clusters from the GCMHD+ simulations. I performed the correlation analyses between the X-ray and radio emissions and contributed to the interpretation of the results. I also conducted consistency and accuracy checks for the science, and assisted the lead author in some parts of the manuscript preparation.

The previous chapter presents the theoretical framework of rotation measure, fluctuations and large-scale magnetic fields. Most of the current methods in rotation 93 measure fluctuation (RMF) analyses rely on a random walk model, in which the standard deviation of RMF provides a statistical measure of the magnetic field correlation length. This chapter discusses the usage of Faraday RM in diagnosing intracluster magnetic fields. Galaxy clusters are the largest virialised objects in the Universe. They are assembled under hierarchical structure formation processes, whereby smaller structures, such as galaxies and galaxy groups, coalesce first, then collapse and aggregate under gravity to build larger structures (e.g. Peebles and Yu 1970; Press and Schechter 1974; see e.g. Kravtsov and Borgani 2012, for review). The merger activities drive shocks and turbulence into the intracluster medium (ICM), which accelerate particles to relativistic energies and also heat up the ICM to temperatures of 107 108 K (e.g. Sarazin 1986; Fabian 1994; Burns 1998; Sanders − et al. 2009a,b; see e.g. Brunetti and Jones 2014, for review). In the presence of µG intracluster magnetic fields, the relativistic electrons emit synchrotron radiation, whereas the turbulent and/or shock heated ICM produces thermal bremmstrahlung X-rays (e.g. Meekins et al. 1971; Sarazin 2002; Zhuravleva et al. 2014, 2018; Hitomi Collaboration et al. 2016; see e.g. Feretti et al. 2012, for review). Over time, the 14 15 clusters relax, reaching typical masses of about 10 10 M and volumes of − approximately 100 Mpc3 (e.g. Zwicky 1933, 1937; see e.g. Ettori et al. 2013, for review).

There is growing evidence in a number of galaxy clusters of the presence of diffuse radio synchrotron sources, typically extending over scales of 100 kpc to > Mpc at the cluster centre (mini-halos or halos) and/or cluster periphery (relics) (e.g. Giovannini et al. 1993; Finoguenov et al. 2010; Zandanel et al. 2014; see e.g. Ferrari et al. 2008; Feretti et al. 2012, for reviews). These diffuse radio sources neither have an optical counterpart nor an obvious connection to galaxies in the cluster and are therefore often associated with the ICM. Their formation and energetics are usually attributed to recent cluster merger processes, with turbulence being the likely driving force behind regular, cluster-centric, mostly unpolarised radio halos; and shock acceleration at cluster outskirts for the elongated, irregular, rather strongly (up to 30%) polarised peripheral relics (e.g. Ensslin et al. 1998; Buote 2001; Govoni et al. 2004; Giacintucci et al. 2008; Iapichino et al. 2011; Kang et al. 2012; Donnert et al. 2013; Iapichino et al. 2017; Burke-Spolaor et al. 2017; see e.g. Brüggen et al. 2012, for review). 4.1. Variance of rotation measure fluctuations 94

Approximately 20 30% of massive, X-ray luminous clusters appear to host − radio halos of low surface brightness, typically of several µJy arcsec−2 at 1.4 GHz (see e.g. van Weeren et al. 2019, for a recent review). There is circumstantial evidence linking cluster merging and radio halos (e.g. Blasi 2001; Brunetti et al. 2001, 2004; Bacchi et al. 2003; Feretti et al. 2004; Brunetti et al. 2008; Cassano et al. 2010; Cantwell et al. 2016). Several clusters, e.g. 1E 0657-56, A2163, Coma, A2255, A2744, A2319 and 1RXS J0603.3+4214, show similar morphological halo features in their radio and X-ray images (e.g. Liang et al. 2000; Govoni et al. 2001a,b; Feretti et al. 2001; Brunetti et al. 2008; Brown and Rudnick 2011; Parekh et al. 2017; Rajpurohit et al. 2018; Cassano et al. 2019). Upon a local point-by-point comparison between the radio and X-ray surface brightness, the correlation is fairly described by a power-law (e.g. Govoni et al. 2001a; Feretti et al. 2001; Giacintucci et al. 2005; Cassano et al. 2006; Rajpurohit et al. 2018; Cova et al. 2019). This chapter is organised as follows. As outlined in Chapter3, Section 4.1 assesses the usage of RMF analyses for magnetic field diagnostics in the framework of polarised radiative transfer. Models of the ICM are generated with various magnetic field configurations and electron density distributions. These synthetic models are used to investigate how density fluctuations could affect the magnetic field correlation length inferred from the conventional RMF analyses. Section 4.2 presents galaxy cluster models from the cosmological MHD simulations, which are used to study the scaling relations and properties of Faraday RM, radio and X-ray emissions. Lastly, Section 4.3 summarises this chapter.

4.1 Variance of rotation measure ﬂuctuations

4.1.1 Assessing the rotation measure ﬂuctuation approach

The expression derived in Section 3.2.2, r L ∆s ne,th Bkrms −2 σR = 0.812 rad m , (4.1) ∆s pc cm−3 µG is commonly used in RM fluctuation analysis for probing the structures in large-scale magnetic fields. Here the formalism is assessed as to when it is justified and when it deserves caution. There are two variables in the expression: ne and B, whose spatial distribution properties are the focus of this assessment. The simulated RM fluctuations are computed using a Monte-Carlo method by considering Mpc cubes of 4.1. Variance of rotation measure fluctuations 95 thermal electrons and magnetic fields. The thermal electron number density (d) and magnetic field strength (b) are assigned independently with a specific distribution: either uniform (U), Gaussian (G), fractal (F) or log-normal (L). Each simulation is designated by a four-letter label. For example, ‘UdGb’ stands for uniformly distributed densities and Gaussian-distributed magnetic field strengths with random orientations. The Mersenne Twister (MT, Matsumoto and Nishimura 1998) is implemented to generate uniformly distributed pseudo-random numbers, Z ( 0, 1 ], ∈ which transform into the G-, F- and L-distributed variates accordingly. The cubes are discretised into 2563 voxels with an equal edge length of ∆s. Thus the statistical properties of the parameters are essentially averaged over a length scale specified by the voxel size. The magnetic fields and the thermal electrons are simulated according to the assigned distributions over the voxels. The distributions are normalised to be consistent with those observed in galaxy clusters: n = 10−3 cm−3, B = 1 µG and L = 1 Mpc (e.g. Cho and Ryu 2009). The h e,thi rms thermal electron number density and the magnetic field strength in the whole simulation cube, regardless of the magneto-ionic distribution, are 16777.216 cm−3 and 16777216 µG, respectively. This ensures consistency between the model cubes and enables meaningful comparisons to be made between the simulations. To compute the RM, the contributions along the lines-of-sight, x, y and z, are summed using the discretised expression of equation (3.13) in terms of lattice units (i, j, k),

X ∆s ne,th(i, j, k) B(i, j, k) −2 ⊥ = 0.812 rad m . (4.2) R pc cm−3 µG k k

The standard deviation ςR⊥ across the simulated sky plane is then computed and compared with the longitudinal standard deviation given in equation (3.28).

Modelling magnetic fields Consider magnetic fields with random orientations and no spatial correlation. The orientation of the magnetic fields is specified by the unit vectors: Bˆx = sin θ cos φ, Bˆ = sin θ sin φ and Bˆ = cos θ, with cos θ ( 1, 1 ] and φ ( 0, 2π ]. The field y z ∈ − ∈ strength, on the other hand, has a uniform, non-solenoidal (Ub*) or a uniform, solenoidal (Ub) or a Gaussian (Gb) distribution. The normalisation is such that the root mean square is Brms = 1 µG. Therefore, the line-of-sight rms component 4.1. Variance of rotation measure fluctuations 96 is B = B /√3 0.577 µG. The Gaussian distribution is generated using the krms rms ' Box-Muller (1958) transform in the usual Monte-Carlo simulations. The simulated magnetic fields are then cleaned in Fourier space with the application of a divergence- free ( B = 0) filter: B (k ) = (δ k k /k2) B˜ (k )(Balsara 1998). ∇ · i m ij − i j j m The divergence-free filter is constructed as follows. Consider a vector k (= ki), defining a reference axis in a vector space. Any other arbitrary vector X can be decomposed into two components, one parallel to and another one perpendicular (k) (k) (k) (k) to k: X = X + X , with k X 0 and k X = 0. Now introduce k ⊥ · k ≥ · ⊥ a projection operator (k), such that, X0 = (k) X = X(k). This projection P P ⊥ operator eliminates the longitudinal component of X, ensuring that k X0 = 0 for · any given X . A non-trivial example of (k) is (I kˆkˆ), where I is an identity P − operator and kˆ = k/ k , such that, | | h i k k k I kˆkˆ X = k δ i j X = 0 . (4.3) · − · i ij − k2 j

Magnetic fields in vacuum are solenoidal, i.e. divergence-free, satisfying B = ∇· ∂ B = 0. In Fourier space, the divergence-free relation is expressed as k B(k) = i i · kiBi = 0, which requires the field component parallel to k to vanish. Thus, the filter (I kˆkˆ) may be applied in Fourier space to build a divergence-free magnetic − field (with designated structural properties) from a generic initial simulated random vector field (with otherwise the same structural properties). The procedure is:

˜ (i) Construct a random ﬁeld B(k) (= B˜i(km)) according to the speciﬁed structural properties in Fourier space.

(ii) Apply the divergence-free ﬁlter, i.e. carry out the projection operation: B (k ) = (δ k k /k2)B˜ (k ). i m ij − i j j m

(iii) Use an inverse-Fourier transform on Bi(km) to obtain Bi(xm) in conﬁguration space.

As the filtering process removes the longitudinal part of the magnetic field in Fourier space, it reduces the total magnetic energy stored in the system. The Parseval’s (energy) Theorem, Z Z 2 2 d3x B(x) = d3k B(k) , (4.4) Vx Vk 4.1. Variance of rotation measure fluctuations 97 requires that the total magnetic energy is reduced by the same amount in configuration space as in Fourier space. With the divergence-free magnetic field given by B(k) = (I kˆkˆ) B˜(k), the energy density of the magnetic field is − 1 1 k k k k B 2 = δ i j δ i m B˜ B˜ 8π | | 8π ij − k2 im − k2 j m 1 1 2 = B˜ B˜ k B˜ 8π i i − k2 i i 1 2 = B˜ (1 µ2) , (4.5) 8π − where µ = kˆ B˜/ B˜ . For a randomly-oriented magnetic field in Fourier space, · | | 1 Z 1 2 1 µ2 = dµ 1 µ2 = . (4.6) − 2 −1 − 3

Hence, one-third of the magnetic energy density is filtered out. This is the expected amount when there is equipartition between the energies in the longitudinal component and the two orthogonal perpendicular components (the solenoidal components) of the initial “magnetic” field B˜. To recover the energy loss by the filtering process, p the field components are rescaled by a factor of 3/2. An inverse Fourier transform is then conducted to obtain the divergence-free (“solenoidal”) magnetic field in the configuration space. Note that the divergence-free filtering process also introduces a residual dipole, which has a preferred orientation, depending on how the filtering process is executed. A quick-fix solution1 to suppress this dipole structure is a superposition of three independent, orthogonal field realisations. The resultant field from the superposition p is then renormalised by a 1/3 scaling factor. Since the realisations prepared as such are divergence free in real (configuration) and Fourier space, their linear superpositions in real and Fourier space2 are also divergence free.

1A more proper treatment requires a superposition of three anti-parallel pairs of independent, orthogonal field realisations to completely remove the dipole. This process would then leave a residual quadrupole. Nonetheless the quick-fix solution is sufficient for the purpose of this demonstrative study. In practice, the divergence filtering is not always necessary before radiative transfer, as the magnetic field outputs by a detailed magneto-hydrodynamic simulation (see e.g. Marinacci et al. 2015, 2018; Barnes et al. 2018) should be divergence free, at least in principle.

2It is impossible to consider an infinite span of the configuration space while executing the Fourier transform. The restriction of the electron number density and magnetic field structure within a finite volume is equivalent to the introduction of a cubic window function to an infinite configuration 4.1. Variance of rotation measure fluctuations 98

Modelling free-electron number density Consider four model electron number density distributions: uniform (Ud), Gaussian (Gd), log-normal (Ld) and fractal (Fd). For the uniform distribution, the electron number density is set to be one unit in each cell. For the Gaussian distribution, a Box-Muller transform is applied with a standard deviation of 0.2 times the mean 10−3 cm−3, so that there is only a few negative numbers, which can be simply converted to positive by taking the absolute. The log-normal distribution is generated by taking the exponential function of Gaussian-distributed random numbers. For a fractal model, random phases (kx, ky, kz) are generated in Fourier space. Then, a power-law filter k −5/3 is applied to mimic a Kolmogorov-like turbulence spec- | | trum (1941b; 1941a). Simulations predict various kinds of turbulence in clusters and cosmic filaments (e.g. Iapichino et al. 2011). Scaling laws originally derived for incompressible media also turn out to be a good approximation for compressible turbulence in subsonic regions of real observed or numerically simulated IGM (see e.g. Schuecker et al. 2004; Miniati 2014; Nakwacki et al. 2016; White et al. 2019).3 In this case, frequency cutoffs as in Saxton et al.(2005) are imposed numerically: kmax = N/2 to prevent excessively sharp contrasts at voxel scale, and kmin = 8 to prevent any single density peak dominating. The inverse Fast Fourier Trans- form yields a fractal-like spatial structure with normally distributed local values . Dimensionless positive densities are obtained by taking exp (α / ), with N N Nmax the contrast factor α = 4 and as a fiducial maximum fluctuation (Elmegreen Nmax et al. 1989; Elmegreen 2002). Lastly various astrophysical configurations of thermal electron number densities are obtained by normalising n of each cube to h e,thi 10−3 cm−3.

RM dependence on the density and magnetic field structures Synthetic RM maps are calculated by integrating along lines-of-sight x, y and z using equation (4.2) through various distributions of thermal electron number densities and magnetic field strengths. The RM maps from the GdGb and UdUb* space. Thus, the density distribution and the magnetic field structure that are obtained in the Fourier representation are actually the convolutions of the cubic window function with the electron number density distribution and the magnetic field structure.

3The alternative extreme, of shock-compressed supersonic turbulence, yields steeper ∼ k−2.1 spectra; e.g. Lee et al.(1991); Federrath(2013). 4.1. Variance of rotation measure ﬂuctuations 99

Fig. 4.1: Synthetic RM maps of one realisation of GdGb (top) and UdUb* (bottom) in the xy (left), xz (middle) and yz (right) planes. distributions are indistinguishable from a simple eyeball test (see Fig. 4.1), even though the maps are generated from distinct distributions of number densities and magnetic field strengths. GdGb is commonly assumed in astrophysical scenarios, whereas UdUb* is simply unrealistic because the magnetic fields are non-solenoidal. The resulting RM maps are similar across all lines-of-sight, implying that it is non trivial to characterise the thermal number densities and the magnetic field strengths from the observed RM fluctuations alone.

Table 4.1 shows a quantitative comparison of the models. Calculating the line-of-sight longitudinal dispersions using equation (3.28), σxy σxz σyz R ' R ' R ' 29.3 rad m−2 is obtained in all cases, indicating that this type of RM fluctuation formula cannot distinguish between the different distributions of number densities and magnetic field strengths. The tiny variations in the least significant figures of xy xz yz σR , σR and σR are due to the numerical noise and random differences in the generated realisations. Table 4.1 also shows that the line-of-sight and sky transverse

fluctuations match reasonably well (σR ςR) in the cases of UdUb*, UdUb and ' UdGb, indicating that the widely-used RM fluctuation formula is applicable for uniformly distributed densities and magnetic field strengths with uniform distributions 4.1. Variance of rotation measure fluctuations 100 and Gaussian distributions. However, for GdUb, GdGb, FdUb, FdGb, LdUb and

LdGb; σR < ςR, meaning that the RM fluctuation formula is inadequate in situations with Gaussian, fractal or log-normal density distributions. The disagreement between σR and ςR is at the level of 2%, 25% and 40%, respectively for G, F ∼ ∼ ∼ and L density models. For a comparison, Bhat and Subramanian(2013) calculated the evolving RM properties of the ICM in fluctuation dynamo simulations, and found that ςR was 10%–15% above some statistical indicators of RMF ( σR). ∼ ≈ In their models, the evolving magnetic features seemed to be more influential than the density variations. Notably, both the RMS model and explicit RT simulation cannot distinguish the difference between solenoidal and non-solenoidal fields, as shown by ∗ −2 σR(UdUb ) σR(UdUb) 29.3 rad m . The calculations also show that the ' ' ∗ sky planar fluctuations: ςR(UdUb ) ςR(UdUb) ςR(UdGb), ςR(GdUb) ' ' ' ςR(GdGb), ςR(FdUb) ςR(FdGb) and ςR(LdUb) ςR(LdGb). The Fourier trans- ' ' form and inverse Fourier transform are part of the divergence cleaning process. Note that the Fourier transform of uniformly distributed fields in a finite volume gives a 3-D sinc function (in the Cartesian coordinate). For a more detailed characterisation of the RM distributions, histograms and cumulative distribution functions (CDFs) of the RM maps are calculated from every line-of-sight through the UdUb*, UdUb, UdGb, GdUb, GdGb, FdUb, FdGb, LdUb and LdGb distributions. For each cube, the histograms and CDFs along lines-of- sight x, y or z coincide, confirming that isotropy is preserved in each cube (also demonstrated by the results in Table 4.1, where σxy σxz σyz for all simulations). R ' R ' R For the sake of model comparisons, GdGb(z) is set as the basis CDF, from which its numerical difference from the CDFs of the rest of the models are calculated. The CDFs at every line-of-sight are almost indistinguishable in each case of uniformly- distributed and Gaussian-distributed densities: GdGb, UdUb*, UdUb, UdGb and GdUb, as shown by the tiny fluctuations in the zero line (Fig. 4.2). In comparison, the CDFs of fractal and log-normal distributed densities: FdUb, FdGb, LdUb and LdGb deviate significantly from the basis CDF. 4.1. Variance of rotation measure fluctuations 101 yz . The last three columns /ς yz R yz ς σ and xz xz R /ς ς , xz σ xy R ς xy /ς xy σ yz R ς , respectively, as light travels through various configurations x , and y xz R , ς z xy R ς yz R σ 1 Mpc. The magnetic fields are strictly divergence-free, except for Ub*. ∼ L xz R , calculated along lines-of-sight σ yz R G and σ µ 1 and ∼ xz R xy R σ rms σ , B xy R , 3 σ − cm 3 − 10 i ∼ Ld UbLd Gb 29.2969 29.2968 29.2975 29.2972 29.2964 48.3524 29.2970 48.3218 48.3058 48.3327 48.3017 0.6059 48.3146 0.6063 0.6065 0.6062 0.6065 0.6064 Fd UbFd Gb 29.2987 29.2965 29.2975 29.2956 29.2966 39.1187 29.2968 39.1392 39.1185 39.1134 39.1357 0.7490 39.1186 0.7485 0.7489 0.7490 0.7486 0.7489 Ud Ub*Ud Ub 29.2971Ud 29.2975 Gb 29.2962 29.2962 29.2861 29.2977 29.2965 29.3151 29.2970 29.2965 29.3042 29.3101 29.2979 1.0004 29.3198 29.3231 29.3278 0.9994 29.2934 0.9995 0.9997 29.3299 0.9993 0.9991 0.9990 1.0001 0.9989 Distribution Longitudinal, eq. ( 3.28 ) Sky transverse, eq.4.2 ) ( Ratio Gd UbGd Gb 29.2982 29.2961 29.2970 29.2966 29.2973 29.9263 29.2966 29.8972 29.9113 29.8994 29.8854 0.9790 29.8902 0.9799 0.9795 0.9798 0.9803 0.9801 th , e n h The dispersion of RM, of magnetised, thermal plasma. The transverse dispersion is calculated from the RM maps over the sky plane: Table 4.1: show the ratios of the longitudinalgalaxy clusters: dispersion to the transverse dispersion. The realisations are normalised to the order of magnitude typically found in 4.1. Variance of rotation measure fluctuations 102

Using the numerical CDF curves, a Kolmogorov-Smirnov (Kolmogorov 1933; Smirnov 1948, hereafter KS) test is performed, with the null hypothesis being that the two RM samples, observed either in the x, y or z direction, are drawn from the same distribution. The KS test is non-parametric and reports the maximum value of absolute (vertical) diﬀerence between two CDFs (see e.g. Press et al. 2007). The KS statistics are calculated and summarised in Table 4.2. The KS statistics D 1 and p-value probabilities are in the range of 0.2 0.6, favouring the null hypothesis − since p > 0.05. The KS tests do not show evidence of anisotropy.

In addition, a fractal medium with two density phases (hereafter referred to as cloudy models)4, mimicking the typical environment in the ICM/ISM, is considered. Astrophysical plasmas are susceptible to form substructures through a variety of thermal, magnetic, and buoyancy instabilities (e.g. Field 1965; Shu et al. 1972; Balbus and Soker 1989; Quataert 2008; McCourt et al. 2012; Sharma et al. 2012; Wareing et al. 2016). An optically thin plasma of nearly solar composition has a temperature-dependent radiative cooling function that incurs thermal instabil- 4 8.5 ity over an interval 10 K . T . 10 K (e.g. Sutherland and Dopita 1993; Lodders 2003; Maio et al. 2007; Sutherland et al. 2018). A homogeneous medium can sponta- neously self-segregate into a quasi-equilibrium of two co-existing phases: the original hot diffuse medium; and a minor component of cooler dense clouds. Due to external 3 isobaric conditions, a density ratio & 10 is implied between the phases, in the absence of any further gravitational collapse. Thermally condensed clouds are endemic in otherwise hot extragalactic media, and can stretch into filaments in upflows and downflows associated with active galaxies (e.g. Ford and Butcher 1979; Saxton et al. 2001; Conselice et al. 2001; McDonald et al. 2010; Voit et al. 2017; Combes 2018; Olivares et al. 2019).

As a test of RMF due to strong density inhomogeneities, two-phase toy models are built, which are capable of approximating the knotty medium of a galaxy cluster core, or the ISM of an elliptical galaxy that acquired clouds via thermal instabil-

4The magnetic field morphology is often complex in a turbulent medium. There is usually a large-scale magnetic field component that is spatially coherent at the scale of the astrophysical object. Meanwhile, on smaller scales, turbulence would stretch and bend the frozen-in magnetic field lines, causing small-scale magnetic fluctuations to occur. However, for the purpose of this study, it is sufficient to consider simpler magnetic field models: Ub*, Ub, and Gb. 4.1. Variance of rotation measure fluctuations 103

Fig. 4.2: Top-left panel shows the CDF of GdGb(z) as a reference to calculate the numerical diﬀerences with the CDFs of GdGb(y), GdGb(x), as well as UdUb*, UdUb, UdGb, GdUb, FdUb, FdGb, LdUb and LdGb for every line-of-sight. The CDFs are almost identical in all cases, except for models with fractal and log-normal density distributions. 4.1. Variance of rotation measure ﬂuctuations 104

Distribution D p-value xy & xz xy & yz xz & yz xy & xz xy & yz xz & yz Ud Ub* 0.0044 0.0048 0.0045 0.5707 0.4763 0.5460 Ud Ub 0.0052 0.0047 0.0052 0.4272 0.5502 0.4364 Ud Gb 0.0052 0.0055 0.0057 0.4255 0.3654 0.3712 Gd Ub 0.0052 0.0048 0.0052 0.4240 0.5123 0.4509 Gd Gb 0.0050 0.0051 0.0051 0.4696 0.4359 0.4275 Fd Ub 0.0047 0.0072 0.0072 0.5110 0.1549 0.1421 Fd Gb 0.0049 0.0068 0.0071 0.5052 0.1696 0.1658 Ld Ub 0.0050 0.0050 0.0050 0.4833 0.4662 0.4644 Ld Gb 0.0049 0.0045 0.0049 0.4806 0.5550 0.4865

Table 4.2: The KS statistics D and p-value probabilities corresponding to various conﬁg- urations in Table 4.1.

ity or a wet-dry merger (e.g. Larson 1974, 1975; Piontek and Ostriker 2004; Sage et al. 2007; Skory et al. 2013). A Gaussian distribution of pseudo-random complex numbers is generated before applying an amplitude filter to obtain a Kolmogorov- like power spectrum. This cube is transformed according to the Elmegreen recipe for imitating lognormal density fluctuations in a turbulent medium, which will represent the diffuse phase. A volume filling factor of clouds (0 < f 1) is then prescribed and the densest ranked voxels are selected down to a suitable threshold. Their densities are multiplied by a uniform constant, set to ensure a mean density ratio of 103 between the cloud and non-cloud phases. Assuming that the clouds are condensing from the hot medium, the mean of the entire cloudy block is normalised to 10−3 cm−3, matching the standard for the single-phased density models.

Models, ranging from a negligible smattering of clouds (f 10−6) to a heavily ≈ obscured overcast case (f 10−2) where a majority ( 0.7) of RM map pixels or ≈ & rays traverse at least one dense cloud, are created and tested. Table 4.3 presents basic global properties of these models. In terms of area, the cloud coverage factors decrease with f, and vary with orientation due to the clouds’ random fractal shapes. Clouds account for only a tiny fraction of the total mass in Cd4–Cd6, and just under 4.1. Variance of rotation measure ﬂuctuations 105

Model log f mc/M ax/A ay/A az/A Cd2 -2.01 0.912 0.713 0.752 0.749 Cd3 -3.01 0.495 0.150 0.155 0.155 Cd4 -4.02 0.0877 0.0188 0.0190 0.0189 Cd5 -5.12 0.00746 0.00171 0.00172 0.00169 Cd6 -6.32 0.000477 0.000122 0.000122 0.000122

Table 4.3: Summary of cloudy model properties: volume filling factors f; cloud mass fraction mc; and area covering factors for the three orthogonal views ax, ay, az. Before radiative transfer calculations, all models are normalised to the same total mass or mean density. half the mass in Cd3. The overcast case Cd2 is dominated by the mass of the dense cold phase, making it unrealistic for the filament-infused core of a galaxy cluster (where the cold fraction is at most a few tens of percents), but perhaps more like the primordial medium of a hypothetical wet protogalaxy. The mean densities of the cubes vary by factors of a few before their normalisations into fiducial ICM units. Various cloud volume filling factors f = 10−2, 10−3, 10−4, 10−5, and 10−6, corresponding to Cd2, Cd3, Cd4, Cd5 and Cd6, respectively, are considered. Fig- ures 4.3 and 4.4 show the log10 cross-sections and column densities of Cd3 and Cd5, respectively. Each cross-section is a slice taken from the cloudy models Cd3 and Cd5 at x = 128, y = 128 and z = 128. The cross-sections and column densities show the non-uniformity of the diffuse media and the cloud phases. The cloudy models are fairly isotropic along every line-of-sight, with Cd3 being more dense than Cd5, as indicated by the larger number of bright specks embedded within the cloudy media. Fig. 4.5 shows that the resulting RM maps of the Cd2Gb and Cd2Ub* distributions are indistinguishable, despite the distinction between the distributions of magnetic field strengths, especially with Ub* being non-solenoidal and unphysical. Moreover, Table 4.4 shows that the RMS statistics are unable to tell the cloudy features apart, in spite of Cd2 – 6 having different volume filling factors. In particular, the RMS −2 statistics for various distributions of Cd and b are σR 29.3 rad m , which is ' similar to the RMS statistics for various distributions of the single-phased models in Table 4.1, indicating that the RMS method cannot distinguish between a range of different clumpy (or smooth) configurations of density and magnetic fields. 4.1. Variance of rotation measure fluctuations 106

Fig. 4.3: Cross-sections of Cd3 (top) and Cd5 (bottom) at x = 128 (left), y = 128 (middle) and z = 128 (right).

Fig. 4.4: Column densities of Cd3 (top) and Cd5 (bottom) in the x (left), y (middle) and z (right) directions. 4.1. Variance of rotation measure ﬂuctuations 107

Fig. 4.5: Synthetic RM maps of Cd2Gb (top) and Cd2Ub* (bottom) in the xy (left), xz (middle) and yz (right) planes.

The sky-transverse standard deviations reveal that, with the exception of the overcast model Cd2, the ςR decreases with decreasing volume filling factor. This is as expected since the scatter should be less with fewer clouds, and approaching the L lognormal models as f 0. Clumpiness always causes ςR > σR, and → often by large multiples (with relative differences up to 94%). Furthermore, comparing the dispersions between the longitudinal direction and the sky transverse direction, the cloudy models in Table 4.4 show a greater scatter than the Ud, Gd, Fd and Ld models did in Table 4.1. The variability of standard deviations may be attributed to the random shapes and orientations of the clouds. Histograms are also calculated to characterise the RM distributions from every line-of-sight through the Cd2Ub*, Cd2Ub, Cd2Gb, Cd3Ub*, Cd3Ub, Cd3Gb, Cd4Ub*, Cd4Ub, Cd4Gb, Cd5Ub*, Cd5Ub, Cd5Gb, Cd6Ub*, Cd6Ub and Cd6Gb distributions. For each cube, the histograms along lines-of-sight x, y and z coincide, confirming that isotropy is preserved in each cube, which is also shown by ςxy ςxz ςyz in Ta- R ' R ' R ble 4.4. The results from the KS-tests are summarised in Table 4.5. The KS statistics do not show evidence of anisotropy. Using GdGb(z) as the basis CDF, its nu- 4.1. Variance of rotation measure fluctuations 108 merical difference from the CDFs of the cloudy, magnetised models are calculated and shown in Fig. 4.6. These panels are almost indistinguishable between different configurations of magnetic fields, for example, the numerical difference trends for Cd3Ub*(x, y, z), Cd3Ub(x, y, z) and Cd3Gb(x, y, z) look similar to each other. This suggests that the RMs are more dependent on the cloud structures, rather than the magnetic field configurations since the density variations are of orders of magnitude within each cube, while the variation of the magnetic field is relatively smaller. Notably, the CDFs of Cd2 show the largest deviation from the basis CDF, whereas the CDFs for the rest of the models, apart from Cd3, are almost indistinguishable. This may be a consequence of a scarcity of clouds in Cd4, 5 and 6.

Hence, from the results above, the widely-used RM fluctuation formula (RMS statistics) is valid when all of the following conditions hold: (i) a random field produces random Faraday rotation, (ii) there exists a meaningful characteristic thermal electron number density, (iii) there exists a uniform or Gaussian distribution of magnetic field strengths, (iv) the field is isotropic, and (v) the density and the magnetic field are not correlated.5 In situations where some of these criteria are not met, the RMS statistics would be inadequate to be used to interpret the magnetic field properties from the RM analyses. More specifically, different statistical indicators can potentially mislead when the line-of-sight longitudinal dispersion is less than the sky transverse dispersion σR < ςR, by tens of percents or by factors of a few, especially if there is unrecognised cloudiness. Discrepancies could, in principle, be large in some environments, such as cluster cores, where multiphase features are obvious in other wavebands (e.g. Conselice et al. 2001) or faint cluster outskirts where clumpiness is conjectured (e.g. Urban et al. 2014).

4.1.2 Subtleties in the interpretation of magnetic ﬁeld properties from polarisation data

Ambiguity in the polarisation angle

The inference of RM from observations of linear polarisation is subjected to an nπ ambiguity in its direction (Ruzmaikin and Sokoloﬀ 1979). For a clean line-of-

5Condition (v) does not hold in the ideal MHD scenario, where the magnetic field lines are assumed to be frozen into the plasma. Therefore, the plasma and the magnetic field lines have to move together. 4.1. Variance of rotation measure fluctuations 109 sight with a single point source, measuring the polarisation angles ϕ at various wavelengths λ satisfies a relationship: ϕ = ϕ + λ2, giving the rotation measure 0 R from the slope and the intrinsic polarisation angle ϕ . The foreground magnetic R 0 field structure can be inferred from the RM if the emission measure is known. In practice, the measured polarisation angle ϕ can only be constrained between 0 and π, leading to an ambiguity of nπ, where n is an integer, thus causing a problem ± in correctly determining ϕ and . 0 R 4.1. Variance of rotation measure fluctuations 110 yz /ς yz σ xz /ς xz σ xy /ς , respectively, as light travels through various xy x σ , and y yz , R ς z xz R ς xy R ς 1 Mpc. The magnetic fields are strictly divergence-free, except for Ub*. ∼ L , calculated along lines-of-sight yz R yz R σ σ G and µ and 1 xz R ∼ σ xz R , rms σ xy R B σ , 3 − cm xy R 3 σ − 10 i ∼ th , e n h Distribution Longitudinal, eq. ( 3.28 )Cd2 Ub*Cd2 Sky transverse, 29.2976 eq. Ub4.2 ) ( Cd2 29.2984 Gb 29.2949 29.2922Cd3 280.6983 29.2979 Ub* 29.2968 280.0177 29.3008Cd3 29.2986 280.4870 29.2976 280.5404 Ub 29.2956Cd3 0.1044 29.2984 279.4889 279.8605 Gb Ratio 29.2949 281.2761 29.2922 0.1046Cd4 281.7843 471.8747 0.1044 29.2979 Ub* 0.1044 280.9985 29.2968 483.3400 29.3008Cd4 0.1048 0.1047 29.2986 473.8896 29.2976 478.3895 Ub 29.2956 0.1042 Cd4 0.0621 29.2984 479.4463 0.1040 466.7194 Gb 29.2949 476.4639 29.2922 0.0606 0.1043 Cd5 480.4994 267.1094 0.0612 29.2979 Ub* 0.0618 475.2179 29.2968 271.5542 29.3008Cd5 0.0611 0.0628 29.2986 265.8259 29.2976 276.8559 Ub 29.2956 0.0615 Cd5 0.1100 29.2984 266.4851 0.0610 257.7923 Gb 29.2949 267.9559 29.2922 0.1079 0.0617 Cd6 271.5684 0.1058 29.2979 93.4944 Ub* 0.1102 268.0812 29.2968 29.3008Cd6 0.1099 0.1136 29.2986 86.3299 29.2976 Ub 98.3658 29.2956 0.1093 Cd6 29.2984 0.1079 89.4997 Gb 88.2455 29.2949 29.2922 83.6243 0.3134 0.1093 29.2979 97.2471 44.5203 83.0571 0.3394 29.2968 29.3008 0.2978 29.2986 43.3651 0.3273 89.1919 46.9690 29.2956 0.3320 0.3503 43.0838 43.8330 0.3013 42.4488 0.6581 0.3528 45.7565 40.6938 0.6756 0.3285 0.6236 0.6800 44.7132 0.6684 0.6902 0.6404 0.7200 0.6552 The standard deviations of RM, configurations of magnetised, two-phase fractal plasma Cd2,found Cd3, in Cd4, galaxy Cd5, clusters: and Cd6. The realisations are normalised to the order of magnitude typically Table 4.4: 4.1. Variance of rotation measure fluctuations 111 yz & yz xz & -value xz xy & yz xy & yz xz -value probabilities corresponding to the configurations in Table 4.4 . & D p p and D xz xy & xy The KS statistics Distribution Cd2 Ub*Cd2 5.7068E-03 UbCd2 1.9516E-02 Gb 1.9180E-02 6.3782E-03Cd3 2.3559E-01 2.0584E-02 Ub* 4.2114E-03 2.7937E-11 2.3743E-02Cd3 1.7532E-02 6.5523E-11 4.5929E-03 1.3851E-01 Ub 2.0309E-02Cd3 7.0648E-03 1.6807E-12 6.0557E-01 Gb 8.1940E-03 1.7189E-16 6.5002E-03Cd4 3.4726E-09 4.9313E-01 5.8136E-03 Ub* 3.5123E-12 3.6011E-03 7.5602E-02 6.1188E-03Cd4 8.2855E-03 2.4407E-02 3.1281E-03 1.2494E-01 Ub 7.5684E-03Cd4 4.2267E-03 2.1739E-01 7.8844E-01 Gb 5.7831E-03 1.7129E-01 4.6844E-03Cd5 2.2106E-02 9.0517E-01 3.0975E-03 Ub* 4.6618E-02 3.4027E-03 6.0094E-01 6.0120E-03Cd5 6.3477E-03 2.2247E-01 3.0823E-03 4.6754E-01 Ub 6.3019E-03Cd5 4.3945E-03 9.1125E-01 8.4190E-01 Gb 5.8289E-03 1.8646E-01 4.7150E-03Cd6 1.4208E-01 9.1421E-01 3.2654E-03 Ub* 1.4758E-01 3.3112E-03 5.5062E-01 6.3171E-03Cd6 6.0883E-03 2.1488E-01 3.0060E-03 4.5916E-01 Ub 5.3864E-03Cd6 4.5624E-03 8.7535E-01 8.6457E-01 Gb 5.9357E-03 1.4573E-01 4.8370E-03 1.7552E-01 9.2823E-01 3.2043E-03 2.9698E-01 3.3112E-03 5.0180E-01 6.2866E-03 5.9357E-03 1.9792E-01 4.2644E-01 5.2948E-03 8.8908E-01 8.6457E-01 1.4945E-01 1.9792E-01 3.1644E-01 Table 4.5: 4.1. Variance of rotation measure fluctuations 112

Early efforts were taken to resolve this ambiguity by imposing a search limit for the best RM from an astrophysical perspective and carrying out observations in several frequencies so to obtain the best fit using a chi-squared minimisation (see e.g. Simard-Normandin et al. 1981; Rand and Lyne 1994). This method assumed that no nπ ambiguity occurs between two closely-spaced wavelengths, such that ∆ϕ < π/2 is fulfilled (Ruzmaikin and Sokoloff 1979). The source is observed across | | a radio broad band with sparsely sampled wavelengths, and near each observed wavelength, combinations of (ϕ nπ) are considered in the fitting process. While ± this method can be applied to Faraday-thin media with a bright background point source, it sometimes gives multiple acceptable solutions. It does not work well for faint sources. In the Faraday-thick regime, the method will break down because the linear relation does not hold. It is also problematic when there are multiple sources along a line-of-sight or when Faraday depolarisation occurs (see e.g. Vallee 1980; Sokoloff et al. 1998; Farnsworth et al. 2011). Recently, alternative methods have been developed, for example, the circular statistical method (Sarala and Jain 2001), the PACERMAN algorithm (Dolag et al. 2005a; Vogt et al. 2005), the RM synthesis/RMCLEAN method (Burn 1966; Brentjens and de Bruyn 2005; Heald et al. 2009), Stokes QU-fitting (e.g. Farnsworth et al. 2011; O’Sullivan et al. 2012), and the dependence on RM of neighbouring sources (Taylor et al. 2009; Ma et al. 2019). The nπ ambiguity is one of the obstacles that must be overcome when analysing large-scale magnetic fields using the RM information. On the other hand, it may be possible to bypass the reliance on RM statistics by carrying out a proper (covariant) polarised radiative transfer, which can directly track the evolution of polarisation along a line-of-sight to naturally resolve the nπ ambiguity without having to make an a priori assumption on the Faraday complexity (see Chan et al. 2019).

Issues in polarisation analyses associated with large-scale astrophysical structures

FRBs and quasars as diagnostics: FRBs and quasars are exceptionally bright, polarised radio point sources, which are observable across cosmological distances. They are therefore useful probes of the intergalactic magnetic ﬁelds (see e.g. Xu and Han 2014b; Zheng et al. 2014; Akahori et al. 2016; Ravi et al. 2016; Vazza et al. 2018; Hackstein et al. 2019) and their evolution (see e.g. Xu and Han 2014b), if their 4.1. Variance of rotation measure ﬂuctuations 113

Fig. 4.6: Top-left panel shows the CDF of GdGb(z) as a reference to calculate the numerical differences with the CDFs of GdGb(y), GdGb(x), as well as Cd2Ub*, Cd2Ub, Cd2Gb, Cd3Ub*, Cd3Ub, Cd3Gb, Cd4Ub*, Cd4Ub, Cd4Gb, Cd5Ub*, Cd5Ub, Cd5Gb, Cd6Ub*, Cd6Ub and Cd6Gb at every line-of-sight. 4.1. Variance of rotation measure fluctuations 114 redshifts and dispersion measures are known (see e.g. Kronberg and Perry 1982; Blasi et al. 1999; Kronberg et al. 2008; Xu and Han 2014a; Petroff et al. 2016). Cir- cular polarisation has been detected in some quasars (see e.g. Roberts et al. 1975; Saikia and Salter 1988; Rayner et al. 2000; O’Sullivan et al. 2013) and FRBs (see e.g. Petroff et al. 2015, 2017), indicating that Faraday conversion (see e.g. Vedan- tham and Ravi 2019; Gruzinov and Levin 2019) or scintillation-induced variations (Macquart and Melrose 2000) might occur. As the number of detections of FRBs and quasars increases (see e.g. Keane 2018), the polarisation properties of their signals can be used to better constrain large-scale magnetic field properties. Apart from the effects of Faraday conversion and scintillation, it is also important to distinguish between the RM contributions from multiple sources along the line-of-sight, consider the effects of traversing multi-phase media, as well as taking into account the structural evolution and stretching of radiation wavelength in an expanding Universe (see e.g. Han 2017). In these situations, RM is no longer sufficient to fully characterise the changes in polarisation and hence a covariant cosmological polarised radiative transfer treatment is necessary (see Chan et al. 2019).

Direct radio emission from large-scale structure: An emissive and Faraday- rotating medium will result in a net depolarisation due to differential Faraday rotation (e.g. Sokoloff et al. 1998; Beck 1999; Shukurov and Berkhuijsen 2003; Fletcher and Shukurov 2006). This effect is particularly important in extended sources such as emitting filaments in the cosmic web. A simple Faraday screen with a bright source behind a Faraday-rotating medium would be insufficient to capture this effect properly. A covariant polarised radiative transfer calculation is therefore essential to evaluate the line-of-sight depolarisation effect from all radiative processes at different redshifts (see Chan et al. 2019).

Contamination in the power spectrum: The power spectrum of the observed polarised intensity may be contaminated by emissions from the medium and embedded sources. Contributions from these sources would lead to apparent higher power in fluctuations at small-scales. It is crucial to assess whether these signatures of spatially separated sources can be distinguished from those imprinted by the true structures of magnetic fields. In addition to these, the interpretation of the power spectrum is complicated by the contributions at various cosmological redshifts. Consider a radio observation of the sky at a fixed frequency νobs. The 4.2. Cosmological MHD simulations of cluster formation 115

(1 + z) νobs

z )| (k νobs ) P z k )| ( (k =0 P P z )| (k P

Fig. 4.7: Schematic of how higher-redshift structures can contaminate the observed power spectrum at a fixed νobs (dashed line), which differs from the theoretical power spectra P (k) from all sources at a single redshift. |z observed power spectrum P (k) is the result of contributions from sources at different redshifts. Hence, at each k, the power spectrum is contaminated by different levels of emission from various sources at higher redshifts (see Fig. 4.7), which differ from the power spectrum of the Universe at every redshift, P (k) . Local P (k) |z |z does not contain any contribution from the higher redshifts, whereas observationally, different components at higher k are picked up at νobs.

4.2 Cosmological MHD simulations of cluster formation

While the use of RMS statistics requires caution, much of what is known about magnetic ﬁelds in galaxy clusters comes from probing the diﬀuse radio sources associated with them and/or measuring the Faraday RMs of embedded and background polarised sources. Many galaxy clusters, based on observations and numerical simulations, have shown similar morphological features in their radio halo and X-ray images, as well as a power-law relation in their radio (or RM) and X-ray brightness (see e.g. Feretti et al. 2001; Dolag et al. 2001; Govoni et al. 2001a; Giacintucci et al. 2005; Dolag et al. 2005b; Govoni et al. 2010; Xie et al. 2020). 4.2. Cosmological MHD simulations of cluster formation 116

The detection of diffuse radio sources in a number of galaxy clusters, either at their centres (radio halos) or peripheries (radio relics), indicates that magnetic fields and relativistic electrons are pervasive in the ICM. The physical acceleration mechanism(s) generating the radio synchrotron emission is still not well understood. Since the diffuse radio sources extend over several Mpcs, cooling time arguments predict that such an active mechanism has to operate in the ICM, for example: in- situ shock acceleration during cluster mergers, ultra-high energy particles emitted from radio galaxies near/in the cluster, and/or secondary electrons of cosmic ray interactions with the ICM (e.g. Miniati 2014, 2015; see e.g. van Weeren et al. 2019, for review).

The local pixel-by-pixel relation between the observed emissions in radio and X-ray (and also gamma-ray) in galaxy clusters may provide a means to distinguish between the possible acceleration mechanisms. The development of magnetic ﬁelds as a cluster evolves should also imprint signatures in observations. In Sections 4.2.3 and 4.2.4, galaxy clusters from the N-body/Smoothed Particle Magnetohydrody- namic (SPMHD) code GCMHD+ are used to quantify the relations and properties of Faraday RM, radio and X-ray emissions. For completeness, the numerical method and cluster simulations are brieﬂy summarised in Sections 4.2.1 and 4.2.2 (see Barnes et al. 2012, 2018, for more details).

4.2.1 Numerical method

The N-body SPMHD cosmological code GCMHD+ is used to simulate the evolution of magnetic fields in galaxy clusters. The hydrodynamical code GCD+ and its parameters are presented in Kawata et al.(2013), with the MHD implementation by Barnes et al.(2012), where the magnetic field is evolved through the induction equation. Further improvements are made in Barnes et al.(2018) by adding an artificial resistivity switch at magnetic discontinuities (Tricco and Price 2013), and implementing a hyperbolic cleaning scheme (Tricco and Price 2012) to maintain a solenoidal magnetic field B = 0. The hyperbolic cleaning scheme significantly ∇ · reduces the divergence error in the magnetic fields, while the artificial resistivity scheme should be carefully chosen in order not to destroy the weak magnetic fields (see Barnes et al. 2018, for details). 4.2. Cosmological MHD simulations of cluster formation 117

4.2.2 Cluster simulations

The cosmological simulation begins at z = 60, where density perturbations are introduced into a low resolution dark matter spherical volume of 200 Mpc3 using the GRAFIC2 code (Bertschinger 2001). Assuming a flat ΛCDM cosmological model with parameters from the WMAP 5-year data (Komatsu et al. 2009), the density perturbations and initial conditions are evolved to z = 0, then a central halo with a virial 14 mass of at least 10 M is chosen using the friend-of-friend algorithm. The halo is iteratively refined to produce a “zoomed-in” simulation, where each particle within (8 r ) and (4 r ) is replaced 8 times for the low resolution simulations (here- × 200 × 200 after prefixed ‘C’), whereas for the high resolution simulations (hereafter prefixed ‘HR’), each particle is replaced 8 times and 64 times, respectively; until the halo is free from lower mass particles within (2 r ). Every highest refined particle × 200 8 has a mass m = 6.20 10 M . Then, the same number of gas particles, each DM × 8 of mass m = 1.26 10 M , are added to the highest resolution region. The gas g × particles are embedded within a seed magnetic field of 10−11 G, which is uniformly distributed in the x-direction to ensure zero divergence initially.

4.2.3 RM and X-ray correlations during the assembly of galaxy clusters

Although numerous studies on the correlation between the RM and X-ray surface brightness of galaxy clusters have been carried out (e.g. Dolag et al. 2001), this is the first time the relation is calculated and examined while the clusters form and evolve. Observationally, galaxy clusters are typically X-ray luminous because of their hot (105.2 K T 107 K) and diffuse (10−6 cm−3 n 10−2 cm−3) intracluster gas. ≤ . . e . By exploiting the presence of radio sources within or behind the galaxy clusters, the intracluster magnetic fields can be inferred from the RMs. How the magnetic fields develop as the cluster evolves should therefore also imprint signatures in the RM and X-ray relation. Here the GCMHD+ code is used to simulate the formation of ten low resolution (C01–10) and two high resolution (HR01–02) magnetised galaxy clusters. The evolution of the density, magnetic field strength and temperature of every cluster is followed from z = 3 to z = 0 (see Tables 4.6–4.15), except for the 4.2. Cosmological MHD simulations of cluster formation 118

z r /Mpc n /cm−3 B /G T /K vir h e,thi h i h i 0.00 1.976 3.419E-05 1.202E-09 1.457E+07 0.25 1.603 5.230E-05 1.350E-09 1.513E+07 0.50 1.401 7.093E-05 1.054E-09 1.547E+07 1.00 1.041 1.324E-04 3.261E-10 1.337E+07 1.50 0.5665 2.628E-04 3.177E-10 9.675E+06 2.00 0.4325 4.311E-04 7.689E-11 7.284E+06 3.00 0.2027 1.030E-03 2.447E-11 4.849E+06

Table 4.6: Summary of cluster C01 properties: redshift z; virial radius rvir; mean thermal number density n ; mean magnetic ﬁeld strength B ; and mean temperature T . h e,thi h i h i

z r /Mpc n /cm−3 B /G T /K vir h e,thi h i h i 0.00 2.322 3.082E-05 1.250E-09 1.914E+07 0.25 1.909 4.675E-05 8.370E-10 1.770E+07 0.50 1.518 6.751E-05 6.874E-10 1.782E+07 1.00 0.8544 1.600E-04 4.858E-10 1.308E+07 1.50 0.6138 2.502E-04 1.605E-10 1.106E+07 2.00 0.3852 4.581E-04 4.402E-11 7.902E+06 3.00 0.1668 1.003E-03 1.169E-11 3.448E+06

Table 4.7: Summary of cluster C02 properties: redshift z; virial radius rvir; mean thermal number density n ; mean magnetic field strength B ; and mean temperature T . h e,thi h i h i higher resolution clusters6 (see Tables 4.16–4.17). Generally the weak, frozen-in seed field gets amplified via a turbulent dynamo mechanism driven by cluster formation. The continuum X-rays are thermal bremssthrahlung radiation, with an emissivity given by

1/2 5 6 dW 2πkBTe,th 2 πe 2 −1 −2 jX = = 3 Z ne,th ni g¯B erg s cm (4.7) dt dV 3me 3hmec

(Rybicki and Lightman 1979), where me is the electron mass, e is the electron charge, c is the speed of light, kB is the Boltzmann constant and h is the Planck constant.

6The higher resolution cluster simulations are more computationally demanding, so their evolution could only be tracked at fewer redshifts. 4.2. Cosmological MHD simulations of cluster formation 119

z r /Mpc n /cm−3 B /G T /K vir h e,thi h i h i 0.00 1.851 3.069E-05 1.210E-09 1.124E+07 0.25 1.521 4.711E-05 1.095E-09 1.202E+07 0.50 1.234 7.320E-05 8.196E-10 1.095E+07 1.00 0.8616 1.356E-04 4.269E-10 1.055E+07 1.50 0.5729 2.543E-04 1.481E-10 8.619E+06 2.00 0.3744 4.676E-04 6.845E-11 8.456E+06 3.00 0.1801 9.823E-04 1.328E-11 3.528E+06

Table 4.8: Summary of cluster C03 properties: redshift z; virial radius rvir; mean thermal number density n ; mean magnetic ﬁeld strength B ; and mean temperature T . h e,thi h i h i

z r /Mpc n /cm−3 B /G T /K vir h e,thi h i h i 0.00 2.432 3.166E-05 1.264E-09 2.118E+07 0.25 2.011 4.613E-05 1.128E-09 2.300E+07 0.50 1.663 6.846E-05 6.508E-10 2.012E+07 1.00 0.9121 1.349E-04 3.476E-10 1.435E+07 1.50 0.5218 2.605E-04 8.982E-11 9.253E+06 2.00 0.3074 5.313E-04 3.791E-11 5.138E+06 3.00 0.1571 9.714E-04 2.523E-11 2.623E+06

Table 4.9: Summary of cluster C04 properties: redshift z; virial radius rvir; mean thermal number density n ; mean magnetic ﬁeld strength B ; and mean temperature T . h e,thi h i h i

z r /Mpc n /cm−3 B /G T /K vir h e,thi h i h i 0.00 1.019 3.333E-05 1.865E-10 4.885E+06 0.25 0.8116 4.785E-05 1.803E-10 4.495E+06 0.50 0.8211 4.662E-05 9.070E-11 4.517E+06 1.00 0.4232 1.400E-04 7.576E-11 2.793E+06 1.50 0.3325 2.351E-04 2.422E-11 2.795E+06 2.00 0.2121 4.378E-04 1.481E-11 2.189E+06

Table 4.10: Summary of cluster C05 properties: redshift z; virial radius rvir; mean thermal number density n ; mean magnetic ﬁeld strength B ; and mean temperature T . h e,thi h i h i 4.2. Cosmological MHD simulations of cluster formation 120

z r /Mpc n /cm−3 B /G T /K vir h e,thi h i h i 0.00 2.144 2.968E-05 4.038E-10 1.413E+07 0.25 1.227 4.960E-05 3.345E-10 1.290E+07 0.50 1.001 6.974E-05 1.998E-10 9.091E+06 1.00 0.6675 1.328E-04 9.889E-11 6.476E+06 1.50 0.4413 2.419E-04 2.822E-11 4.880E+06 2.00 0.3268 4.163E-04 3.808E-11 5.292E+06 3.00 0.1735 9.884E-04 1.495E-11 3.519E+06

Table 4.11: Summary of cluster C06 properties: redshift z; virial radius rvir; mean thermal number density n ; mean magnetic ﬁeld strength B ; and mean temperature T . h e,thi h i h i

z r /Mpc n /cm−3 B /G T /K vir h e,thi h i h i 0.00 2.455 3.103E-05 1.052E-09 2.047E+07 0.25 1.857 4.746E-05 9.188E-10 1.942E+07 0.50 1.514 6.884E-05 3.841E-10 1.482E+07 1.00 0.6637 1.320E-04 7.286E-11 7.749E+06 1.50 0.5169 2.436E-04 1.595E-10 7.043E+06 2.00 0.3390 4.324E-04 6.288E-11 5.549E+06 3.00 0.1851 1.016E-03 2.054E-11 3.852E+06

Table 4.12: Summary of cluster C07 properties: redshift z; virial radius rvir; mean thermal number density n ; mean magnetic ﬁeld strength B ; and mean temperature T . h e,thi h i h i

z r /Mpc n /cm−3 B /G T /K vir h e,thi h i h i 0.00 3.197 3.175E-05 1.208E-09 3.981E+07 0.25 2.512 4.695E-05 8.275E-10 3.072E+07 0.50 1.839 6.771E-05 7.626E-10 2.759E+07 1.00 1.069 1.422E-04 2.753E-10 1.822E+07 1.50 0.6599 2.504E-04 8.814E-11 1.232E+07 2.00 0.3826 4.550E-04 3.210E-11 8.100E+06 3.00 0.1991 9.839E-04 7.603E-12 4.295E+06

Table 4.13: Summary of cluster C08 properties: redshift z; virial radius rvir; mean thermal number density n ; mean magnetic ﬁeld strength B ; and mean temperature T . h e,thi h i h i 4.2. Cosmological MHD simulations of cluster formation 121

z r /Mpc n /cm−3 B /G T /K vir h e,thi h i h i 0.00 2.364 3.175E-05 1.324E-09 1.995E+07 0.25 1.987 4.716E-05 1.008E-09 2.223E+07 0.50 1.631 6.853E-05 6.702E-10 1.734E+07 1.00 0.9860 1.374E-04 4.098E-10 1.296E+07 1.50 0.6202 2.656E-04 4.063E-10 1.131E+07 2.00 0.4728 4.303E-04 1.232E-10 1.108E+07 3.00 0.2204 9.980E-04 3.543E-11 5.940E+06

Table 4.14: Summary of cluster C09 properties: redshift z; virial radius rvir; mean thermal number density n ; mean magnetic ﬁeld strength B ; and mean temperature T . h e,thi h i h i

z r /Mpc n /cm−3 B /G T /K vir h e,thi h i h i 0.00 1.349 2.619E-05 3.655E-10 6.008E+06 0.25 1.098 4.162E-05 2.886E-10 6.633E+06 0.50 0.8976 6.450E-05 2.709E-10 5.632E+06 1.00 0.6415 1.325E-04 1.835E-10 6.918E+06 1.50 0.4322 2.615E-04 6.797E-11 5.030E+06 2.00 0.2977 4.392E-04 3.072E-11 4.924E+06 3.00 0.1420 1.016E-03 7.198E-12 1.808E+06

Table 4.15: Summary of cluster C10 properties: redshift z; virial radius rvir; mean thermal number density n ; mean magnetic ﬁeld strength B ; and mean temperature T . h e,thi h i h i

z r /Mpc n /cm−3 B /G T /K vir h e,thi h i h i 0.00 1.450 2.226E-05 3.634E-08 6.704E+06 0.50 0.9781 5.340E-05 2.597E-08 6.213E+06 1.00 0.6556 1.230E-04 1.554E-08 6.409E+06 2.00 0.3142 3.721E-04 4.532E-10 4.765E+06

Table 4.16: Summary of cluster HR01 properties: redshift z; virial radius rvir; mean thermal number density n ; mean magnetic ﬁeld strength B ; and mean temperature h e,thi h i T . h i 4.2. Cosmological MHD simulations of cluster formation 122

z r /Mpc n /cm−3 B /G T /K vir h e,thi h i h i 0.00 1.444 2.288E-05 1.559E-08 6.588E+06

Table 4.17: Summary of cluster HR02 properties: redshift z; virial radius rvir; mean thermal number density n ; mean magnetic ﬁeld strength B ; and mean temperature h e,thi h i T . h i

The number density and temperature of the thermal electrons are ne,th and Te,th, respectively. Without losing generality, the velocity-averaged Gaunt factorg ¯B is set to 1. Assuming a fully ionised hydrogen plasma, Z = 1 and ne,th = ni. In addition 5.2 to that, only particles with temperatures higher than Te,th = 10 K are taken into account. It is essential to only consider emission from regions below the critical −3 density to form stars, i.e. ρ 0.01 M pc (Cox et al. 2006; Geng et al. 2012). ≤ With all of these assumptions, the X-ray intensity is calculated by integrating the emissivity over the interval ds along the line-of-sight Z I ds n2 T 1/2 erg s−1 cm−2. (4.8) X ∝ e,th e,th

Comparing the X-ray intensity to the standard deviation of RM allows the relation between magnetic fields and gas density in galaxy clusters to be constrained. The standard deviation of RM,σ ¯R, at each pixel of the simulated sky plane is the square root of the variance of contributing RMs along that line-of-sight. At z = 0, the simulated galaxy clusters reach mean number densities of 10−5 cm−3 and mean temperatures of 106 107 K (see Tables 4.6–4.17). Mean- ∼ ∼ − while, the magnetic field strengths of high-resolution clusters HR01 and HR02 reach a maximum of 10−7 10−6 G, which reasonably agree with typical peak µG ob- ∼ − servations in galaxy clusters. The peak magnetic field strengths of lower-resolution clusters C01–10, on the other hand, are several orders of magnitude lower than expected. This is because the magnetic field amplification is sensitive to the numerical resolution limit. Lowering the resolution smooths out the small-scale velocity fluctuations that would otherwise further amplify the seed magnetic field during structure formation (e.g. Barnes et al. 2018). The synthetic X-ray images and RM maps of all clusters at various redshifts are shown in Figures 4.8–4.19. Notably, the differences in X-ray morphologies as each cluster evolves from z = 3 to z = 0 reveal different formation histories. At z = 0, 4.2. Cosmological MHD simulations of cluster formation 123

Fig. 4.8: Properties of cluster C01 at z = 3.0, 2.0, 1.5, 1.0, 0.5, 0.25 and 0.0, where panels from left to right correspond to the log of X-ray surface brightness, sin(λ 2 ), correlation 10 0 R 4 between IX andσ ¯ in the sky plane with cut-oﬀs applied at 10− of the maxima, and corre- R lation between IX andσ ¯ without any cut-oﬀs, respectively. Here the reference wavelength R is chosen as λ0 = 100 cm. 4.2. Cosmological MHD simulations of cluster formation 124

Fig. 4.9: Properties of cluster C02 at z = 3.0, 2.0, 1.5, 1.0, 0.5, 0.25 and 0.0, where panels from left to right correspond to the log of X-ray surface brightness, sin(λ 2 ), correlation 10 0 R 4 between IX andσ ¯ in the sky plane with cut-oﬀs applied at 10− of the maxima, and corre- R lation between IX andσ ¯ without any cut-oﬀs, respectively. Here the reference wavelength R is chosen as λ0 = 100 cm. 4.2. Cosmological MHD simulations of cluster formation 125

Fig. 4.10: Properties of cluster C03 at z = 3.0, 2.0, 1.5, 1.0, 0.5, 0.25 and 0.0, where panels from left to right correspond to the log of X-ray surface brightness, sin(λ 2 ), 10 0 R 4 correlation between IX andσ ¯ in the sky plane with cut-oﬀs applied at 10− of the maxima, R and correlation between IX andσ ¯ without any cut-oﬀs, respectively. Here the reference R wavelength is chosen as λ0 = 100 cm. 4.2. Cosmological MHD simulations of cluster formation 126

Fig. 4.11: Properties of cluster C04 at z = 3.0, 2.0, 1.5, 1.0, 0.5, 0.25 and 0.0, where panels from left to right correspond to the log of X-ray surface brightness, sin(λ 2 ), 10 0 R 4 correlation between IX andσ ¯ in the sky plane with cut-oﬀs applied at 10− of the maxima, R and correlation between IX andσ ¯ without any cut-oﬀs, respectively. Here the reference R wavelength is chosen as λ0 = 100 cm. 4.2. Cosmological MHD simulations of cluster formation 127

Fig. 4.12: Properties of cluster C05 at z = 2.0, 1.5, 1.0, 0.5, 0.25 and 0.0, where panels from left to right correspond to the log of X-ray surface brightness, sin(λ 2 ), correlation 10 0 R 4 between IX andσ ¯ in the sky plane with cut-oﬀs applied at 10− of the maxima, and corre- R lation between IX andσ ¯ without any cut-oﬀs, respectively. Here the reference wavelength R is chosen as λ0 = 100 cm. 4.2. Cosmological MHD simulations of cluster formation 128

Fig. 4.13: Properties of cluster C06 at z = 3.0, 2.0, 1.5, 1.0, 0.5, 0.25 and 0.0, where panels from left to right correspond to the log of X-ray surface brightness, sin(λ 2 ), 10 0 R 4 correlation between IX andσ ¯ in the sky plane with cut-oﬀs applied at 10− of the maxima, R and correlation between IX andσ ¯ without any cut-oﬀs, respectively. Here the reference R wavelength is chosen as λ0 = 100 cm. 4.2. Cosmological MHD simulations of cluster formation 129

Fig. 4.14: Properties of cluster C07 at z = 3.0, 2.0, 1.5, 1.0, 0.5, 0.25 and 0.0, where panels from left to right correspond to the log of X-ray surface brightness, sin(λ 2 ), 10 0 R 4 correlation between IX andσ ¯ in the sky plane with cut-oﬀs applied at 10− of the maxima, R and correlation between IX andσ ¯ without any cut-oﬀs, respectively. Here the reference R wavelength is chosen as λ0 = 100 cm. 4.2. Cosmological MHD simulations of cluster formation 130

Fig. 4.15: Properties of cluster C08 at z = 3.0, 2.0, 1.5, 1.0, 0.5, 0.25 and 0.0, where panels from left to right correspond to the log of X-ray surface brightness, sin(λ 2 ), 10 0 R 4 correlation between IX andσ ¯ in the sky plane with cut-oﬀs applied at 10− of the maxima, R and correlation between IX andσ ¯ without any cut-oﬀs, respectively. Here the reference R wavelength is chosen as λ0 = 100 cm. 4.2. Cosmological MHD simulations of cluster formation 131

Fig. 4.16: Properties of cluster C09 at z = 3.0, 2.0, 1.5, 1.0, 0.5, 0.25 and 0.0, where panels from left to right correspond to the log of X-ray surface brightness, sin(λ 2 ), 10 0 R 4 correlation between IX andσ ¯ in the sky plane with cut-oﬀs applied at 10− of the maxima, R and correlation between IX andσ ¯ without any cut-oﬀs, respectively. Here the reference R wavelength is chosen as λ0 = 100 cm. 4.2. Cosmological MHD simulations of cluster formation 132

Fig. 4.17: Properties of cluster C10 at z = 3.0, 2.0, 1.5, 1.0, 0.5, 0.25 and 0.0, where panels from left to right correspond to the log of X-ray surface brightness, sin(λ 2 ), 10 0 R 4 correlation between IX andσ ¯ in the sky plane with cut-oﬀs applied at 10− of the maxima, R and correlation between IX andσ ¯ without any cut-oﬀs, respectively. Here the reference R wavelength is chosen as λ0 = 100 cm. 4.2. Cosmological MHD simulations of cluster formation 133

Fig. 4.18: Properties of cluster HR01 at z = 2.0, 1.0, 0.5 and 0.0, where panels from left to right correspond to the log of X-ray surface brightness, sin(λ 2 ), correlation between 10 0 R 4 IX andσ ¯ in the sky plane with cut-oﬀs applied at 10− of the maxima, and correlation R between IX andσ ¯ without any cut-oﬀs, respectively. Here the reference wavelength is R chosen as λ0 = 100 cm.

Fig. 4.19: Properties of cluster HR02 at z = 0.0, where panels from left to right correspond 2 to the log10 of X-ray surface brightness, sin(λ0 ), correlation between IX andσ ¯ in the R R 4 sky plane with cut-oﬀs applied at 10− of the maxima, and correlation between IX andσ ¯ R without any cut-oﬀs, respectively. Here the reference wavelength is chosen as λ0 = 100 cm. 4.2. Cosmological MHD simulations of cluster formation 134

Fig. 4.20: The change of slope in theσ ¯ IX relation with redshift for 12 simulated R − 4 clusters, with cut-oﬀs applied at 10− of the maxima (left) and without (right). their X-ray intensity peaks at 10−4 erg s−1 cm−2. In general, the X-ray images ∼ and RM maps show similar morphological features, hinting at a possible relation between the line-of-sight magnetic ﬁeld strength Bk and the thermal electron number density ne,th. A local point-by-point comparison betweenσ ¯R and IX of every snap- shot image as the cluster evolves is used to quantify this relation. It appears that theσ ¯R and IX of each cluster follow a power-law relation

b σ¯R = I , (4.9) A X where is the normalisation constant and b is the slope of theσ ¯R-I relation. A X Using the method of least-squares, the relation

log (¯σR) = log ( ) + b log (I ) , (4.10) 10 10 A 10 X is ﬁtted and the results are shown in Figures 4.8–4.19.

While some scatter is visible, theσ ¯R–IX relation seems to be universal in virialised clusters. The linear best ﬁts are performed across all data and compared to those where the central region of the cluster is excised at I 10−4 erg s−1 cm−2. X ≥ The slopes have values of b 1, indicating that locally, the magnetic ﬁeld essentially ∼ scales with the gas density. Generally, the power-law is not steady and it appears to co-evolve with the cluster formation and dynamics. To investigate the time dependence of the power-law relation, the power indices of every evolving cluster are plotted against their redshifts in Figure 4.20. 4.2. Cosmological MHD simulations of cluster formation 135

Overall, there is no obvious trend in the slope evolution with redshift in the centre-excised clusters as shown by the left panel of Figure 4.20. Most of the slopes have values lower than one. However, when all data is being considered in the right panel of Figure 4.20, there is a noticeable bump at z = 1. This bump coincides with the period of active cluster formation at z = 1, during which a substantial amount of materials collapse to form larger-scale structures, thus amplifying ne,th and B. Furthermore, there is a noticeable decline in the overall slopes approaching the higher-redshift end, where little magnetic field is present. There is a considerable number of slopes which are larger than unity in Figure 4.20. It can be deduced that, in these instances,σ ¯R > IX, indicating that the X-rays are perhaps emitted after sufficient magnetic field amplification. It is worth noting that the slope is sensitive to how the cut-offs are applied. The centre-excised scenario where b . 1 is more reliable, as it gets rid of the overdense, central region of the cluster resulting from the simulation being adiabatic and lacking of e.g., radiative cooling and AGN feedback. Without the cut-offs, the scatter in theσ ¯R-IX relation may be better fitted with a power-law of order 2, suggesting that the RMs double with every X-ray photon count rate. This may make sense if the hot gas gets re-accelerated by the threading magnetic fields to re-emit in the radio wavelengths. Apart from this, the resolution level of the simulations has quite an effect on the overall magnetic field amplification, when comparing the final magnetic field strengths of C01–10 with HR01–02. This is clear from Figure 4.20 where the slopes of C01–10 are generally lower than those of HR01–02. The HR clusters are generated with an improved divergence cleaning scheme at higher resolution, −2 reportingσ ¯R 100 rad m which is consistent with literature. ∼ A future direction would be to address whether a similar correlation exists in larger-scale structures, such as superclusters, which would further improve the understanding on their physical and magnetic field properties. It would also be worthwhile to investigate whether the power-law relation still holds in X-ray emitting clusters with radio relics. This would potentially shed some light as to whether there exists a common origin for radio halos and relics.

4.2.4 Properties of cluster X-ray and radio emissions

Apart from Faraday RMs, it would also be interesting to look into the properties of cluster X-ray and radio emissions. The high-resolution cleaned GCMHD+ simulation 4.2. Cosmological MHD simulations of cluster formation 136

(see AppendixA) produces a cluster with a central peak magnetic ﬁeld of 2.5 µG, ≈ which is in reasonable agreement with e.g., Govoni and Feretti(2004); Bonafede et al.(2010). This cluster is however an order of magnitude smaller in mass than clusters with observed radio halos. At redshift z = 0, the X-ray and radio emissions are calculated and examined whether a power-law relation naturally arises. The X-ray emission is calculated using equations 4.7 and 4.8. The radio emission is synchrotron radiation from energetic electrons gyrating around the cluster magnetic ﬁelds. Assuming relativistic electrons with a power-law energy spectrum

n (γ) dγ = γ−p dγ , (4.11) re C where p is the electron energy spectral index and nre(γ) dγ is the number density of relativistic electrons in the energy range of γ and γ + dγ, the energy density of the relativistic electrons is therefore Z ∞ 2 1−p re = (mec ) dγ γ . (4.12) E C 1 Treating as 1% of the thermal energy density (Kotera and Olinto 2011; Geng Ere et al. 2012), the normalisation constant becomes −3 nekBTe = 3 10 2 . (4.13) C × mec As such, the speciﬁc radio emissivity is

(p−1) 3 − 2 √3e B⊥ mec2πν p 19 p 1 −1 −3 −1 jR = 2 C Γ + Γ erg s cm Hz mec (p + 1) 3eB⊥ 4 12 4 − 12 (4.14)

(Rybicki and Lightman 1979) at frequency ν, where B⊥ is the magnetic ﬁelds component perpendicular to the line-of-sight and Γ(...) is the Gamma function. The frequency is set to ν = 1.4 GHz, which is commonly used in radio observations, such as the VLA (see e.g. Xu et al. 2003). The electron spectral index is set to p = 2.2, corresponding to electrons that are freshly accelerated by shocks (see e.g. Achterberg et al. 2001). Integrating the radio emissivity over the interval ds along the line-of-sight gives the intensity Z ds B(p+1)/2 erg s−1 cm−2 Hz−1. (4.15) IR ∝ C ⊥

The resulting X-ray and radio intensities are shown in the left panels of Fig. 4.21. Each synthetic map/image of the cluster spans roughly 2.9 Mpc 2.9 Mpc × 4.2. Cosmological MHD simulations of cluster formation 137

4 5 6 Fig. 4.21: Left: Contours of radio intensities IR at 10 , 10 and 10 Jy, superimposed over the coloured X-ray intensities log10 IX of the higher resolution cluster generated using the cleaned implementation at present-day configuration, viewed in the xy (top), xz (middle) and yz (bottom) planes. The X-ray and radio intensities have similar morphologies. Right: A point-by-point comparison of the X-ray and radio intensities in the xy, xz and yz planes, 4 1 2 with a best-fit performed at I 10− erg s− cm− (red) and across all intensities (blue). X ≤ The correlation follows a power-law with a slope close to one. 4.2. Cosmological MHD simulations of cluster formation 138 and is divided into 256 256 square pixels. The synthetic X-ray colour images × of the cluster are overlaid with radio contours in the xy, xz and yz planes. −6 −1 −2 Here, a minimum cut-off at IX = 10 erg s cm is applied below which the X-ray emission would be too faint to be detected. The cut-off corresponds to a flux lower-limit F min 2.37 10−17 erg s−1 cm−2, after accounting for the spa- X ≈ × tial isotropy of the emission, following the method by Kotarba et al.(2011). The imposed minimum flux threshold is lower than the detection limit of ROSAT at F (0.1 2.4 keV) 10−12 erg s−1 cm−2 (e.g. Böhringeret al. 2013), but similar to X − ∼ that of Athena at F (0.5 2 keV) 10−17 erg s−1 cm−2 (e.g. Barret et al. 2013). The X − ∼ X-ray images show irregular and asymmetric substructures, hinting that the cluster is likely to be disturbed and may have experienced a recent merger. The cluster is also X-ray luminous, reaching a maximum intensity of I 10−3 erg s−1 cm−2 at the X ≈ centre. The overall temperature and density of the cluster are traced out to its virial radius, leading to lower values than those typically measured within r500 in X-ray observations (e.g. Mantz et al. 2010). Moreover, as mentioned before, the cluster 13 7 is less massive (m 10 M ) and rather small (r 1.44 Mpc) compared to gas ∼ vir ≈ many of the clusters that have been observed using Chandra and ROSAT. Superimposed over the X-ray images are synthetic radio contours in white, where the levels define intensities decreasing outwards at 106, 105 and 104 Jy, respectively. These values of intensities8 are independent of distance from the source and correspond to artificial fluxes of several µJy per pixel at 1.4 GHz, which are consistent with various observations (e.g. Govoni et al. 2001a; Murgia et al. 2009; Vacca et al. 2010; Murgia et al. 2010; Farnsworth et al. 2013) and numerical simulations (e.g. Vacca et al. 2010; Kotarba et al. 2011; Geng et al. 2012; Govoni et al. 2013). Following the method by Geng et al.(2012), the expected flux arriving on Earth is estimated by multiplying the calculated total intensities with a factor f = πr2 /(4πD2), assuming D 99 Mpc for the distance to the cluster (e.g. beam ≈ Coma, see Reiprich and Böhringer 2002), and treating rbeam as the beam radius corresponding to an angular resolution of 500, if observed with, e.g. the VLA ∼ B-configuration at 1.4 GHz (see e.g. Condon et al. 2003).

7 Note that r500 ≈ 0.7r200 ≈ 0.5rvir.

8The conversion between intensity and flux/surface brightness does not affect the value of the best-fit slope. 4.2. Cosmological MHD simulations of cluster formation 139

Qualitatively, the X-ray images and radio contours show similar morphological features, indicating a spatial coincidence between the thermal and relativistic components in the cluster. To quantify this, a local point-by-point comparison between the X-ray and radio intensities projected along the lines-of-sight: x, y and z is performed; adopting a similar approach to Dolag et al.(2001). This method is un- aﬀected by morphological diﬀerences and possible misalignments between the radio and X-ray maxima (Govoni et al. 2001a). The simulated cluster has a bolometric X-ray luminosity of L = 3.11 1044 erg s−1, which is marginally higher than X,500 × typically observed (e.g. Maughan et al. 2012) and likely due to the simulations being adiabatic. A positive correlation is found between the radio and X-ray intensities in the cluster, which appears to satisfy a power-law relation

= AIb , (4.16) IR X where A is the normalisation constant and b is the slope of the -I relation. The IR X method of least squares is used to ﬁt the linear relation

log ( ) = log (A) + b log (I ) , (4.17) 10 IR 10 10 X and the results are shown in the right panels of Fig. 4.21. Clearly, there is some scatter around the lines of best fit, indicating that the cluster is rather dynamically disturbed. Further investigations of the effects on the scaling relation is carried out by performing a best-fit across all data (blue) and comparing it to that where the central region of the cluster is excised (red). The latter omits the anomalous, overdense X-ray bright region of the cluster, which is a consequence of the simulation being adiabatic and lacking of additional physics, such as radiative cooling and AGN −4 −1 −2 feedback. By setting the upper limit at IX = 10 erg s cm , the best fit slopes are calculated, obtaining 0.998, 1.013 and 1.024 along the z, y and x directions, respectively. These values are slightly larger than those derived from an all-data fit: 0.914, 0.926 and 0.934; implying that the centre-excised -I relation is tighter, IR X and that the central region of the cluster does not follow a universal behaviour, possibly due to the missing physics in the simulations. Overall, the high resolution cleaned simulation is capable of reproducing results that are consistent with observations. The main results are as follows: 4.2. Cosmological MHD simulations of cluster formation 140

(i) The simulated cluster reaches the observed magnetic field amplitude at µG levels with a radially declining profile. (ii) The simulated cluster has a bolometric X-ray luminosity of L 3.11 X ≈ × 1044 erg s−1, which is an order of magnitude larger than observed for a cluster of this low mass (e.g. Giodini et al. 2013). This is likely due to the over-cooling of adiabatic simulations. (iii) At 1.4 GHz, the radio surface brightness of the cluster is calculated to be several µJy per pixel, comparable to that found by observations and other numerical simulations. (iv) The morphological similarities between the X-ray and radio emissions from the cluster correspond well with observations of several massive, merging clusters (e.g. Liang et al. 2000; Feretti et al. 2001; Govoni et al. 2001a; Kempner and David 2004; Storm et al. 2015). (v) At every line-of-sight, a local point-by-point radio to X-ray comparison shows a positive relation, which can be adequately described by Ib with b 1. These IR ∝ X ≈ results are consistent with b 1 derived from observed relations (Feretti et al. 2001; ≤ Govoni et al. 2001a). (vi) The observables in the radio and X-ray wavebands are related to the intracluster gas density, temperature and magnetic fields. Given the initial choice of the energy density of the relativistic electrons, the ratio between the radio and X-ray intensities yields the relation9:

Z Z − ds B(p+1)/2 ds n T 1/2 . (4.18) ⊥ ∝ e,th e,th

There are several caveats to take into consideration. First of all, the simulated cluster presented here is at least ten times smaller than the Coma cluster. Observa- tionally, radio halos have only been found in some massive clusters, and so far, they have not been detected in smaller clusters. Hence, whether or not small clusters could harbour a weak halo is debatable. The two key ingredients for producing synchrotron radio emission are energetic electrons and magnetic ﬁelds. Unless there is a strong reason to exclude their presence, radio halos could exist in low mass

√ 9 R 1 R 1 2 It is worth noting that, while −1 dx f(x) ∝ −1 dx f( 1 − x ), f(x) is not necessarily propor- √ tional to f( 1 − x2). Furthermore, the result may be obtained by assuming the frozen-in condition and neglecting temperature dependence (Dolag 2018, private communication). 4.3. Summary 141 clusters. Given that clusters grow in mass through accretion and mergers, on top of radio-emitting relativistic electrons having a shorter lifetime than the dynamical time of clusters and their major merging intervals, as well as larger clusters having more charged particles available to be accelerated, it is natural to have a higher probability in ﬁnding radio halos in more massive clusters.

Whilst the magnetic fields are relatively weak and frozen into the hot plasma, their interactions with the relativistic plasma is not known from the simulations. Assumptions have to be made where the relativistic energy density is scaled with the thermal energy density, as well as treating the energy spectrum of the relativistic electrons as a simple power-law. As a result, the spatial coincidence between the radio halo emission and the hot gas regions is somewhat expected. The calculations also show hints that the power-law may not even hold in the -I relation, IR X as the slope is particularly sensitive to how the cut-offs are applied. While it is difficult to rule out whether these may be artefacts resulting from the models being prescribed in this work, it could also be due to the MHD interactions between the bulk of magnetic flows and the non-trivial interactions heating the gas particles. Hence it is premature to draw any conclusions on discriminating between the possible re-acceleration mechanisms or the magnetic field distribution (see e.g. Planck Collaboration 2013, for various models) for radio halo generation based on the value of the slope itself using the simulations shown in this work.

4.3 Summary

This chapter discusses the applications of Faraday rotation measure in the diagnosis of galaxy cluster magnetism. Firstly, the usage of rotation measure fluctuation (RMF) analyses for large-scale magnetic field diagnostics is assessed in the context of polarised radiative transfer. By ray-tracing through various magnetic field configurations and electron density distributions, the conventional random walk approach to modelling RMF is shown to be not universally valid, breaking down for densities with fractal and log-normal structures. This implies that density fluctuations can significantly obscure the effect of magnetic field fluctuations, and hence affect the correlation length of magnetic fields determined by the conventional random walk model. Furthermore, it is difficult to disentangle the signals from the density and magnetic field fluctuations based on the value of the standard deviation of RM itself. 4.3. Summary 142

Even without degeneracy between the signals from density and magnetic field fluctuations, radiative processes such as absorption and emission can confuse and ambiguate the interpretation of the RM. In addition to the thermal electrons, non- thermal electrons can also contribute to the Faraday rotation. The interpretations of RMF analyses are cautioned against when a characteristic density is ill defined. The inference of observational data has to be clarified as the spatial correlations are generally dissimilar along the line-of-sight and across the sky plane. A more proper treatment based on covariant polarised radiative transfer is necessary to develop solid theoretical models and predictions in preparation for the SKA. Next, the scaling relations between the dispersion of RM and X-ray intensity are calculated for galaxy clusters assembling from z = 3 to z = 0 in the GCMHD+ simulations. The relation between the dispersion of RM and X-ray intensity varies unsteadily during cluster formation with slopes of b 1 for I 10−4 erg s−1 cm−2, . X ≥ indicating that, locally, the magnetic field scales with the gas density. The centre- excised relation is more reliable than the cluster-centric relation as the cluster centre tends to be anomalous from the lack of additional physics in the simulations. Since the correlation length of magnetic fields can be derived from the dispersion of RM, this relation allows X-ray observations by e.g. Athena to determine the evolution of magnetic fields at galaxy cluster scales, which can be independently compared to that inferred from radio observations by e.g. the SKA. Lastly, the radio and X-ray intensities of a high resolution cluster are calculated and found to exhibit similar morphological features. A local point-by-point comparison between the radio and X-ray images of the simulated cluster reveals a power-law relation with a best-fit slope of b 1, which is consistent with several ∼ observed halo clusters. This suggests, even though the simulated cluster is of lower mass than those seen to host halos, the power-law relation can still naturally arise from having the weak magnetic field frozen into the gas. Additionally, it is ascertained from the radiative transfer calculations that cluster halo observations can be reproduced by assuming that a fraction of existing thermal electrons in the ICM is accelerated to re-emit in the radio synchrotron. 143

Chapter 5

Cosmic rays in magnetised molecular clouds

This chapter presents results which have been submitted in whole or in part in: Ellis R. Owen, Alvina Y. L. On, Shih-Ping Lai and Kinwah Wu, “Interactions of energetic cosmic rays in molecular clouds: astrophysical implications and observational signatures”, submitted March 2020 to MNRAS. I am a co-author of this paper. I formulated the application of structure functions to characterise the magnetic fields and cosmic ray propagation in molecular clouds. I contributed to the calculations of angular dispersion function that was used to derive the cosmic ray diffusion coefficient. I cleaned the IC 5146 polarisation data and calculated the frequency distributions and power spectra. I also conducted consistency and accuracy checks for the science, and assisted the lead author in parts of the manuscript preparation.

The previous chapter assesses how Faraday rotation measure fluctuation analyses are used in the diagnosis of galaxy cluster magnetism. These large-scale magnetic fields leave observational signatures in the radio synchrotron emission, whose total intensity reveals the field strength and polarisation traces the field orientation. On smaller scales, magnetic fields are also known to pervade molecular clouds (MCs) by leaving observable imprints through Zeeman (1897) splitting, the Goldreich- Kylafis (GK) effect (Goldreich and Kylafis 1981; Kylafis 1983), as well as in the sub-millimetre dust polarisation. The Zeeman and GK detections are scarce, particularly in the denser regions of the MCs (e.g. Crutcher et al. 1993; Ward-Thompson 144 et al. 1994; Crutcher et al. 1999; Lai et al. 2003; Crutcher 2012). Whether or not dust polarisation can effectively trace magnetic fields in denser regions of the cloud is still unclear. The interpretation of dust polarisation in terms of magnetic fields relies on the understanding of grain alignment theories (Davis and Greenstein 1951; Gold 1952; Dolginov and Mitrofanov 1976; Purcell 1979; see also, e.g. Lazarian 1995; Draine and Weingartner 1996b; Lazarian et al. 1997; Roberge and Lazarian 1999; Cho and Lazarian 2005; Lazarian and Hoang 2007, 2008; Hoang and Lazarian 2008; Hoang et al. 2018; Lazarian and Hoang 2019). Although these theories are complex and still a work-in-progress, they are often used to derive magnetic field information from sub-millimetre observations in the MCs (see e.g. Ward-Thompson et al. 2000, 2002; Crutcher et al. 2004; Kirk et al. 2006; Andersson and Potter 2007; Li et al. 2013; Wang et al. 2019). More specifically, dust grains in denser cloud regions may not efficiently align with the magnetic fields (Lazarian et al. 1997; Lazarian and Hoang 2007; Hoang and Lazarian 2009), although there has been recent evidence suggesting otherwise (e.g. Cho and Lazarian 2005; Jones et al. 2016; Wang et al. 2017).

The dense cores of MCs within the Milky Way are expected to be shielded from much of the ionising (particularly ultraviolet, UV) interstellar radiation by dust and molecular Hydrogen (Draine 2011). However, observations have revealed sustained ionisation rates of up to ζH = 10−17 10−15 s−1 in dense cores (Caselli − et al. 1998; van der Tak and van Dishoeck 2000; Doty et al. 2002), compared to ζH = 10−16 s−1 in diﬀuse interstellar clouds (Black et al. 1978; van Dishoeck and Black 1986; Federman et al. 1996). The cause of this ionisation is widely attributed to cosmic rays – CRs (Goldsmith and Langer 1978; Goldsmith 2001; Lequeux 2005; Draine 2011), which would also act to regulate the temperature (e.g. Spitzer and Tomasko 1968) and chemical evolution (e.g. Desch et al. 2004; Indriolo et al. 2015; Bisbas et al. 2017; Padovani et al. 2018; Albertsson et al. 2018; Gaches et al. 2019) of MCs. CRs are energetic, charged particles which can penetrate into a MC (e.g. Ivlev et al. 2018) and ionise its shielded dense core. CR protons in the MeV-GeV range and CR electrons in the 10 keV-10 MeV range are believed to contribute to the bulk of this ionisation (e.g. Spitzer and Tomasko 1968; Padovani and Galli 2011; Yamamoto 2017).

In a Galactic setting, CR ionisation rates can vary from ζH 6.8 10−18 s−1 to ≈ × 145

ζH 1.2 10−15 s−1 (Spitzer and Tomasko 1968), with a rate of around 10−16 s−1 ≈ × typically adopted (see Glassgold and Langer 1974; Hartquist et al. 1978; Black et al. 1978; van Dishoeck and Black 1986; Federman et al. 1996; Geballe et al. 1999; Indriolo et al. 2007; Geballe et al. 2007, and Padovani et al. 2009; Draine 2011 for overviews). CRs with energies above a GeV can also play a role. These are associated with star-forming activities which yield massive stellar end products, e.g. supernova remnants (see Hillas 1984; Kotera and Olinto 2011, for reviews). Such environments can accelerate particles to relativistic energies through, e.g. Fermi (1949) acceleration in diffusive shocks (e.g. Krymskii 1977; Bell 1978a,b; Axford et al. 1977; Blandford and Ostriker 1978). At these high energies, pion-producing (π0, π±) pp interactions between CR protons and the dense MC gas can arise (cf. Owen et al. 2018; Kafexhiu et al. 2014). The decay of π± yields secondary MeV CR electrons which may be able to contribute to the ionisation rate (e.g. Padovani et al. 2009). Higher-energy CRs can also engage with the ambient magnetic field, and drive a gas heating rate via Alfvénwave excitation (Wentzel 1971; Wiener et al. 2013b).

The ionisation level of a dense cloud governs the degree to which it is coupled to its ambient magnetic field. This, in turn, regulates its stability against fragmentation and/or gravitational collapse (Mestel and Spitzer 1956; Price and Bate 2008) and influences its subsequent star-forming activities. The propagation of charged CRs is affected by the local magnetic field, which can develop a complicated structure (Padovani and Galli 2013) with field strengths being amplified during co-evolution with its host cloud (Crutcher 2012). In this scenario, CRs may be focused by the field morphology, causing a convergence in their diffusive propagation. On the other hand, CRs may also be reflected/deflected as their pitch angles increase due to the stronger magnetic field. These antagonistic processes generally act concurrently (Cesarsky and Volk 1978; Ko 1992; Chandran 2000; Desch et al. 2004; Padoan and Scalo 2005), with mirroring/deflection effect usually dominating over focusing (Padovani and Galli 2011, 2013).

This chapter investigates the ionisation and heating of MCs in the presence of CR irradiation and magnetic ﬁelds. Firstly, the propagation of CRs in MC environments is formulated, accounting for their injection/absorption via hadronic interactions, ionisations, and MHD wave excitations in a self-consistent manner. 5.1. Cosmic rays in molecular clouds 146

The formulation is then applied to a model MC in a typical Galactic interstellar environment. Gas temperatures and molecular abundances are derived from the + + + calculations of the chemical balances of species including HCO ,H3 , OH and + H2O . Finally, the formulation is also applied to the interstellar cloud IC 5146, whose polarisation observations are used to trace its magnetic fields, from which the CR propagation parameters and ionisation/heating patterns are determined. This chapter is organised as follows. Section 5.1.1 outlines the characteristics of MC environments based on studies of the Milky Way and nearby galaxies, whereby a model is introduced to describe the density and magnetic structures of MCs. The dominant components engaging with CRs are identified, and the key features of the Galactic IC 5146 region are described. Sections 5.1.2 and 5.1.3 present the relevant CR physics – the CR propagation in MC environments and the interaction channels for both hadronic and leptonic CRs. Section 5.2 details the analysis of magnetic field structures in MC environments and demonstrates how the propagation of CRs in IC 5146 can be determined from this. Section 5.3 presents the results of the CR heating and ionisation rates in a model MC and in the dense filamentary structures of IC 5146. Lastly, Section 5.4 summarises this chapter.

5.1 Cosmic rays in molecular clouds

5.1.1 Molecular clouds

The interstellar medium is multi-phase, with cold, dense neutral MCs inter-mingled with hot tenuous gases in pressure equilibrium. MCs are believed to have formed via condensation from ISM gas, as the latter cools and collapses under gravity. The size of a MC varies from several to tens of pc. The cloud density and temperature are typically around 102 cm−3 and 10 K, respectively. Observations indicate that MCs are permeated by magnetic fields with strengths of a few µG. These magnetic fields are presumably swept in from the ISM by the collapsing material to create a “pinching” in the field orientation. The structure of the magnetic field vectors has been observed to resemble an “hourglass” (Girart et al. 2006; Rao et al. 2009; Tang et al. 2009), where field lines are drawn closer together in regions of higher cloud density. This produces a much stronger magnetic field within the MC compared to the surrounding ISM (Crutcher et al. 1999; Basu 2000; Basu et al. 2009). Further gravitational collapse within MCs to form clumps and cores is mediated by 5.1. Cosmic rays in molecular clouds 147 pressure support, either due to magnetic fields (Mouschovias 1991; Mouschovias and Ciolek 1999) or turbulence (Padoan and Nordlund 1999; Mac Low and Klessen 2004; Zhang and Li 2017; Coudéet al. 2019). Recent studies suggest that both have an important role (Kudoh and Basu 2008; Vázquez-Semadeniet al. 2011; Seifried and Walch 2015; Federrath 2016; Planck Collaboration 2016). This leads to a hierarchical substructure, wherein clouds host clumps of size 0.3 – 3 pc and density 103 104 cm−3. Moreover, cores on scales of 0.03 – 0.2 pc may develop within these − clumps. Cores have higher densities of around 104 105 cm−3, with some cases even − reaching 106 107 cm−3 (see e.g. Bergin and Tafalla 2007, for a review). Note that − different distinctions between the components are also proposed in literature and are equally valid (see e.g. Myers 1995). This is because an exact description of the continuous substructure of MCs cannot be fully captured by a simple hierarchy of just a few distinct elements (Rodr´ıguez 2005). Polarisation observations have revealed complex structures in the magnetic fields towards the dense cores of a number of MCs (see e.g. Hull et al. 2017). This indicates that turbulence is important in the cloud cores and plays a role in giving rise to the complexity in the magnetic structures of these MCs. There are, however, some massive cores where the magnetic fields are ordered, on 0.1 – 0.01 pc scales, and mostly parallel or perpendicular to the ordered fields on larger scales (Zhang et al. 2014) and to the observed orientation of their host cloud (Li et al. 2015a). In these clouds, the magnetic fields are of dynamical importance globally, as they regulate the initial cloud collapse/fragmentation processes (see also Li et al. 2009; Koch et al. 2014), despite that turbulence may be arguably more dominant on the sub-pc scales (see Ching et al. 2017; Tang et al. 2019). Being of higher density and smaller size, the cores are observed to sustain typical magnetic field strengths of around 10 15 µG(Crutcher et al. 1999),1 while the mean line-of-sight and total − magnetic field strengths in the inter-core regions of MCs have been observed to be 7.4 µG and 14.8 µG, respectively, via Zeeman effect (Thompson et al. 2019). A three-zone model for the structure of a MC and its core is constructed to calculate the interactions and propagation of energetic particles. The model is illustrated in Fig. 5.1. In Zone 1, which is an external ISM region outside the

1Note that observations show substantial spread in this value, which seems to be sensitive to the type of core observed (Crutcher et al. 2010). 5.1. Cosmic rays in molecular clouds 148

MC, CRs propagate at the local Alfvénspeed v (= v = B/√4πρ, with ρ as the A | A| plasma density) in the almost fully ionised medium (i.e. ionisation fraction, x 1), i ≈ due to the streaming instability (see Wentzel 1974; Kulsrud 2005). The generalised modified Alfvénspeed of a medium which is not fully ionised (i.e. xi < 1) is given by

−1/2 vA,i = vA(1 + εi) (5.1)

(Mart´ınez-Gómezet al. 2018), where ε = n /n . Here, n = n (1 x )/x is i H ions H ions − i i the density of neutral molecules (regardless of whether they are H or H2), and nions is the density of ions (assuming their abundance is equivalent to that of thermal electrons). It follows that v v √x . From Zone 2 to 3, the magnetic field pinching A,i ≈ A i effect resulting from the cloud’s evolution, combined with the associated increased density/neutral fraction, would begin to cause CR propagation to become increasingly diffusive as the higher neutral fraction damps the Alfvénwaves (Kulsrud and Pearce 1969; Zweibel and Shull 1982). As long as the magnetic field orientations and density structure are known at any given location, this approach can be adapted to account for any likely MC structure. It is not only limited to the “hourglass” magnetic field configuration as illustrated in Fig. 5.1. In this simplified model, only a single core (Zone 3) is considered within the MC, but this can be extended to any number of cores as required. Unless otherwise specified, nH refers to the number density of all gas within the cloud, whether atomic or molecular Hydrogen.

5.1.2 Cooling and absorption of cosmic rays

Electrons

In each interaction event, a CR electron only transfers a small fraction of its energy to the medium. The cumulative energy loss of the electron is a sum of many small decrements. In practice, the sequence of energy loss can be approximated as smooth and continuous. Thus, a continuous model, with averaging properties of energy loss in interactions, can be constructed and this model is adequate for the purpose of this study. Although CR electrons and positrons have opposite charges, their energy loss mechanism is practically identical in the astrophysical environments considered here. The rate derived for energy loss (such as cooling due to Coulomb interaction) applies to both electrons and positrons and also to both primary and secondary particles. Hence, both e+ and e− are referred to as “electrons” hereafter without 5.1. Cosmic rays in molecular clouds 149

ZONE 1 ZONE 3 ZONE 1 Dense clumps/cores

Incoming Incoming CRs from CRs from ISM ISM

Diffuse cloud (between clumps)

ZONE 2

Fig. 5.1: Schematic of the different CR propagation regions within the density and magnetic field profiles of an idealised molecular cloud, characterising observed features (e.g. Rao et al. 2009). Zone 1 is the ISM region (x 1). In Zone 2, the magnetic field pinching effect i ≈ and increased density/neutral fraction of the cloud begins to affect the propagation of CRs, which becomes increasingly diffusive. Zone 3 is the dense core of the MC where magnetic fields are the strongest and densities the highest. The gas is almost completely neutral 7 (x 10− or less, see Draine 2011) except for the ionisation caused by CRs. Magnetic i ≈ focusing/mirroring effects become important in Zone 3. losing generality. In a low-density plasma, the rate of cooling via electron-Coulomb interactions2 for ionisation fraction xi is

b m c2 n x c σ ln Λ (5.2) C ≈ e H i T

(see Dermer and Menon 2009), where me is the rest mass of an electron, c is the speed of light and σT is the Thomson cross-section. The Coulomb logarithm is taken as ln Λ 30. The rate of cooling due to electron bremsstrahlung (free-free) process3 ' is b (γ ) α c σ n x γ m c2 (5.3) ﬀ e ≈ f T H i e e 2See e.g. equation 8.12 (Dermer and Menon 2009) for the full description.

3See e.g. equation 8.30 (Dermer and Menon 2009) for the full description. 5.1. Cosmic rays in molecular clouds 150

(see Dermer and Menon 2009), where αf is the ﬁne-structure constant and γe is the Lorentz factor of the electrons. Average ionisation losses due to interactions with Hydrogen per CR electron are given by

b (γ ) = [n /n ] (1 x )ζion(γ )γ m c2 (5.4) ion e H e − i e e e e (Spitzer and Tomasko 1968). Here, the diﬀerential (per energy interval) direct ionisation rate by electrons4 is deﬁned as

ion X ion ζe (γe) = ne(γe)c σe,x (γe) , (5.5) x ion where σe,x is the cross section associated with each relevant ionisation process x between electrons and the neutral medium. Strictly, the composition of MCs is dominated by molecular Hydrogen, not atomic Hydrogen, and this changes the available channels through which ionisation may proceed. Together with direct ionisation (e + H e + H+ + e), dissociative ionisation (e + H e CR 2 → CR 2 CR 2 → CR + H + H+ + e) and double ionisation (e + H e + 2H+ + 2e) may also CR 2 → CR occur. This work follows the approach by Padovani et al.(2009) to model the ionisation cross sections.5 The rates for Compton and synchrotron cooling are

4 γ 1 4 b (γ ) = σ c β2γ 2 e σ c γ 2 (5.6) rad e 3 T e i −−−→ 3 T e i

(see e.g. Blumenthal 1970; Dermer and Menon 2009), where i is the energy density 2 of the radiation field ph (Compton cooling) or magnetic field B (= B /8π, synchrotron cooling), respectively. In MC environments, , and hence Compton ph B cooling is insignificant.6 The total cooling rate (at a position s) is the sum of all the contributing processes:

be(γe, s) = bC(s) + bﬀ (γe, s) + bion(γe, s) + brad(γe, s) . (5.7)

4This does not account for so-called “knock-on” ionisations caused by energetic electrons ejected by a primary ionisation process – these are only relevant for the eﬃciency of thermalisation/total ionisation rate experienced by the cloud.

5Padovani et al.(2009) adopts the semi-empirical expression by Rudd(1991) for the direct ionisation process, the polynomial ﬁt to data from Straub et al.(1996) (see also Liu and Shemansky 2004) for the dissociative ionisation process, and the ﬁt to the data from Kossmann et al.(1990) for the double ionisation process.

6Other eﬀects, e.g. triplet pair-production processes can arise at higher energies, but are not important in the weak radiation ﬁelds inside MCs (Schlickeiser 2002) 5.1. Cosmic rays in molecular clouds 151

Protons

Protons interact with their surrounding medium either by ionisation or, at kinetic th energies above a threshold of Ep = 0.28 GeV (Kafexhiu et al. 2014), by pion- producing hadronic interactions. This hadronic threshold is the minimum energy for the production of two neutral pions, being the lowest energy particle generated th 2 during the cascade, where Ep = 0.28 GeV = 2mπ0 + mπ0 /2mp, mπ0 is the neutral pion rest mass, and mp is the proton rest mass (see Owen et al. 2019a, for more details).

The average proton cooling rate (per CR proton) due to collision-induced ionisations follows that for electrons, and is modelled as a cooling process arising at a rate given by

b (γ ) = [n /n ] (1 x )ζion(γ )γ m c2 (5.8) ion p H p − i p p p p

(Spitzer and Tomasko 1968), where np is the number density of protons and γp is the Lorentz factor of the particles. The total diﬀerential ionisation rate in the cloud is

ion X ion ζp (γp) = np(γp)c σp,x(γp) , (5.9) x

ion where σp,x is the cross section associated with each relevant ionisation process x between protons and neutral H2. The possible channels are direct ionisation (pCR + H p + H+ + e), electron capture ionisation (p + H H + H+, which 2 → CR 2 CR 2 → 2 is eﬀectively a charge exchange process), dissociative ionisation (p + H p CR 2 → CR + H + H+ + e), and double ionisation (p + H p + 2H+ + 2e, which are CR 2 → CR modelled by adopting the cross sections in Padovani et al.(2009)). 7

The pion-producing events are modelled as an absorption process because a large fraction of the CR energy is transferred in a single interaction. These proceed

7Padovani et al.(2009) used the empirical ﬁt to the cross section for direct ionisation from Rudd et al.(1985), and the ﬁt by Rudd et al.(1983) for the electron capture (charge exchange) ionisation cross section. The cross sections for dissociative ionisation and double ionisation are taken to be equivalent to those for the corresponding electron interactions. 5.1. Cosmic rays in molecular clouds 152 via the major channels     0 0 + −   ppπ ξ0(π )ξ±(π π )      + + − 0 + −  p∆ ppπ π ξ0(π )ξ±(π π )  →     + 0 + − p + p  pnπ ξ0(π )ξ±(π π ) , (5.10) →      + 0 + −   npπ ξ0(π )ξ±(π π )  ++  n∆  →  + 0 + −   nn2π ξ0(π )ξ±(π π ) where ∆+ and ∆++ baryons are the resonances (Almeida et al. 1968; Skorodko et al.

2008), and ξ0 and ξ± are the multiplicities of the neutral and charged pions respectively, which are increasingly formed at higher energies. The hadronic products continue their interaction processes until their energies fall below the interaction th energy threshold Ep , which generally occur within a few interaction events (see e.g. Owen et al. 2018). The neutral pions decay rapidly into two γ-rays, with a branching ratio of 98.8% (Patrignani et al. 2016), on timescales of 8.5 10−17 s. × The charged pions undergo a weak interaction, either via π+ µ+ν e+ν ν¯ ν → µ → e µ µ or π− µ−ν¯ e−ν¯ ν ν¯ , with a branching ratio of 99.9% (Patrignani et al. → µ → e µ µ 2016), on a timescale of 2.6 10−8 s. The rate per unit volume at which protons × are absorbed by the pp process is

Sp(γp, s) = nH(s)np(γp)cσpπ(γp) , (5.11) where σpπ, the total inelastic pp interaction cross section, is parametrised as

3 σ = 30.7 0.96 ln(χ ) + 0.18(ln χ )2 1 χ−1.9 mb (5.12) pπ − E E − E (Kafexhiu et al. 2014), where χ = E/Eth = (γ 1)/(γth 1). Here, E is the E p p − p − th proton kinetic energy, Ep is the threshold kinetic energy for the pp interaction and th γp is the Lorentz factor of a proton at this threshold energy. 5.1.3 Cosmic ray propagation

A charged particle gyrates along a magnetic ﬁeld line, with a gyro-radius 1.07 10−5 E B −1 r = × pc , (5.13) L Z 100 MeV µG | | where E is the particle energy, Z is the magnitude of the particle charge and B is | | the strength of the magnetic ﬁeld. The gyro-frequency of the particle is ω c/r L ≈ L (in the relativistic limit). 5.1. Cosmic rays in molecular clouds 153

For a MC, the magnetic field usually exhibits an ordered structure on the length scale of the cloud itself (cf. section 5.1.1). Within the MC, however, the CR particles may propagate through disordered magnetic fields on various length scales, and as a result, the particles are deflected randomly. The multiple stochastic deflections of the CR particles can be regarded as a series of random scatterings in the magnetic field domains. From a phenomenological perspective, this may be approximated as a diffusion process with a characteristic length scale set by rL (see also Ivlev et al. 2018, for a self-consistent formulation of CR propagation into MCs). The size of the gyro-radius for particles in typical MC environments is many orders of magnitude smaller than the cloud size: the gyro-radius of a 100 MeV CR in a 100 µG magnetic field is around 10−7 pc, being even smaller at lower energies. This compares with a length scale of a few 10s of pc for a MC, a few pc for clumps (with n 103 cm−3) or a few tenths of a pc for the dense cores (with H ∼ n 104 105 cm−3) i.e. many orders of magnitude larger than r in all cases. As H ∼ − L such, within MCs, this leads to strongly directed, anisotropic CR diffusion in the direction of the magnetic field vector, which facilitates magnetic focusing and CR entrapment. Magnetic mirroring and focusing

Since CRs are constrained to gyrate and propagate along the magnetic field lines, their flux (per unit area) and any change in their number density (per unit volume) must be proportional to the density of the magnetic field lines per unit area. This is quantified using a magnetic concentration parameter, χ( B/B ), as in Desch ≡ 0 et al.(2004). This parameter is the ratio of the magnetic field strength B measured at some location within the MC (including the core region) compared to the mean

ISM value, B0. This follows from the magnetic field strength being defined as the magnetic flux through a point, which is proportional to the concentration of magnetic field lines through that point. As a CR propagates along a curved magnetic field vector into a MC, kinetic energy and magnetic moment must both be conserved. If it enters the cloud with a pitch angle (the angle between the incoming particle’s velocity and the orientation of the magnetic field vector) θin and speed vin, the component of its velocity along the field vector would be v|| = vinµ˜in, whereµ ˜in = cos θin. The CR gyration velocity around the magnetic field would then follow as 2 1/2 v⊥ = v (1 µ˜ ) . When a CR has propagated into a cloud where magnetic field in − in 5.1. Cosmic rays in molecular clouds 154

strength has attained B = χB0, the cosine of its corresponding pitch angle must be

µ˜2 = 1 χ + χµ˜2 , (5.14) − in in order to ensure that kinetic energy v2 + v2 and magnetic moment v2 /B are ∝ ⊥ || ∝ ⊥ conserved. It therefore follows thatµ ˜2 < 0 (i.e. the CR will be deflected and unable to propagate into the core) unlessµ ˜2 > 1 1/χ. This effect reduces the overall in − flux of CRs penetrating into the core of a MC, with CRs generally being reflected out of regions where B χB unless the pitch angle at which they enter the cloud ≥ 0 is small. Averaging over pitch angles within and outside the cloud can characterise this effect. The ratio of the internal and external fluxes demonstrates the degree to which magnetic mirroring/focusing has modified the CR flux at a given point where the magnetic field strength is known. Within the ISM, the angle-averaged CR flux (for n as CR particle density) may be expressed as

Z Z π Z 1 ∂n dΩ ∂n ∂n dφ 1 ∂n = = d˜µ = , (5.15) ∂t Ω 4π ∂t ∂t −π 4π 0 2 ∂t in assuming that the external ISM CR ﬂux is isotropic, and only the fraction of CRs 8 directed within a solid angle of Ωin towards the cloud are able to enter it . The same analysis through some boundary within the MC (where the propagation is now anisotropic) yields

Z ∂n ∂n = dΩ ξ(˜µin, µ˜) ∂t Ω ∂t MC ∂n Z π Z 1 = dφ d˜µ ξ(˜µin, µ˜) , (5.16) ∂t −π 0 where the term ξ(˜µin, µ˜) accounts for the combined focusing and mirroring effects, with the effective CR flux being amplified by focusing through a factor of B0χµ˜ and reduced by mirroring through a factor of B0χµ˜in. By substituting equation 5.14, ξ may be expressed as χµ˜ ξ(˜µin, µ˜) = p , (5.17) χµ˜2 χ + χ2 −

8Note that the magnetic concentration parameter may be small at some points of the cloud boundary. This occurs when B0 > B, e.g. if the external magnetic ﬁeld is not well-ordered, such that 0 < χ < 1. As a consequence, equation (5.18) becomes unphysical, indicating that the magnetic mirroring and focusing eﬀects at these points would be negligible. 5.1. Cosmic rays in molecular clouds 155 such that equation 5.16 can be written as

Z 1 ∂n ∂n χµ˜ d˜µ = 2π p ∂t ∂t 0 χµ˜2 χ + χ2 MC − h p i ∂n = 2π χ χ2 χ . (5.18) − − ∂t The magnetic scaling factor applicable to the CR ﬂux and/or number density is then the ratio of the angle averaged result inside the cloud (equation 5.18) compared to that outside the cloud (equation 5.15), deﬁned as

h p i η(χ) = 4π χ χ2 χ , (5.19) − − which is adopted in the subsequent calculations. Cosmic ray spectrum

Consider splitting the irradiating incident CR spectrum into two components (e.g. Padovani et al. 2009; hereafter P09), where the component above a GeV is largely responsible for heating and hadronic interactions, while that below a GeV is more important in driving ionisation processes. The diﬀerential spectrum of CRs above a GeV observed in the Milky Way follows a distinctive power-law

−ΓHE,i dni(E) E ni(E) = =n ˜HE,i , (5.20) dE E0

9 for species i being electrons or protons, and ni being their volume density. The spectral index in the 1 GeV to 1 PeV regime (largely attributed to internal Galactic 10 CR sources) may be characterised by ΓHE,p = 2.7, being appropriate for primary protons (e.g. Kotera and Olinto 2011), or the steeper index of ΓHE,e = 3.3 for primary electrons (e.g. Hillas 2006). The spectral index of secondary electrons is determined by the solution of the injection/transport equation rather than being adopted as an intrinsic boundary condition. The normalisation of the high-energy componentn ˜HE,i is speciﬁed as

−ΓHE,i CR,i fU (2 ΓHE,i)E0 n˜HE,i = − , (5.21) 2−ΓHE,i 2−ΓHE,i Emax E − 0 9While beyond the scope of this work, it would be useful to also consider the contribution of helium. The helium intensity is comparable or larger than that of electrons above 1 GeV.

10Note that the index would be less steep in regions of “fresh” CR acceleration, e.g. in the Galactic ridge or a starburst galaxy (e.g. Aharonian et al. 2006; Gaggero et al. 2017; H.E.S.S. Collaboration 2018b,a). 5.1. Cosmic rays in molecular clouds 156

with Emax = 1 PeV, and the reference energy taken as E0 = 1 GeV. CR,i is the CR energy density across all components, which takes a value of around 1.8 eV cm−3 for the Galactic ISM,11 of which (roughly) around 0.4 eV cm−3 may be attributed to the component below 1 GeV and the rest to higher energy CRs (e.g. Webber 1998;

Ferri`ere 2001). The parameter fU = 1.4/1.8 is the fraction of CR energy density attributed to the high-energy component of the spectrum, with the remainder in the lower energies described by

E −ΓLE,i ni(E) =n ˜LE,i . (5.22) E0

The normalisationn ˜LE,i is speciﬁed by the high-energy spectrum at 1 GeV (i.e at

E0) to ensure continuity across the spectral break. The spectral continuity is a more physically meaningful condition than maintaining an exact energy density ratio between the two CR components, which is subjected to substantial uncertainties as a result of e.g. modulation effects in the solar neighbourhood, and spatial inhomogeneities (e.g. Webber 1998; Cummings et al. 2016). The ratio ne/np at 1 GeV is set to be 1%, in line with observations (e.g., see Hillas 2006). Two values are set for each of ΓLE,e and ΓLE,p to account for the broad range of spectral indices considered in literature. The “minimum” spectra use Γ = 0.95 (Webber 1998, LE,p − hereafter W98) and Γ = 0.08 (the “conventional” model C of Strong et al. LE,e − 2000 – hereafter C00, following the notation of P09). The “maximum” spectra use 12 ΓLE,p = 1 (Moskalenko et al. 2002, hereafter M02) and ΓLE,e = 1 (model SE in Strong et al. 2000, hereafter E00 by the P09 convention). A CR flux model j(E) based on the above treatment is adopted, normalised to that of P09 for Milky Way conditions to provide an appropriate flux boundary condition when later solving the transport equations 5.25 and 5.26.

11Voyager 1 and Voyager 2 recently estimated the Galactic CR energy density to be 0.83 − 1.02 eV cm−3 (Cummings et al. 2016; Stone et al. 2019). The lower values compared to that in Webber(1998) could be due to diﬀerent interstellar energy spectra being used.

12Note that this index is ﬂatter than that obtained by Voyager 1. The electron intensity is also found to be larger than the proton intensity at E < 50 MeV. These new estimates are likely to aﬀect the lower-energy CR ionisation rate and heating rate. 5.1. Cosmic rays in molecular clouds 157

The transport equation

The propagation of cosmic rays can be modelled as a transport process, and the cosmic-ray transport equation takes the form:

∂n ∂ [D(E, s) n] + [vn] + [b(E, s)n] ∂t − ∇ · ∇ ∇ · ∂E = Q(E, s) S(E, s) (5.23) −

(e.g. Schlickeiser 2002), where n = n(E, s) is the differential number density of CRs (number of CR particles per unit volume per energy interval between E and E+dE) at a location s. The diffusive process is specified by the term [D(E, s) n]. The ∇ · ∇ diffusion coefficient, D(E, s), is determined by the gyro-scattering radius of the CRs, which depends on the CR energy E, the interaction of the CR with the local magnetic field (cf. equation 5.13), and the turbulence and the magnetohydrody- namical (MHD) perturbations along the local magnetic field vectors. This is dealt with empirically in section 5.2. [vn] is essentially an advection term which ∇ · describes the propagation of CRs in the bulk flow of a magnetised medium (e.g. inflow/outflow). However, in this work, it describes the forward propagation of CRs through a magnetised ISM (Zone 1 in Fig. 5.1), which is restricted to the Alfvén 13 speed vA(= B/√4πρ) due to the CR streaming instability. The mechanical and radiative cooling term b(E, s) can also be considered as an advection of the CR ensemble in energy space due to cooling processes arising along their propagations.14 The injection of CRs by the source term is given by Q(E, s), while CR absorption/attenuation is encoded in the sink term S(E, s). As with the cooling term, the exact form of these source/sink terms depends on the CR species in question.

The transport equation takes a diﬀerent form for both CR protons (denoted as np) and electrons (denoted ne – although, where necessary, primary and secondary electrons are diﬀerentiated as ne,1 and ne,2, respectively, with the total electron number 15 density ne = ne,1 + ne,2).

13 P Here, ρ = i ni mi is the mass density, for number density ni of particles with mass mi. 14Cooling eﬀects are most severe for CR electrons. The CR protons, being of larger mass, have a substantially smaller Thomson cross section than the CR electrons. As such, proton cooling in this work – apart from ionisation losses – is neglected.

15It is argued that both primary and secondary CRs contribute to the CR electron component of Galactic CRs and also in ISM environments of nearby galaxies. Of these, as much as 60-80% 5.1. Cosmic rays in molecular clouds 158

The transport equation for protons and electrons are considered separately, where the absorption of the protons is dominated by pp losses (see Owen et al. 2018, 2019b). This process injects some of the secondary CRs into the source term of the transport equation for electrons (others being provided by ionisations). A Cartesian geometry is adopted, with a coordinate s for the distance into a MC from the boundary where the cloud meets the ISM (similar to, e.g. Morlino and Gabici 2015 and Phan et al. 2018). Primary protons

For simplicity, consider the absence of additional sources of CR protons within or in the vicinity of the MC environment, and ignore CR acceleration in the system. The absorption is therefore dominated by the hadronic (pp) interaction. Thus, the transport equation is ∂ [η n ] ∂ ∂ [η n ] ∂ [η n ] p D(E , s) p + v p ∂t − ∂s p ∂s A ∂s ∂ + [bp(Ep, s)η np ] ∂Ep = Q (E , s) n (s)σ (E )η n c , (5.24) p p − H pπ p p where σpπ is the hadronic (pp) interaction cross section for pion production (see section 5.1.2), and the cooling term is given by ionisation loss (see equation 5.9). In the steady-state, equation 5.24 becomes ∂ ∂ [η n ] ∂ [η n ] D p + v p −∂s ∂s A ∂s ∂ = [bp η np ] nH σpπ η np c , (5.25) − ∂Ep − representing a balance between cooling loss, absorption, diﬀusion and advection of the energy of the CR particles. The source term can be replaced by introducing a boundary condition, given by spectra 5.20 and 5.22, at s = 0. This may be solved numerically subject to appropriate boundary conditions, as outlined in AppendixB.

Note that the same boundary condition applies symmetrically at s = sc, where sc is the size of the cloud. A further condition is required on D ∂[η np]/∂s, which is taken to be the CR flux through the boundary as estimated by j(E) (see section 5.1.3). could be secondary CRs (e.g. Torres 2004; Thompson et al. 2007; Lacki et al. 2010; Lacki and Beck 2013). In this work, primary CRs are regarded as those which enter the MC through the boundary, so both contributions are taken into account in this definition of the primary flux. Secondary CRs are considered as those produced within the MC. 5.1. Cosmic rays in molecular clouds 159

Primary electrons

Primary electrons are deﬁned as those which enter the cloud through some model boundary. The form of the electron transport equation in this case is similar to that for the protons, with the exception that there is now no absorption term. Electrons cool much more rapidly than protons and are not subject to catastrophic processes like the pp interaction. In the steady state, this gives ∂ ∂ [η ne,1] ∂ [η ne,1] ∂ D + vA = [be η ne,1 ] , (5.26) −∂s ∂s ∂s −∂Ee where D, η and vA are the same as for protons (see, e.g. Kulsrud 2005), and the electron cooling term is the sum of all relevant contributions, given by equation 5.7 (see section 5.1.2 for details). Equation 5.26 may be solved numerically (see AppendixB) subject to boundary conditions at s = 0 and s = sc. Secondary electrons

Secondary electrons are defined as those which are injected within the MC environment, at a rate encoded by the source term Qe(Ee, s). In this case, the transport equation reduces to ∂ ∂ [η ne,2] ∂ [η ne,2] ∂ D +vA = [be η ne,2 ] + Qe . (5.27) −∂s ∂s ∂s −∂Ee The cooling terms retain their definitions from equation 5.26, while the injection term is mediated by the solution to the proton transport equation. The injection of had CR electron secondaries Qe is related to the local number density and interaction rates of the CR protons (see Owen et al. 2019b, for discussions on multiplicities, by- products and energy transfer efficiencies from primary to secondary species). Note that the pp injection term is converted into differential units of electron energy (assuming, for simplicity, that multiple secondaries produced in a given interaction have adopted an equal share of energy), instead of in terms of the energy of the initiating proton flux. The inclusive pion formation cross sections are adopted from Blattnig et al.(2000). Moreover, to simplify the computation, the pion and muon decay processes are assumed to yield secondaries of equal energies. Electrons may also be injected by so-called “knock-on” production, where the ionisation of the ambient MC gas leads to the emission of an electron of sufficiently high energy to cause further ionisations. The knock-on injection term is Z K ion Qe (Ee) = 1.75 nH(s) dE1 σH (Ee; E1)cn1(E1) (5.28) E1 5.1. Cosmic rays in molecular clouds 160

(Brunstein 1965; Brown and Marscher 1977), where n1 is the differential number density (i.e. per energy interval) of CRs of energy E1 which initiate the first ionisation – either protons or electrons (including those provided by the pp process for completeness, although from comparison of cross sections, this would be substantially less important in a MC environment except at very high energies when ion CR fluxes would be relatively small) – and σH (Ee; E1) is the cross section for the production of knock-on electrons of energy Ee due to an initial CR energy E1 (see Abraham et al. 1966, although the energy-integrated form is used here). This gives the contribution of knock-on secondary electrons per unit volume per energy interval between Ee and Ee + dEe, with the total secondary CR electron injection term had K Qe = Qe + Qe . Equation 5.27 can then be solved (see AppendixB) subject to the boundary conditions that both ne and D ∂[η ne]/∂s = 0 at s = 0 and s = sc for all energies (as no secondary electrons would be expected to be flowing through or be present at the boundaries).

5.1.4 Eﬀects of heating and ionisation

Ionisation and chemical tracers

The ionisation rates of CRs can be constrained observationally through their impact on the astrochemistry in the MCs. The observed column density of molecular species along the line-of-sight gives their relative abundances, and the CR ionisation rate can be inferred from the abundances of the molecules. The molecular species of particular interest are those most sensitive to CR ionisation, three of which are CO+, OH+ and C+ being considered in this work.

Analysing chemical formation and destruction channels and their rates allows abundances to be modelled in a steady-state chemical system (see AppendixC). These would typically be probed using the abundance ratios, integrated over a line- of-sight, between two chemically related species.16 The ratios of closely related + + + species, N(CO )/N(H2), N(OH )/N(H2) and N(C )/N(C), (via reactions P6, P4 of Table1 and carbon ionisation, respectively), are considered by invoking a sim- pliﬁed astrochemical network (see Table1) for key reaction chains involving C, O and molecular/atomic Hydrogen as initiated by CR ionisations. This network is

16The use of chemically related species ensures that the ratios are probing abundances at roughly similar locations along each line-of-sight. 5.1. Cosmic rays in molecular clouds 161 valid in cold dense regions, on scales of less than 1 pc, where the gas temperature is below 10 K. For this proof-of-concept study, a binary ionisation approximation is adopted, such that CRs either ionise or do not ionise a species in an interaction event. The excited atomic/molecular states and the associated modifications required for the rate coefficients of reactions are neglected, thus relaxing the limit on the temperature. + + While H3 and HCO are conventional probes of CR ionisation rates in certain systems (e.g. van der Tak and van Dishoeck 2000), the former is hampered by foreground contamination issues in infrared (IR) bands and the latter is strongly affected by photo-ionisation. As such, alternative species have to be considered. OH+ is a more direct measure of CR ionisation, as it relies on fewer interaction steps in its formation (Hollenbach et al. 2012). The strongest OH+ transition is at 972 GHz, with previous detection in Herschel observations (e.g. Neufeld et al. 2010). The ionisation potential for the formation of OH+ is around 13.0 eV, so any CR ionisation signal could be easily contaminated by photo-ionisation in diffuse, non- shielded cloud regions. Thus, OH+ is only appropriate as a tracer of CR ionisation in dense, well-shielded clumps and cores. The abundance ratio

+ H N(OH ) (4/fH ) + 2 ζ 2 (5.29) N(H2) ≈ nH(kP4 fH2 + 2xikT1) is derived in AppendixC, using equation 16 for the OH + balance, assuming that + the rate of process P2 and the H3 dissociative recombination rate are negligible compared to the ionisation rates of H and H2. Here, fH2 is the molecular Hydrogen 17 fraction, and the rate constants kT1 and kP4 relate to the processes T1 and P4 (see AppendixC). CO+ is another plausible tracer of CR ionisation. It has accessible observable lines at 236.1 GHz (strongest), 235.8 GHz, 353.7 GHz and 354.0 GHz, which are all within the detection range of the Atacama Large Millimeter/Submillimeter Array (ALMA). The ionisation potential in this case is greater than 13.6 eV, so CO+ would not be susceptible to substantial ionisation by interstellar radiation ﬁelds, making it a more robust probe of CR ionisation. Even though the strongest 236 GHz line is easily confused with emission from complex molecular species (in particular

17 This is deﬁned as the number of H nuclei in H2 molecules as a fraction of the total number of H nuclei per unit volume. 5.1. Cosmic rays in molecular clouds 162

13 CH3OH), several detections have now been reported (Latter et al. 1993; Fuente and Mart´ın-Pintado 1997; Ceccarelli et al. 1998). The ratio

+ H N(CO ) xCζ = , (5.30) N(H2) nH kP5 + kP6fH2 /2 + xikT5

+ is derived using the chemical balance for CO (eq. 17), where xC is the Carbon abundance fraction. Another ratio

N(C+) 2ζH = , (5.31) N(C) fH2 nHxOHkI3 may be further considered from the chemical balance of C+ from C ionisation in equation 14. Here, the rate constants kT5, kP5, kP6 and kI3 are given in Table1, and + xOH is the OH abundance fraction. Although N(C )/N(C) would be contaminated by photo-ionisation in photon-dominated regions and environments that are not well-shielded from ionising interstellar radiation ﬁelds, it is a direct measure of CR ionisation rate in the denser clumps and cores, which is complementary to the other line ratios above.

Note that all of these abundance ratios require an estimate for the ambient volume density of the region being probed. Typically, estimates for density nH can be found via rotational excitation analysis of observed C2 lines (Sonnentrucker et al.

2007), analysis of H and the J = 4 level of H2 (Jura 1975), or from thermal pressure analysis of C i (Jenkins et al. 1983) – see also Indriolo and McCall(2012). Moreover, estimates for the fractional abundance of C and OH are required. For this proof-of- concept case, ﬁducial values of f = 0.053 (Indriolo et al. 2015)18, x = 1.6 10−4 H2 C × (Soﬁa et al. 2004) and x = 1.0 10−9 (Hollenbach et al. 2012)19 are adopted. OH × The model for xi, the fractional ionisation, is discussed in section 5.3.1.

18This is a mean value from a Milky Way sample. See Indriolo et al.(2015) for details.

19 Typically, variation of the xOH fraction would be expected throughout a MC due to its formation + + by the recombination of H3O with electrons or destruction by reactions with C or C . Studies indicate that this generally would lead to a build-up of OH in some parts of the cloud compared to the adopted ﬁducial limit (Hollenbach et al. 2012). As such, the estimated N(C+)/N(C) ratio in equation 5.31 should be regarded as an upper limit to constrain ζH. Care must be taken if using this ratio alone as strong contamination by photo-ionisation in the non-shielded regions would be expected to boost the C+ abundance far above the levels driven by CR ionisation processes. 5.1. Cosmic rays in molecular clouds 163

Heating

CR heating has been argued to be important in the ISM of galaxies (Field et al. 1969; Wiener et al. 2013b; Walker 2016; Owen et al. 2018, 2019b), in their circumgalactic environments (e.g. Salem et al. 2016; Owen et al. 2019a,b), and even in the intracluster medium between galaxies (e.g. Loewenstein et al. 1991; Wiener et al. 2013a; Ruszkowski et al. 2017). Its power is mediated by the thermalisation mechanism(s) at work, as governed by the local conditions (e.g. density, ionisation fraction, magnetic field). Traditionally, the excitation and subsequent non-linear Landau damping, ion- neutral collisional damping and/or turbulent damping of Alfvénwaves in magnetised environments, e.g. the intracluster medium (Loewenstein et al. 1991; Farmer and Goldreich 2004; Fujita and Ohira 2011; Wiener et al. 2013a; Fujita et al. 2013; Jacob and Pfrommer 2017; Ruszkowski et al. 2017), has been regarded as the main process by which CR thermalise, and it is likely that this would be dominant in a MC. CR heating by this channel would arise at a rate of n σv s−1 (Kulsrud and 0h i Pearce 1969; Zweibel and Shull 1982) for σv as the rate per particle of momentum h i exchange between ions and neutral particles, averaged over a thermal distribution. This indicates a damping timescale of around 3 103 yr, which would suggest that × Alfvénwaves could not propagate far (less than 0.01 pc) into a MC (cf. Martin et al. 1997), even if considering artificially-favourable conditions for wave propagation, i.e. low-density clouds (n 102 cm−3) appropriate for peripheral regions yet a relatively ≈ strong 0.1 mG magnetic field more suitable for a dense core (e.g. Crutcher 2012). At higher temperatures, this damping length would be even shorter, so thermalisation via this mechanism deep within a MC would depend on Alfvénamplification by those CRs able to reach the heavily shielded cores. Amplification could be either driven by streaming instabilities, or CR anisotropies within the cloud (Kulsrud and Pearce 1969; Wentzel 1969; Zweibel and Shull 1982), or any mechanical processes (e.g. motions due to gravitational collapse/condensation within the cloud) or turbulence that may act as a wave source (Carlberg and Pudritz 1990; McKee and Zweibel 1995; Gammie and Ostriker 1996; Martin et al. 1997; Falceta-Gon¸calves et al. 2003). The timescale of MHD wave excitation via the streaming instability is ∼ 106 s (Ginzburg and Syrovatskii 1964; Kulsrud and Cesarsky 1971), which is shorter than the corresponding damping rate. Alfvénwaves are therefore expected to per- 5.1. Cosmic rays in molecular clouds 164 sist, build up and thermalise within a MC, with a power

= v P (5.32) QA | A,i · ∇ c|

(Wentzel 1971; Wiener et al. 2013b), where P is the (local) CR pressure gradient ∇ c and vA,i is the Alfv´envelocity (eq. 5.1). Given that CRs propagate preferentially along the magnetic ﬁeld vectors, P in this direction can be estimated from |∇ c| the number density and spectrum derived from the relevant transport equation

(accounting for both proton and electron contributions), with vA,i also calculated according to the local conditions. The heating power at a location s is then Z

A(s) = (γA 1) vA,i dE kn(E) E . (5.33) Q − ∇ s

Here the adiabatic index is γA = 4/3 for the CRs. The integral over n(E) E dE gives the CR energy density, where n denotes the total contribution from both CR protons (np) and electrons (ne). For energies above a GeV, the number density of protons far exceeds that of electrons. Therefore, the contribution of ne to the total number density is negligibly small in the calculations. CR can also heat a medium by collisional ionisation and subsequent thermalisation. In Spitzer and Tomasko(1968), this was found to arise at a rate of roughly 6.3 10−27 n ergcm−3 s−1, where the heating power may be fully calculated using × H Z X 0 (s) = f n (s) dE (E)ζH (E) (5.34) QI i i Eh i

20 H0 for fi as the abundance fraction of species i with number density ni, and ζ (E) is the total diﬀerential (per energy interval) rate of the CR ionisation process. The energy a “knock-on” electron contributes towards heating its ambient gas is , Eh which is a function of the energy of the CR initiating the ionisation event. Below a threshold of the excitation energy of an atom, being 3/4 of its binding energy EB (13.6 eV for Hydrogen), the full CR knock-on electron energy is available for heating: collisional excitations cannot act to absorb its energy (to re-radiate as photons), while further secondary ionisations are also not possible. At higher energies between

3EB/4 < E < EB, a fraction of an electron energy 3EB/4 in any collision may now be lost to an excitation event, with only the remainder (E = E 3E /4) h − B 20It is important to distinguish between molecular and atomic Hydrogen fraction in this calculation. 5.1. Cosmic rays in molecular clouds 165

available for heating. At energies between EB < E < 3EB/2, a single ionisation or excitation process can proceed, with the probability of each determined by the relative weighting of the respective cross sections:

σex [E g E ] + σion [E E ] (E) = − B − B , (5.35) Eh σex + σion where g = 3/4 (Spitzer and Tomasko 1968), σex is the collisional excitation cross section, proportional to the energy-specific collision strength (Osterbrock 1989) and σion is the ionisation cross section for the process in question (cf. Padovani et al. 2009). Given the energies of interest in this work, collisions would overwhelmingly lead to an ionisation, meaning equation 5.35 reduces to (E) = E E . In the Eh − B range 3EB/2 < E < 7EB/4, only one ionisation may arise. However, if instead an excitation occurs then so must another and equation 5.35 still applies, but with g = 3/2. When E 7E /4, the situation is governed by the exact energy spectrum ≥ B of the CRs, where (E) is determined numerically by Dalgarno and McCray(1972). Eh This work adopts the numerical approximation in Draine(2011) for computational efficiency. Overall, an ionisation efficiency reduction would also arise to account for energy losses to other channels, which is calculated by weighting according to the respective timescales of each process. Other mechanisms have also been proposed, but are unlikely to be as important as MHD wave damping – in particular, see Colafrancesco and Marchegiani (2008); Ruszkowski et al.(2017); Owen et al.(2018, 2019b) where, on 0.1 kpc scales, thermalisation by Coulomb interactions in an ionised ISM is considered. In this process, secondary CRs are injected by pp interactions of CR primary protons (primary electrons would cool too quickly to propagate far from their source) to provide a channel by which CR protons can thermalise. This process can be the most effective in ionised media (see Owen et al. 2019b) but, in a predominantly neutral cloud, Coulomb thermalisation would only be able to attain levels of around −31 −3 −1 4 −3 −8 10 erg cm s for nH = 10 cm and xi = 10 (scaled from the result in Owen et al. 2019b). Gas cooling and equilibrium temperature

Cooling processes would also operate within MC environments, allowing an equilibrium temperature to be reached at which cooling and heating rates are comparable. Under typical dense molecular cloud conditions, gas cooling is dominated by 5.2. Propagation of cosmic rays in magnetised media 166

CO and dust, with the latter only becoming important above densities of around 3 104 cm−3 – below this, only CO cooling would operate effectively (e.g. Galli × et al. 2002; Goldsmith 2001). The analytic approximation in Whitworth and Jaffa(2018) and data from Gold- smith and Langer(1978) are used to calculate the CO cooling rates. Either a uniform dust temperature of 10 K through the cloud or the treatment in Goldsmith (2001) are assumed to calculate the cooling rates in this work. This determines the equilibrium temperature profiles of MCs subject to different intensities of CR heating.

5.2 Propagation of cosmic rays in magnetised media 5.2.1 Empirical characterisation of cosmic-ray propagation

To characterise the CR propagation in MCs, approaches similar to Schlickeiser and Achatz 1993a,b (see also Schlickeiser 2002; Kulsrud 2005) are adopted. The background cloud-scale magnetic field structure is taken to vary on length scales substantially larger than both the magnetic-field fluctuations and gyrating/scattering radii of CRs. Using a quasi-linear approximation (Jokipii 1966; Schlickeiser 2002), the Fokker-Planck (FP) equation (Kirk et al. 1988) can be greatly simplified, assuming that (i) the turbulence driving the field fluctuations is purely magnetic21 and of low- frequency, (ii) the turbulence components on different scales are uncorrelated and non-interacting i.e. v c, and (iii) the CR pitch angle is small. When the flow A of CRs and the orientation of the fluctuations are largely parallel to the orientation of the background cloud-scale magnetic field vector and independent of that in the perpendicular direction, the only non-vanishing FP coefficient is then

2 (λ1) ωL B0 Pµµ J ⊥ . (5.36) ≈ vA λ1 B I

Here λ = λ ( ω /ω ) is the CR resonant scattering length scale parallel to the 1 td | L| p,0 background magnetic ﬁeld line. The turbulent decay length scale is λ v τ , td ≈ A td where a turbulent decay timescale τtd = 2 Myr is adopted in this calculation (see

Gao et al. 2015; Larson et al. 2015). The (relativistic) CR gyro-frequency is ωL (cf. equation 5.13) with a sign convention set by the charge (in units of proton

21Essentially, this means neglecting density perturbations as well as the electric ﬁeld component of associated Alfv´enwaves. 5.2. Propagation of cosmic rays in magnetised media 167 charge), such that ω = (101/γ )(B/1 mG) s−1 for protons, and ω = (1.8 L p L × 4 −1 10 /γe)( B/1 mG) s for electrons (see e.g. Kulsrud 2005). The normalisation

ωp,0 is taken to be the gyro-frequency of a CR proton at a reference energy of

100 MeV. The magnetic field strength normalisation is taken as B0 = 1 mG to be comparable to the field strength outside the densest parts of clumps/cores (see e.g. Crutcher et al. 2010; Li et al. 2015a). Hence what remains to be evaluated are the two variables (λ ) and ⊥ in the above equation. J 1 I The dimensionless variable (λ ) characterises the magnetic field fluctuations J 1 along the direction of the background large-scale magnetic field vector and is defined as

Z λ1 Z ∞ λ λ1 (λ1) dλ Pˆk(kcλ) + dλ Pˆk(kcλ) , (5.37) J ≡ 0 λ1 λ1 λ where Pˆk is the power spectrum of the ﬂuctuations along the large-scale magnetic

ﬁeld vector. The wavenumber normalisation is deﬁned as kc = ωp,0/vA. In terms of the dimensionless variable κk = λkc,

Z λ1kc κ −1 k ˆ (λ1) = kc dκk Pk(κk) J 0 kcλ1 Z ∞ −1 kcλ1 ˆ + kc dκk Pk(κk) . (5.38) λ1kc κk

The variable ⊥ specifies the contribution from the orthogonal components of the I magnetic field fluctuations. Note that ⊥ is not dimensionless. Without losing I generality, two orthogonal components y and z of the perpendicular wave vector are denoted, corresponding to the wave vectors ky and kz, respectively. Therefore, 2 2 2 k⊥ = ky + kz, with k⊥ = ky + kz . In terms of these wave vectors and the normalisation kc,

Z ∞ Z ∞ dky dkz ⊥ Pˆ⊥(ky, kz; kc) I ≡ −∞ kc −∞ kc Z d2k ⊥ ˆ k = 2 P⊥( ⊥; kc) , (5.39) ⊥ kc where Pˆ⊥ is the power spectrum of the perpendicular component of the magnetic field fluctuations, which effectively takes the form of a scalar delta function, i.e. 2 2 Pˆ⊥(k⊥; k ) = δ(k⊥ k ) (cf. the “slab” approximation, Hasselmann and Wibberenz c − c 5.2. Propagation of cosmic rays in magnetised media 168

1968). For isotropic ﬂuctuations in the y-z plane,

Z ∞ dk⊥k⊥ 2 2 ⊥ = 2π 2 δ(k⊥ kc ) I 0 kc − Z ∞ dk⊥k⊥ h i = π 3 δ(k⊥ + kc) + δ(k⊥ kc) . (5.40) 0 kc −

As k = 0, k⊥ + k > 0, and hence, c 6 c Z ∞ dk⊥k⊥ π ⊥ = π 3 δ(k⊥ kc) = 2 . (5.41) I 0 kc − kc

Finally, the spatial diﬀusion coeﬃcient is

c2 1 Z 1 (1 µ2)2 D 1 2 dµ − . (5.42) ≈ 8 − γ −1 Pµµ

Since the pitch angles are small, Pµµ is not strongly dependent of µ (see Schlick- eiser and Achatz 1993b), and can be taken outside the integral. Thus, D ≈ 2c2/15P 22, for γ 1. µµ 5.2.2 Diﬀusion estimation Angular dispersion function

Dust polarisation can be used to probe the magnetic ﬁeld structure in dense MC environments over a range of scales. The ﬁeld structure is often characterised by the angular dispersion function

Npair 1 X (`) = [ϕ (s + `) ϕ (s)]d (5.43) Sd N i − i pair i=1 (e.g Redaelli et al. 2019), which is also referred to as the “structure function” (e.g. Schulz-Dubois and Rehberg 1981). The order number d = 2 is adopted to constrain the power spectrum, whereas the normalisation N = N (N 1)/2 is the number pair P P − of unique pairs in a data set of NP individual points. The angular dispersion function has often been used to study astrophysical magnetic ﬁelds, using e.g. rotation measure (RM) (e.g. Minter and Spangler 1996; Lazarian and Pogosyan 2016; Xu and Zhang 2016), as well as polarisation angle (PA) measurements, including in the analysis of MC environments (e.g. Hildebrand et al. 2009; Houde et al. 2009; Planck Collaboration 2016; Wang et al. 2019; Redaelli et al.

22Note that this formulation would be invalid if the MC is not in a steady-state, i.e. if there are large-scale flows of the cloud medium, resulting from e.g. ongoing gravitational collapse, inflows or outflows. 5.2. Propagation of cosmic rays in magnetised media 169

2019). In this work, differences between pairs of measured dust polarisation angles ϕ over separations ` are computed to quantify the similarity in the orientation of magnetic field fluctuations on different scales, with the intention of encoding the deviation of local perturbations from the background mean field vector.23 In practice, to calculate (`) from a set of N polarisation angles, every unique pair in Sn P that set must be identified and binned according to their angular separation distance

`. This would yield Npair unique pairs, indexed sequentially as j2 j N = j(N 1) − (N i) , (5.44) pair P − − 2 − P − for i and j as the indices of contributing NP data points to that pair in the original data set (where i = j). Equation 5.43 is then applied to all points within that bin 6 to give an estimate for the angular dispersion function at that scale. Diﬀusion parameter estimation

The application of equation 5.42 to empirical data requires the computation of the parallel fluctuation term (λ ). This depends on the power spectrum Pˆ(k) of fluc- J 1 tuations along the large-scale magnetic field vector which, arguably, is characterised by the fluctuations in the measured PAs. The power spectrum Pˆ(k) and angular dispersion function (`) are related via the Wiener-Khinchin theorem (Wiener S2 1930; Percival and Walden 1993) by 1 Pˆ(k) = [ (`)] (5.45) 2F S2 (see AppendixD for details), where [...] denotes a Fourier transform (FT). When F applied to a discrete data set separated into Nbins bins according to scale (e.g. measured PA difference between a pair of points, binned according to the angular 24 separation of each pair – cf. section 5.3.2), the discrete FT of S2(`n) can be taken for each scale-bin `n to find P (κn). The fluctuation statistic then follows a discretised form of equation 5.38:

ib Nbins −1 X κn −1 X kcλ1 (λ1) kc P (κn) + kc P (κn) , (5.46) J ≈ kcλ1 κn n=1 n=ib where ib is the bin index corresponding to the (normalised) resonant length scale

λ1 kc, and P (κn) is the discrete FT of S2(`n) for `n as the characteristic separation

23The strength of the local ﬁeld is estimated separately – see section 5.2.3.

24The Python packages SciPy (Virtanen et al. 2020) and NumPy (Press et al. 2007) are used. 5.3. Semi-quantitative analysis: model and results 170 length between data pairs in the bin (taken simply as the bin centre-point in `).

κn is the normalised wavenumber associated with `n. An empirical estimation for the diﬀusion parameter D within the observed system is obtained by substituting equation 5.46 into 5.36.

5.2.3 Observing magnetic ﬁelds in molecular clouds

Polarisation of starlight by dust (as well as polarised sub-millimetre emission from the dust itself) is often used to probe the strength25, orientation, and structure of interstellar magnetic fields on the plane of the sky (see e.g. Crutcher 2012, for a review). This polarisation arises from selective absorption (or re-emission) by dust grains, whose magnetic moments tend to align themselves with the local interstellar magnetic field due to radiative torque alignment, leading to a preferred perpendicular orientation of the grains to ambient magnetic fields in low-extinction regions (Dolginov and Mitrofanov 1976; Draine and Weingartner 1996a; Lazarian et al. 1997; Draine and Weingartner 1997). As a result, a larger grain extinction cross section occurs perpendicular to the background magnetic field vector, causing linear polarisation. While there is some debate over whether this alignment mechanism could operate effectively in denser high-extinction regions where radiation fields are lacking – e.g. in MCs and the cores therein – results are so far largely consistent with the radiative torque model, but with substantial variation in alignment efficiency between individual sources (see Whittet et al. 2008; Cashman and Clemens 2014).

5.3 Semi-quantitative analysis: model and results

5.3.1 Molecular cloud model

In the first instance, an idealised MC model is used to assess the propagation and interactions of CRs. The model has a Plummer-like density profile with a frozen-in magnetic field and a simplified CR diffusion coefficient. The frozen-in condition suffices in the idealised scenario, although in principle, the magnetic field is only frozen into a fraction of the molecular cloud that is ionised.

25The magnetic ﬁeld strength in MCs is usually estimated using the DCF method (see Section 2.4.3 for more details). 5.3. Semi-quantitative analysis: model and results 171

Fractional ionisation and density proﬁle

The ionisation of molecular gas in diﬀerent cloud components may largely be considered as a balance between the CR ionisation rate and recombination processes.

The resulting ionisation fraction per H2 molecule may be approximated by

1/2 Y 1/2 ζH n(s) −1/2 xi(s) = xi,0 −17 −1 5 −3 (5.47) Y 10 s 10 cm

−8 (Elmegreen 1979), where x = 8.7 10 for small dust grains and Y is the solar i,0 × metallicity. It is sufficient for the purpose of this study to adopt the “standard” CR ionisation rate of ζH = 10−16 s−1, being typical of the diffuse inter-clump medium (Black et al. 1978; van Dishoeck and Black 1986; Federman et al. 1996).26 The density profile n(s) comprises a clump and a core, described by two superimposed Plummer profiles (Whitworth and Ward-Thompson 2001; Lee et al. 2003; Dib et al. 2010), with n(s) = n [1+(s/s )2]a, where 2.5 a 1.5 for different clouds (Fed- 0 0 − ≤ ≤ − errath and Klessen 2013), including IC 5146 (see Arzoumanian et al. 2011). The calculations assume gravitational and thermal pressures to be in equilibrium. The parameters are set to a = 2, where n = 103 cm−3 and s = 2 pc for the clump − 0 0 5 −3 region, and n0 = 10 cm and s0 = 0.2 pc for the core (e.g. Bergin and Tafalla 2007). Magnetic field and diffusion coefficient

Consider that the magnetic ﬁeld strength scales with the cloud density as   BICM (n n1); B(n) = ≤ (5.48)  q¯  BICM (n/n1) (n > n1) ,

−3 where n1 = 300 cm , BICM = 30 µG for a background magnetic field strength of the inter-clump medium,27, andq ¯ 2/3 assuming spherical collapse and magnetic ≈ flux-freezing (see Mestel 1966; Crutcher et al. 2010). In this first approach, the orientation and structure of the magnetic field are ignored. Diffusion is isotropic,

26A more rigorous but computationally-expensive treatment demands the ionisation fraction to be calculated dynamically.

27This is estimated from line-of-sight field strengths in Crutcher et al.(2010) by omitting the diffuse cloud components; see also Thompson et al.(2019) which specifically considers inter-core regions of MCs (with 30 µG as a line-of-sight magnetic field strength falling well within the suggested range, albeit stronger than average for their sample). 5.3. Semi-quantitative analysis: model and results 172 with a coefficient

δ rL (E, B s) D(E, s) = D0 h| |i| , (5.49) rL,0 where B = B(s) is the characteristic mean magnetic field strength at a position h| |i|s | | s, and D = 3.0 1028 cm2 s−1 is an effective, empirical normalisation to the diffusion 0 × coefficient, based on the ISM in the Milky Way (Berezinskii et al. 1990; Aharonian et al. 2012; Gaggero 2012), as would be appropriate for a 1-GeV CR proton in a

5 µG magnetic ﬁeld with gyro-radius rL,0. The index δ = 1/2 (see also Berezinskii et al. 1990; Strong et al. 2007) accounts for the cloud turbulence spectrum which is set here to be the same as the broader ISM. A more physical approach is presented in section 5.3.2.

Results

The transport equations 5.25, 5.26 and 5.27 are solved with a numerical scheme described in AppendixB to calculate the distribution of CRs for the cloud model above. Fig. 5.2 shows the distribution of primary CRs. The number density grad- ually increases towards the cloud centre, which is also reflected in the CR energy density distribution, as shown in Fig. 5.3. The latter arises from the mirroring effects, coupled with a stronger magnetic field hampering CR propagation through the central region. The increase in the CR abundance is relatively moderate, roughly double ISM levels, suggesting that the effects of CR containment and focusing by the MC environments are not particularly severe. Note that the results in Fig. 5.2 may be substantially modified if assuming that the model has a spherical geometry or is in a more realistic environment, e.g. a medium that is strongly inhomogenous (see e.g. Ivlev et al. 2018).

Comments on the secondary electrons

The impact of the secondary CRs in this study is found to be negligible. Their abundance is several orders of magnitude lower than the primary CRs and their impacts are correspondingly small. A low secondary abundance would also be consistent with the view that γ-rays should dominate high-energy emission from MCs (Brown and Marscher 1977; Gabici et al. 2009; Casanova et al. 2010), and any synchrotron emission would be dominated by the primary electrons in the CRs (Strong et al. 2014; 5.3. Semi-quantitative analysis: model and results 173

9 10 3

10 10 cm CENTRE / CR n LOUD C 11 10

Electrons Protons

0.5 1.0 1.5 2.0 2.5 s/pc

Fig. 5.2: Distribution of energetic primary CR protons and electrons above a GeV propagating into the MC model prescribed in section 5.3.1. The small increase in the CR number density towards the core is due to increased magnetic containment.

Padovani et al. 2018).28 This is in line with expectations: knock-on production and pion-production processes injecting secondary electrons operate fairly competitively with one another (with knock-on processes slightly dominating at lower energies).

This occurs at a rate ofn ˙ n cσ ± n , (as protons are more abundant than elec- e ∼ H [π ] p −29 2 trons in the primary flux), where σ ± 10 cm at a GeV (Blattnig et al. 2000) is [π ] ≈ the effective pion-production cross section (note that this is different from the total pp inelastic cross section σpπ defined in equation 5.12). This can be balanced against the “loss” of secondary electrons via cooling over a timescale of t (σion c n )−1. e ≈ H Radiative processes would be more important at higher energies, but the power-law nature of the spectrum means that the number of these higher energy electrons makes a negligible contribution to their total number density. So, lower-energy ionisation losses can be regarded as the dominant secondary electron cooling process.

28Secondaries could begin to dominate at very high densities on much smaller scales, e.g. in circumstellar discs (Padovani et al. 2018). Very compact non-thermal sources in MCs may be a signature of synchrotron emission from secondary electrons (Jones 2014), although these may depend on in-situ re-acceleration in, e.g. proto-stellar jets (Padovani et al. 2015, 2016; C´ecereet al. 2016). Note that this is in tension with earlier studies, which argued that synchrotron radiation from secondary electrons in dark clouds could actually dominate Galactic radio emission (Brown and Marscher 1977); synchrotron emission from secondaries was also considered in Dogel and Sharov (1990) and Jones(2014), but these studies did not compare with the emission from primary electrons. 5.3. Semi-quantitative analysis: model and results 174

3.0 Total Protons Electrons 50 Background 2.5 ⇥ 3

2.0

1.5 ISM background level CENTRE eV cm / 1.0 CR LOUD CR C U AAAB+nicbZC7TsMwFIZPyq2UWwoji0WFxFQlXARjRRfGguhFaqLKcZ3WquNEtgOqQh+FhQGEWHkSNt4Gt80ALb9k6dN/ztE5/oOEM6Ud59sqrKyurW8UN0tb2zu7e3Z5v6XiVBLaJDGPZSfAinImaFMzzWknkRRHAaftYFSf1tsPVCoWi3s9Tqgf4YFgISNYG6tnlz2aKMYNZp6MUP1u0rMrTtWZCS2Dm0MFcjV69pfXj0kaUaEJx0p1XSfRfoalZoTTSclLFU0wGeEB7RoUOKLKz2anT9CxcfoojKV5QqOZ+3siw5FS4ygwnRHWQ7VYm5r/1bqpDq/8jIkk1VSQ+aIw5UjHaJoD6jNJieZjA5hIZm5FZIglJtqkVTIhuItfXobWadU9q17cnldq13kcRTiEIzgBFy6hBjfQgCYQeIRneIU368l6sd6tj3lrwcpnDuCPrM8fN4CT+Q== ✏ 0.5

0.0 0.5 1.0 1.5 2.0 2.5 s/pc

Fig. 5.3: Energy densities of primary CRs throughout the cloud under Galactic conditions. The background interstellar CR energy density is indicated by the blue line, while the contribution from the secondary CRs is negligible (not shown). The primary electron component is multiplied by 50 in this plot for clarity.

Typically, σion 10−20 cm2 is the electron-ionisation cross section (Padovani et al. ≈ 2009) for a secondary electron generated by a 1 GeV hadronic primary (retaining a few percent of the primary’s energy – see Owen et al. 2018). It then follows that ion −9 n n˙ t , which may be re-arranged to give the ratio n /n σ ± /σ 10 , e ≈ e e e p ≈ [π ] ∼ thus conﬁrming the negligible level of secondaries.

Observable and astrophysical impacts

CRs can influence their host environment by ionisation, heating, and modifying the chemical balance of certain species. The ionisation profiles due to the CR distributions in section 5.3.1 are shown in Fig. 5.4. The top panel indicates the impact of the high-energy CR component (HECR), above a GeV, with rates of ζH = 0.6 1.1 10−18 s−1, which are somewhat lower than the widely-adopted values − × of ζH = 10−17 10−15 s−1 in the dense cores (Caselli et al. 1998; van der Tak and − van Dishoeck 2000; Doty et al. 2002), and ζH = 10−16 s−1 in the diffuse inter-clump medium (Black et al. 1978; van Dishoeck and Black 1986; Federman et al. 1996). This difference can be attributed to the low-energy CR (LECR) population, below a GeV, as shown in the lower panel of Fig. 5.4. Although their energy density is less than that of the HECRs, LECRs engage more strongly in the ionisation processes. Hence, they can elevate the rate to ζH = 10−17 s−1, when adopting the “minimum” LECR species spectra in the W98 (Webber 1998) and C00 (Strong 5.3. Semi-quantitative analysis: model and results 175

18 18 1010 1.01.0 ⇥⇥ HECRS 0.9

0.90.8 1 s

0.7 CENTRE / 24 10 H 1.4 ⇥ ⇣

0.80.6 LOUD

1.3 C 1 1.20.5

s 1.1 1 0.70.4 /

s 0.5 1.0 1.5 2.0 2.5 1.0 / H

H s/pc ⇣ ⇣ 0.9 14 10 00.86

15 010.7

1 LECRS

0.6

s 16 10 0.5 1.0 1.5 2.0 2.5 CENTRE

/ 0.5 H

⇣ s/pc 17 10 LOUD C

18 010.4 0.5 HECRS 1.0 1.5 2.0 2.5

0.5 1.0 1.5 2.0 2.5 s/pc s/pc

Fig. 5.4: Ionisation proﬁle due to the HECR component (above 1 GeV) under Galactic conditions (top) and both HECR and LECR (below 1 GeV) components (bottom). et al. 2000) models, or to ζH = 10−14 s−1 if adopting “maximum” spectra in the M02 (Moskalenko et al. 2002) and E00 (Strong et al. 2000) models.

These ionisation profiles29 (again, accounting for the variation between the maximum and minimum LECR contribution to the ionisation rates) can also be used to estimate abundance ratios for the species considered in section 5.1.4, i.e. + + + N(CO )/N(H2), N(OH )/N(H2) and N(C )/N(C). These species are sensitive to probe the CR ionisation, irrespective of whether the ionisation is caused by LECRs or HECRs. The resulting profiles for the three ratios are different, as shown in Fig. 5.5, where the shaded ranges are due to LECR ionisation and the dashed lines due to HECR ionisation. The propagation of higher energy CRs may differ substantially from their lower energy counterparts due to, e.g. energy-dependent

29The ionisation rates are sensitive to both the spectrum and available energy of LECRs. 5.3. Semi-quantitative analysis: model and results 176

+ 0 R = N(C )/N (C)

AAACBnicbVDLSgMxFM34rPU16lKEYBFahDrjA90IxW5cSRX7gHYsmTTThiYzQ5IRyjArN/6KGxeKuPUb3Pk3pu0g2nrgwsk595J7jxsyKpVlfRkzs3PzC4uZpezyyuraurmxWZNBJDCp4oAFouEiSRj1SVVRxUgjFARxl5G62y8P/fo9EZIG/q0ahMThqOtTj2KktNQ2d27gObzKxy3BYTm5i/eTwsHPs9A2c1bRGgFOEzslOZCi0jY/W50AR5z4CjMkZdO2QuXESCiKGUmyrUiSEOE+6pKmpj7iRDrx6IwE7mmlA71A6PIVHKm/J2LEpRxwV3dypHpy0huK/3nNSHlnTkz9MFLEx+OPvIhBFcBhJrBDBcGKDTRBWFC9K8Q9JBBWOrmsDsGePHma1A6L9lHx5Po4V7pI48iAbbAL8sAGp6AELkEFVAEGD+AJvIBX49F4Nt6M93HrjJHObIE/MD6+AbMAlr0=

)

R +

( R = N(OH )/N (H ) AAACCXicbVDLSgMxFM34rPU16tJNsAgtQp2pim6EopuutIp9QDuWTJppQ5OZIckIZZitG3/FjQtF3PoH7vwb03YW2nrgwsk595J7jxsyKpVlfRtz8wuLS8uZlezq2vrGprm1XZdBJDCp4YAFoukiSRj1SU1RxUgzFARxl5GGO7gc+Y0HIiQN/Ds1DInDUc+nHsVIaaljwlt4Dq/ycVtweF1J7uODpHCYvitJp1TomDmraI0BZ4mdkhxIUe2YX+1ugCNOfIUZkrJlW6FyYiQUxYwk2XYkSYjwAPVIS1MfcSKdeHxJAve10oVeIHT5Co7V3xMx4lIOuas7OVJ9Oe2NxP+8VqS8Myemfhgp4uPJR17EoArgKBbYpYJgxYaaICyo3hXiPhIIKx1eVodgT588S+qlon1UPLk5zpUv0jgyYBfsgTywwSkogwqoghrA4BE8g1fwZjwZL8a78TFpnTPSmR3wB8bnD6Gnl8U= 2 10

log R = N(CO+)/N (H ) 15 AAACCXicbVDLSgMxFM3UV62vUZdugkVoEepMVXQjFLvpSqvYB7TjkEkzbWjmQZIRyjBbN/6KGxeKuPUP3Pk3pu0stPXAhZNz7iX3HidkVEjD+NYyC4tLyyvZ1dza+sbmlr690xRBxDFp4IAFvO0gQRj1SUNSyUg75AR5DiMtZ1gd+60HwgUN/Ds5Conlob5PXYqRVJKtw1t4Aa8KcZd7sHqd3MeHSfEofdcSu1y09bxRMiaA88RMSR6kqNv6V7cX4MgjvsQMCdExjVBaMeKSYkaSXDcSJER4iPqko6iPPCKseHJJAg+U0oNuwFX5Ek7U3xMx8oQYeY7q9JAciFlvLP7ndSLpnlsx9cNIEh9PP3IjBmUAx7HAHuUESzZSBGFO1a4QDxBHWKrwcioEc/bkedIsl8zj0unNSb5ymcaRBXtgHxSACc5ABdRAHTQABo/gGbyCN+1Je9HetY9pa0ZLZ3bBH2ifP5mol8A= 2

25 0.5 1.0 1.5 2.0 2.5 s/pc

Fig. 5.5: Abundance ratios due to CR ionisation within the MC model, showing + + + N(C )/N(C) (red), N(CO )/N(H2) (blue) and N(OH )/N(H2) (black). The shaded band represents the range of possible values from LECR ionisations only, while the single dashed line below each band represents the signature expected if only HECRs are present. Typically, any signal would therefore be dominated by the low-energy CR ionisations. diffusion coefficients and the greater loss channels open to the higher energy particles. It is therefore worthwhile to consider the signature of both components, even though the LECRs would usually dominate the signal, with the higher-energy ratios typically around an order of magnitude lower. While a single ratio should not be used independently to infer CR ionisation rates, the ratio N(C+)/N(C) (in red) is + much less sensitive to the underlying MC model than both N(CO )/N(H2) (blue) + and N(OH )/N(H2) (black). This would make it a less stringent quantity to constrain CR ionisation rates alone, and less powerful to diagnose the presence of CRs if alternative ratios are available. Indeed, this is in addition to possible contamination issues the N(C+)/N(C) ratio would be susceptible to. For e.g., intervening non-shielded regions in the cloud vicinity may boost the C+ abundance far above the levels driven by CR ionisation processes. Despite being more effective ionisers, the amount of energy available in LECRs to subsequently drive ionisation-mediated heating effects is lower30 than in their higher-energy counterparts. Moreover, HECRs are more likely to excite Alfvénic fluctuations in the warm ionised medium. These waves subsequently drive heating in

30Note that the energy in LECRs may be more than 25% of that in HECRs (e.g. Webber 1998; Stone et al. 2019). 5.3. Semi-quantitative analysis: model and results 177

25 10− Total Alfv´enicCR heating

1 CR ionisation heating −

s 26 10− 3 −

erg cm 27 10− / H

28 10− 0.5 1.0 1.5 2.0 2.5 s/pc

Fig. 5.6: Total specific CR heating rates with contributions from Alfvénicand ionisation heating in the idealised molecular cloud model under Galactic conditions. the magnetised plasma, making HECRs more effective than LECRs in this capacity. The specific heating rate due to HECRs in the MC is shown in Fig. 5.6. Generally, ionisation-driven heating dominates. In central regions, the Alfvénicspecific heating rate is boosted by the stronger magnetic field31. The peak specific heating rate reaches almost 10−25 erg cm−3 s−1 in the core, while the outer regions of the cloud attain levels of 10−26 ergcm−3 s−1, both being slightly lower than less sophisticated estimates of 6.3 10−27 (n /0.83 cm−3) erg cm−3 s−1 (Spitzer and Tomasko 1968; × H Goldsmith 2001), but around two orders of magnitude higher than the estimated specific CR heating rate in the warm ionised medium outside the cloud (Wiener et al. 2013b). If the CR energy density is increased, the corresponding specific heating rates are boosted proportionally (see Fig. 5.7). This indicates that the CR impact on the evolution of MCs could be strongly influenced by their surrounding conditions, e.g. due to the impact of a nearby SN event (see also Gabici et al. 2009), or if located in a star-forming or high-redshift galaxy (see section 5.4). By balancing CR heating rates against a simple cooling function (cf. section 5.1.4), an equilibrium temperature can be estimated under different levels of CR irradiation. Fig. 5.8 suggests that, in a Galactic scenario, the effect of an enhanced

31This may not hold in a more realistic 3D calculation. Assuming spherical symmetry, the CR pressure gradient would approach zero towards the core. If so, the CRs are unlikely able to excite the waves, according to equation 5.32. Hence, Alfv´enicheating should be negligible at the core, unless other sources of Alfv´enwaves exist. 5.3. Semi-quantitative analysis: model and results 178

232 1010

✏ AAAB6XicbVBNS8NAEJ34WetX1aOXxSJ4KklV9Fj04rGK/YA2lM120y7dbMLuRCih/8CLB0W8+o+8+W/ctjlo64OBx3szzMwLEikMuu63s7K6tr6xWdgqbu/s7u2XDg6bJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LR7dRvPXFtRKwecZxwP6IDJULBKFrpoer2SmW34s5AlomXkzLkqPdKX91+zNKIK2SSGtPx3AT9jGoUTPJJsZsanlA2ogPesVTRiBs/m106IadW6ZMw1rYUkpn6eyKjkTHjKLCdEcWhWfSm4n9eJ8Xw2s+ESlLkis0XhakkGJPp26QvNGcox5ZQpoW9lbAh1ZShDadoQ/AWX14mzWrFO69c3l+Uazd5HAU4hhM4Aw+uoAZ3UIcGMAjhGV7hzRk5L8678zFvXXHymSP4A+fzB+w7jPc= AAAB/nicbZDLSsNAFIYnXmu9RcWVm8EiuJCSeEGXxW7cCFXsBZpQJtOTduhMEmYmQgkFX8WNC0Xc+hzufBunbRba+sPAx3/O4Zz5g4QzpR3n21pYXFpeWS2sFdc3Nre27Z3dhopTSaFOYx7LVkAUcBZBXTPNoZVIICLg0AwG1XG9+QhSsTh60MMEfEF6EQsZJdpYHXvfg0QxbjDzpMDV+xN82xx17JJTdibC8+DmUEK5ah37y+vGNBUQacqJUm3XSbSfEakZ5TAqeqmChNAB6UHbYEQEKD+bnD/CR8bp4jCW5kUaT9zfExkRSg1FYDoF0X01Wxub/9XaqQ6v/IxFSaohotNFYcqxjvE4C9xlEqjmQwOESmZuxbRPJKHaJFY0IbizX56HxmnZPStf3J2XKtd5HAV0gA7RMXLRJaqgG1RDdURRhp7RK3qznqwX6936mLYuWPnMHvoj6/MHUneVEQ== UCR,MW 30 20✏U UMW CR,MWMilkyWay 20 AAAB/nicbZDLSsNAFIYnXmu9RcWVm8EiuJCSeEGXxW7cCFXsBZpQJtOTduhMEmYmQgkFX8WNC0Xc+hzufBunbRba+sPAx3/O4Zz5g4QzpR3n21pYXFpeWS2sFdc3Nre27Z3dhopTSaFOYx7LVkAUcBZBXTPNoZVIICLg0AwG1XG9+QhSsTh60MMEfEF6EQsZJdpYHXvfg0QxbjDzpMDV+xN82xx17JJTdibC8+DmUEK5ah37y+vGNBUQacqJUm3XSbSfEakZ5TAqeqmChNAB6UHbYEQEKD+bnD/CR8bp4jCW5kUaT9zfExkRSg1FYDoF0X01Wxub/9XaqQ6v/IxFSaohotNFYcqxjvE4C9xlEqjmQwOESmZuxbRPJKHaJFY0IbizX56HxmnZPStf3J2XKtd5HAV0gA7RMXLRJaqgG1RDdURRhp7RK3qznqwX6936mLYuWPnMHvoj6/MHUneVEQ== CRCR,,MWMW ⇥ ⇥ ✏10 UMW 50 UMW 10 AAAB/nicbZDLSsNAFIYnXmu9RcWVm8EiuJCSeEGXxW7cCFXsBZpQJtOTduhMEmYmQgkFX8WNC0Xc+hzufBunbRba+sPAx3/O4Zz5g4QzpR3n21pYXFpeWS2sFdc3Nre27Z3dhopTSaFOYx7LVkAUcBZBXTPNoZVIICLg0AwG1XG9+QhSsTh60MMEfEF6EQsZJdpYHXvfg0QxbjDzpMDV+xN82xx17JJTdibC8+DmUEK5ah37y+vGNBUQacqJUm3XSbSfEakZ5TAqeqmChNAB6UHbYEQEKD+bnD/CR8bp4jCW5kUaT9zfExkRSg1FYDoF0X01Wxub/9XaqQ6v/IxFSaohotNFYcqxjvE4C9xlEqjmQwOESmZuxbRPJKHaJFY0IbizX56HxmnZPStf3J2XKtd5HAV0gA7RMXLRJaqgG1RDdURRhp7RK3qznqwX6936mLYuWPnMHvoj6/MHUneVEQ== UCR,MW 50 ✏U ⇥ CR⇥,MW ⇥ AAAB/nicbZDLSsNAFIYnXmu9RcWVm8EiuJCSeEGXxW7cCFXsBZpQJtOTduhMEmYmQgkFX8WNC0Xc+hzufBunbRba+sPAx3/O4Zz5g4QzpR3n21pYXFpeWS2sFdc3Nre27Z3dhopTSaFOYx7LVkAUcBZBXTPNoZVIICLg0AwG1XG9+QhSsTh60MMEfEF6EQsZJdpYHXvfg0QxbjDzpMDV+xN82xx17JJTdibC8+DmUEK5ah37y+vGNBUQacqJUm3XSbSfEakZ5TAqeqmChNAB6UHbYEQEKD+bnD/CR8bp4jCW5kUaT9zfExkRSg1FYDoF0X01Wxub/9XaqQ6v/IxFSaohotNFYcqxjvE4C9xlEqjmQwOESmZuxbRPJKHaJFY0IbizX56HxmnZPStf3J2XKtd5HAV0gA7RMXLRJaqgG1RDdURRhp7RK3qznqwX6936mLYuWPnMHvoj6/MHUneVEQ== ⇥CRCR,,MWMW 1 24 10 s 3

/K 25 10 eq T 1 erg cm 10 / 26 10 H

27 10 0.5 1.0 1.5 2.0 2.5 s/pc

Fig. 5.7: Total specific CR heating rates in the cloud under increasing ambient CR energy densities (as labelled), demonstrating the impact of different environments on the cloud’s thermal evolution, even when shielded from the interstellar radiation field. The baseline case adopts the Milky Way CR energy density (CR,MW), which is then scaled by the factors indicated.

CR energy density would be minimal in the core regions of a MC where the temperature is only boosted marginally compared to the surrounding clump. Moreover, the core region only experiences modest increases in its equilibrium temperature, even when faced with substantial enhancements of the CR ﬂux. Instead, the surrounding clump and inter-clump medium are more strongly impacted. The core temperatures estimated here are somewhat lower than expected, given that Galactic observations indicate core temperatures of around 8 12 K (Bergin and Tafalla 2007). However, − some starless cores are found to have temperatures of T < 7 K (Pagani et al. 2007, 2009, 2015; Lin et al. 2020), while theoretical studies have also found temperatures comparable to those calculated here in idealised cases (e.g. Juvela and Ysard 2011).

These results are consistent with several extreme cases where core temperatures may be as low as 6 K or even less (e.g. Harju et al. 2008). It may indicate that the adopted cooling function is overstated, or other processes, e.g. turbulent heating (Pan and Padoan 2009), residual unattenuated interstellar radiation ﬁelds and/or dust reprocessing/heating (Goldsmith 2001), may be operating alongside CRs to maintain higher core temperatures. If other heating mechanisms operate in competition with CRs, CR heating would only dominate the regulation of MC core temperatures in environments where the CR energy density is substantially greater 5.3. Semi-quantitative analysis: model and results 179

102 ✏ AAAB/nicbZDLSsNAFIYnXmu9RcWVm8EiuJCSeEGXxW7cCFXsBZpQJtOTduhMEmYmQgkFX8WNC0Xc+hzufBunbRba+sPAx3/O4Zz5g4QzpR3n21pYXFpeWS2sFdc3Nre27Z3dhopTSaFOYx7LVkAUcBZBXTPNoZVIICLg0AwG1XG9+QhSsTh60MMEfEF6EQsZJdpYHXvfg0QxbjDzpMDV+xN82xx17JJTdibC8+DmUEK5ah37y+vGNBUQacqJUm3XSbSfEakZ5TAqeqmChNAB6UHbYEQEKD+bnD/CR8bp4jCW5kUaT9zfExkRSg1FYDoF0X01Wxub/9XaqQ6v/IxFSaohotNFYcqxjvE4C9xlEqjmQwOESmZuxbRPJKHaJFY0IbizX56HxmnZPStf3J2XKtd5HAV0gA7RMXLRJaqgG1RDdURRhp7RK3qznqwX6936mLYuWPnMHvoj6/MHUneVEQ== UCR,MW 30 ✏U AAAB/nicbZDLSsNAFIYnXmu9RcWVm8EiuJCSeEGXxW7cCFXsBZpQJtOTduhMEmYmQgkFX8WNC0Xc+hzufBunbRba+sPAx3/O4Zz5g4QzpR3n21pYXFpeWS2sFdc3Nre27Z3dhopTSaFOYx7LVkAUcBZBXTPNoZVIICLg0AwG1XG9+QhSsTh60MMEfEF6EQsZJdpYHXvfg0QxbjDzpMDV+xN82xx17JJTdibC8+DmUEK5ah37y+vGNBUQacqJUm3XSbSfEakZ5TAqeqmChNAB6UHbYEQEKD+bnD/CR8bp4jCW5kUaT9zfExkRSg1FYDoF0X01Wxub/9XaqQ6v/IxFSaohotNFYcqxjvE4C9xlEqjmQwOESmZuxbRPJKHaJFY0IbizX56HxmnZPStf3J2XKtd5HAV0gA7RMXLRJaqgG1RDdURRhp7RK3qznqwX6936mLYuWPnMHvoj6/MHUneVEQ== CR,MW CR,MW ⇥ CR,MW ✏

10 AAAB/nicbZDLSsNAFIYnXmu9RcWVm8EiuJCSeEGXxW7cCFXsBZpQJtOTduhMEmYmQgkFX8WNC0Xc+hzufBunbRba+sPAx3/O4Zz5g4QzpR3n21pYXFpeWS2sFdc3Nre27Z3dhopTSaFOYx7LVkAUcBZBXTPNoZVIICLg0AwG1XG9+QhSsTh60MMEfEF6EQsZJdpYHXvfg0QxbjDzpMDV+xN82xx17JJTdibC8+DmUEK5ah37y+vGNBUQacqJUm3XSbSfEakZ5TAqeqmChNAB6UHbYEQEKD+bnD/CR8bp4jCW5kUaT9zfExkRSg1FYDoF0X01Wxub/9XaqQ6v/IxFSaohotNFYcqxjvE4C9xlEqjmQwOESmZuxbRPJKHaJFY0IbizX56HxmnZPStf3J2XKtd5HAV0gA7RMXLRJaqgG1RDdURRhp7RK3qznqwX6936mLYuWPnMHvoj6/MHUneVEQ== UCR,MW 50 ✏U

CR,MW AAAB/nicbZDLSsNAFIYnXmu9RcWVm8EiuJCSeEGXxW7cCFXsBZpQJtOTduhMEmYmQgkFX8WNC0Xc+hzufBunbRba+sPAx3/O4Zz5g4QzpR3n21pYXFpeWS2sFdc3Nre27Z3dhopTSaFOYx7LVkAUcBZBXTPNoZVIICLg0AwG1XG9+QhSsTh60MMEfEF6EQsZJdpYHXvfg0QxbjDzpMDV+xN82xx17JJTdibC8+DmUEK5ah37y+vGNBUQacqJUm3XSbSfEakZ5TAqeqmChNAB6UHbYEQEKD+bnD/CR8bp4jCW5kUaT9zfExkRSg1FYDoF0X01Wxub/9XaqQ6v/IxFSaohotNFYcqxjvE4C9xlEqjmQwOESmZuxbRPJKHaJFY0IbizX56HxmnZPStf3J2XKtd5HAV0gA7RMXLRJaqgG1RDdURRhp7RK3qznqwX6936mLYuWPnMHvoj6/MHUneVEQ== CRCR,,MWMW ⇥ ⇥ Only CR heating operates /K CENTRE eq T 1

10 LOUD C

0.5 1.0 1.5 2.0 2.5 s/pc

Fig. 5.8: Equilibrium temperature proﬁle arising from CR heating balanced against gas cooling under Galactic conditions, and with CR energy densities boosted by factors of 10, 30 and 50. Other processes may operate to maintain higher temperatures of around 10 K observed in most of the Galactic MCs. than that typically estimated in the Galaxy. A more thorough assessment of the competition between various heating eﬀects throughout the MC sub-regions merits a further dedicated study. 5.3.2 Illustrative case The IC 5146 cloud region and data set

The IC 5146 molecular cloud complex is located in Cygnus, exhibiting a converg- ing filamentary system of dark clouds and elongated sub-structures extending from a main filament as seen with Herschel observations (Arzoumanian et al. 2011). Throughout the cloud, structures are in various stages of their evolution: the main filament and nearby Cocoon Nebula appear to be at different stages of ongoing star-formation episodes (Harvey et al. 2008; Dunham et al. 2015), while some other dark cloud regions remain quiescent (Arzoumanian et al. 2011). The formation of the system is believed to be driven by large-scale turbulence (Arzoumanian et al. 2011), which would have introduced perturbations into the otherwise well-ordered large-scale magnetic field morphology (Wang et al. 2017, 2019), thus making it a useful test-case. There is some debate over the distance to the system (see e.g. Lada et al. 1999; Harvey et al. 2008), and whether it is one system or two clouds along the same line-of-sight at different distances (Wang et al. 2020a). Such matters would 5.3. Semi-quantitative analysis: model and results 180 impact the conversion between angular separations and physical distances in this work. A recent re-analysis (if assuming it to be a single cloud) using Gaia DR2 data (Gaia Collaboration 2018) indicates it to be located 813 106 pc away (Dzib ± et al. 2018). This distance is adopted here, following Wang et al.(2019). The data in this study is obtained from the optical and near infra-red stellar polarisation observations towards IC 5146 (Wang et al. 2017). The data set is compiled by matching polarisation data to the positions of 2022 independent background stars to within 0.5” (corresponding to 0.002 pc at a distance of 813 pc) from the 2MASS all-sky survey (Skrutskie et al. 2006) in at least one of the Rc-, i’, H- and K- bands (see Wang et al. 2017, 2019). In total, only 3 stars were present in all four bands, with around 71% of the stars being detected in the H-band, 24% in the Rc-band, 10% in the i’-band and 8% in the K-band (Wang et al. 2017). The angular dispersion function in every band is computed using a bin size of 90” (corresponding to a physical size of 0.35 pc) to determine the diffusion coefficient of CRs through the region. This bin size gives a reasonable signal-to-noise ratio over the length scales of interest. The uncertainties are estimated using 10,000 Gaussian Monte Carlo perturbations to the Stokes parameters Q and U (see Fig. 5.9, where 1σ errors bars are shown). By assessing the distribution of angular separations, they do not reflect the features seen in Fig. 5.9. The structures evident in this dispersion analysis are likely to be physical in origin rather than due to instrumental or sampling effects. The FT of the dispersion function is calculated with the diffusion coefficient for the region following equation 5.42. Spatial variation of the diffusion parameter D arises from variations in the magnetic field strength alone, while spatial variations due to the field structure are not significant32 – see AppendixE for details. These findings are applied to the subsequent analyses.

Representation of the ﬁlamentary structures

The IC 5146 region consists of a network of filamentary structures. Arzoumanian et al.(2011) identified 27 filaments, to which they fit a cylindrical density profile of the form ( )−p/2 s 2 nH(s) = nc 1 + , (5.50) Rflat

32There is also no compelling support for a signiﬁcant variation in the empirical value of D between the diﬀerent wavebands. 5.3. Semi-quantitative analysis: model and results 181

`/arcsec 0 1000 2000 3000 4000 5000 6000 7000 50 Rc band i band 40 H- band K- band

30 ) ` ( 2

S 20

0 0 5 10 15 20 25 `/pc

Fig. 5.9: Dispersion functions calculated for the 4 bands. The x and y error bars indicate the bin size, and 1σ Gaussian errors, respectively. The latter is estimated by a Monte Carlo approach with 10,000 perturbations.

with nc being the density of the filament ridge and Rflat as the characteristic length scale of the flat innermost portion of the profile. In observations, the density of diffuse material is often characterised by the column density along a line-of-sight. Here the column density along the line-of-light to a radial position s within the projected filament is given by

( )− p−1 s 2 2 NH(s) = nc Rflat 1 + , (5.51) P Rflat where 1/ cos ψ for ψ is the angle of the filament to the sky plane. The variation P ∝ of this parameter due to the inclination of IC 5146 to the sky plane is small. It is within a factor of 2 if assuming random orientations, and less than many other sources of uncertainty in the model. Hence, this system may be assumed to be oriented at ψ = 0◦ to the sky plane. The magnetic field that permeates this cloud is observed to be arranged predominantly perpendicular to the filament orientations on the sky plane (Wang et al. 2017, 2020a). The mean magnetic field strength across the filaments can be well described by a simple power-law:

n p¯ B (n) = B . (5.52) mean 0 150 cm−3

+0.50 +0.12 The best-fit values are B0 = 2.46−0.51 µG andp ¯ = 0.50−0.13 (Wang et al. 2020a). This prescription is specifically based on the analysis of IC 5146 and is different 5.3. Semi-quantitative analysis: model and results 182 from the idealised case following Crutcher et al.(2010) (see equation 5.48). The filamentary structures are therefore useful test-cases, where the heating and ionisation rate profiles can be calculated through each filament by adopting a natural boundary condition at the filament edge. Results

15 of the 27 distinct filamentary structures identified in Arzoumanian et al.(2011) have sufficient parametric information (column density NH and inner length scale

Rflat) available for the calculations (see Table 5.1). Filaments 16, 17, 18 and 19 are located in a photon-dominated region, and are therefore omitted, as the temperature estimates and comparisons would not be reliable for these cases. Filaments 22 and 27 are excluded for their unresolved width. Filaments 3, 15, 16, 18, 22 and 24 are also excluded, due to the lack of data on the filament width Rflat, the p index in the density profile and/or a single best-fit value. Filaments 14, 23 and 26 are excluded due to an asymmetric profile. The filament ridge volume density, nc, is estimated from the measured column densities N using n N /(2 R ) following Wang H c ≈ H flat et al.(2020a). This assumes each filament has a cylindrical geometry. Note that

Rflat has been adjusted to account for the updated distance to IC 5146 of 813 pc (compared to a distance of 460 pc used in Arzoumanian et al. 2011). For each case, the ionisation rate and specific heating rate are computed to deduce the equilibrium temperature. More specifically, the ionisation rates are calculated from the mini- H,min H,max mum W98 LECR spectrum (ζLECRs), maximum M02 LECR spectrum (ζLECRs) and HECRs above a GeV (ζH ). The resulting heating rate and ionisation rates HECRs H in the filament ridge (being their densest point) are shown in Table 5.1, together with an estimate of the temperature increase due to the CR processes ∆Teq,CR/K. 5.3. Semi-quantitative analysis: model and results 183 K / 0 0 0 0 0 0 0 0 ...... CR 1 1 1 1 1 1 1 1 , eq < < < < < < < < T ∆ 1 − s 19 − 10 / H HECRs ζ 1 − s 14 − 10 / max , H LECRs ζ 1 − s 20 − 10 / min , H LECRs ζ 1 − s 3 − ergcm 27 − 10 / H 3 − cm 4 10 / c n pc / flat Specific rates of CR heating and ionisation in 15 out of the 27 filaments identified by Arzoumanian et al. ( 2011 ) in the IC 5146 region. p R ID 1 2.1 0.09 0.3 3.2 6.9 1.5 9.4 2 1.9 0.1 0.7 12 6.8 1.4 9.3 4 1.4 0.04 0.7 7.8 6.8 1.4 9.3 5 1.56 1.7 0.027 1.6 0.078 1.5 0.059 7 1.5 0.0910 4 2.1 0.0711 2 1.9 0.112 0.4 1.5 0.0713 0.8 22 1.6 0.0520 0.5 51 1.5 0.0421 1 19 1.7 0.0525 11 4 1.5 0.09 15 3 0.05 0.2 6.8 6.4 0.3 6.8 9.2 0.7 6.8 46 6.6 20 6.7 2.5 1.5 6.8 6.1 1.5 6.9 8.5 1.5 1.4 6.8 1.4 6.8 9.4 6.8 1.5 9.3 6.7 1.5 9.4 6.8 9.1 1.4 1.1 9.2 1.5 1.3 1.4 9.4 1.0 1.4 9.4 1.0 1.4 9.3 1.0 9.4 9.3 9.2 1.3 9.3 1.0 Table 5.1: Filament IDs correspond to those used invalues, that being paper. dominated by Uncertainties in the calculated uncertainty quantities in are the estimated empirical to CR be diffusion within parameter. an order of magnitude of the stated 5.3. Semi-quantitative analysis: model and results 184

Stronger heating is found to occur in the filaments with larger volume densities, but the heating efficiency is not linearly proportional to the density (e.g. compare filaments 4 and 5). This can be explained as follows. An increase in density will also lead to an increase in magnetic field strength inside a filament. This, in turn, increases the amount of deflection experienced by the CRs, which reduces their heating efficiency. As shown, the strongest heating occurs in filaments 6 and 12. These have a high density but not a particularly steep density profile, thus reducing magnetic mirroring/deflection effects and allowing a substantial amount of CRs to penetrate inside. CR ionisation rates due to both low and high energy CRs are much less susceptible to density variations deep inside the cloud. This would imply relatively consistent CR ionisation rates throughout Galactic MCs, regardless of their internal configuration or exact filament/clump properties.

5.3.3 Additional remarks

A clear implication of the results in Table 5.1 is that CR-driven processes alone, while able to sustain relatively high rates of ionisation within the cloud and heating rates comparable to literature estimates (see, e.g. Goldsmith 2001; Wiener et al. 2013b and section 5.1.4), do not yield substantial increases in core/filament temperatures if operating alone. Although caveats in the model would offer scope for larger values, this result taken by itself would suggest that other, mechanical, processes (e.g. heating from turbulence dissipation driven by gravitational collapse – see Carl- berg and Pudritz 1990; McKee and Zweibel 1995; Gammie and Ostriker 1996; Martin et al. 1997; Falceta-Gon¸calves et al. 2003) may be more important in maintaining clump/filament temperatures at the expected level of around 8 12 K (Bergin and − Tafalla 2007) under Galactic conditions. This would point towards clump destruction being reliant on the emergence of proto-stars heating and ionising dense regions from within, or photo-evaporation/conduction from the external environment (i.e. outside-in destruction) rather than CR heating. However, the models described in this work rely on a number of assumptions. Chief among these is the use of dust polarisation of stellar radiation to trace the magnetic fields: the reliability of this indirect means of probing magnetic fields remains under debate – particularly towards the dense higher-extinction regions of principal interest in this work, and variation in alignment efficiency between individual sources (see Whittet et al. 2008; Cashman and Clemens 2014) is well known. 5.3. Semi-quantitative analysis: model and results 185

The observed PA dispersion is used to estimate the diffusion coefficient, however in high density regions with poor radiative alignment efficiencies, the correspondence between dust alignment and magnetic field vectors could be weak – or timescales for radiative grain alignment to operate could become very long and unable to fully reflect rapidly-varying turbulent magnetic field structures or fluctuations on the very small scales. Moreover, CR heating of dust grains themselves (Kalv¯ans 2018) could also influence torque alignment efficiencies (cf. thermal wobbling, see Lazarian and Hoang 2007), and hence dust polarisation. This would decrease the diffusion parameter and could substantially increase the CR heating rate felt in the densest regions. Future observations probing a wide range of magnetic field structures in MCs to higher resolution and avoiding the need to rely on dust alignment mechanisms may be possible with the upcoming Square Kilometer Array (SKA) (see e.g. Strong et al. 2014). Moreover, an assessment of the impact of various magnetic field configurations on the CR diffusion parameter and subsequent heating effect via simulations is worthy of a dedicated future study.

A further assumption is that the cooling is dominated by CO and dust (Galli et al. 2002; Goldsmith 2001). This is an analytical approximation (Whitworth and Jaffa 2018; Goldsmith and Langer 1978), which assumes dust temperatures of around 10 K (a compromise value between warmer regions in the cloud peripheries and cooler regions within the denser parts). While suitable under typical galactic molecular cloud conditions, variations in dust fraction and/or composition may result in different cooling rates in alternative settings – e.g. in dusty galaxies or more/less chemically pristine or primordial environments (Jaacks et al. 2019). Moreover, substantial variation in dust and gas temperatures throughout a cloud would likely yield different cooling rates: presumably the rates adopted in this work are somewhat overstated in the densest core/filament regions where gas temperatures and subsequent cooling rates would be lower and less able to balance CR heating.

Variations in the irradiating CR flux or spectrum are also plausible. While this work has considered the effect of scaling the CR energy density and maintaining the same spectral form (see Figs. 5.7 and 5.8), there is wide-ranging evidence in literature for different CR spectral indices at high energies in different environments. For e.g., there could be a harder spectrum in the vicinity of the galactic ridge, where more 5.3. Semi-quantitative analysis: model and results 186

103

/K 102 eq T

NGC 253 M82 101 Arp 220

0.5 1.0 1.5 2.0 2.5 s/pc

Fig. 5.10: Equilibrium temperature as a result of only CR heating of a cloud in diﬀerent starburst environments. A Galactic CR spectral shape is considered. The CR energy density is estimated from Yoast-Hull et al.(2016).

CRs are freshly accelerated (Aharonian et al. 2006; Gaggero et al. 2017; H.E.S.S. Collaboration 2018b,a).

Implications

Although relatively well-shielded by the magnetic field configuration, the modest level of CR heating experienced in a filamentary clump or core would have substantial implications for subsequent star-formation. Fig. 5.8 explores the effect of different irradiating energy densities of CRs, scaled according to the estimated energy densities in nearby star-forming galaxies. In the exterior clump regions and inter-clump medium, a roughly linear relationship between CR energy density and resulting heating and equilibrium temperature is evident (see also Figs. 5.7 and 5.8), but the effect on the core temperature is much more modest. Aside from magnetic deflection effects, CR heating is much more important in the inter-clump (diffuse cloud) medium in general where the ionisation fraction is higher and CRs are deflected less. This would favour clump heating via conduction rather than direct internal heating by the CRs.

Despite the core temperature increases remaining modest even when CR energy densities are enhanced substantially, the astrophysical implications can still be signiﬁcant. The Jeans mass, for example, is dependent on temperature (as well as volume density of gas, nH) as this sets the level of thermal pressure against 5.4. Summary 187 gravitational collapse:

3/2 −1/2 2 T nH M = 1.9 10 M . (5.53) J × 10 K 103 cm−3

While this is a crude measure of the maximum stable mass against collapse, neglecting cloud/clump fragmentation, small-scale magnetic/turbulent support and the micro-physics of the stellar initial mass function, it can still give an idea of the size of MCs and the stellar clusters they develop into. Without CR heating, Galactic clouds could reach around 200 M before they become unstable and gravitationally collapse. However, with Arp 220 levels of CR irradiation (see Fig. 5.10 for equilibrium temperatures resulting from CR heating in Arp 220, M82 and NGC 253 – 3 three nearby starburst galaxies), this could increase to around 4 10 M . The × mass distribution of clouds/clumps in systems with higher CR energy densities (e.g. in star-forming galaxies) can therefore become distorted, as higher star-formation rates would yield larger interstellar cloud sizes. This may favour more stochastic burst-like star-forming episodes. More importantly, the larger mass threshold required for gravitational collapse would likely lead to a period of quenching in a galaxy rich in CRs, as it would take longer for suﬃciently large MCs to gain enough mass to begin their collapse.

5.4 Summary

This chapter investigates the effects of CR heating and ionisation in magnetised molecular clouds. First of all, a CR transport formulation is constructed to determine how CRs propagate and deposit their energy within molecular cloud models. For an irradiating CR flux comparable to that in the Milky Way, the specific heating rate is able to reach a level of 10−26 erg cm−3 s−1 in the densest regions of the ∼ cloud. This heating is somewhat suppressed by magnetic deflection effects, but is still higher than in the surrounding inter-clump medium by a factor of around 3. On the other hand, the CR ionisation rates are in the range of ζH = 10−18 10−16 s−1, − with ionisation being dominated by the contribution from low-energy CRs below a GeV. Higher energy CRs are more engaged with the heating processes. Next, the effect of magnetic fields in the IC 5146 star-forming region in Cygnus is quantified, assuming that the fields are traceable via optical and near-infrared starlight polarisations towards IC 5146. The CR diffusion coefficient across the 5.4. Summary 188 cloud is estimated from the underlying fluctuations in the local magnetic field, and then used to compute the CR heating and ionisation rates in 15 of the filamentary structures identified in IC 5146. There is a broad variation of specific heating rates, whereas the ionisation rates remain fairly uniform between different filaments. While the calculated specific heating rate of 10−27 10−25 erg cm−3 s−1 is unlikely to cause − a large temperature increase in the filaments, molecular clouds in environments with a strong CR flux (e.g. in star-forming galaxies) can be substantially heated. This may result in an increase of the Jeans mass and subsequently lead to larger ISM clump sizes. Hence, it could arguably yield a greater tendency for stochastic, burst- like star-formation activities to occur, and even periods of quenching to arise. 189

Chapter 6

Summary and outlook

Magnetic fields are omnipresent across all scales of the Universe. Their origins and evolution are still open questions. On wider scales, observational signatures imprinted in the radio synchrotron emission and Faraday rotation measures (RMs) offer a unique window to infer the magnetic field properties in galaxy clusters and larger-scale structures. More specifically, the spatial correlation of the RMs is often used to characterise the length scale of the magnetic field variation. Firstly, the work in Chapter3 shows how the RM is derived from the polarised radiative transfer equations subject to restrictive conditions, and assesses the merit of RM fluctuations (RMF) for large-scale magnetic field diagnostics (On et al. 2019; Chan et al. 2019). The conditions are that: (i) the circular polarisation is negligible; (ii) the emission and absorption processes are absent; and (iii) only thermal electrons contribute to the RM. As a result, there are two conventional expressions (3.27) or (3.28) to quantify the RMF. However they are not always explicitly distinguished in many RM analyses of intergalactic and intracluster magnetic fields. The work in Chapter3 shows that these two expressions are derived based on the theory of simple random-walk, but with subtly distinct assumptions. The general assumptions are for having a constant electron number density and/or a non-correlation between the density and the magnetic field. However these assumptions do not usually hold in real astrophysical environments. It also matters whether the local quantities are multiplied before spatial averaging, or averaged independently before multiplication. Furthermore, the fluctuations along the line- of-sight and across the sky plane are generally different. The spatial properties of the magnetic fields and the magneto-ionic plasma may not be independent from 190 their temporal properties as the Universe evolves.

It is further shown in Chapter4 that the interpretation of RMF analyses is ambiguous for a poorly-defined characteristic density, such as that with lognormal- distributed or fractal-like structures. This implies that density fluctuations can significantly obscure magnetic field fluctuations. In practice, it is also difficult to disentangle the signals from the density and magnetic field fluctuations, based solely on the standard deviation of RM itself. As a consequence, it would be unreliable to determine the correlation length of large-scale magnetic fields using the conventional RMF analyses.

Even without degeneracy between the signals from the density and magnetic field fluctuations, the RMF approach also falls short under the presence of non- thermal electrons and other radiative processes, such as absorption, emission, and Faraday mixing. Hence it is important to bear in mind which underlying assumptions have been made before using the statistics of RMs to interpret the properties of magnetic fields. A more robust formulation based on covariant polarised radiative transfer (Chan et al. 2019) is essential to build solid theoretical models and make predictions in preparation for the SKA.

Apart from the RMF approach, magnetic fields in galaxy clusters can also be quantified from the local scaling relations between the cluster radio and X-ray observables. As such, the work in Chapter4 calculates the relations between the standard deviation of RMσ ¯R, and X-ray intensity IX, for galaxy clusters assembling between z = 3 and z = 0 in the GCMHD+ simulations. Theσ ¯R I relation varies − X unsteadily during cluster formation with slopes of b . 1 for the centre-excised clusters. This indicates that the magnetic field scales with the gas density locally.

It can be generalised that theσ ¯R I relation at different redshifts would allow X- − X ray observations by, e.g. Athena, to determine the evolution of magnetic fields in the cluster scales. These can be independently compared to the magnetic field properties inferred from radio observations. While the simulations are purely adiabatic here, a more definitive calculation of intracluster magnetic fields would require additional physics to be taken into consideration, such as radiative cooling, star formation, and AGN feedback.

The following work calculates the radio and X-ray intensities of a high resolution galaxy cluster from the GCMHD+ simulations (Barnes et al. 2018). The radio and 191

X-ray images reveal similar morphological features. Upon a local point-by-point comparison between the radio and X-ray images, a power-law relation with a best- ﬁt slope of b 1 is obtained. This relation is consistent with a few observed halo ∼ clusters, even though the simulated cluster is of a lower mass. It may be a natural consequence from having the weak magnetic ﬁeld frozen into the gas. The radiative transfer results are also suggestive of well reproducing radio halo observations by assuming a fraction of fossil, thermal electrons being re-accelerated to emit in the radio synchrotron within the ICM. It is cautioned that the simulations do not explicitly track the non-thermal electron properties and here they are assumed to be freshly shock accelerated. Therefore, this work can be extended by implementing a self-consistent model in the simulations for the injection of relativistic electrons by nearby SNRs or AGN and following their transport during cluster evolution.

On smaller scales, magnetic fields may also play a crucial role in the dynamics of the interstellar medium, particularly during the formation and evolution of molecular clouds (MCs). Notably, magnetic fields can focus cosmic rays (CRs) into the cores and clumps of MCs, which may lead to ionisation and heating of these dense regions. As such, the work in Chapter5 investigates the effects of CR interactions in magnetised MCs (Owen et al., submitted). Firstly, a formulation on CR transport is constructed to determine how CRs propagate and deposit their energy within magnetised MC models. For an irradiating CR flux typical of that observed in the Milky Way, the specific heating rate can attain up to 10−26 erg cm−3 s−1 in the densest regions of the cloud. Although the heating is suppressed by magnetic deflections, it is still approximately three times higher than that in the ambient inter-clump regions. On the contrary, the CR ionisation rates are around ζH = 10−18 10−16 s−1, − which are presumably dominated by the contribution from low-energy CRs below a GeV. Higher energy CRs are more involved with heating effects. While these assumptions are taken in the first instance to simplify the semi-analytical calculations, the heating and ionisation processes are, in fact, intimately connected. It would therefore be more physically meaningful to extend this work by calculating the effects of heating and ionisation for an incident CR flux with both high-energy and low-energy components taken into account.

Next, the effect of magnetic fields in the IC 5146 star-forming region is quantified, assuming that the fields are traceable via optical and near-infrared starlight 192 polarisations towards the cloud. The angular dispersion function of the observed polarisation angles is used to constrain the power spectrum of the magnetic fields and determine the CR diffusion coefficient across the cloud. As a result, the specific heating rates are shown to vary between 10−27 10−25 erg cm−3 s−1, whereas the − ionisation rates remain fairly constant throughout the cloud. This may suggest that MCs in environments with a strong CR flux, such as within a starburst galaxy, can be sufficiently heated and consequently experience an increase in their Jeans mass. This may lead to more stochastic, starburst activities to occur, and even periods of quenching to arise. It is important to note that several assumptions have to be made to simplify the representation of the IC 5146 filamentary structures, the physics of CR transport and the magnetic field properties, in order to carry out this work. These limitations can be improved by performing a numerical simulation of a star-forming MC, where the CR propagation and interaction effects can be traced as the cloud evolves under gravity, turbulence, and the presence of magnetic fields. Furthermore, it may be possible to derive information on the local CR population and diffusion coefficient from future gamma-ray observations by the CTA. Hence, this work brings a new perspective to understanding CRs and magnetic fields in MCs, from sub-millimetre to gamma-ray observations. 193

Appendices A. Errata in the GCMHD+ analysis code output.f 194

For completeness, appendicesB,C,D andE present results which have been submitted in whole or in part in: Ellis R. Owen, Alvina Y. L. On, Shih-Ping Lai and Kinwah Wu, “Interactions of energetic cosmic rays in molecular clouds: astrophysical implications and observational signatures”, submitted March 2020 to MNRAS. I am a co-author of this paper. I formulated the application of structure functions to characterise the magnetic fields and cosmic ray propagation in molecular clouds. I contributed to the calculations of angular dispersion function that was used to derive the cosmic ray diffusion coefficient. I cleaned the IC 5146 polarisation data and calculated the frequency distributions and power spectra. I also conducted consistency and accuracy checks for the science, and assisted the lead author in parts of the manuscript preparation.

A Errata in the GCMHD+ analysis code output.f

(i) Conversion from simulation units to physical units By deﬁnition, ρ n = , (1) µmH where µ is the mean molecular weight, n is the number density of gas particles in −3 −3 units of cm , ρ is the gas mass density in g cm , and mH is the hydrogen mass in g. Assuming a fully ionised gas with primordial composition (whose mass is roughly made up of 70.6% hydrogen), np = ne and µ = 0.6. Therefore, equation1 can be written as ρ ne = . (2) µmp However, line 32 of output.f reads as

ρµ ne = , (3) mp so the number density output is corrected by a factor of 1/µ2 before carrying out the radiative transfer calculations in Chapter4. (ii) Misalignment of axes From lines 82+ of output.f, the y and z axes have been mistakenly swapped while writing the cluster outputs. This is corrected for by applying a transpose to the cluster outputs before carrying out the radiative transfer calculations in Chapter4. B. Numerical scheme for solving the transport equation 195

B Numerical scheme for solving the transport equation

Consider the variables X(E, s) = η(s) n(E, s) and U(s) = D(E, s) ∂X/∂s, as well as the indices q and r for the energy Eq and spatial sr grid points, respectively, in discretising the partial differential equations 5.25, 5.26 and 5.27. In each case, a numerical solution for Xq,r is calculated. Owing to the large differences between the variations of each term (especially in sections B.2 and B.3), an implicit Runge- Kutta (RK) 4/5 scheme is required to handle the inherent stiffness of the differential equations. The RADAU5 implementation from Hairer and Wanner(1996) is adopted. 1

B.1 Primary protons

Equation 5.25 is discretised as follows, when rewritten as a system of two diﬀerence equations:

∂U Uq,r+1 = Uq,r + ∆s ; (4) ∂s q,r Uq,r Xq,r+1 = Xq,r + ∆s , (5) D(Eq) where ∆s is the step-size in the s direction into the molecular cloud (MC). To solve them, consider a two-step scheme:

( ∂U ∂U ∂X = + ∆s v (s ) ∂s ∂s A r ∂s q,r+1 q,r q,r ) X b X − b − + q+1,r q+1,r − q 1,r q 1,r + n (s )X σ (E )c ; (6) 2∆E H r q,r pπ q

∂X Uq,r = . (7) ∂s q,r D(Eq, sr)

The boundary conditions are

X1,1 = X1,rmax = np(E1, s1)η(s1) ; (8)

U1,1 = U1,rmax = j(E1, s1)η(s1) , (9)

where rmax is the maximum index on the spatial grid, and the CR inﬂux, j, is estimated from Padovani et al.(2009).

1The RK-Fehlberg implementation from Press et al.(1992) also gives equivalent results. C. Abundance ratios 196

B.2 Primary electrons

The same iterative scheme as set out in equations4 and5 is used to solve equation 5.26, but with equation6 replaced with: ( ∂U ∂U ∂X = + ∆s v (s ) ∂s ∂s A r ∂s q,r+1 q,r q,r ) X b X − b − + q+1,r q+1,r − q 1,r q 1,r , (10) 2∆E and an appropriate choice of cooling function b. The boundary conditions are outlined in section 5.1.3. B.3 Secondary electrons

The scheme for secondary electrons follows from the discretisation of equation 5.27. Again, the form is the same as in section B.1, but where the inner equation6 is replaced by: ( ∂U ∂U ∂X = + ∆s v (s ) ∂s ∂s A r ∂s q,r+1 q,r q,r ) X b X − b − + q+1,r q+1,r − q 1,r q 1,r Q , (11) 2∆E − e|q,r with an appropriate choice of cooling function b. This time, the boundary conditions on X are X1,1 = X1,rmax = 0, with those on U following similarly as U1,1 = U1,rmax = 0. The source term depends on the primary proton/electron solutions above, with the Q injection term as prescribed in section 5.1.3. e|q,r C Abundance ratios

CR ionisation rates can be inferred from the astrochemistry of molecular clouds. The derived rates account for direct CR ionisations together with the knock-on events caused by electrons being released in each ionisation process. The chemical processes can be split into three categories: (1) initiation steps arise when a species ionised by a CR subsequently reacts with species expected to be present in abundance in the MC environment (e.g. H, H2 or C); (2) propagation steps yield the formation of readily observable species, or facilitate their destruction in a way which would impact on the chemical balance/abundance of that species, and (3) termination steps result in the neutralisation of an ionised species within the chain, without leading to the production of a further species of interest – the ‘products’ in Table1, C. Abundance ratios 197 where the key reactions resulting from a CR ionisation event in a MC are listed together with their associated rate coeﬃcient. CR ionisation in a MC proceeds as

H + CR H+ + CR0 ; → H + CR H + + CR0 ; 2 → 2 C + CR C+ + CR0 , → which lead directly to steps I1, I2 and I3, respectively (see ID keys in Table1). In a steady state the reactions in Table1 give the abundance ratios for the species of interest, OH+, CO+ and C+ (see section 5.1.4). The key reactions yield

n(O+)n(H )k + n(H+)n(O)k n(OH+) = 2 P1 3 P3 ; (12) n(H2)kP4 + ne kT1 n(C+)n(OH)k n(CO+) = I3 ; (13) n(H)kP5 + n(H2)kP6 + nekT5 n(C)ζH n(C+) = , (14) n(OH)kI3 which may be used together with

H + n(H2)ζ n(H3 ) = (15) n(CO)kP2 + n(O)kP3 + ne(kT2 + kT3 + kT4) to assess the chemical balances of the required species. The rate coeﬃcients for the relevant processes are given in Table1. These assume that atomic and molecular hydrogen and atomic carbon ionisation rates by CRs are all equivalent. Equation 12 can be reduced to

[2 + f ] ζH n(OH+) H2 , (16) ≈ kP4 fH2 + 2xikT1

+ assuming that the value of n(H3 )n(CO)kP2 (i.e. the rate of process P2) and the + H3 dissociative recombination rate are negligible compared to the ionisation rates of H and H2 (which would presumably be valid in a neutral MC with low ionisation fraction). Moreover, equation 13 becomes

x ζH n(CO+) = C , (17) (2kP5/fH2 ) + kP6 + (2xikT5/fH2 ) where xC = n(C)/n(H2). These expressions, together with equation 14, then yield equations 5.29, 5.30 and 5.31 in section 5.1.4, where ratios are taken to allow the abundances to be expressed as column densities. C. Abundance ratios 198

ID Reaction Rate coeﬃcient(a) c /cm3 s−1, c , c /K { 1 2 3 } Reference(s)(b) I1 H+ + O O+ + H k = 7.0 10−10, 0.26, 224.3 → I1 { × } Stancil et al.(1999) I2 H + + H H + + H k = 2.1 10−9, 0.0, 0.0 2 2 → 3 I2 { × } Theard and Huntress(1974) I3 C+ + OH CO+ + H k = 7.7 10−10, 0.5, 0.0 → I3 { × − } Prasad and Huntress(1980) P1 O+ + H OH+ + H k = 1.7 10−9, 0.0, 0.0 2 → P1 { × } Adams et al.(1980) P2 H + + CO HCO+ + H k = 1.4 10−9, 0.14, 3.4 (d) 3 → 2 P2 { × − − } Klippenstein et al.(2010) P3 H + + O OH+ + H k = 8.0 10−10, 0.16, 1.4 3 → 2 P3 { × − } Bettens et al.(1999) P4 OH+ + H H O+ + H k = 1.0 10−9, 0.0, 0.0 2 → 2 P4 { × } Jones et al.(1981) P5 CO+ + H CO + H+ k = 7.5 10−10, 0.0, 0.0 → P5 { × } Federer et al.(1984) (c) P6 CO+ + H HCO+ + H k = 1.8 10−9, 0.0, 0.0 2 → P6 { × } Adams et al.(1978) T1 OH+ + e− O + H k = 3.8 10−8, 0.50, 0.0 → T1 { × − } Mitchell(1990) T2 H + + e− H + H k = 2.3 10−8, 0.52, 0.0 3 → 2 T2 { × − } McCall et al.(2004) T3 H + + e− 3H k = 4.4 10−8, 0.52, 0.0 3 → T3 { × − } McCall et al.(2004) T4 H + + e− 2H k = 1.6 10−8, 0.43, 0.0 2 → T4 { × − } Mitchell(1990) T5 CO+ + e− C + O k = 1.0 10−7, 0.46, 0.0 → T5 { × − } Mitchell(1990)

Table 1: Rate coeﬃcients for principal formation/destruction reactions of chemical tracers in MCs as collated in the UMIST RATE95(Millar et al. 1997) and updated UMIST RATE12(McElroy et al. 2013) databases. Reaction IDs indicate initiation steps (I), propagation steps (P) and termination steps (T). D. Angular dispersion function and power spectrum 199

Table 1: Notes: (a) Rate coefficients are given in terms of the parameters c , c , c , for k = { 1 2 3} xx c (T/300 K)c2 exp( c /T ), where xx denotes the process ID. 1 − 3 (b) Reference(s) of the original source of the rate coefficients in the RATE12 database (McEl- roy et al. 2013). (c) CO production via CH and CH2 channels can also become important when the metallicity is high (i.e. large CH/OH ratio). This channel is not considered in the reaction network as it is sub-dominant for the objects of interest in this work. This should, however, be included in the analyses of the high metallicity (above solar) regions. (d) The sign of the parameter c3 in process P2 differs from other processes due to the molecular geometry. The charge-dipole, charge-quadrupole, and the charge-induced-dipole interactions influence the rate coefficient differently at different temperatures for the capture of reacting species in forming the transition state – see Klippenstein et al.(2010) for details.

D Angular dispersion function and power spectrum

The empirical CR diffusion parameter is computed from the power spectrum Pˆ(k) of the magnetic fields permeating a MC (cf. section 5.2.2). CR diffusion is determined by the properties and structures of the magnetic field. If the effects due to the magnetic field dominate, the dispersion function (`) can be calculated from the S2 magnetic field power spectrum Pˆ(k) under a slab approximation, i.e. where it only depends on the parallel component of k (Hasselmann and Wibberenz 1968). Therefore, Pˆ(k) can be related to the dispersion function (`) (more generally S2 referred to as the second order structure function) of observed polarisation angles through the cloud via a Fourier transform, denoted here as [...]. From the Wiener- F Khinchin theorem (Wiener 1930; Percival and Walden 1993),

Z ∞ Pˆ(k) = [ (`)] = d` (`) exp(ik`) , (18) F A −∞ A where the autocorrelation function (`) in the case of a statistically homogeneous A and isotropic ﬁeld can be expressed in terms of (`) = (0) (`)/2 (e.g. Schulz- A A − S2 Dubois and Rehberg 1981). Therefore,

Z ∞ 1 Pˆ(k) = d` (0) 2(`) exp(ik`) −∞ A − 2S 1 = (0) δ(k) [ (`)] , (19) A − 2F S2 E. Spatial dependence of the empirical diffusion parameter in IC 5146 200 where the first term on the right hand side is unphysical when k = 0 and vanishes when k = 0. Hence, 6 1 Pˆ(k) = [ (`)] , (20) 2F S2 since (`) is real and the Fourier Transform of (`) has a Hermitian symmetry. S2 S2 E Spatial dependence of the empirical diffusion parameter in IC 5146

The analyses in section 5.2.2 assumes that the spatial dependence of the diffusion parameter is derived only from the spatial variation in the magnetic field strength. There is no clear empirical evidence for variation of the diffusion parameter within the IC 5146 region due to the magnetic fluctuations. Here four sub-regions of IC 5146 are analysed to demonstrate the lack of evidence in large spatial variations of the diffusion parameter, if adopting a fixed magnetic field strength and gas density. Four circular regions (of radius 10’, labelled A, B, C and D) around the hub/core filament structures of the region, shown in Arzoumanian et al.(2011) and Wang et al.(2017), are selected. The Rc-, i’-, H- and K- bands are analysed separately to determine whether there is any variation between observational bands. The number of data points in each region is summarised in Table2. The diffusion parameter is computed for each region and band according to section 5.2.2, for which a magnetic field strength on the plane of the sky is estimated from the density using equation 5.52 (a relation appropriate specifically for the IC 5146 region), and for a CR energy of 1 GeV. Given the large ranges in vol-

Region RA Dec Number of points A 21h 53m 0s 47o 140 000 34 (Rc), – (i), 120 (H-), 15 (K-) B 21h 50m 0s 47o 300 000 53 (Rc), – (i), 87 (H-), 15 (K-) C 21h 45m 0s 47o 400 000 48 (Rc), 41 (i), 165 (H-), 14 (K-) D 21h 48m 0s 48o 100 000 70 (Rc), 92 (i), 187 (H-), 10 (K-)

Table 2: Location and number of points in each analysis region. Each region selects all points within a radius of 10’ from the centre. Note that only regions C and D intersect the smaller observation window for the i-band data. Region A roughly corresponds to the Cocoon Nebula. E. Spatial dependence of the empirical diffusion parameter in IC 5146 201 ume density throughout each region, a characteristic value of 103 cm−3 is used for this comparative estimate. Note that future dedicated work should more carefully quantify the structures of magnetic and density fields with higher resolution. The results are shown in Fig.1, where error bars are at 1 σ confidence level. There is a slight tension in the Rc-band data between the values derived for the four regions., However there is no evidence of variation in any of the other bands. Moreover, the K-band yields slightly lower values for the diffusion coefficient, which presumably results from each of the bands being sensitive to slightly different scales (see also Wang et al. 2019) and subject to different opacities into the cloud. Hence, there is insufficient evidence to motivate the calculations of spatially-varying diffusion coefficients in this study. E. Spatial dependence of the empirical diffusion parameter in IC 5146 202

Rc band 33 Rc band

32 ] 1 s

2 31 cm D/

[ 30 10 log 29

28 A B I band C D 33 Region i' band

32 ] 1 s

2 31 cm D/

[ 30 10 log 29 No data No data

28 A B H band C D 33 Region H- band 32 ] 1 s

2 31 cm D/

[ 30 10 log 29

28 A B K band C D 33 Region K- band 32 ] 1 s

2 31 cm D/

[ 30 10 log 29

28 A B C D Region

Fig. 1: Estimated empirical diﬀusion parameter values for each of the regions A, B, C and D in each of the four bands. 1σ error bars are shown. 203

Bibliography

Abel, T., Bryan, G. L. and Norman, M. L. (2002), The Formation of the First Star in the Universe, Science 295, 93–98.

Abraham, P. B., Brunstein, K. A. and Cline, T. L. (1966), Production of Low-Energy Cosmic-Ray Electrons, Physical Review 150(4), 1088–1103.

Achterberg, A., Gallant, Y. A., Kirk, J. G. and Guthmann, A. W. (2001), Par- ticle acceleration by ultrarelativistic shocks: theory and simulations, MNRAS 328, 393–408.

Ackermann, M., Ajello, M., Allafort, A. and et al. (2013), Detection of the Charac- teristic Pion-Decay Signature in Supernova Remnants, Science 339(6121), 807– 811.

Acuna, M. H., Connerney, J. E. P., Ness, N. F., Lin, R. P., Mitchell, D., Carl- son, C. W., McFadden, J., Anderson, K. A., Reme, H., Mazelle, C., Vignes, D., Wasilewski, P. and Cloutier, P. (1999), Global Distribution of Crustal Magneti- zation Discovered by the Mars Global Surveyor MAG/ER Experiment, Science 284, 790.

+ Adams, N. G., Smith, D. and Grief, D. (1978), Reactions of HnCO ions with molecules at 300 K, International Journal of Mass Spectrometry and Ion Processes 26, 405–415.

Adams, N. G., Smith, D. and Paulson, J. F. (1980), An experimental survey of + the reactions of NHn ions (n = 0 to 4) with several diatomic and polyatomic molecules at 300 K, The Journal of Chemical Physics 72(1), 288–297.

Aharonian, F., Akhperjanian, A. G., Bazer-Bachi, A. R., Beilicke, M., Benbow, W., Berge, D., Bernl¨ohr,K., Boisson, C., Bolz, O., Borrel, V., Braun, I., Breitling, Bibliography 204

F., Brown, A. M., Chadwick, P. M., Chounet, L. M., Cornils, R., Costamante, L., Degrange, B., Dickinson, H. J., Djannati-Ata¨ı,A., Drury, L. O., Dubus, G., Emmanoulopoulos, D., Espigat, P., Feinstein, F., Fontaine, G., Fuchs, Y., Funk, S., Gallant, Y. A., Giebels, B., Gillessen, S., Glicenstein, J. F., Goret, P., Had- jichristidis, C., Hauser, D., Hauser, M., Heinzelmann, G., Henri, G., Hermann, G., Hinton, J. A., Hofmann, W., Holleran, M., Horns, D., Jacholkowska, A., de Jager, O. C., Khélifi,B., Klages, S., Komin, N., Konopelko, A., Latham, I. J., Le Gallou, R., Lemière,A., Lemoine-Goumard, M., Leroy, N., Lohse, T., Marcowith, A., Martin, J. M., Martineau-Huynh, O., Masterson, C., McComb, T. J. L., de Naurois, M., Nolan, S. J., Noutsos, A., Orford, K. J., Osborne, J. L., Ouchrif, M., Panter, M., Pelletier, G., Pita, S., Pühlhofer,G., Punch, M., Raubenheimer, B. C., Raue, M., Raux, J., Rayner, S. M., Reimer, A., Reimer, O., Ripken, J., Rob, L., Rolland, L., Rowell, G., Sahakian, V., Saugé,L., Schlenker, S., Schlickeiser, R., Schuster, C., Schwanke, U., Siewert, M., Sol, H., Spangler, D., Steenkamp, R., Stegmann, C., Tavernet, J. P., Terrier, R., Théoret,C. G., Tluczykont, M., van Eldik, C., Vasileiadis, G., Venter, C., Vincent, P., Völk,H. J. and Wagner, S. J. (2006), Discovery of very-high-energy γ-rays from the Galactic Centre ridge, Nature 439(7077), 695–698.

Aharonian, F., Bykov, A., Parizot, E., Ptuskin, V. and Watson, A. (2012), Cosmic Rays in Galactic and Extragalactic Magnetic Fields, Space Sci. Rev. 166(1), 97.

Akahori, T., Gaensler, B. M. and Ryu, D. (2014), Statistical Techniques for Detect- ing the Intergalactic Magnetic Field from Large Samples of Extragalactic Faraday Rotation Data, ApJ 790(2), 123.

Akahori, T. and Ryu, D. (2010), Faraday Rotation Measure Due to the Intergalactic Magnetic Field, ApJ 723, 476–481.

Akahori, T., Nakanishi, H., Sofue, Y., Fujita, Y., Ichiki, K., Ideguchi, S., Kameya, O., Kudoh, T., Kudoh, Y., Machida, M., Miyashita, Y., Ohno, H., Ozawa, T., Takahashi, K., Takizawa, M. and Yamazaki, D. G. (2018), Cosmic magnetism in centimeter- and meter-wavelength radio astronomy, PASJ 70(1), R2.

Akahori, T., Ryu, D. and Gaensler, B. M. (2016), Fast Radio Bursts as Probes of Magnetic Fields in the Intergalactic Medium, ApJ 824, 105. Bibliography 205

Albertsson, T., Kauﬀmann, J. and Menten, K. M. (2018), Atlas of Cosmic-Ray- induced Astrochemistry, ApJ 868, 40.

Alfv´en, H. (1942), Existence of Electromagnetic-Hydrodynamic Waves, Nature 150, 405–406.

Almeida, S. P., Rushbrooke, J. G., Scharenguivel, J. H., Behrens, M., Blobel, V., Borecka, I., Dehne, H. C., Dfaz, J., Knies, G., Schmitt, A., Str¨omer,K. and Swanson, W. P. (1968), pp Interactions at 10 GeV/c, Phys. Rev. 174, 1638.

Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A. and Sorensen, D. (1999), LA- PACK Users’ Guide (Third Ed.), Society for Industrial and Applied Mathematics, Philadelphia, PA, United States.

Anderson, T. W. (1976), Panels and time series analysis: Markov Chains and au- toregressive processes, Technical Report No.24, Department of Statistics, Stanford University, California.

Andersson, B.-G. (2015), Interstellar Grain Alignment: Observational Status, in Magnetic Fields in Diﬀuse Media, Vol. 407 of Astrophysics and Space Science Library, Springer, Berlin, Heidelberg.

Andersson, B. G. and Potter, S. B. (2007), Observational Constraints on Interstellar Grain Alignment, ApJ 665(1), 369–389.

Angel, J. R. P. and Landstreet, J. D. (1974), A Determination by the Zeeman Eﬀect of the Magnetic Field Strength in the White Dwarf G99-37, ApJ 191, 457–464.

Arzoumanian, D., André,P., Didelon, P., Könyves, V., Schneider, N., Men’shchikov, A., Sousbie, T., Zavagno, A., Bontemps, S., di Francesco, J., Griffin, M., Hen- nemann, M., Hill, T., Kirk, J., Martin, P., Minier, V., Molinari, S., Motte, F., Peretto, N., Pezzuto, S., Spinoglio, L., Ward-Thompson, D., White, G. and Wil- son, C. D. (2011), Characterizing interstellar filaments with Herschel in IC 5146, A&A 529, L6.

Axford, W. I., Leer, E. and Skadron, G. (1977), The Acceleration of Cosmic Rays by Shock Waves, in International Cosmic Ray Conference, Vol. 11 of International Cosmic Ray Conference, p.132. Bibliography 206

Babcock, H. W. (1947), Zeeman Eﬀect in Stellar Spectra., ApJ 105, 105.

Bacchi, M., Feretti, L., Giovannini, G. and Govoni, F. (2003), Deep images of cluster radio halos, A&A 400, 465–476.

Balbus, S. A. and Soker, N. (1989), Theory of local thermal instability in spherical systems, ApJ 341, 611–630.

Balsara, D. S. (1998), Total Variation Diminishing Scheme for Adiabatic and Isothermal Magnetohydrodynamics, ApJS 116, 133–153.

Balsara, D. S. and Kim, J. (2005), Ampliﬁcation of Interstellar Magnetic Fields and Turbulent Mixing by Supernova-driven Turbulence. II. The Role of Dynamical Chaos, ApJ 634(1), 390–406.

Balsara, D. S., Kim, J., Mac Low, M.-M. and Mathews, G. J. (2004), Ampliﬁcation of Interstellar Magnetic Fields by Supernova-driven Turbulence, ApJ 617(1), 339– 349.

Banerjee, R. and Jedamzik, K. (2004), Evolution of cosmic magnetic ﬁelds: From the very early Universe, to recombination, to the present, Phys. Rev. D 70(12), 123003.

Barnes, D. J., Kawata, D. and Wu, K. (2012), Cosmological simulations using GCMHD+, MNRAS 420, 3195–3212.

Barnes, D. J., On, A. Y. L., Wu, K. and Kawata, D. (2018), SPMHD simulations of structure formation, MNRAS 476(3), 2890–2904.

Barret, D., Nandra, K., Barcons, X., Fabian, A., den Herder, J. W., Piro, L., Watson, M., Aird, J., Branduardi-Raymont, G., Cappi, M., Carrera, F., Comastri, A., Costantini, E., Croston, J., Decourchelle, A., Done, C., Dovciak, M., Ettori, S., Finoguenov, A., Georgakakis, A., Jonker, P., Kaastra, J., Matt, G., Motch, C., O’Brien, P., Pareschi, G., Pointecouteau, E., Pratt, G., Rauw, G., Reiprich, T., Sanders, J., Sciortino, S., Willingale, R. and Wilms, J. (2013), Athena+: The ﬁrst Deep Universe X-ray Observatory, in SF2A-2013: Proceedings of the Annual meeting of the French Society of Astronomy and Astrophysics, pp.447–453. Bibliography 207

Basu, S. (2000), Magnetic Fields and the Triaxiality of Molecular Cloud Cores, ApJ 540(2), L103–L106.

Basu, S., Ciolek, G. E., Dapp, W. B. and Wurster, J. (2009), Magnetically- regulated fragmentation induced by nonlinear ﬂows and ambipolar diﬀusion, New A 14, 483–495.

Bazavov, A., Bhattacharya, T., DeTar, C., Ding, H. T., Gottlieb, S., Gupta, R., Hegde, P., Heller, U. M., Karsch, F., Laermann, E., Levkova, L., Mukherjee, S., Petreczky, P., Schmidt, C., Schroeder, C., Soltz, R. A., Soeldner, W., Sugar, R., Wagner, M., Vranas, P. and HotQCD Collaboration (2014), Equation of state in (2 + 1)-ﬂavor QCD, Phys. Rev. D 90(9), 094503.

Beck, A. M., Hanasz, M., Lesch, H., Remus, R.-S. and Stasyszyn, F. A. (2013a), On the magnetic ﬁelds in voids, MNRAS 429, L60–L64.

Beck, R. (1999), Depolarization eﬀects in galaxies, in E. M. Berkhuijsen (ed.), Galactic Foreground Polarization, p.3.

Beck, R. (2001), Galactic and Extragalactic Magnetic Fields, Space Sci. Rev. 99, 243–260.

Beck, R. (2009a), Galactic and extragalactic magnetic ﬁelds - a concise review, Astrophysics and Space Sciences Transactions 5, 43–47.

Beck, R. (2009b), Magnetic Visions: Mapping Cosmic Magnetism with LOFAR and SKA, in Revista Mexicana de Astronom´ıay Astrof´ısica Conference Series, Vol. 36 of Revista Mexicana de Astronom´ıay Astrof´ısica Conference Series, pp.1–8.

Beck, R. (2011), Future Observations of Cosmic Magnetic Fields with the SKA and its Precursors, in M. Soida, K. Otmianowska-Mazur, E. M. de Gouveia dal Pino and A. Lazarian (eds.), Magnetic Fields in the Universe III: From Laboratory and Stars to Primordial Structures.

Beck, R. (2012), Magnetic Fields in Galaxies, Space Sci. Rev. 166(1-4), 215–230.

Beck, R. (2015), Future Observations of Cosmic Magnetic Fields with LOFAR, SKA and Its Precursors, in A. Lazarian, E. M. de Gouveia Dal Pino and C. Melioli Bibliography 208

(eds.), Magnetic Fields in Diﬀuse Media, Vol. 407 of Astrophysics and Space Science Library, p.3.

Beck, R. (2016), Magnetic ﬁelds in spiral galaxies, A&A Rev. 24, 4.

Beck, R., Anderson, J., Heald, G., Horneﬀer, A., Iacobelli, M., K¨ohler,J., Mulcahy, D., Pizzo, R., Scaife, A., Wucknitz, O. and LOFAR Magnetism Key Science Project Team (2013b), The LOFAR view of cosmic magnetism, Astronomische Nachrichten 334, 548–557.

Beck, R., Berkhuijsen, E. M., Gießübel, R. and Mulcahy, D. D. (2020), Magnetic fields and cosmic rays in M 31. I. Spectral indices, scale lengths, Faraday rotation, and magnetic field pattern, A&A 633.

Beck, R. and Wielebinski, R. (2013), Magnetic Fields in Galaxies, in T. D. Oswalt and G. Gilmore (eds.), Planets, Stars and Stellar Systems. Volume 5: Galactic Structure and Stellar Populations, Vol. 5, Springer, Berlin, p.641.

Bell, A. R. (1978a), The acceleration of cosmic rays in shock fronts - I., MNRAS 182, 147–156.

Bell, A. R. (1978b), The acceleration of cosmic rays in shock fronts - II., MNRAS 182, 443–455.

Berezinskii, V. S., Bulanov, S. V., Dogiel, V. A. and Ptuskin, V. S. (1990), Astro- physics of cosmic rays, Amsterdam: North-Holland.

Berger, E. (2014), Short-Duration Gamma-Ray Bursts, ARA&A 52, 43–105.

Bergin, E. A. and Tafalla, M. (2007), Cold Dark Clouds: The Initial Conditions for Star Formation, ARA&A 45(1), 339–396.

Bern´e,O., Marcelino, N. and Cernicharo, J. (2014), IRAM 30 m Large Scale Sur- vey of 12CO(2-1) and 13CO(2-1) Emission in the Orion Molecular Cloud, ApJ 795(1), 13.

Bertone, S., Vogt, C. and Enßlin, T. (2006), Magnetic ﬁeld seeding by galactic winds, MNRAS 370, 319–330. Bibliography 209

Bertschinger, E. (2001), Multiscale Gaussian Random Fields and Their Application to Cosmological Simulations, ApJS 137, 1–20.

Bettens, R. P. A., Hansen, T. A. and Collins, M. A. (1999), Interpolated 3 + 1 0 potential energy surface and reaction dynamics for O( P)+H3 ( A1) and + 3 − 1 + OH ( Σ )+H2( Σg ), The Journal of Chemical Physics 111(14), 6322–6332.

Bhat, P. and Subramanian, K. (2013), Fluctuation dynamos and their Faraday rotation signatures, MNRAS 429(3), 2469–2481.

Bhat, S. (2002), Faraday: Father of electromagnetism, Resonance 7(3), 46–50.

Biermann, L. (1950), Uber¨ den Ursprung der Magnetfelder auf Sternen und im inter- stellaren Raum (miteinem Anhang von A. Schl¨uter), Zeitschrift Naturforschung Teil A 5, 65.

Bisbas, T. G., van Dishoeck, E. F., Papadopoulos, P. P., Sz˝ucs,L., Bialy, S. and Zhang, Z.-Y. (2017), Cosmic-ray Induced Destruction of CO in Star-forming Galaxies, ApJ 839, 90.

Black, J. H., Hartquist, T. W. and Dalgarno, A. (1978), Models of interstellar clouds. II - The Zeta Persei cloud, ApJ 224, 448–452.

Blandford, R. D. and Ostriker, J. P. (1978), Particle acceleration by astrophysical shocks, ApJ 221, L29–L32.

Blasi, P. (2001), The non-thermal radiation-cluster merger connection, Astroparticle Physics 15(2), 223–240.

Blasi, P. (2013), The origin of galactic cosmic rays, A&A Rev. 21, 70.

Blasi, P., Burles, S. and Olinto, A. V. (1999), Cosmological Magnetic Field Limits in an Inhomogeneous Universe, ApJ 514, L79–L82.

Blattnig, S. R., Swaminathan, S. R., Kruger, A. T., Ngom, M., Norbury, J. W. and Tripathi, R. K. (2000), Parameterized Cross Sections for Pion Production in Proton-Proton Collisions, Technical Report, NASA/TP-2000-210640; L-18021; NAS 1.60:210640 01, 18568. Bibliography 210

Blumenthal, G. R. (1970), Energy Loss of High-Energy Cosmic Rays in Pair- Producing Collisions with Ambient Photons, Phys. Rev. D 1, 1596–1602.

Bohren, C. F. and Huﬀman, D. R. (2008), Absorption and scattering of light by small particles, John Wiley & Sons.

B¨ohringer,H., Chon, G., Collins, C. A., Guzzo, L., Nowak, N. and Bobrovskyi, S. (2013), The extended ROSAT-ESO ﬂux limited X-ray galaxy cluster survey (REFLEX II) II. Construction and properties of the survey, A&A 555, A30.

Bonafede, A., Feretti, L., Murgia, M., Govoni, F., Giovannini, G., Dallacasa, D., Dolag, K. and Taylor, G. B. (2010), The Coma cluster magnetic ﬁeld from Faraday rotation measures, A&A 513, A30.

Bonafede, A., Vazza, F., Br¨uggen,M., Murgia, M., Govoni, F., Feretti, L., Giovan- nini, G. and Ogrean, G. (2013), Measurements and simulation of Faraday rotation across the Coma radio relic, MNRAS 433(4), 3208–3226.

Bond, J. R., Kofman, L. and Pogosyan, D. (1996), How ﬁlaments of galaxies are woven into the cosmic web, Nature 380, 603.

Box, G. E. P. and Jenkins, G. M. (1976), Time series analysis. Forecasting and control, Holden-Day, San Francisco.

Box, G. E. P. and Muller, M. E. (1958), A Note on the Generation of Random Normal Deviates, Ann. Math. Statist. 29(2), 610–611.

Bracewell, R. N., Cooper, B. F. C. and Cousins, T. E. (1962), Polarization in the Central Component of Centaurus A, Nature 195(4848), 1289–1290.

Brandenburg, A. (2015), Simulations of Galactic Dynamos, in A. Lazarian, E. M. de Gouveia Dal Pino and C. Melioli (eds.), Magnetic Fields in Diﬀuse Media, Vol. 407 of Astrophysics and Space Science Library, Springer, Berlin, Heidelberg, p.529.

Brandenburg, A., Donner, K. J., Moss, D., Shukurov, A., Sokoloﬀ, D. D. and Tuomi- nen, I. (1993), Vertical Magnetic Fields above the Discs of Spiral Galaxies, A&A 271, 36. Bibliography 211

Brandenburg, A. and Subramanian, K. (2005), Astrophysical magnetic ﬁelds and nonlinear dynamo theory, Phys. Rep. 417, 1–209.

Brentjens, M. A. and de Bruyn, A. G. (2005), Faraday rotation measure synthesis, A&A 441, 1217–1228.

Brown, J. C., Haverkorn, M., Gaensler, B. M., Taylor, A. R., Bizunok, N. S., McClure-Griﬃths, N. M., Dickey, J. M. and Green, A. J. (2007), Rotation Mea- sures of Extragalactic Sources behind the Southern Galactic Plane: New Insights into the Large-Scale Magnetic Field of the Inner Milky Way, ApJ 663, 258–266.

Brown, J. C., Taylor, A. R. and Jackel, B. J. (2003), Rotation Measures of Compact Sources in the Canadian Galactic Plane Survey, ApJS 145, 213–223.

Brown, R. L. and Marscher, A. P. (1977), Energetic secondary electrons in dense interstellar clouds, ApJ 212, 659–663.

Brown, S. and Rudnick, L. (2011), Diﬀuse radio emission in/around the Coma cluster: beyond simple accretion, MNRAS 412(1), 2–12.

Brown, S., Vernstrom, T., Carretti, E., Dolag, K., Gaensler, B. M., Staveley- Smith, L., Bernardi, G., Haverkorn, M., Kesteven, M. and Poppi, S. (2017), Limiting magnetic ﬁelds in the cosmic web with diﬀuse radio emission, MNRAS 468(4), 4246–4253.

Br¨uggen,M., Bykov, A., Ryu, D. and R¨ottgering,H. (2012), Magnetic Fields, Rel- ativistic Particles, and Shock Waves in Cluster Outskirts, Space Sci. Rev. 166(1- 4), 187–213.

Br¨uggen,M., Ruszkowski, M., Simionescu, A., Hoeft, M. and Dalla Vecchia, C. (2005), Simulations of Magnetic Fields in Filaments, ApJ 631, L21–L24.

Brunetti, G. (2009), Non-thermal emission from Galaxy Clusters and future observations with the FERMI gamma-ray telescope and LOFAR, in Revista Mexicana de Astronom´ıay Astrof´ısica Conference Series, Vol. 36 of Revista Mexicana de Astronom´ıay Astrof´ısica Conference Series, pp.201–208.

Brunetti, G., Blasi, P., Cassano, R. and Gabici, S. (2004), Alfv´enicreacceleration Bibliography 212

of relativistic particles in galaxy clusters: MHD waves, leptons and hadrons, MNRAS 350(4), 1174–1194.

Brunetti, G., Giacintucci, S., Cassano, R., Lane, W., Dallacasa, D., Venturi, T., Kassim, N. E., Setti, G., Cotton, W. D. and Markevitch, M. (2008), A low- frequency radio halo associated with a cluster of galaxies, Nature 455(7215), 944– 947.

Brunetti, G. and Jones, T. W. (2014), Cosmic Rays in Galaxy Clusters and Their Nonthermal Emission, International Journal of Modern Physics D 23(4), 1430007–98.

Brunetti, G., Setti, G., Feretti, L. and Giovannini, G. (2001), Particle reacceleration in the Coma cluster: radio properties and hard X-ray emission, MNRAS 320(3), 365–378.

Brunstein, K. A. (1965), Possible Source Mechanism for Low-Energy Galactic Elec- trons, Physical Review 137(3B), 757–759.

Buote, D. A. (2001), On the Origin of Radio Halos in Galaxy Clusters, ApJ 553(1), L15–L18.

Burke-Spolaor, S., G¨ultekin,K., Postman, M., Lauer, T. R., Taylor, J. M., Lazio, T. J. W. and Moustakas, L. A. (2017), A Radio Relic and a Search for the Central Black Hole in the Abell 2261 Brightest Cluster Galaxy, ApJ 849(1), 59.

Burn, B. J. (1966), On the depolarization of discrete radio sources by Faraday dispersion, MNRAS 133, 67.

Burns, J. O. (1998), Stormy Weather in Galaxy Clusters, Science 280(5362), 400– 404.

Bushby, P. and Mason, J. (2004), Understanding the solar dynamo, Astronomy & Geophysics 45(4), 4.7–4.13.

Cantwell, T. M., Scaife, A. M. M., Oozeer, N., Wen, Z. L. and Han, J. L. (2016), A newly discovered radio halo in merging cluster MACS J2243.3-0935, MNRAS 458(2), 1803–1814. Bibliography 213

Carilli, C. L. and Taylor, G. B. (2002), Cluster Magnetic Fields, ARA&A 40, 319– 348.

Carlberg, R. G. and Pudritz, R. E. (1990), Magnetic Support and Fragmentation of Molecular Clouds, MNRAS 247, 353.

Casanova, S., Aharonian, F. A., Fukui, Y., Gabici, S., Jones, D. I., Kawamura, A., Onishi, T., Rowell, G., Sano, H., Torii, K. and Yamamoto, H. (2010), Molecular Clouds as Cosmic-Ray Barometers, PASJ 62, 769.

Caselli, P., Walmsley, C. M., Terzieva, R. and Herbst, E. (1998), The Ionization Fraction in Dense Cloud Cores, ApJ 499, 234–249.

Cashman, L. R. and Clemens, D. P. (2014), The Magnetic Field of Cloud 3 in L204, ApJ 793, 126.

Cassano, R., Botteon, A., Di Gennaro, G., Brunetti, G., Sereno, M., Shimwell, T. W., van Weeren, R. J., Br¨uggen,M., Gastaldello, F., Izzo, L., Bˆırzan, L., Bonafede, A., Cuciti, V., de Gasperin, F., R¨ottgering,H. J. A., Hardcastle, M., Mechev, A. P. and Tasse, C. (2019), LOFAR Discovery of a Radio Halo in the High-redshift Galaxy Cluster PSZ2 G099.86+58.45, ApJ 881(1), L18.

Cassano, R., Brunetti, G. and Setti, G. (2006), Statistics of giant radio haloes from electron reacceleration models, MNRAS 369(4), 1577–1595.

Cassano, R., Ettori, S., Giacintucci, S., Brunetti, G., Markevitch, M., Venturi, T. and Gitti, M. (2010), On the Connection Between Giant Radio Halos and Cluster Mergers, ApJ 721(2), L82–L85.

Catalano, A., Adam, R., Ade, P., André,P., Aussel, H., Beelen, A., Benoˆıt,A., Bideaud, A., Billot, N., Bourrion, O., Calvo, M., Coiffard, G., Comis, B., Désert, F. X., Doyle, S., Goupy, J., Kramer, C. F., Lagache, G., Leclercq, S., Lestrade, J. F., Mac´ıas-Pérez,J. F., Maury, A., Mauskopf, P., Mayet, F., Monfardini, A., Pajot, F., Pascale, E., Perotto, L., Pisano, G., Ponthieu, N., Revéret,V., Ritacco, A., Rodriguez, L., Romero, C., Roussel, H., Ruppin, F., Schuster, K., Sievers, A., Soler, J., Triqueneaux, S., Tucker, C. and Zylka, R. (2016), The NIKA2 commissioning campaign: performance and first results, arXiv: 1605.08628 p.11. Bibliography 214

Ceccarelli, C., Caux, E., Wolﬁre, M., Rudolph, A., Nisini, B., Saraceno, P. and White, G. J. (1998), ISO Detection of CO+ toward the protostar IRAS 16293- 2422, A&A 331, L17–L20.

Cécere,M., Velázquez,P. F., Araudo, A. T., De Colle, F., Esquivel, A., Carrasco- González,C. and Rodr´ıguez, L. F. (2016), A Study of Radio Polarization in Protostellar Jets, ApJ 816(2), 64.

Cesarsky, C. J. and Volk, H. J. (1978), Cosmic Ray Penetration into Molecular Clouds, A&A 70, 367.

Chan, J. Y. H. and Wu, K. (2014), private communication.

Chan, J. Y. H., Wu, K., On, A. Y. L., Barnes, D. J., McEwen, J. D. and Kitching, T. D. (2019), Covariant polarized radiative transfer on cosmological scales for investigating large-scale magnetic ﬁeld structures, MNRAS 484, 1427–1455.

Chandran, B. D. G. (2000), Conﬁnement and Isotropization of Galactic Cosmic Rays by Molecular-Cloud Magnetic Mirrors When Turbulent Scattering Is Weak, ApJ 529, 513–535.

Chandrasekhar, S. and Fermi, E. (1953), Magnetic Fields in Spiral Arms., ApJ 118, 113.

Chanmugam, G. (1992), Magnetic ﬁelds of degenerate stars, ARAA 30, 143–184.

Cheng, K. S., Chernyshov, D. O., Dogiel, V. A., Ko, C. M., Ip, W. H. and Wang, Y. (2012), The Fermi Bubble as a Source of Cosmic Rays in the Energy Range > 1015 eV, ApJ 746(2), 116.

Chernyshov, D., Cheng, K.-S., Dogiel, V. and Ko, C.-M. (2017), Fermi bubbles as sources of cosmic rays above 1 PeV, in European Physical Journal Web of Conferences, Vol. 145 of European Physical Journal Web of Conferences, p.04004.

Ching, T.-C., Lai, S.-P., Zhang, Q., Girart, J. M., Qiu, K. and Liu, H. B. (2017), Magnetic Fields in the Massive Dense Cores of the DR21 Filament: Weakly Mag- netized Cores in a Strongly Magnetized Filament, ApJ 838(2), 121.

Cho, J. (2019), A Technique for Removing Large-scale Variations in Regularly and Irregularly Spaced Data, ApJ 874(1), 75. Bibliography 215

Cho, J. and Lazarian, A. (2005), Grain Alignment by Radiation in Dark Clouds and Cores, ApJ 631(1), 361–370.

Cho, J. and Ryu, D. (2009), Characteristic Lengths of Magnetic Field in Magneto- hydrodynamic Turbulence, ApJ 705, L90.

Cho, J. and Yoo, H. (2016), A Technique for Constraining the Driving Scale of Turbulence and a Modiﬁed Chandrasekhar-Fermi Method, ApJ 821(1), 21.

Christensen, U. R. (2006), A deep dynamo generating Mercury’s magnetic ﬁeld, Nature 444(7122), 1056–1058.

Chy˙zy,K. T., Drzazga, R. T., Beck, R., Urbanik, M., Heesen, V. and Bomans, D. J. (2016), The Magnetized Galactic Wind and Synchrotron Halo of the Starburst Dwarf Galaxy IC 10, ApJ 819(1), 39.

Chy˙zy,K. T. and Beck, R. (2004), Magnetic ﬁelds in merging spirals - the Antennae, AAP 417, 541–555.

Clarke, T. E. (2004), Faraday Rotation Observations of Magnetic Fields in Galaxy Clusters, Journal of Korean Astronomical Society 37, 337.

Clarke, T. E., Kronberg, P. P. and B¨ohringer,H. (2001), A New Radio-X-Ray Probe of Galaxy Cluster Magnetic Fields, ApJ 547(2), L111–L114.

Colafrancesco, S. and Marchegiani, P. (2008), Warming rays in cluster cool cores, A&A 484(1), 51–65.

Coleman, S. and Weinberg, E. (1973), Radiative Corrections as the Origin of Spon- taneous Symmetry Breaking, Phys. Rev. D 7, 1888–1910.

Collett, E. (2005), Field Guide to Polarization, Field Guide Series, SPIE Press.

Combes, F. (2018), Molecular gas ﬁlamentary structures in galaxy clusters, in K. Asada, E. de Gouveia dal Pino, M. Giroletti, H. Nagai and R. Nemmen (eds.), Perseus in Sicily (IAU S342): From Black Hole to Cluster Outskirts (Proceedings of the International Astronomical Union Symposia and Colloquia), Cambridge University Press; 1st edition, pp.77–84. Bibliography 216

Condon, J. J., Cotton, W. D., Yin, Q. F., Shupe, D. L., Storrie-Lombardi, L. J., Helou, G., Soifer, B. T. and Werner, M. W. (2003), The SIRTF First-Look Survey. I. VLA Image and Source Catalog, AJ 125, 2411–2426.

Condon, J. J., Darling, J., Kovalev, Y. Y. and Petrov, L. (2017), A Nearly Naked Supermassive Black Hole, ApJ 834(2), 184.

Connerney, J. E. P. (1993), Magnetic ﬁelds of the outer planets, J. Geophys. Res. 98(E10), 18659–18680.

Conselice, C. J., Gallagher, III, J. S. and Wyse, R. F. G. (2001), On the Nature of the NGC 1275 System, AJ 122, 2281–2300.

Cordes, J. M. and Chatterjee, S. (2019), Fast Radio Bursts: An Extragalactic Enigma, ARA&A 57, 417–465.

Cortes, P. and Crutcher, R. M. (2006), Interferometric Mapping of Magnetic Fields: G30.79 FIR 10, ApJ 639, 965–968.

Coud´e, S., Bastien, P., Houde, M., Sadavoy, S., Friesen, R., Di Francesco, J., Johnstone, D., Mairs, S., Hasegawa, T. and Kwon, W. (2019), The JCMT BISTRO Survey: The Magnetic Field of the Barnard 1 Star-forming Region, ApJ 877(2), 88.

Cova, F., Gastaldello, F., Wik, D. R., Boschin, W., Botteon, A., Brunetti, G., Buote, D. A., De Grand i, S., Eckert, D., Ettori, S., Feretti, L., Gaspari, M., Ghizzardi, S., Giovannini, G., Girardi, M., Govoni, F., Molendi, S., Murgia, M., Rossetti, M. and Vacca, V. (2019), A joint XMM-NuSTAR observation of the galaxy cluster Abell 523: Constraints on inverse Compton emission, A&A 628, A83.

Cox, T. J., Di Matteo, T., Hernquist, L., Hopkins, P. F., Robertson, B. and Springel, V. (2006), X-Ray Emission from Hot Gas in Galaxy Mergers, ApJ 643, 692–706.

Crutcher, R. (2014), CN Zeeman Mapping of Magnetic Fields in Dense Molecular Cores, in Dense Cores: Origin, Evolution, and Collapse, p.30601.

Crutcher, R. M. (2012), Magnetic Fields in Molecular Clouds, ARA&A 50, 29–63.

Crutcher, R. M. and Kemball, A. J. (2019), Review of Zeeman Eﬀect Observations of Regions of Star Formation, Frontiers in Astronomy and Space Sciences 6, 66. Bibliography 217

Crutcher, R. M., Nutter, D. J., Ward-Thompson, D. and Kirk, J. M. (2004), SCUBA Polarization Measurements of the Magnetic Field Strengths in the L183, L1544, and L43 Prestellar Cores, ApJ 600(1), 279–285.

Crutcher, R. M., Troland, T. H., Goodman, A. A., Heiles, C., Kazes, I. and Myers, P. C. (1993), OH Zeeman Observations of Dark Clouds, ApJ 407, 175.

Crutcher, R. M., Troland, T. H., Lazareff, B., Paubert, G. and Kazès,I. (1999), Detection of the CN Zeeman Effect in Molecular Clouds, ApJ 514, L121–L124.

Crutcher, R. M., Wandelt, B., Heiles, C., Falgarone, E. and Troland, T. H. (2010), Magnetic Fields in Interstellar Clouds from Zeeman Observations: Inference of Total Field Strengths by Bayesian Analysis, ApJ 725, 466–479.

Cummings, A. C., Stone, E. C., Heikkila, B. C., Lal, N., Webber, W. R., J´ohannesson,G., Moskalenko, I. V., Orlando, E. and Porter, T. A. (2016), Galac- tic Cosmic Rays in the Local Interstellar Medium: Voyager 1 Observations and Model Results, ApJ 831(1), 18.

Dainotti, M. G. and Del Vecchio, R. (2017), Gamma Ray Burst afterglow and prompt-afterglow relations: An overview, New A Rev. 77, 23–61.

Dalgarno, A. and McCray, R. A. (1972), Heating and Ionization of HI Regions, ARA&A 10, 375.

Daly, R. A. and Loeb, A. (1990), A possible origin of galactic magnetic ﬁelds, ApJ 364, 451–455.

Davis, Jr., L. and Greenstein, J. L. (1951), The Polarization of Starlight by Aligned Dust Grains., ApJ 114, 206.

Davis, L. (1951), The Strength of Interstellar Magnetic Fields, Physical Review 81(5), 890–891. de Souza, R. S. and Opher, R. (2010), Origin of 1015-1016 G magnetic ﬁelds in the central engine of gamma ray bursts, J. Cosmology Astropart. Phys. 2010(2), 022.

Demorest, P. B., Pennucci, T., Ransom, S. M., Roberts, M. S. E. and Hessels, J. W. T. (2010), A two-solar-mass neutron star measured using Shapiro delay, Nature 467(7319), 1081–1083. Bibliography 218

Deng, W. and Zhang, B. (2014), Cosmological Implications of Fast Radio Burst/Gamma-Ray Burst Associations, ApJ 783, L35.

Dermer, C. D., Cavadini, M., Razzaque, S., Finke, J. D., Chiang, J. and Lott, B. (2011), Time Delay of Cascade Radiation for TeV Blazars and the Measurement of the Intergalactic Magnetic Field, ApJ 733, L21.

Dermer, C. D. and Menon, G. (2009), High Energy Radiation from Black Holes: Gamma Rays, Cosmic Rays, and Neutrinos, Princeton University Press.

Desch, S. J., Connolly, Jr., H. C. and Srinivasan, G. (2004), An Interstellar Origin for the Beryllium 10 in Calcium-rich, Aluminum-rich Inclusions, ApJ 602, 528– 542.

Dib, S., Shadmehri, M., Padoan, P., Maheswar, G., Ojha, D. K. and Khajenabi, F. (2010), The IMF of stellar clusters: eﬀects of accretion and feedback, MNRAS 405(1), 401–420.

Dogel, V. A. and Sharov, G. S. (1990), Manifestations of cosmic ray acceleration in giant molecular clouds., A&A 229, 259–271.

Dolag, K., Grasso, D., Springel, V. and Tkachev, I. (2005a), Constrained simulations of the magnetic ﬁeld in the local Universe and the propagation of ultrahigh energy cosmic rays, J. Cosmology Astropart. Phys. 1, 009.

Dolag, K. and Enßlin, T. A. (2000), Radio halos of galaxy clusters from hadronic secondary electron injection in realistic magnetic ﬁeld conﬁgurations, A&A 362, 151– 157.

Dolag, K., Schindler, S., Govoni, F. and Feretti, L. (2001), Correlation of the magnetic ﬁeld and the intra-cluster gas density in galaxy clusters, A&A 378, 777–786.

Dolag, K., Vogt, C. and Enßlin, T. A. (2005b), PACERMAN- I. A new algorithm to calculate Faraday rotation maps, MNRAS 358(3), 726–731.

Dolginov, A. Z. and Mitrofanov, I. G. (1976), Orientation of Cosmic Dust Grains, Ap&SS 43(2), 291–317. Bibliography 219

Dom´ınguez-Fernández,P., Vazza, F., Brüggen,M. and Brunetti, G. (2019), Dynam- ical evolution of magnetic fields in the intracluster medium, MNRAS 486(1), 623– 638.

Donati, J.-F. and Landstreet, J. D. (2009), Magnetic Fields of Nondegenerate Stars, ARAA 47, 333–370.

Donnert, J., Dolag, K., Brunetti, G., Cassano, R. and Bonafede, A. (2010), Radio haloes from simulations and hadronic models - I. The Coma cluster, MNRAS 401(1), 47–54.

Donnert, J., Dolag, K., Brunetti, G. and Cassano, R. (2013), Rise and fall of radio haloes in simulated merging galaxy clusters, MNRAS 429(4), 3564–3569.

Donnert, J., Dolag, K., Lesch, H. and Müller,E. (2009), Cluster magnetic fields from galactic outflows, MNRAS 392, 1008–1021.

Dormand, J. R. and Prince, P. J. (1980), A family of embedded Runge-Kutta formulae, Journal of computational and applied mathematics 6(1), 19–26.

Doty, S. D., van Dishoeck, E. F., van der Tak, F. F. S. and Boonman, A. M. S. (2002), Chemistry as a probe of the structures and evolution of massive star- forming regions, A&A 389, 446–463.

Draine, B. T. (2011), Physics of the Interstellar and Intergalactic Medium, Princeton University Press.

Draine, B. T. and Weingartner, J. C. (1996a), Alignment of Interstellar Grains by Starlight, in American Astronomical Society Meeting Abstracts, Vol. 28 of Bulletin of the American Astronomical Society, p.1294.

Draine, B. T. and Weingartner, J. C. (1996b), Radiative Torques on Interstellar Grains. I. Superthermal Spin-up, ApJ 470, 551.

Draine, B. T. and Weingartner, J. C. (1997), Radiative Torques on Interstellar Grains. II. Grain Alignment, ApJ 480(2), 633–646.

Dunham, M. M., Allen, L. E., Evans, Neal J., I., Broekhoven-Fiene, H., Cieza, L. A., Di Francesco, J., Gutermuth, R. A., Harvey, P. M., Hatchell, J., Heiderman, A., Bibliography 220

Huard, T. L., Johnstone, D., Kirk, J. M., Matthews, B. C., Miller, J. F., Peterson, D. E. and Young, K. E. (2015), Young Stellar Objects in the Gould Belt, ApJS 220(1), 11.

Durrer, R. and Neronov, A. (2013), Cosmological magnetic ﬁelds: their generation, evolution and observation, A&A Rev. 21, 62.

Dzhatdoev, T. A., Podlesnyi, E. I. and Vaiman, I. A. (2020), Can we constrain the extragalactic magnetic ﬁeld from very high energy observations of GRB 190114C?, Phys. Rev. D 102(12), 123017.

Dzib, S. A., Loinard, L., Ortiz-Le´on,G. N., Rodr´ıguez,L. F. and Galli, P. A. B. (2018), Distances and Kinematics of Gould Belt Star-forming Regions with Gaia DR2 Results, ApJ 867(2), 151.

Elmegreen, B. G. (1979), Magnetic diﬀusion and ionization fractions in dense molecular clouds: the role of charged grains., ApJ 232, 729–739.

Elmegreen, B. G. (2002), A Fractal Origin for the Mass Spectrum of Interstellar Clouds. II. Cloud Models and Power-Law Slopes, ApJ 564, 773–781.

Elmegreen, B. G., Efremov, Y., Pudritz, R. E. and Zinnecker, H. (2000), Observa- tions and Theory of Star Cluster Formation, Protostars and Planets IV p.179.

Elmegreen, B. G., Seiden, P. E. and Elmegreen, D. M. (1989), Spiral arm amplitude variations and pattern speeds in the grand design galaxies M51, M81, and M100, ApJ 343, 602.

Elsasser, W. M. (1939), On the Origin of the Earth’s Magnetic Field, Phys. Rev. 55, 489–498.

Ensslin, T. A., Biermann, P. L., Klein, U. and Kohle, S. (1998), Cluster radio relics as a tracer of shock waves of the large-scale structure formation, A&A 332, 395– 409.

Ettori, S., Donnarumma, A., Pointecouteau, E., Reiprich, T. H., Giodini, S., Lovis- ari, L. and Schmidt, R. W. (2013), Mass Proﬁles of Galaxy Clusters from X-ray Analysis, Space Sci. Rev. 177(1-4), 119–154. Bibliography 221

Fabian, A. C. (1994), Cooling Flows in Clusters of Galaxies, ARA&A 32, 277–318.

Falceta-Gon¸calves, D., de Juli, M. C. and Jatenco-Pereira, V. (2003), Dusty Molec- ular Cloud Collapse in the Presence of Alfv´enWaves, ApJ 597, 970–974.

Farmer, A. J. and Goldreich, P. (2004), Wave Damping by Magnetohydrodynamic Turbulence and Its Eﬀect on Cosmic-Ray Propagation in the Interstellar Medium, ApJ 604, 671–674.

Farnsworth, D., Rudnick, L., Brown, S. and Brunetti, G. (2013), Discovery of Megaparsec-scale, Low Surface Brightness Nonthermal Emission in Merging Galaxy Clusters Using the Green Bank Telescope, ApJ 779, 189.

Farnsworth, D., Rudnick, L. and Brown, S. (2011), Integrated Polarization of Sources at λ 1 m and New Rotation Measure Ambiguities, AJ 141, 191. ∼

Federer, W., Villinger, H., Howorka, F., Lindinger, W., Tosis, P., Bassi, D. and Ferguson, E. (1984), Reaction of O+, CO+, and CH+ Ions with Atomic Hydrogen, Physical Review Letters 52, 2084–2086.

Federman, S. R., Weber, J. and Lambert, D. L. (1996), Cosmic Ray–induced Chem- istry toward Perseus OB2, ApJ 463, 181.

Federrath, C. (2013), On the universality of supersonic turbulence, MNRAS 436, 1245–1257.

Federrath, C. (2016), The role of turbulence, magnetic ﬁelds and feedback for star formation, in Journal of Physics Conference Series, Vol. 719 of Journal of Physics Conference Series, p.012002.

Federrath, C. and Klessen, R. S. (2013), On the Star Formation Eﬃciency of Tur- bulent Magnetized Clouds, ApJ 763, 51.

Fender, R., Wu, K., Johnston, H., Tzioumis, T., Jonker, P., Spencer, R. and van der Klis, M. (2004), An ultra-relativistic outﬂow from a neutron star accreting gas from a companion, Nature 427(6971), 222–224.

Feretti, L., Fusco-Femiano, R., Giovannini, G. and Govoni, F. (2001), The giant radio halo in Abell 2163, A&A 373, 106–112. Bibliography 222

Feretti, L., Giovannini, G., Govoni, F. and Murgia, M. (2012), Clusters of galaxies: observational properties of the diﬀuse radio emission, A&A Rev. 20, 54.

Feretti, L., Orr`u, E., Brunetti, G., Giovannini, G., Kassim, N. and Setti, G. (2004), Spectral index maps of the radio halos in Abell 665 and Abell 2163, A&A 423, 111–119.

Fermi, E. (1949), On the Origin of the Cosmic Radiation, Physical Review 75, 1169– 1174.

Fermi, E. (1954), Galactic Magnetic Fields and the Origin of Cosmic Radiation., ApJ 119, 1.

Fern´andezAlonso, M., Supanitsky, A. D. and Rovero, A. C. (2018), Probing the IGMF with the Next Generation of Cherenkov Telescopes, ApJ 869(1), 43.

Ferrari, C., Govoni, F., Schindler, S., Bykov, A. M. and Rephaeli, Y. (2008), Obser- vations of Extended Radio Emission in Clusters, Space Sci. Rev. 134(1-4), 93–118.

Ferri`ere,K. M. (2001), The interstellar environment of our galaxy, Reviews of Mod- ern Physics 73(4), 1031–1066.

Fialkov, A. and Loeb, A. (2017), A Fast Radio Burst Occurs Every Second throughout the Observable Universe, ApJ 846(2), L27.

Field, G. B. (1965), Thermal Instability, ApJ 142, 531.

Field, G. B., Goldsmith, D. W. and Habing, H. J. (1969), Cosmic-Ray Heating of the Interstellar Gas, ApJ 155, L149.

Field, G. B. and Carroll, S. M. (2000), Cosmological magnetic ﬁelds from primordial helicity, Phys. Rev. D 62(10), 103008.

Finoguenov, A., Sarazin, C. L., Nakazawa, K., Wik, D. R. and Clarke, T. E. (2010), XMM-Newton Observation of the Northwest Radio Relic Region in A3667, ApJ 715(2), 1143–1151.

Fletcher, A. and Shukurov, A. (2006), Canals in Milky Way radio polarization maps, MNRAS 371(1), L21–L25. Bibliography 223

Ford, H. C. and Butcher, H. (1979), The system of ﬁlaments in M87: evidence for matter falling into an active nucleus, ApJS 41, 147–172.

Fosalba, P., Lazarian, A., Prunet, S. and Tauber, J. A. (2002), Statistical Properties of Galactic Starlight Polarization, ApJ 564, 762–772.

Frail, D. A., Jagannathan, P., Mooley, K. P. and Intema, H. T. (2016), Known Pulsars Identiﬁed in the GMRT 150 MHz All-sky Survey, ApJ 829(2), 119.

Fuente, A. and Mart´ın-Pintado, J. (1997), Detection of CO+ toward the Reﬂection Nebula NGC 7023, ApJ 477(2), L107–L109.

Fujita, Y. and Ohira, Y. (2011), Stable Heating of Cluster Cooling Flows by Cosmic- Ray Streaming, ApJ 738, 182.

Fujita, Y., Kimura, S. and Ohira, Y. (2013), Stability analysis for cosmic-ray heating of cool cores in galaxy clusters, MNRAS 432, 1434–1443.

Fujita, Y., Takizawa, M., Yamazaki, R., Akamatsu, H. and Ohno, H. (2015), Turbu- lent Cosmic-Ray Reacceleration at Radio Relics and Halos in Clusters of Galaxies, ApJ 815(2), 116.

Gabici, S., Aharonian, F. A. and Casanova, S. (2009), Broad-band non-thermal emission from molecular clouds illuminated by cosmic rays from nearby supernova remnants, MNRAS 396(3), 1629–1639.

Gaches, B. A. L., Oﬀner, S. S. R. and Bisbas, T. G. (2019), The Astrochemical Impact of Cosmic Rays in Protoclusters. I. Molecular Cloud Chemistry, ApJ 878(2), 105.

Gaensler, B., Haverkorn, M., Staveley-Smith, L., Dickey, J., McClure-Griﬃths, N., Dickel, J. and Wolleben, M. (2005), The Magnetic Field of the Large Magel- lanic Cloud: A New Way of Studying Galactic Magnetism, in K. T. Chyzy, K. Otmianowska-Mazur, M. Soida and R.-J. Dettmar (eds.), The Magnetized Plasma in Galaxy Evolution, pp.209–216.

Gaensler, B. M., Beck, R. and Feretti, L. (2004), The origin and evolution of cosmic magnetism, New Astron. Rev. 48(11-12), 1003–1012. Bibliography 224

Gaensler, B. M., Landecker, T. L., Taylor, A. R. and POSSUM Collaboration (2010), Survey Science with ASKAP: Polarization Sky Survey of the Universe’s Mag- netism (POSSUM), in American Astronomical Society Meeting Abstracts #215, Vol. 42 of Bulletin of the American Astronomical Society, p.515.

Gaggero, D. (2012), Cosmic Ray Diﬀusion in the Galaxy and Diﬀuse Gamma Emis- sion, Springer Theses, Springer: Berlin.

Gaggero, D., Grasso, D., Marinelli, A., Taoso, M. and Urbano, A. (2017), Diﬀuse Cosmic Rays Shining in the Galactic Center: A Novel Interpretation of H.E.S.S. and Fermi-LAT γ -Ray Data, Physical Review Letters 119(3), 031101.

Gaia Collaboration (2018), Gaia Data Release 2. Summary of the contents and survey properties, A&A 616, A1.

Gaisser, T. K. and Stanev, T. (2006), High-energy cosmic rays, Nucl. Phys. A 777, 98–110.

Galli, D., Walmsley, M. and Gon¸calves, J. (2002), The structure and stability of molecular cloud cores in external radiation ﬁelds, A&A 394, 275–284.

Gammie, C. F. and Ostriker, E. C. (1996), Can Nonlinear Hydromagnetic Waves Support a Self-gravitating Cloud?, ApJ 466, 814.

Gao, Y., Xu, H. and Law, C. K. (2015), Turbulence Decay and Cloud Core Relax- ation in Molecular Clouds, ApJ 799(2), 227.

Gasperini, M. and Veneziano, G. (1994), Dilaton production in string cosmology, Phys. Rev. D 50, 2519–2540.

+ Geballe, T. R., Indriolo, N., McCall, B. J. and Oka, T. (2007), Abundant H3 and High Values of ζ: Common Properties of Diﬀuse Clouds, in American Astronom- ical Society Meeting Abstracts, Vol. 39 of Bulletin of the American Astronomical Society, p.985.

+ Geballe, T. R., McCall, B. J., Hinkle, K. H. and Oka, T. (1999), Detection of H3 in the Diﬀuse Interstellar Medium: The Galactic Center and Cygnus OB2 Number 12, ApJ 510, 251–257. Bibliography 225

Gendre, B., Stratta, G., Atteia, J. L., Basa, S., Bo¨er,M., Coward, D. M., Cutini, S., D’Elia, V., Howell, E. J., Klotz, A. and Piro, L. (2013), The Ultra-long Gamma- Ray Burst 111209A: The Collapse of a Blue Supergiant?, ApJ 766(1), 30.

Geng, A., Beck, A. M., Dolag, K., B¨urzle,F., Beck, M. C., Kotarba, H. and Nielaba, P. (2012), Synthetic X-ray and radio maps for two diﬀerent models of Stephan’s Quintet, MNRAS 426, 3160–3177.

Ghosh, A., Bharadwaj, S., Ali, S. S. and Chengalur, J. N. (2011), GMRT observation towards detecting the post-reionization 21-cm signal, MNRAS 411(4), 2426–2438.

Giacintucci, S., Venturi, T., Brunetti, G., Bardelli, S., Dallacasa, D., Ettori, S., Finoguenov, A., Rao, A. P. and Zucca, E. (2005), Spectral properties and origin of the radio halo in A3562, A&A 440(3), 867–879.

Giacintucci, S., Venturi, T., Macario, G., Dallacasa, D., Brunetti, G., Markevitch, M., Cassano, R., Bardelli, S. and Athreya, R. (2008), Shock acceleration as origin of the radio relic in A 521?, A&A 486(2), 347–358.

Gilbert, W. and Wright, E. (1600), De magnete, magneticisque corporibus, et de magno magnete tellure: physiologia noua, plurimis & argumentis, & experimentis demonstrata, excudebat Short.

Ginzburg, V. L. and Syrovatskii, S. I. (1964), The Origin of Cosmic Rays, New York: Macmillan.

Ginzburg, V. L. and Syrovatskii, S. I. (1968), On the Spectrum of Relativistic Electrons accelerated in Cosmic-Ray Sources, Ap&SS 1, 442–450.

Giodini, S., Lovisari, L., Pointecouteau, E., Ettori, S., Reiprich, T. H. and Hoek- stra, H. (2013), Scaling Relations for Galaxy Clusters: Properties and Evolution, Space Sci. Rev. 177, 247–282.

Giovannini, G., Bonafede, A., Brown, S., Feretti, L., Ferrari, C., Gitti, M., Govoni, F., Murgia, M. and Vacca, V. (2015), Mega-parsec scale magnetic ﬁelds in low density regions in the SKA era: ﬁlaments connecting galaxy clusters and groups, in Advancing Astrophysics with the Square Kilometre Array (AASKA14), p.104. Bibliography 226

Giovannini, G., Bonafede, A., Feretti, L., Govoni, F. and Murgia, M. (2010), The diﬀuse radio ﬁlament in the merging system ZwCl 2341.1+0000, A&A 511, L5.

Giovannini, G., Feretti, L., Venturi, T., Kim, K. T. and Kronberg, P. P. (1993), The Halo Radio Source Coma C and the Origin of Halo Sources, ApJ 406, 399.

Girart, J. M., Rao, R. and Marrone, D. P. (2006), Magnetic Fields in the Formation of Sun-Like Stars, Science 313, 812–814.

Glassgold, A. E. and Langer, W. D. (1974), Model calculations for diﬀuse molecular clouds, ApJ 193, 73–91.

Gold, T. (1952), Polarization of Starlight, Nature 169(4295), 322.

Goldreich, P. and Kylaﬁs, N. D. (1981), On mapping the magnetic ﬁeld direction in molecular clouds by polarization measurements, ApJ 243, L75–L78.

Goldsmith, P. F. (2001), Molecular Depletion and Thermal Balance in Dark Cloud Cores, ApJ 557(2), 736–746.

Goldsmith, P. F. and Langer, W. D. (1978), Molecular cooling and thermal balance of dense interstellar clouds, ApJ 222, 881–895.

Govoni, F., Dolag, K., Murgia, M., Feretti, L., Schindler, S., Giovannini, G., Boschin, W., Vacca, V. and Bonafede, A. (2010), Rotation measures of radio sources in hot galaxy clusters, A&A 522, A105.

Govoni, F., Enßlin, T. A., Feretti, L. and Giovannini, G. (2001a), A comparison of radio and X-ray morphologies of four clusters of galaxies containing radio halos, A&A 369, 441–449.

Govoni, F., Feretti, L., Giovannini, G., B¨ohringer,H., Reiprich, T. H. and Murgia, M. (2001b), Radio and X-ray diﬀuse emission in six clusters of galaxies, A&A 376, 803–819.

Govoni, F. and Feretti, L. (2004), Magnetic Fields in Clusters of Galaxies, Interna- tional Journal of Modern Physics D 13, 1549–1594.

Govoni, F., Markevitch, M., Vikhlinin, A., van Speybroeck, L., Feretti, L. and Giovannini, G. (2004), Chandra Temperature Maps for Galaxy Clusters with Radio Halos, ApJ 605, 695–708. Bibliography 227

Govoni, F., Murgia, M., Xu, H., Li, H., Norman, M. L., Feretti, L., Giovannini, G. and Vacca, V. (2013), Polarization of cluster radio halos with upcoming radio interferometers, A&A 554, A102.

Govoni, F., Orrù,E., Bonafede, A., Iacobelli, M., Paladino, R., Vazza, F., Murgia, M., Vacca, V., Giovannini, G., Feretti, L., Loi, F., Bernardi, G., Ferrari, C., Pizzo, R. F., Gheller, C., Manti, S., Brüggen,M., Brunetti, G., Cassano, R., de Gasperin, F., Enßlin, T. A., Hoeft, M., Horellou, C., Junklewitz, H., Röttgering, H. J. A., Scaife, A. M. M., Shimwell, T. W., van Weeren, R. J. and Wise, M. (2019), A radio ridge connecting two galaxy clusters in a filament of the cosmic web, Science 364(6444), 981–984.

Greenstein, J. L. and Schmidt, M. (1964), The Quasi-Stellar Radio Sources 3C 48 and 3C 273, ApJ 140, 1.

Greisen, K. (1966), End to the Cosmic-Ray Spectrum?, Phys. Rev. Lett. 16(17), 748– 750.

Grenier, I. A., Black, J. H. and Strong, A. W. (2015), The Nine Lives of Cosmic Rays in Galaxies, ARA&A 53, 199–246.

Gressel, O., Elstner, D., Ziegler, U. and R¨udiger,G. (2008), Direct simulations of a supernova-driven galactic dynamo, A&A 486(3), L35–L38.

Grunwald, G. K., Hyndman, R. J. and Tedesco, L. M. (1996), A uniﬁed view of linear AR(1) models, Technical Report, Department of Statistics, University of Melbourne, Australia and Department of Mathematics, Monash University, Aus- tralia.

Gruzinov, A. and Levin, Y. (2019), Conversion Measure of Faraday Rotation- Conversion with Application to Fast Radio Bursts, ApJ 876(1), 74.

Gusdorf, A., G¨usten,R., Anderl, S., Hezareh, T. and Wiesemeyer, H. (2014), The molecular emission from old supernova remnants, in A. Ray and R. A. McCray (eds.), IAU Symposium, Vol. 296 of IAU Symposium, pp.178–182.

Guth, A. H. (1981), Inﬂationary universe: A possible solution to the horizon and ﬂatness problems, Phys. Rev. D 23, 347–356. Bibliography 228

Hackstein, S., Br¨uggen,M., Vazza, F., Gaensler, B. M. and Heesen, V. (2019), Fast radio burst dispersion measures and rotation measures and the origin of intergalactic magnetic ﬁelds, MNRAS 488(3), 4220–4238.

Hairer, E. and Wanner, G. (1996), Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Vol. 14 of Springer Series in Computational Mathematics, Springer-Verlag Berlin Heidelberg.

Hairer, E., Nørsett, S. and Wanner, G. (2008), Solving Ordinary Diﬀerential Equa- tions I: Nonstiﬀ Problems, Vol. 8 of Springer Series in Computational Mathemat- ics, Springer-Verlag Berlin Heidelberg.

Hale, G. E. (1908), On the Probable Existence of a Magnetic Field in Sun-Spots, ApJ 28, 315.

Hale, G. E., Ellerman, F., Nicholson, S. B. and Joy, A. H. (1919), The Magnetic Polarity of Sun-Spots, ApJ 49, 153.

Hale, G. E. and Nicholson, S. B. (1925), The Law of Sun-Spot Polarity, ApJ 62, 270.

Hall, J. S. (1949), Observations of the Polarized Light from Stars, Science 109, 166– 167.

Han, J. L. (2017), Observing Interstellar and Intergalactic Magnetic Fields, ARA&A 55, 111–157.

Han, J. L., Beck, R., Ehle, M., Haynes, R. F. and Wielebinski, R. (1999), Magnetic ﬁelds in the spiral galaxy NGC 2997, A&A 348, 405–417.

Han, J. L., van Straten, W., Lazio, T. J. W., Deller, A., Sobey, C., Xu, J., Schnitzeler, D., Imai, H., Chatterjee, S., Macquart, J.-P., Kramer, M. and Cordes, J. M. (2015), Three-dimensional Tomography of the Galactic and Extragalactic Magnetoionic Medium with the SKA, in Advancing Astrophysics with the Square Kilometre Array (AASKA14), p.41.

Hanasz, M., Kowal, G., Otmianowska-Mazur, K. and Lesch, H. (2004), Ampli- ﬁcation of Galactic Magnetic Fields by the Cosmic-Ray-driven Dynamo, ApJ 605(1), L33–L36. Bibliography 229

Hanasz, M., Woltanski, D. and Kowalik, K. (2013), Interstellar and intergalactic dynamos, in A. G. Kosovichev, E. de Gouveia Dal Pino and Y. Yan (eds.), Solar and Astrophysical Dynamos and Magnetic Activity, Vol. 294 of IAU Symposium, pp.225–236.

Harju, J., Juvela, M., Schlemmer, S., Haikala, L. K., Lehtinen, K. and Mattila, K. (2008), Detection of 6 K gas in Ophiuchus D, A&A 482(2), 535–539.

Hartquist, T. W., Black, J. H. and Dalgarno, A. (1978), Cosmic ray ionization and the deuterium abundance, MNRAS 185, 643–646.

Harvey, P. M., Huard, T. L., Jørgensen, J. K., Gutermuth, R. A., Mamajek, E. E., Bourke, T. L., Mer´ın, B., Cieza, L., Brooke, T., Chapman, N., Alcal´a,J. M., Allen, L. E., Evans, Neal J., I., Di Francesco, J. and Kirk, J. M. (2008), The Spitzer Survey of Interstellar Clouds in the Gould Belt. I. IC 5146 Observed With IRAC and MIPS, ApJ 680(1), 495–516.

Haslam, C. G. T., Salter, C. J., Stoﬀel, H. and Wilson, W. E. (1982), A 408 MHz all-sky continuum survey. II. The atlas of contour maps., A&AS 47, 1–143.

Hasselmann, K. and Wibberenz, G. (1968), Scattering of charged particles by random electromagnetic ﬁelds, Z. Geophys 328, 269.

Haverkorn, M., Brown, J. C., Gaensler, B. M. and McClure-Griﬃths, N. M. (2008), The Outer Scale of Turbulence in the Magnetoionized Galactic Interstellar Medium, ApJ 680, 362–370.

Hazard, C., Mackey, M. B. and Shimmins, A. J. (1963), Investigation of the Radio Source 3C 273 By The Method of Lunar Occultations, Nature 197(4872), 1037– 1039.

Heald, G., Braun, R. and Edmonds, R. (2009), The Westerbork SINGS survey. II Polarization, Faraday rotation, and magnetic ﬁelds, A&A 503, 409–435.

Herrnstein, J. R., Moran, J. M., Greenhill, L. J., Diamond, P. J., Inoue, M., Nakai, N., Miyoshi, M., Henkel, C. and Riess, A. (1999), A geometric distance to the galaxy NGC4258 from orbital motions in a nuclear gas disk, Nature 400(6744), 539–541. Bibliography 230

H.E.S.S. Collaboration (2018a), Characterising the VHE diﬀuse emission in the central 200 parsecs of our Galaxy with H.E.S.S., A&A 612, A9.

H.E.S.S. Collaboration (2018b), The H.E.S.S. Galactic plane survey, A&A 612, A1.

Hewitt, J. N., Turner, E. L., Schneider, D. P., Burke, B. F. and Langston, G. I. (1988), Unusual radio source MG1131+0456: a possible Einstein ring, Nature 333(6173), 537–540.

Hildebrand, R. H., Kirby, L., Dotson, J. L., Houde, M. and Vaillancourt, J. E. (2009), Dispersion of Magnetic Fields in Molecular Clouds. I, ApJ 696, 567–573.

Hillas, A. M. (1984), The Origin of Ultra-High-Energy Cosmic Rays, ARA&A 22, 425–444.

Hillas, A. M. (2006), Cosmic Rays: Recent Progress and some Current Questions, in proceedings of “Cosmology, Galaxy Formation and Astroparticle Physics on the pathway to the SKA”, Oxford, p.12.

Hiltner, W. A. (1949), Polarization of Radiation from Distant Stars by the Inter- stellar Medium, Nature 163, 283.

Hitomi Collaboration, Aharonian, F., Akamatsu, H., Akimoto, F., Allen, S. W., Anabuki, N., Angelini, L., Arnaud, K., Audard, M. and Awaki, H. (2016), The quiescent intracluster medium in the core of the Perseus cluster, Nature 535(7610), 117–121.

Hoang, T., Cho, J. and Lazarian, A. (2018), Alignment of Irregular Grains by Mechanical Torques, ApJ 852(2), 129.

Hoang, T. and Lazarian, A. (2008), Radiative torque alignment: essential physical processes, MNRAS 388, 117–143.

Hoang, T. and Lazarian, A. (2009), Radiative Torques Alignment in the Presence of Pinwheel Torques, ApJ 695(2), 1457–1476.

Hollenbach, D., Kaufman, M. J., Neufeld, D., Wolﬁre, M. and Goicoechea, J. R. + + + (2012), The Chemistry of Interstellar OH ,H2O , and H3O : Inferring the Cosmic-Ray Ionization Rates from Observations of Molecular Ions, ApJ 754, 105. Bibliography 231

Houde, M., Rao, R., Vaillancourt, J. E. and Hildebrand, R. H. (2011), Dispersion of Magnetic Fields in Molecular Clouds. III., ApJ 733, 109.

Houde, M., Vaillancourt, J. E., Hildebrand, R. H., Chitsazzadeh, S. and Kirby, L. (2009), Dispersion of Magnetic Fields in Molecular Clouds. II., ApJ 706, 1504– 1516.

Huang, L. and Shcherbakov, R. V. (2011), Faraday conversion and rotation in uniformly magnetized relativistic plasmas, MNRAS 416(4), 2574–2592.

Hull, C. L. H., Mocz, P., Burkhart, B., Goodman, A. A., Girart, J. M., Cort´es, P. C., Hernquist, L., Springel, V., Li, Z.-Y. and Lai, S.-P. (2017), Unveiling the Role of the Magnetic Field at the Smallest Scales of Star Formation, ApJ 842, L9.

Hurwitz, H. and Jones, R. C. (1941), A New Calculus for the Treatment of Optical Systems II. Proof of Three General Equivalence Theorems, J. Opt. Soc. Am. 31(7), 493–499.

Iapichino, L., Federrath, C. and Klessen, R. S. (2017), Adaptive mesh reﬁnement simulations of a galaxy cluster merger - I. Resolving and modelling the turbulent ﬂow in the cluster outskirts, MNRAS 469, 3641–3655.

Iapichino, L., Schmidt, W., Niemeyer, J. C. and Merklein, J. (2011), Turbulence production and turbulent pressure support in the intergalactic medium, MNRAS 414(3), 2297–2308.

Ichiki, K. and Takahashi, K. (2006), Cosmological magnetic ﬁeld: a fossil of density perturbations in the early Universe, Astronomical Herald 99, 568–575.

Ichiki, K., Takahashi, K., Ohno, H., Hanayama, H. and Sugiyama, N. (2006), Cos- mological Magnetic Field: A Fossil of Density Perturbations in the Early Universe, Science 311(5762), 827–829.

+ Indriolo, N., Geballe, T. R., Oka, T. and McCall, B. J. (2007), H3 in Diﬀuse Interstellar Clouds: A Tracer for the Cosmic-Ray Ionization Rate, ApJ 671, 1736– 1747.

Indriolo, N. and McCall, B. J. (2012), Investigating the Cosmic-Ray Ionization Bibliography 232

+ Rate in the Galactic Diﬀuse Interstellar Medium through Observations of H3 , ApJ 745, 91.

Indriolo, N., Neufeld, D. A., Gerin, M., Schilke, P., Benz, A. O., Winkel, B., Menten, K. M., Chambers, E. T., Black, J. H., Bruderer, S., Falgarone, E., Godard, B., Goicoechea, J. R., Gupta, H., Lis, D. C., Ossenkopf, V., Persson, C. M., Sonnentrucker, P., van der Tak, F. F. S., van Dishoeck, E. F., Wolﬁre, M. G. + + + and Wyrowski, F. (2015), Herschel Survey of Galactic OH ,H2O , and H3O : Probing the Molecular Hydrogen Fraction and Cosmic-Ray Ionization Rate, ApJ 800, 40.

Inoue, T., Yamazaki, R. and Inutsuka, S.-I. (2009), Turbulence and Magnetic Field Ampliﬁcation in Supernova Remnants: Interactions Between a Strong Shock Wave and Multiphase Interstellar Medium, ApJ 695(2), 825–833.

Ivlev, A. V., Dogiel, V. A., Chernyshov, D. O., Caselli, P., Ko, C. M. and Cheng, K. S. (2018), Penetration of Cosmic Rays into Dense Molecular Clouds: Role of Diﬀuse Envelopes, ApJ 855, 23.

Jaacks, J., Finkelstein, S. L. and Bromm, V. (2019), Legacy of star formation in the pre-reionization universe, MNRAS 488, 2202–2221.

Jacob, S. and Pfrommer, C. (2017), Cosmic ray heating in cool core clusters - I. Diversity of steady state solutions, MNRAS 467, 1449–1477.

Jacoby, B. A., Chandler, A. M., Backer, D. C., Anderson, S. B. and Kulkarni, S. R. (2002), Pulsars in M62, IAU Circ. 7783, 1.

Jansky, K. G. (1933), Radio Waves from Outside the Solar System, Nature 132(3323), 66.

Jeans, J. H. (1905a), A Comparison between Two Theories of Radiation, Nature 72(1865), 293–294.

Jeans, J. H. (1905b), The Dynamical Theory of Gases, Nature 71(1852), 607.

Jeans, J. H. (1905c), The Dynamical Theory of Gases and of Radiation, Nature 72(1857), 101–102. Bibliography 233

Jelić,V., de Bruyn, A. G., Pandey, V. N., Mevius, M., Haverkorn, M., Brentjens, M. A., Koopmans, L. V. E., Zaroubi, S., Abdalla, F. B., Asad, K. M. B., Bus, S., Chapman, E., Ciardi, B., Fernandez, E. R., Ghosh, A., Harker, G., Iliev, I. T., Jensen, H., Kazemi, S., Mellema, G., Offringa, A. R., Patil, A. H., Vedan- tham, H. K. and Yatawatta, S. (2015), Linear polarization structures in LOFAR observations of the interstellar medium in the 3C 196 field, A&A 583, A137.

Jenkins, E. B., Jura, M. and Loewenstein, M. (1983), Copernicus observations of C I : pressures and Carbon abundances in diﬀuse interstellar clouds, ApJ 270, 88– 104.

Jia, X., Kivelson, M., Khurana, K. and Walker, R. (2010), Magnetic Fields of the Satellites of Jupiter and Saturn, Space Sci. Rev. 152, 271–305.

Johnston-Hollitt, M., Govoni, F., Beck, R., Dehghan, S., Pratley, L., Akahori, T., Heald, G., Agudo, I., Bonafede, A., Carretti, E., Clarke, T., Colafrancesco, S., Ensslin, T. A., Feretti, L., Gaensler, B., Haverkorn, M., Mao, S. A., Oppermann, N., Rudnick, L., Scaife, A., Schnitzeler, D., Stil, J., Taylor, A. R. and Vacca, V. (2015), Using SKA Rotation Measures to Reveal the Mysteries of the Mag- netised Universe, in Advancing Astrophysics with the Square Kilometre Array (AASKA14), p.92.

Jokipii, J. R. (1966), Cosmic-Ray Propagation. I. Charged Particles in a Random Magnetic Field, ApJ 146, 480.

Jones, C. A. (2011), Planetary Magnetic Fields and Fluid Dynamos, Annual Review of Fluid Mechanics 43(1), 583–614.

Jones, D. I. (2014), Prospects for Detection of Synchrotron Emission from Secondary Electrons and Positrons in Starless Cores: Application to G0.216+0.016, ApJ 792(1), L14.

Jones, J., Birkinshaw, K. and Twiddy, N. (1981), Rate coeﬃcients and product ion + + distributions for the reactions of OH and H2O with N2,O2, NO.N2O, Xe, CO,

CO2,H2S and H2 at 300 K, Chemical Physics Letters 77(3), 484 – 488.

Jones, R. C. (1941a), A New Calculus for the Treatment of Optical Systems I. Description and Discussion of the Calculus, J. Opt. Soc. Am. 31(7), 488–493. Bibliography 234

Jones, R. C. (1941b), A New Calculus for the Treatment of Optical Systems III. The Sohncke Theory of Optical Activity, J. Opt. Soc. Am. 31(7), 500–503.

Jones, R. C. (1942), A New Calculus for the Treatment of Optical Systems. IV., J. Opt. Soc. Am. 32(8), 486–493.

Jones, T. J., Gordon, M., Shenoy, D., Gehrz, R. D., Vaillancourt, J. E. and Krejny, M. (2016), SOFIA Mid-infrared Imaging1 and CSO Submillimeter Polarimetry Observations of G034.43+00.24 MM1, AJ 151(6), 156.

Jones, T. W. and O’Dell, S. L. (1977), Transfer of polarized radiation in self- absorbed synchrotron sources. I. Results for a homogeneous source, ApJ 214, 522.

Jura, M. (1975), Interstellar clouds containing optically thick H2, ApJ 197, 581–586.

Juvela, M. and Ysard, N. (2011), On the Gas Temperature of Molecular Cloud Cores, ApJ 739(2), 63.

Kaczmarek, J. F., Purcell, C. R., Gaensler, B. M., McClure-Griﬃths, N. M. and Stevens, J. (2017), Detection of a Coherent Magnetic Field in the Magellanic Bridge through Faraday Rotation, MNRAS 467(2), 1776–1794.

Kafexhiu, E., Aharonian, F., Taylor, A. M. and Vila, G. S. (2014), Parametriza- tion of gamma-ray production cross sections for pp interactions in a broad proton energy range from the kinematic threshold to PeV energies, Phys. Rev. D 90(12), 123014.

Kahniashvili, T., Tevzadze, A. G., Brandenburg, A. and Neronov, A. (2013), Evolution of primordial magnetic ﬁelds from phase transitions, Phys. Rev. D 87(8), 083007.

Kalv¯ans,J. (2018), Temperature Spectra of Interstellar Dust Grains Heated by Cosmic Rays. II. Dark Cloud Cores, ApJS 239(1), 6.

Kandus, A., Kunze, K. E. and Tsagas, C. G. (2011), Primordial magnetogenesis, Phys. Rep. 505, 1–58.

Kang, H., Ryu, D. and Jones, T. W. (2012), Diﬀusive Shock Acceleration Simula- tions of Radio Relics, ApJ 756(1), 97. Bibliography 235

Katz, H., Martin-Alvarez, S., Devriendt, J., Slyz, A. and Kimm, T. (2019), Magne- togenesis at Cosmic Dawn: tracing the origins of cosmic magnetic ﬁelds, MNRAS 484(2), 2620–2631.

Kawabata, K. (1964), Transfer of the Gyro-Resonance Radiation, PASJ 16, 30.

Kawata, D., Okamoto, T., Gibson, B. K., Barnes, D. J. and Cen, R. (2013), Calibrat- ing an updated smoothed particle hydrodynamics scheme within GCD+, MNRAS 428, 1968–1979.

Kazes, I. and Baan, W. A. (1991), OH megamasers above Z = 0.07, A&A 248, L15– L18.

Keane, E. F. (2018), The future of fast radio burst science, Nature Astronomy 2, 865–872.

Kemp, J. C., Swedlund, J. B., Landstreet, J. D. and Angel, J. R. P. (1970), Discovery of Circularly Polarized Light from a White Dwarf, ApJ 161, L77.

Kempner, J. C. and David, L. P. (2004), A Chandra view of the multiple merger in Abell 2744, MNRAS 349, 385–392.

Kerr, M., Reardon, D. J., Hobbs, G., Shannon, R. M., Manchester, R. N., Dai, S., Russell, C. J., Zhang, S., van Straten, W., Oslowski, S., Parthasarathy, A., Spiewak, R., Bailes, M., Bhat, N. D. R., Cameron, A. D., Coles, W. A., Dempsey, J., Deng, X., Goncharov, B., Kaczmarek, J. F., Keith, M. J., Lasky, P. D., Lower, M. E., Preisig, B., Sarkissian, J. M., Toomey, L., Wang, H., Wang, J., Zhang, L. and Zhu, X. (2020), The Parkes Pulsar Timing Array project: second data release, PASA 37, e020.

Keshet, U. and Loeb, A. (2010), Using Radio Halos and Minihalos to Measure the Distributions of Magnetic Fields and Cosmic Rays in Galaxy Clusters, ApJ 722(1), 737–749.

Kirk, J. G., Schlickeiser, R. and Schneider, P. (1988), Cosmic-ray transport in ac- celerating ﬂows, ApJ 328, 269–274.

Kirk, J. M., Ward-Thompson, D. and Crutcher, R. M. (2006), SCUBA polarization Bibliography 236

observations of the magnetic ﬁelds in the pre-stellar cores L1498 and L1517B, MNRAS 369(3), 1445–1450.

Klebesadel, R. W., Strong, I. B. and Olson, R. A. (1973), Observations of Gamma- Ray Bursts of Cosmic Origin, ApJ 182, L85.

Klein, U. (2011), Magnetization of the IGM by Starburst Dwarf Galaxies?, in M. Koleva, P. Prugniel and I. Vauglin (eds.), EAS Publications Series, Vol. 48 of EAS Publications Series, pp.473–474.

Klippenstein, S. J., Georgievskii, Y. and McCall, B. J. (2010), Temperature De- + 3 + + pendence of Two Key Interstellar Reactions of H3 : O( P) + H3 and CO + H3 , Journal of Physical Chemistry A 114, 278–290.

Knudsen, O. (1976), The Faraday eﬀect and physical theory, 1845–1873, Archive for History of Exact Sciences 15, 235–281. 10.1007/BF00412258.

Knutson, L. and Burikham, P. (2003), The Zeeman Eﬀect, in Physics 244 Lecture Notes, Department of Physics, University of Wisconsin-Madison, USA.

Ko, C.-M. (1992), A note on the hydrodynamical description of cosmic ray propagation, A&A 259, 377–381.

Koch, P. M., Tang, Y.-W., Ho, P. T. P., Zhang, Q., Girart, J. M., Chen, H.-R. V., Frau, P., Li, H.-B., Li, Z.-Y., Liu, H.-Y. B., Padovani, M., Qiu, K., Yen, H.-W., Chen, H.-H., Ching, T.-C., Lai, S.-P. and Rao, R. (2014), The Importance of the Magnetic Field from an SMA-CSO-combined Sample of Star-forming Regions, ApJ 797, 99.

Kolmogorov, A. N. (1933), Sulla determinazione emp´ırica di uma legge di dis- tribuzione (On the empirical determination of a distribution law), Giornale dell’Istituto Italiano degli Attuari 4, 83–91.

Kolmogorov, A. N. (1941a), Dissipation of Energy in Locally Isotropic Turbulence, Akademiia Nauk SSSR Doklady 32, 16.

Kolmogorov, A. N. (1941b), The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds’ Numbers, Akademiia Nauk SSSR Doklady 30, 301. Bibliography 237

Komatsu, E., Dunkley, J., Nolta, M. R., Bennett, C. L., Gold, B., Hinshaw, G., Jarosik, N., Larson, D., Limon, M., Page, L., Spergel, D. N., Halpern, M., Hill, R. S., Kogut, A., Meyer, S. S., Tucker, G. S., Weiland, J. L., Wollack, E. and Wright, E. L. (2009), Five-Year Wilkinson Microwave Anisotropy Probe Obser- vations: Cosmological Interpretation, ApJS 180, 330–376.

Kossmann, H., Schwarzkopf, O. and Schmidt, V. (1990), Absolute ionisation cross

sections for electron impact on H2, Journal of Physics B Atomic Molecular Physics 23, 301–313.

Kotarba, H., Lesch, H., Dolag, K., Naab, T., Johansson, P. H., Donnert, J. and Stasyszyn, F. A. (2011), Galactic ménageàtrois: simulating magnetic fields in colliding galaxies, MNRAS 415, 3189–3218.

Kotera, K. and Olinto, A. V. (2011), The Astrophysics of Ultrahigh-Energy Cosmic Rays, ARA&A 49(1), 119–153.

Koyama, K., Petre, R., Gotthelf, E. V., Hwang, U., Matsuura, M., Ozaki, M. and Holt, S. S. (1995), Evidence for shock acceleration of high-energy electrons in the supernova remnant SN1006, Nature 378(6554), 255–258.

Krause, M. (2015), Magnetic ﬁelds in spiral galaxies, Highlights of Astronomy 16, 399–399.

Kravtsov, A. V. and Borgani, S. (2012), Formation of Galaxy Clusters, ARA&A 50, 353–409.

Kronberg, P. P. (2010), Cosmic magnetic ﬁelds and implications for HE particle anisotropies, in the XVI International Symposium on Very High Energy Cosmic Ray Interactions (ISVHECRI 2010), Batavia, IL, USA, p.9.

Kronberg, P. P. (2016), Magnetic ﬁeld outﬂow from dwarf and starburst galaxies, in Cosmic Magnetic Fields, Cambridge Astrophysics, Cambridge University Press, p.100–116.

Kronberg, P. P., Bernet, M. L., Miniati, F., Lilly, S. J., Short, M. B. and Higdon, D. M. (2008), A Global Probe of Cosmic Magnetic Fields to High Redshifts, ApJ 676(1), 70–79. Bibliography 238

Kronberg, P. P., Dufton, Q. W., Li, H. and Colgate, S. A. (2001), Magnetic Energy of the Intergalactic Medium from Galactic Black Holes, ApJ 560, 178–186.

Kronberg, P. P. and Perry, J. J. (1982), Absorption lines, Faraday rotation, and magnetic ﬁeld estimates for QSO absorption-line clouds, ApJ 263, 518–532.

Kronberg, P. P., Lesch, H. and Hopp, U. (1999), Magnetization of the Intergalactic Medium by Primeval Galaxies, ApJ 511(1), 56–64.

Krymskii, G. F. (1977), A regular mechanism for the acceleration of charged particles on the front of a shock wave, Akademiia Nauk SSSR Doklady 234, 1306–1308.

Kuchar, P. and Enßlin, T. A. (2011), Magnetic power spectra from Faraday rotation maps. REALMAF and its use on Hydra A, A&A 529, A13.

Kudoh, T. and Basu, S. (2008), Three-dimensional Simulation of Magnetized Cloud Fragmentation Induced by Nonlinear Flows and Ambipolar Diﬀusion, ApJ 679, L97.

Kulsrud, R. and Pearce, W. P. (1969), The Eﬀect of Wave-Particle Interactions on the Propagation of Cosmic Rays, ApJ 156, 445.

Kulsrud, R. M. (2005), Plasma physics for astrophysics, Princeton University Press.

Kulsrud, R. M., Cen, R., Ostriker, J. P. and Ryu, D. (1997), The Protogalactic Origin for Cosmic Magnetic Fields, ApJ 480(2), 481–491.

Kulsrud, R. M. and Cesarsky, C. J. (1971), The Eﬀectiveness of Instabilities for the Conﬁnement of High Energy Cosmic Rays in the Galactic Disk, Astrophys. Lett. 8, 189.

Kulsrud, R. M. and Zweibel, E. G. (2008), On the origin of cosmic magnetic ﬁelds, Reports on Progress in Physics 71(4), 046901.

Kumar, P. and Zhang, B. (2015), The physics of gamma-ray bursts and relativistic jets, Phys. Rep. 561, 1–109.

Kutta, W. (1901), Beitrag zur naherungsweisen integration totaler diﬀerentialgle- ichungen, Z. Math. Phys. 46, 435–453. Bibliography 239

Kylaﬁs, N. D. (1983), Linear polarization of interstellar radio-frequency absorption lines and magnetic ﬁeld direction, ApJ 275, 135–144.

Lacki, B. C. and Beck, R. (2013), The equipartition magnetic ﬁeld formula in starburst galaxies: accounting for pionic secondaries and strong energy losses, MN- RAS 430, 3171–3186.

Lacki, B. C., Thompson, T. A. and Quataert, E. (2010), The Physics of the Far- infrared-Radio Correlation. I. Calorimetry, Conspiracy, and Implications, ApJ 717, 1–28.

Lada, C. J., Alves, J. and Lada, E. A. (1999), Infrared Extinction and the Structure of the IC 5146 Dark Cloud, ApJ 512(1), 250–259.

Lai, S.-P., Girart, J. M. and Crutcher, R. M. (2003), Interferometric Mapping of Magnetic Fields in Star-forming Regions. III. Dust and CO Polarization in DR 21(OH), ApJ 598(1), 392–399.

Landi Degl’Innocenti, E. (2003), The Zeeman eﬀect: applications to solar physics, Astronomische Nachrichten 324(4), 393–396.

Lang, K. R. (1974), Astrophysical formulae : a compendium for the physicist and astrophysicist, Springer-Verlag Berlin Heidelberg.

Larson, R. B. (1974), Eﬀects of supernovae on the early evolution of galaxies, MN- RAS 169, 229–246.

Larson, R. B. (1975), Models for the formation of elliptical galaxies., MNRAS 173, 671–699.

Larson, R. L., Evans, Neal J., I., Green, J. D. and Yang, Y.-L. (2015), Evidence for Decay of Turbulence by MHD Shocks in the ISM via CO Emission, ApJ 806(1), 70.

Latter, W. B., Walker, C. K. and Maloney, P. R. (1993), Detection of the Carbon Monoxide Ion (CO+) in the Interstellar Medium and a Planetary Nebula, ApJ 419, L97.

Lazarian, A. (1995), Davis-Greenstein Alignment of Nonspherical Grains, ApJ 453, 229. Bibliography 240

Lazarian, A., Goodman, A. A. and Myers, P. C. (1997), On the Eﬃciency of Grain Alignment in Dark Clouds, ApJ 490(1), 273–280.

Lazarian, A. and Beresnyak, A. (2006), Cosmic ray scattering in compressible turbulence, MNRAS 373(3), 1195–1202.

Lazarian, A. and Hoang, T. (2007), Radiative torques: analytical model and basic properties, MNRAS 378(3), 910–946.

Lazarian, A. and Hoang, T. (2008), Alignment of Dust with Magnetic Inclusions: Radiative Torques and Superparamagnetic Barnett and Nuclear Relaxation, ApJ 676(1), L25.

Lazarian, A. and Hoang, T. (2019), Magnetic Properties of Dust Grains, Eﬀect of Precession, and Radiative Torque Alignment, ApJ 883(2), 122.

Lazarian, A. and Pogosyan, D. (2016), Spectrum and Anisotropy of Turbulence from Multi-frequency Measurement of Synchrotron Polarization, ApJ 818, 178.

Lee, C.-F., Rao, R., Ching, T.-C., Lai, S.-P., Hirano, N., Ho, P. T. P. and Hwang, H.-C. (2014), Magnetic Field Structure in the Flattened Envelope and Jet in the Young Protostellar System HH 211, ApJ 797, L9.

Lee, J.-E., Evans, Neal J., I., Shirley, Y. L. and Tatematsu, K. (2003), Chemistry and Dynamics in Pre-protostellar Cores, ApJ 583(2), 789–808.

Lee, S., Lele, S. K. and Moin, P. (1991), Eddy shocklets in decaying compressible turbulence, Physics of Fluids A 3(4), 657–664.

Lequeux, J. (2005), The Interstellar Medium, Springer.

Li, H.-B., Dowell, C. D., Goodman, A., Hildebrand, R. and Novak, G. (2009), Anchoring Magnetic Field in Turbulent Molecular Clouds, ApJ 704, 891–897.

Li, H.-B., Fang, M., Henning, T. and Kainulainen, J. (2013), The link between magnetic ﬁelds and ﬁlamentary clouds: bimodal cloud orientations in the Gould Belt, MNRAS 436, 3707–3719.

Li, H.-B. and Henning, T. (2011), The alignment of molecular cloud magnetic ﬁelds with the spiral arms in M33, Nature 479, 499–501. Bibliography 241

Li, H.-B. and Houde, M. (2008), Probing the Turbulence Dissipation Range and Magnetic Field Strengths in Molecular Clouds, ApJ 677(2), 1151–1156.

Li, H.-B., Yuen, K. H., Otto, F., Leung, P. K., Sridharan, T. K., Zhang, Q., Liu, H., Tang, Y.-W. and Qiu, K. (2015a), Self-similar fragmentation regulated by magnetic ﬁelds in a region forming massive stars, Nature 520(7548), 518–521.

Li, P. S., McKee, C. F. and Klein, R. I. (2015b), Magnetized interstellar molecular clouds - I. Comparison between simulations and Zeeman observations, MNRAS 452, 2500–2527.

Liang, H., Hunstead, R. W., Birkinshaw, M. and Andreani, P. (2000), A Powerful Radio Halo in the Hottest Known Cluster of Galaxies 1E 0657-56, ApJ 544, 686– 701.

Lin, S.-J., Pagani, L., Lai, S.-P., Lef`evre,C. and Lique, F. (2020), Physical and chemical modeling of the starless core L 1512, A&A 635, A188.

Liu, X. and Shemansky, D. E. (2004), Ionization of Molecular Hydrogen, ApJ 614, 1132–1142.

Lloyd, N. M. and Petrosian, V. (2000), Synchrotron Radiation as the Source of Gamma-Ray Burst Spectra, ApJ 543(2), 722–732.

Lodders, K. (2003), Solar System Abundances and Condensation Temperatures of the Elements, ApJ 591(2), 1220–1247.

Loewenstein, M., Zweibel, E. G. and Begelman, M. C. (1991), Cosmic-ray heating of cooling ﬂows - A critical analysis, ApJ 377, 392–402.

Loi, S. T., Murphy, T., Cairns, I. H., Menk, F. W., Waters, C. L., Erickson, P. J., Trott, C. M., Hurley-Walker, N., Morgan, J., Lenc, E., Oﬀringa, A. R., Bell, M. E., Ekers, R. D., Gaensler, B. M., Lonsdale, C. J., Feng, L., Hancock, P. J., Kaplan, D. L., Bernardi, G., Bowman, J. D., Briggs, F., Cappallo, R. J., Deshpande, A. A., Greenhill, L. J., Hazelton, B. J., Johnston-Hollitt, M., McWhirter, S. R., Mitchell, D. A., Morales, M. F., Morgan, E., Oberoi, D., Ord, S. M., Prabu, T., Shankar, N. U., Srivani, K. S., Subrahmanyan, R., Tingay, S. J., Wayth, R. B., Webster, R. L., Williams, A. and Williams, C. L. (2015), Real-time imaging Bibliography 242

of density ducts between the plasmasphere and ionosphere, Geophys. Res. Lett. 42(10), 3707–3714.

Lorimer, D. R., Bailes, M., McLaughlin, M. A., Narkevic, D. J. and Crawford, F. (2007), A Bright Millisecond Radio Burst of Extragalactic Origin, Science 318(5851), 777.

Lyne, A. G., Burgay, M., Kramer, M., Possenti, A., Manchester, R., Camilo, F., McLaughlin, M. A., Lorimer, D. R., D’Amico, N., Joshi, B. C., Reynolds, J. and Freire, P. C. C. (2004), A Double-Pulsar System: A Rare Laboratory for Relativistic Gravity and Plasma Physics, Science 303(5661), 1153–1157.

Ma, Y. K., Mao, S. A., Stil, J., Basu, A., West, J., Heiles, C., Hill, A. S. and Betti, S. K. (2019), A broad-band spectro-polarimetric view of the NVSS rotation measure catalogue - I. Breaking the nπ-ambiguity, MNRAS 487(3), 3432–3453.

Mac Low, M.-M. and Klessen, R. S. (2004), Control of star formation by supersonic turbulence, Reviews of Modern Physics 76, 125–194.

Machida, M. N., Inutsuka, S.-i. and Matsumoto, T. (2007), Magnetic Fields and Rotations of Protostars, ApJ 670, 1198–1213.

Macquart, J. P. and Melrose, D. B. (2000), Scintillation-induced Circular Polariza- tion in Pulsars and Quasars, ApJ 545(2), 798–806.

Maeda, S., Takahashi, K. and Ichiki, K. (2011), Primordial magnetic ﬁelds generated by the non-adiabatic ﬂuctuations at pre-recombination era, J. Cosmology Astropart. Phys. 2011(11), 045.

Maio, U., Dolag, K., Ciardi, B. and Tornatore, L. (2007), Metal and molecule cooling in simulations of structure formation, MNRAS 379(3), 963–973.

Mantz, A., Allen, S. W., Ebeling, H., Rapetti, D. and Drlica-Wagner, A. (2010), The observed growth of massive galaxy clusters - II. X-ray scaling relations, MNRAS 406, 1773–1795.

Mao, S. A., Carilli, C., Gaensler, B. M., Wucknitz, O., Keeton, C., Basu, A., Beck, R., Kronberg, P. P. and Zweibel, E. (2017), Detection of microgauss coherent magnetic ﬁelds in a galaxy ﬁve billion years ago, Nature Astronomy 1, 621–626. Bibliography 243

Mao, S. A., Gaensler, B. M., Haverkorn, M., Zweibel, E. G., Madsen, G. J., McClure- Griﬃths, N. M., Shukurov, A. and Kronberg, P. P. (2010), A Survey of Extra- galactic Faraday Rotation at High Galactic Latitude: The Vertical Magnetic Field of the Milky Way Toward the Galactic Poles, ApJ 714, 1170–1186.

Marchwinski, R. C., Pavel, M. D. and Clemens, D. P. (2012), Resolved Magnetic Field Mapping of a Molecular Cloud Using GPIPS, ApJ 755(2), 130.

Marinacci, F., Vogelsberger, M., Mocz, P. and Pakmor, R. (2015), The large-scale properties of simulated cosmological magnetic ﬁelds, MNRAS 453(4), 3999–4019.

Marinacci, F., Vogelsberger, M., Pakmor, R., Torrey, P., Springel, V., Hernquist, L., Nelson, D., Weinberger, R., Pillepich, A., Naiman, J. and Genel, S. (2018), First results from the IllustrisTNG simulations: radio haloes and magnetic ﬁelds, MNRAS 480(4), 5113–5139.

Martin, C. E., Heyvaerts, J. and Priest, E. R. (1997), Alfven wave support of a dwarf molecular cloud. I. an isothermal model, A&A 326, 1176–1186.

Martin, J. and Yokoyama, J. (2008), Generation of large scale magnetic fields in single-field inflation, J. Cosmology Astropart. Phys. 1, 25.

Mart´ınez-G´omez,D., Soler, R. and Terradas, J. (2018), Multi-ﬂuid Approach to High-frequency Waves in Plasmas. III. Nonlinear Regime and Plasma Heating, ApJ 856(1), 16.

Mathewson, D. S., Cleary, M. N. and Murray, J. D. (1974), The Magellanic Stream, in J. R. Shakeshaft (ed.), The Formation and Dynamics of Galaxies, Vol. 58 of IAU Symposium, p.367.

Matsumoto, M. and Nishimura, T. (1998), Mersenne Twister: A 623-dimensionally Equidistributed Uniform Pseudo-random Number Generator, ACM Trans. Model. Comput. Simul. 8(1), 3–30.

Maughan, B. J., Giles, P. A., Randall, S. W., Jones, C. and Forman, W. R. (2012), Self-similar scaling and evolution in the galaxy cluster X-ray luminosity- temperature relation, MNRAS 421(2), 1583–1602. Bibliography 244

Mayer, C. H., McCullough, T. P. and Sloanaker, R. M. (1957), Evidence for Polar- ized Radio Radiation from the Crab Nebula, ApJ 126, 468.

Mayer, C. H., McCullough, T. P. and Sloanaker, R. M. (1962), Evidence for Polar- ized 3 15-cm Radiation from the Radio Galaxy Cygnus A, ApJ 135, 656.

McBride, J. and Heiles, C. (2013), An Arecibo Survey for Zeeman Splitting in OH Megamaser Galaxies, ApJ 763, 8.

McBride, J., Quataert, E., Heiles, C. and Bauermeister, A. (2014), The Role of Mag- netic Fields in Starburst Galaxies as Revealed by OH Megamasers, ApJ 780, 182.

McCall, B. J., Huneycutt, A. J., Saykally, R. J., Djuric, N., Dunn, G. H., Semaniak, J., Novotny, O., Al-Khalili, A., Ehlerding, A., Hellberg, F., Kalhori, S., Neau, A., Thomas, R. D., Paal, A., Osterdahl,¨ F. and Larsson, M. (2004), Dissociative + recombination of rotationally cold H3 , Phys. Rev. A 70(5), 052716.

McCourt, M., Sharma, P., Quataert, E. and Parrish, I. J. (2012), Thermal instability in gravitationally stratiﬁed plasmas: implications for multiphase structure in clusters and galaxy haloes, MNRAS 419(4), 3319–3337.

McDonald, M., Veilleux, S., Rupke, D. S. N. and Mushotzky, R. (2010), On the Ori- gin of the Extended Hα Filaments in Cooling Flow Clusters, ApJ 721(2), 1262– 1283.

McElroy, D., Walsh, C., Markwick, A. J., Cordiner, M. A., Smith, K. and Millar, T. J. (2013), The UMIST database for astrochemistry 2012, A&A 550, A36.

McKee, C. F. and Zweibel, E. G. (1995), Alfven Waves in Interstellar Gasdynamics, ApJ 440, 686.

Meekins, J. F., Fritz, G., Chubb, T. A. and Friedman, H. (1971), Physical Sciences: X-rays from the Coma Cluster of Galaxies, Nature 231, 107–108.

Meggitt, S. M. A. and Wickramasinghe, D. T. (1982), The polarization properties of magnetic accretion columns, MNRAS 198, 71–82.

Mestel, L. (1966), The Magnetic Field of a Contracting Gas Cloud: I. Strict Flux- freezing, MNRAS 133, 265. Bibliography 245

Mestel, L. and Spitzer, Jr., L. (1956), Star formation in magnetic dust clouds, MNRAS 116, 503.

Michel, F. C. and Yahil, A. (1973), Galactic Magnetic Fields: Cellular or Filamen- tary Structure?, ApJ 179, 771–780.

Millar, T. J., Farquhar, P. R. A. and Willacy, K. (1997), The UMIST Database for Astrochemistry 1995, A&AS 121, 139–185.

Miniati, F. (2014), The Matryoshka Run: A Eulerian Reﬁnement Strategy to Study the Statistics of Turbulence in Virialized Cosmic Structures, ApJ 782(1), 21.

Miniati, F. (2015), The Matryoshka Run. II. Time-dependent Turbulence Statistics, Stochastic Particle Acceleration, and Microphysics Impact in a Massive Galaxy Cluster, ApJ 800(1), 60.

Miniati, F., Ryu, D., Kang, H. and Jones, T. W. (2001), Cosmic-Ray Protons Accelerated at Cosmological Shocks and Their Impact on Groups and Clusters of Galaxies, ApJ 559(1), 59–69.

Minter, A. H. and Spangler, S. R. (1996), Observation of Turbulent Fluctuations in the Interstellar Plasma Density and Magnetic Field on Spatial Scales of 0.01 to 100 Parsecs, ApJ 458, 194.

Mitchell, J. B. A. (1990), The dissociative recombination of molecular ions, Phys. Rep. 186, 215–248.

Miyoshi, M., Moran, J., Herrnstein, J., Greenhill, L., Nakai, N., Diamond, P. and Inoue, M. (1995), Evidence for a black hole from high rotation velocities in a sub-parsec region of NGC4258, Nature 373(6510), 127–129.

Morison, I. (2007), 50 years of the Lovell Telescope, Astronomy and Geophysics 48(5), 5.23–5.29.

Morlino, G. and Gabici, S. (2015), Cosmic ray penetration in diﬀuse clouds, MNRAS 451, L100–L104.

Moskalenko, I. V., Strong, A. W., Ormes, J. F. and Potgieter, M. S. (2002), Sec- ondary Antiprotons and Propagation of Cosmic Rays in the Galaxy and Helio- sphere, ApJ 565(1), 280–296. Bibliography 246

Moss, D., Brandenburg, A. and Tuominen, I. (1991), Properties of mean ﬁeld dynamos with non-axisymmetric alpha-eﬀect, A&A 247(2), 576–579.

Moss, D., Sokoloﬀ, D., Beck, R. and Krause, M. (2010), Galactic winds and the symmetry properties of galactic magnetic ﬁelds, A&A 512, A61.

Mouschovias, T. C. (1991), Cosmic Magnetism and the Basic Physics of the Early Stages of Star Formation, in C. J. Lada and N. D. Kylaﬁs (eds.), NATO Advanced Science Institutes (ASI) Series C, Vol. 342, p.61.

Mouschovias, T. C. and Ciolek, G. E. (1999), Magnetic Fields and Star Formation: A Theory Reaching Adulthood, in C. J. Lada and N. D. Kylaﬁs (eds.), NATO Advanced Science Institutes (ASI) Series C, Vol. 540, p.305.

Mueller, H. (1943), Memorandum on the polarization optics of the photoelastic shutter, Report of the OSRD project (Massachusetts Institute of Technology) .

Murgia, M., Govoni, F., Feretti, L. and Giovannini, G. (2010), A double radio halo in the close pair of galaxy clusters Abell 399 and Abell 401, A&A 509, A86.

Murgia, M., Govoni, F., Markevitch, M., Feretti, L., Giovannini, G., Taylor, G. B. and Carretti, E. (2009), Comparative analysis of the diﬀuse radio emission in the galaxy clusters A1835, A2029, and Ophiuchus, A&A 499(3), 679–695.

Myers, P. C. (1995), Molecular Clouds and Star Formation, Singapore: World Sci- entiﬁc.

Nagase, F., Dotani, T., Tanaka, Y., Makishima, K., Mihara, T., Sakao, T., Tsunemi, H., Kitamoto, S., Tamura, K., Yoshida, A. and Nakamura, H. (1991), Cyclotron line features in the spectrum of the transient X-ray pulsar X0115+634, ApJ 375, L49–L52.

Nakwacki, M. S., Kowal, G., Santos-Lima, R., de Gouveia Dal Pino, E. M. and Falceta-Gon¸calves, D. A. (2016), Features of collisionless turbulence in the intracluster medium from simulated Faraday Rotation maps, MNRAS 455, 3702–3723.

Narayan, R., Paczynski, B. and Piran, T. (1992), Gamma-Ray Bursts as the Death Throes of Massive Binary Stars, ApJ 395, L83. Bibliography 247

Neronov, A. and Vovk, I. (2010), Evidence for Strong Extragalactic Magnetic Fields from Fermi Observations of TeV Blazars, Science 328, 73–.

Neronov, A., Semikoz, D. V., Vovk, I. and Mirzoyan, R. (2016), Cosmic-ray composition measurements and cosmic ray background-free γ -ray observations with Cherenkov telescopes, Phys. Rev. D 94(12), 123018.

Ness, N. F., Acu˜na,M. H., Burlaga, L. F., Connerney, J. E. P., Lepping, R. P. and Neubauer, F. M. (1989), Magnetic Fields at Neptune, Science 246(4936), 1473– 1478.

Neufeld, D. A., Goicoechea, J. R., Sonnentrucker, P., Black, J. H., Pearson, J., Yu, S., Phillips, T. G., Lis, D. C., de Luca, M., Herbst, E., Rimmer, P., Gerin, M., Bell, T. A., Boulanger, F., Cernicharo, J., Coutens, A., Dartois, E., Kazmierczak, M., Encrenaz, P., Falgarone, E., Geballe, T. R., Giesen, T., Godard, B., Goldsmith, P. F., Gry, C., Gupta, H., Hennebelle, P., Hily-Blant, P., Joblin, C., Ko los, R., Krelowski, J., Mart´ın-Pintado, J., Menten, K. M., Monje, R., Mookerjea, B., Perault, M., Persson, C., Plume, R., Salez, M., Schlemmer, S., Schmidt, M., Stutzki, J., Teyssier, D., Vastel, C., Cros, A., Klein, K., Lorenzani, A., Philipp, S., Samoska, L. A., Shipman, R., Tielens, A. G. G. M., Szczerba, R. and Zmuidzinas, + + J. (2010), Herschel/HIFI observations of interstellar OH and H2O towards W49N: a probe of diﬀuse clouds with a small molecular fraction, A&A 521, L10.

Ohman,¨ Y. (1942), A polarigraphic study of obscuring clouds in the Great An- dromeda Nebula M 31, Stockholms Observatoriums Annaler 14, 4.1–4.34.

Olivares, V., Salome, P., Combes, F., Hamer, S., Guillard, P., Lehnert, M. D., Polles, F. L., Beckmann, R. S., Dubois, Y., Donahue, M., Edge, A., Fabian, A. C., McNamara, B., Rose, T., Russell, H. R., Tremblay, G., Vantyghem, A., Canning, R. E. A., Ferland, G., Godard, B., Peirani, S. and Pineau des Forets, G. (2019), Ubiquitous cold and massive ﬁlaments in cool core clusters, A&A 631, A22.

On, A. Y. L., Chan, J. Y. H., Wu, K., Saxton, C. J. and van Driel-Gesztelyi, L. (2019), Polarized radiative transfer, rotation measure ﬂuctuations, and large-scale magnetic ﬁelds, MNRAS 490(2), 1697–1713.

Oppermann, N., Junklewitz, H., Robbers, G., Bell, M. R., Enßlin, T. A., Bonafede, Bibliography 248

A., Braun, R., Brown, J. C., Clarke, T. E., Feain, I. J., Gaensler, B. M., Ham- mond, A., Harvey-Smith, L., Heald, G., Johnston-Hollitt, M., Klein, U., Kron- berg, P. P., Mao, S. A., McClure-Griﬃths, N. M., O’Sullivan, S. P., Pratley, L., Robishaw, T., Roy, S., Schnitzeler, D. H. F. M., Sotomayor-Beltran, C., Stevens, J., Stil, J. M., Sunstrum, C., Tanna, A., Taylor, A. R. and Van Eck, C. L. (2012), An improved map of the Galactic Faraday sky, A&A 542, A93.

O’Shea, B. W. and Norman, M. L. (2007), Population III Star Formation in a ΛCDM Universe. I. The Eﬀect of Formation Redshift and Environment on Protostellar Accretion Rate, ApJ 654(1), 66–92.

Osterbrock, D. E. (1989), Astrophysics of gaseous nebulae and active galactic nuclei, University Science Books.

O’Sullivan, S. P., Brown, S., Robishaw, T., Schnitzeler, D. H. F. M., McClure- Griﬃths, N. M., Feain, I. J., Taylor, A. R., Gaensler, B. M., Landecker, T. L., Harvey-Smith, L. and Carretti, E. (2012), Complex Faraday depth structure of active galactic nuclei as revealed by broad-band radio polarimetry, MNRAS 421(4), 3300–3315.

O’Sullivan, S. P., McClure-Griﬃths, N. M., Feain, I. J., Gaensler, B. M. and Sault, R. J. (2013), Broad-band radio circular polarization spectrum of the relativistic jet in PKS B2126-158, MNRAS 435(1), 311–319.

Owen, E. R., Jacobsen, I. B., Wu, K. and Surajbali, P. (2018), Interactions between ultra-high-energy particles and protogalactic environments, MNRAS 481, 666– 687.

Owen, E. R., Jin, X., Wu, K. and Chan, S. (2019a), Hadronic interactions of energetic charged particles in protogalactic outﬂow environments and implications for the early evolution of galaxies, MNRAS 484(2), 1645–1671.

Owen, E. R., Wu, K., Jin, X., Surajbali, P. and Kataoka, N. (2019b), Starburst and post-starburst high-redshift protogalaxies. The feedback impact of high energy cosmic rays, A&A 626, A85.

Owen, F. N., Rudnick, L., Eilek, J., Rau, U., Bhatnagar, S. and Kogan, L. (2014), Bibliography 249

Wideband Very Large Array Observations of A2256. I. Continuum, Rotation Mea- sure, and Spectral Imaging, ApJ 794(1), 24.

Pacholczyk, A. G. (1977), Radio galaxies: Radiation transfer, dynamics, stability and evolution of a synchrotron plasmon, Pergamon Press, Oxford.

Pacholczyk, A. G. and Swihart, T. L. (1975), Polarization of radio sources. VI - an oscillatory behavior of the intensity in a general solution of the radiation transfer problem in a plasma, ApJ 196, 125–127.

Padoan, P. and Nordlund, A.˚ (1999), A Super-Alfv´enicModel of Dark Clouds, ApJ 526, 279–294.

Padoan, P. and Scalo, J. (2005), Conﬁnement-driven Spatial Variations in the Cosmic-Ray Flux, ApJ 624, L97–L100.

Padovani, M., Galli, D. and Glassgold, A. E. (2009), Cosmic-ray ionization of molecular clouds, A&A 501, 619–631.

Padovani, M., Galli, D., Ivlev, A. V., Caselli, P. and Ferrara, A. (2018), Production of atomic hydrogen by cosmic rays in dark clouds, A&A 619, A144.

Padovani, M. and Galli, D. (2011), Eﬀects of magnetic ﬁelds on the cosmic-ray ionization of molecular cloud cores, A&A 530, A109.

Padovani, M. and Galli, D. (2013), Cosmic-Ray Propagation in Molecular Clouds, in D. F. Torres and O. Reimer (eds.), Cosmic Rays in Star-Forming Environments, Vol. 34, p.61.

Padovani, M., Hennebelle, P., Marcowith, A. and Ferri`ere,K. (2015), Cosmic-ray acceleration in young protostars, A&A 582, L13.

Padovani, M., Marcowith, A., Hennebelle, P. and Ferri`ere,K. (2016), Protostars: Forges of cosmic rays?, A&A 590, A8.

Pagani, L., Bacmann, A., Cabrit, S. and Vastel, C. (2007), Depletion and low gas temperature in the L183 (=L134N) prestellar core: the N H+ N D+ tool, A&A 2 − 2 467(1), 179–186. Bibliography 250

Pagani, L., Lef`evre,C., Juvela, M., Pelkonen, V. M. and Schuller, F. (2015), Can we trace very cold dust from its emission alone?, A&A 574, L5.

Pagani, L., Vastel, C., Hugo, E., Kokoouline, V., Greene, C. H., Bacmann, A., Bayet, E., Ceccarelli, C., Peng, R. and Schlemmer, S. (2009), Chemical modeling

of L183 (= L134N): an estimate of the ortho/para H2 ratio, A&A 494(2), 623–636.

Pan, L. and Padoan, P. (2009), The Temperature of Interstellar Clouds from Tur- bulent Heating, ApJ 692(1), 594–607.

Parekh, V., Dwarakanath, K. S., Kale, R. and Intema, H. (2017), A study of diﬀuse radio sources and X-ray emission in six massive clusters, MNRAS 464(3), 2752– 2765.

Parker, E. N. (1955), Hydromagnetic Dynamo Models, ApJ 122, 293.

Parker, E. N. (1958), Dynamics of the Interplanetary Gas and Magnetic Fields, ApJ 128, 664.

Parker, E. N. (1993), A Solar Dynamo Surface Wave at the Interface between Con- vection and Nonuniform Rotation, ApJ 408, 707.

Parnell, C. E. (2000), Magnetic reconnection and some solar applications, Philo- sophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 358(1767), 669–688.

Pasetto, A., Carrasco-Gonz´alez,C., Bruni, G., Basu, A., O’Sullivan, S., Kraus, A. and Mack, K.-H. (2016), JVLA Wideband Polarimetry Observations on a Sample of High Rotation Measure Sources, Galaxies 4(4), 66.

Patrignani, C. et al. (2016), Review of Particle Physics, Chin. Phys. C40(10), 100001.

Peebles, P. J. E. and Yu, J. T. (1970), Primeval Adiabatic Perturbation in an Expanding Universe, ApJ 162, 815.

Percival, D. B. and Walden, A. T. (1993), Spectral analysis for physical applications: multitaper and conventional univariate techniques, Cambridge University Press. Bibliography 251

Petroﬀ, E., Bailes, M., Barr, E. D., Barsdell, B. R., Bhat, N. D. R., Bian, F., Burke-Spolaor, S., Caleb, M., Champion, D. and Chandra, P. (2015), A real-time fast radio burst: polarization detection and multiwavelength follow-up, MNRAS 447(1), 246–255.

Petroﬀ, E., Barr, E. D., Jameson, A., Keane, E. F., Bailes, M., Kramer, M., Morello, V., Tabbara, D. and van Straten, W. (2016), FRBCAT: The Fast Radio Burst Catalogue, PASA 33, e045.

Petroﬀ, E., Burke-Spolaor, S., Keane, E. F., McLaughlin, M. A., Miller, R., An- dreoni, I., Bailes, M., Barr, E. D., Bernard, S. R. and Bhandari, S. (2017), A polarized fast radio burst at low Galactic latitude, MNRAS 469(4), 4465–4482.

Pfrommer, C., Enßlin, T. A. and Springel, V. (2008), Simulating cosmic rays in clusters of galaxies - II. A uniﬁed scheme for radio haloes and relics with predictions of the γ-ray emission, MNRAS 385(3), 1211–1241.

Phan, V. H. M., Morlino, G. and Gabici, S. (2018), What causes the ionization rates observed in diﬀuse molecular clouds? The role of cosmic ray protons and electrons, MNRAS 480, 5167–5174.

Piontek, R. A. and Ostriker, E. C. (2004), Thermal and Magnetorotational Insta- bility in the Interstellar Medium: Two-dimensional Numerical Simulations, ApJ 601(2), 905–920.

Planck Collaboration (2013), Planck intermediate results. X. Physics of the hot gas in the Coma cluster, A&A 554, A140.

Planck Collaboration (2015), Planck intermediate results. XIX. An overview of the polarized thermal emission from Galactic dust, A&A 576, A104.

Planck Collaboration (2016), Planck intermediate results. XXXV. Probing the role of the magnetic ﬁeld in the formation of structure in molecular clouds, A&A 586, A138.

Prasad, S. S. and Huntress, Jr., W. T. (1980), A model for gas phase chemistry in interstellar clouds. I - The basic model, library of chemical reactions, and chemistry among C, N, and O compounds, ApJS 43, 1–35. Bibliography 252

Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T. (1992), Nu- merical Recipes in FORTRAN 77: The Art of Scientiﬁc Computing, 2nd edition, Cambridge University Press.

Press, W. H. and Schechter, P. (1974), Formation of Galaxies and Clusters of Galax- ies by Self-Similar Gravitational Condensation, ApJ 187, 425–438.

Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (2007), Numer- ical Recipes 3rd Edition: The Art of Scientiﬁc Computing, 3rd edition, Cambridge University Press, New York, NY, USA.

Price, D. J. and Bate, M. R. (2008), The eﬀect of magnetic ﬁelds on star cluster formation, MNRAS 385, 1820–1834.

Ptuskin, V., Zirakashvili, V. and Seo, E.-S. (2010), Spectrum of Galactic Cosmic Rays Accelerated in Supernova Remnants, ApJ 718(1), 31–36.

Purcell, E. M. (1979), Suprathermal rotation of interstellar grains, ApJ 231, 404– 416.

Quashnock, J. M., Loeb, A. and Spergel, D. N. (1989), Magnetic Field Generation during the Cosmological QCD Phase Transition, ApJ 344, L49.

Quataert, E. (2008), Buoyancy Instabilities in Weakly Magnetized Low- Collisionality Plasmas, ApJ 673(2), 758–762.

Rajpurohit, K., Hoeft, M., van Weeren, R. J., Rudnick, L., R¨ottgering,H. J. A., Forman, W. R., Br¨uggen,M., Croston, J. H., Andrade-Santos, F., Dawson, W. A., Intema, H. T., Kraft, R. P., Jones, C. and Jee, M. J. (2018), Deep VLA Obser- vations of the Cluster 1RXS J0603.3+4214 in the Frequency Range of 1-2 GHz, ApJ 852(2), 65.

Rand, R. J. and Lyne, A. G. (1994), New Rotation Measures of Distant Pulsars in the Inner Galaxy and Magnetic Field Reversals, MNRAS 268, 497.

Rao, R., Girart, J. M., Marrone, D. P., Lai, S.-P. and Schnee, S. (2009), IRAS 16293: A “Magnetic” Tale of Two Cores, ApJ 707, 921–935.

Ratra, B. (1992), Cosmological ’seed’ magnetic ﬁeld from inﬂation, ApJ 391, L1–L4. Bibliography 253

Ravi, V., Shannon, R. M., Bailes, M., Bannister, K., Bhandari, S., Bhat, N. D. R., Burke-Spolaor, S., Caleb, M., Flynn, C., Jameson, A., Johnston, S., Keane, E. F., Kerr, M., Tiburzi, C., Tuntsov, A. V. and Vedantham, H. K. (2016), The magnetic ﬁeld and turbulence of the cosmic web measured using a brilliant fast radio burst, Science 354(6317), 1249–1252.

Rayleigh (1900), LIII. Remarks upon the law of complete radiation, The Lon- don, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 49(301), 539–540.

Rayleigh (1905a), The Dynamical Theory of Gases, Nature 71(1850), 559.

Rayleigh (1905b), The Dynamical Theory of Gases and of Radiation, Nature 72(1855), 54–55.

Rayner, D. P., Norris, R. P. and Sault, R. J. (2000), Radio circular polarization of active galaxies, MNRAS 319(2), 484–496.

Redaelli, E., Alves, F. O., Santos, F. P. and Caselli, P. (2019), Magnetic properties of the protostellar core IRAS 15398-3359, A&A 631, A154.

Rees, M. J. (1987), The origin and cosmogonic implications of seed magnetic ﬁelds., QJRAS 28, 197–206.

Reiprich, T. H. and B¨ohringer, H. (2002), The Mass Function of an X-Ray Flux- limited Sample of Galaxy Clusters, ApJ 567, 716–740.

Roberge, W. G. and Lazarian, A. (1999), Davis-Greenstein alignment of oblate spheroidal grains, MNRAS 305(3), 615–630.

Roberts, J. A., Roger, R. S., Ribes, J. C., Cooke, D. J., Murray, J. D., Cooper, B. F. C. and Biraud, F. (1975), Measurements of the circular polarization of radio sources at frequencies of 0.63, 1.4, 5.0 and 8.9 GHz, Australian Journal of Physics 28, 325–351.

Robinson, R. D., Worden, S. P. and Harvey, J. W. (1980), Observations of magnetic ﬁelds on two late-type dwarf stars, ApJ 236, L155–L158. Bibliography 254

Robishaw, T. and Heiles, C. (2005), GBT Measurements of the Zeeman Eﬀect, in American Astronomical Society Meeting, Vol. 37 of Bulletin of the American Astronomical Society, pp.1493–.

Robishaw, T., Quataert, E. and Heiles, C. (2008), Extragalactic Zeeman Detections in OH Megamasers, ApJ 680, 981–998.

Rodr´ıguez,L. F. R. (2005), Molecular Clouds: Fragmentation, Modeling and Ob- servations, in C. Lidman and D. Alloin (eds.), The Cool Universe: Observing Cosmic Dawn, Vol. 344 of Astronomical Society of the Paciﬁc Conference Series, p.146.

Rudd, M. E. (1991), Diﬀerential and total cross sections for ionization of helium and hydrogen by electrons, Phys. Rev. A 44, 1644–1652.

Rudd, M. E., Goﬀe, T. V., Dubois, R. D., Toburen, L. H. and Ratcliﬀe, C. A. (1983), Cross sections for ionization of gases by 5 4000 keV protons and for electron − capture by 5 150 keV protons, Phys. Rev. A 28, 3244–3257. −

Rudd, M. E., Kim, Y. K., Madison, D. H. and Gallagher, J. W. (1985), Electron production in proton collisions: total cross sections, Rev. Mod. Phys. 57, 965–994.

Runge, C. (1895), Uber¨ die numerische Auflösungvon Differentialgleichungen, Math- ematische Annalen 46(2), 167–178.

Ruszkowski, M., Yang, H.-Y. K. and Reynolds, C. S. (2017), Cosmic-Ray Feedback Heating of the Intracluster Medium, ApJ 844, 13.

Ruzmaikin, A. A. and Sokoloﬀ, D. D. (1979), The calculation of Faraday rotation measures of cosmic radio sources, A&A 78(1), 1–6.

Ruzmaikin, A., Sokolov, D. and Shukurov, A. (1989), The dynamo origin of magnetic ﬁelds in galaxy clusters, MNRAS 241, 1.

Rybicki, G. B. and Lightman, A. P. (1979), Radiative Processes in Astrophysics, Wiley, New York.

Ryu, D., Kang, H., Cho, J. and Das, S. (2008), Turbulence and Magnetic Fields in the Large-Scale Structure of the Universe, Science 320(5878), 909–912. Bibliography 255

Ryu, D., Kang, H. and Biermann, P. L. (1998), Cosmic magnetic ﬁelds in large scale ﬁlaments and sheets, A&A 335, 19–25.

Ryu, D., Schleicher, D. R. G., Treumann, R. A., Tsagas, C. G. and Widrow, L. M. (2012), Magnetic Fields in the Large-Scale Structure of the Universe, Space Sci. Rev. 166, 1–35.

Sage, L. J., Welch, G. A. and Young, L. M. (2007), The Cool ISM in Elliptical Galaxies. I. A Survey of Molecular Gas, ApJ 657(1), 232–240.

Saikia, D. J. and Salter, C. J. (1988), Polarization properties of extragalactic radio sources, ARA&A 26, 93–144.

Salem, M., Bryan, G. L. and Corlies, L. (2016), Role of cosmic rays in the circumgalactic medium, MNRAS 456, 582–601.

Sanders, J. S., Fabian, A. C. and Taylor, G. B. (2009a), Feedback through multiple outbursts in the cluster 2A 0335+096, MNRAS 396(3), 1449–1459.

Sanders, J. S., Fabian, A. C. and Taylor, G. B. (2009b), Giant cavities, cooling and metallicity substructure in Abell 2204, MNRAS 393(1), 71–82.

Sanna, A., Reid, M. J., Dame, T. M., Menten, K. M. and Brunthaler, A. (2017), Mapping spiral structure on the far side of the Milky Way, Science 358(6360), 227–230.

Santangelo, A., Segreto, A., Giarrusso, S., Dal Fiume, D., Orlandini, M., Parmar, A. N., Oosterbroek, T., Bulik, T., Mihara, T., Campana, S., Israel, G. L. and Stella, L. (1999), A BEPPOSAX Study of the Pulsating Transient X0115+63: The First X-Ray Spectrum with Four Cyclotron Harmonic Features, ApJ 523, L85.

Sarala, S. and Jain, P. (2001), A circular statistical method for extracting rotation measures, MNRAS 328, 623–634.

Sarazin, C. L. (1986), X-ray emission from clusters of galaxies, Reviews of Modern Physics 58(1), 1–115.

Sarazin, C. L. (2002), The Physics of Cluster Mergers, Vol. 272 of Astrophysics and Space Science Library, Springer Netherlands, pp.1–38. Bibliography 256

Sarma, A. P., Momjian, E., Troland, T. H. and Crutcher, R. M. (2005), Very Large Array H I Zeeman Observations of NGC 1275 (Perseus A), The Astronomical Journal 130(6), 2566.

Sato, K. (1981), First-order phase transition of a vacuum and the expansion of the Universe, MNRAS 195, 467–479.

Saxton, C. J., Bicknell, G. V., Sutherland, R. S. and Midgley, S. (2005), Interactions of jets with inhomogeneous cloudy media, MNRAS 359(2), 781–800.

Saxton, C. J., Sutherland, R. S. and Bicknell, G. V. (2001), The Centaurus A Northern Middle Lobe as a Buoyant Bubble, ApJ 563(1), 103–117.

Sazonov, V. N. (1969), Generation and Transfer of Polarized Synchrotron Radia- tion., Soviet Ast. 13, 396.

Scarrott, S. M., Ward-Thompson, D. and Warren-Smith, R. F. (1987), Evidence for a spiral magnetic ﬁeld conﬁguration in the galaxy M51, MNRAS 224, 299–305.

Schilizzi, R. T. (2007), Highlights of Astronomy, Proceedings of the International Astronomical Union 14, 539.

Schleicher, D. R. G., Banerjee, R., Sur, S., Arshakian, T. G., Klessen, R. S., Beck, R. and Spaans, M. (2010), Small-scale dynamo action during the formation of the ﬁrst stars and galaxies. I. The ideal MHD limit, A&A 522, A115.

Schlickeiser, R. (2002), Cosmic Ray Astrophysics, Springer.

Schlickeiser, R. and Achatz, U. (1993a), Cosmic-ray particle transport in weakly turbulent plasmas. Part 1. Theory, Journal of Plasma Physics 49, 63–77.

Schlickeiser, R. and Achatz, U. (1993b), Cosmic-ray particle transport in weakly turbulent plasmas. Part 2. Mean free path of cosmic-ray protons, Journal of Plasma Physics 50, 85–107.

Schmidt, M. (1963), 3C 273: A Star-Like Object with Large Red-Shift, Nature 197(4872), 1040.

Schober, J., Schleicher, D. R. G. and Klessen, R. S. (2013), Magnetic ﬁeld ampliﬁ- cation in young galaxies, A&A 560, A87. Bibliography 257

Schuecker, P., Finoguenov, A., Miniati, F., B¨ohringer,H. and Briel, U. G. (2004), Probing turbulence in the Coma galaxy cluster, A&A 426, 387–397.

Schulz-Dubois, E. O. and Rehberg, I. (1981), Structure function in lieu of correlation function, Applied Physics 24, 323–329.

Seifried, D. and Walch, S. (2015), The impact of turbulence and magnetic ﬁeld orientation on star-forming ﬁlaments, MNRAS 452, 2410–2422.

Sharma, P., McCourt, M., Quataert, E. and Parrish, I. J. (2012), Thermal instability and the feedback regulation of hot haloes in clusters, groups and galaxies, MNRAS 420(4), 3174–3194.

Shu, F. H., Milione, V., Gebel, W., Yuan, C., Goldsmith, D. W. and Roberts, W. W. (1972), Galactic Shocks in an Interstellar Medium with Two Stable Phases, ApJ 173, 557.

Shukurov, A. and Berkhuijsen, E. M. (2003), Faraday ghosts: depolarization canals in the Galactic radio emission, MNRAS 342(2), 496–500.

Siejkowski, H., Soida, M. and Chy˙zy,K. T. (2018), Magnetic ﬁeld evolution in dwarf and Magellanic-type galaxies, A&A 611, A7.

Silk, J. and Langer, M. (2006), On the ﬁrst generation of stars, MNRAS 371, 444– 450.

Silva, R., Martins, H., Nascimento, I., Baptista, J., Lobo Ribeiro, A., Santos, J., Jorge, P. and Fraz˜ao,O. (2012), Optical Current Sensors for High Power Systems: A Review, Applied Sciences 2, 602–628.

Simard-Normandin, M. and Kronberg, P. P. (1980), Rotation measures and the galactic magnetic ﬁeld, ApJ 242, 74–94.

Simard-Normandin, M., Kronberg, P. P. and Button, S. (1981), The Faraday rotation measures of extragalactic radio sources, ApJS 45, 97–111.

Skorodko, T., Bashkanov, M., Bogoslawsky, D., Calen, H., Cappellaro, F., Clement, H., Demiroers, L., Doroshkevich, E., Duniec, D., Ekstr¨om,C., Franssen, K., Gustafsson, L., H¨oistad,B., Ivanov, G., Jacewicz, M., Jiganov, E., Johansson, Bibliography 258

T., Khakimova, O., Kaskulov, M., Keleta, S., Koch, I., Kren, F., Kullander, S., Kup´sć,A., Kuznetsov, A., Marciniewski, P., Meier, R., Morosov, B., Pauly, C., Petterson, H., Petukhov, Y., Povtorejko, A., Schönning,K., Scobel, W., Shwartz, B., Sopov, V., Stepeniak, J., Thörngren-Engblom,P., Tikhomirov, V., Wagner, G. J., Wolke, M., Yamamoto, A., Zabierowski, J. and Z lomanczuk,J. (2008), Ex- citation of the Roper resonance in single- and double-pion production in nucleon- nucleon collisions, Eur. Phys. J. A 35, 317–319.

Skory, S., Hallman, E., Burns, J. O., Skillman, S. W., O’Shea, B. W. and Smith, B. D. (2013), On The Road To More Realistic Galaxy Cluster Simulations: The Eﬀects of Radiative Cooling and Thermal Feedback Prescriptions on the Obser- vational Properties of Simulated Galaxy Clusters, ApJ 763(1), 38.

Skrutskie, M. F., Cutri, R. M., Stiening, R., Weinberg, M. D., Schneider, S., Carpen- ter, J. M., Beichman, C., Capps, R., Chester, T., Elias, J., Huchra, J., Liebert, J., Lonsdale, C., Monet, D. G., Price, S., Seitzer, P., Jarrett, T., Kirkpatrick, J. D., Gizis, J. E., Howard, E., Evans, T., Fowler, J., Fullmer, L., Hurt, R., Light, R., Kopan, E. L., Marsh, K. A., McCallon, H. L., Tam, R., Van Dyk, S. and Wheelock, S. (2006), The Two Micron All Sky Survey (2MASS), AJ 131(2), 1163–1183.

Smirnov, N. (1948), Table for Estimating the Goodness of Fit of Empirical Distri- butions, Ann. Math. Statist. 19(2), 279–281.

Soﬁa, U. J., Lauroesch, J. T., Meyer, D. M. and Cartledge, S. I. B. (2004), Inter- stellar Carbon in Translucent Sight Lines, ApJ 605(1), 272–277.

Sokoloﬀ, D. D., Bykov, A. A., Shukurov, A., Berkhuijsen, E. M., Beck, R. and Poezd, A. D. (1998), Depolarization and Faraday eﬀects in galaxies, MNRAS 299, 189–206.

Sonnentrucker, P., Welty, D. E., Thorburn, J. A. and York, D. G. (2007), Abun- 12 13 dances and Behavior of CO, CO, and C2 in Translucent Sight Lines, ApJS 168(1), 58–99.

Spitzer, Jr., L. and Tomasko, M. G. (1968), Heating of HI Regions by Energetic Particles, ApJ 152, 971. Bibliography 259

Stancil, P. C., Schultz, D. R., Kimura, M., Gu, J. P., Hirsch, G. and Buenker, R. J. (1999), Charge transfer in collisions of O+ with H and H+ with O, A&AS 140, 225–234.

Stanley, S. (2014), Chapter 6 - Magnetic Field Generation in Planets, in T. Spohn, D. Breuer and T. V. Johnson (eds.), Encyclopedia of the Solar System (Third Edition), Elsevier, Boston, pp.121 – 136.

Stanley, S. and Bloxham, J. (2006), Numerical dynamo models of Uranus’ and Neptune’s magnetic ﬁelds, Icarus 184(2), 556–572.

Stasyszyn, F. A. and de los Rios, M. (2019), Faraday rotation measure dependence on galaxy cluster dynamics, MNRAS 487(4), 4768–4774.

Staubert, R., Tr¨umper, J., Kendziorra, E., Klochkov, D., Postnov, K., Kretschmar, P., Pottschmidt, K., Haberl, F., Rothschild, R. E., Santangelo, A., Wilms, J., Kreykenbohm, I. and F¨urst, F. (2019), Cyclotron lines in highly magnetized neutron stars, A&A 622, A61.

Staveley-Smith, L., Briggs, D. S., Rowe, A. C. H., Manchester, R. N., Reynolds, J. E., Tzioumis, A. K. and Kesteven, M. J. (1993), Structure of the radio remnant of supernova 1987A, Nature 366(6451), 136–138.

Stepanov, R., Arshakian, T. G., Beck, R., Frick, P. and Krause, M. (2008), Magnetic ﬁeld structures of galaxies derived from analysis of Faraday rotation measures, and perspectives for the SKA, A&A 480(1), 45–59.

Stevenson, D. J. (1983), Planetary magnetic ﬁelds, Reports on Progress in Physics 46(5), 555–620.

Stevenson, D. J. (2003), Planetary magnetic ﬁelds, Earth and Planetary Science Letters 208(1-2), 1–11.

Stone, E. C., Cummings, A. C., Heikkila, B. C. and Lal, N. (2019), Cosmic ray measurements from Voyager 2 as it crossed into interstellar space, Nature Astronomy 3, 1013–1018.

Storm, E., Jeltema, T. E. and Rudnick, L. (2015), A radio and X-ray study of the merging cluster A2319, MNRAS 448(3), 2495–2503. Bibliography 260

Straub, H. C., Renault, P., Lindsay, B. G., Smith, K. A. and Stebbings, R. F.

(1996), Absolute partial cross sections for electron-impact ionization of H2,N2,

and O2 from threshold to 1000 eV, Phys. Rev. A 54, 2146–2153.

Strong, A. W., Dickinson, C. and Murphy, E. J. (2014), Synchrotron radiation from molecular clouds, in Proceedings of Conference “Cosmic Rays and their Inter- Stellar Medium Environment (CRISM-2014)”, June 24-27, 2014, Montpellier, France. PoS(CRISM2014)036, p.8.

Strong, A. W., Moskalenko, I. V. and Ptuskin, V. S. (2007), Cosmic-Ray Propaga- tion and Interactions in the Galaxy, Ann. Rev. Nucl. and Part. Sci. 57(1), 285.

Strong, A. W., Moskalenko, I. V. and Reimer, O. (2000), Diﬀuse Continuum Gamma Rays from the Galaxy, ApJ 537(2), 763–784.

Subramanian, K. (2010), Magnetic ﬁelds in the early Universe, Astronomische Nachrichten 331, 110.

Subramanian, K. (2016), The origin, evolution and signatures of primordial magnetic ﬁelds, Reports on Progress in Physics 79(7), 076901.

Subramanian, K. (2019), From Primordial Seed Magnetic Fields to the Galactic Dynamo, Galaxies 7(2), 47.

Subramanian, K., Shukurov, A. and Haugen, N. E. L. (2006), Evolving turbulence and magnetic ﬁelds in galaxy clusters, MNRAS 366, 1437–1454.

Sur, S. (2019), Decaying turbulence and magnetic ﬁelds in galaxy clusters, MNRAS 488(3), 3439–3445.

Sur, S., Schleicher, D. R. G., Banerjee, R., Federrath, C. and Klessen, R. S. (2010), The Generation of Strong Magnetic Fields During the Formation of the First Stars, ApJ 721, L134–L138.

Sutherland, R., Dopita, M., Binette, L. and Groves, B. (2018), MAPPINGS V: As- trophysical plasma modeling code, Astrophysics Source Code Library p.1807.005.

Sutherland, R. S. and Dopita, M. A. (1993), Cooling Functions for Low-Density Astrophysical Plasmas, ApJS 88, 253. Bibliography 261

Swain, M. R., Bridle, A. H. and Baum, S. A. (1998), Internal Structure of the Jets in 3C 353, ApJ 507(1), L29–L33.

Takahashi, K., Ichiki, K., Ohno, H. and Hanayama, H. (2005), Magnetic Field Gen- eration from Cosmological Perturbations, Physical Review Letters 95(12), 121301.

Takahashi, K., Inoue, S., Ichiki, K. and Nakamura, T. (2011), Probing early cosmic magnetic ﬁelds through pair echoes from high-redshift GRBs, MNRAS 410(4), 2741–2748.

Takahashi, K., Mori, M., Ichiki, K., Inoue, S. and Takami, H. (2013), Lower Bounds on Magnetic Fields in Intergalactic Voids from Long-term GeV-TeV Light Curves of the Blazar Mrk 421, ApJ 771, L42.

Tang, Y.-W., Ho, P. T. P., Koch, P. M., Girart, J. M., Lai, S.-P. and Rao, R. (2009), Evolution of Magnetic Fields in High-Mass Star Formation: Linking Field Geometry and Collapse for the W51 e2/e8 Cores, ApJ 700, 251–261.

Tang, Y.-W., Koch, P. M., Peretto, N., Novak, G., Duarte-Cabral, A., Chapman, N. L., Hsieh, P.-Y. and Yen, H.-W. (2019), Gravity, Magnetic Field, and Turbu- lence: Relative Importance and Impact on Fragmentation in the Infrared Dark Cloud G34.43+00.24, ApJ 878(1), 10.

Taylor, A. M., Vovk, I. and Neronov, A. (2011), Extragalactic magnetic ﬁelds constraints from simultaneous GeV-TeV observations of blazars, A&A 529, A144.

Taylor, A. R., Stil, J. M. and Sunstrum, C. (2009), A Rotation Measure Image of the Sky, ApJ 702, 1230–1236.

Tevzadze, A. G., Kisslinger, L., Brand enburg, A. and Kahniashvili, T. (2012), Magnetic Fields from QCD Phase Transitions, ApJ 759(1), 54.

The Pierre Auger Collaboration (2018), An Indication of Anisotropy in Arrival Directions of Ultra-high-energy Cosmic Rays through Comparison to the Flux Pattern of Extragalactic Gamma-Ray Sources, ApJ 853(2), L29.

The Telescope Array Collaboration (2014), Indications of Intermediate-scale Anisotropy of Cosmic Rays with Energy Greater Than 57 EeV in the North- Bibliography 262

ern Sky Measured with the Surface Detector of the Telescope Array Experiment, ApJ 790(2), L21.

The Telescope Array Collaboration (2018), Testing a Reported Correlation between Arrival Directions of Ultra-high-energy Cosmic Rays and a Flux Pattern from nearby Starburst Galaxies using Telescope Array Data, ApJ 867(2), L27.

Theard, L. P. and Huntress, W. T. (1974), Ion-molecule reactions and vibrational + deactivation of H2 ions in mixtures of hydrogen and helium, J. Chem. Phys. 60, 2840–2848.

Thompson, C. and Duncan, R. C. (1993), Neutron star dynamos and the origins of pulsar magnetism, ApJ 408, 194–217.

Thompson, K. L., Troland, T. H. and Heiles, C. (2019), A Survey of Magnetic Field Strengths in the Envelopes of Molecular Clouds via the 18 cm OH Zeeman Eﬀect, ApJ 884(1), 49.

Thompson, T. A., Quataert, E. and Waxman, E. (2007), The Starburst Contribution to the Extragalactic γ-Ray Background, ApJ 654, 219–225.

Torres, D. F. (2004), Theoretical Modeling of the Diﬀuse Emission of Gamma Rays from Extreme Regions of Star Formation: The Case of ARP 220, ApJ 617, 966– 986.

Tricco, T. S. and Price, D. J. (2012), Constrained hyperbolic divergence cleaning for smoothed particle magnetohydrodynamics, Journal of Computational Physics 231, 7214–7236.

Tricco, T. S. and Price, D. J. (2013), A switch to reduce resistivity in smoothed particle magnetohydrodynamics, MNRAS 436, 2810–2817.

Trimble, V. and Ceja, J. A. (2010), Productivity and impact of astronomical facili- ties: A recent sample, Astronomische Nachrichten 331(3), 338.

Truemper, J., Pietsch, W., Reppin, C., Voges, W., Staubert, R. and Kendziorra, E. (1978), Evidence for strong cyclotron line emission in the hard X-ray spectrum of Hercules X-1., ApJ 219, L105–L110. Bibliography 263

Tsouros, A. and Kylaﬁs, N. D. (2017), The energy distribution of electrons in radio jets, A&A 603, L4.

Turk, M. J., Abel, T. and O’Shea, B. (2009), The Formation of Population III Binaries from Cosmological Initial Conditions, Science 325, 601–.

Turner, M. S. and Widrow, L. M. (1988), Inﬂation-produced, large-scale magnetic ﬁelds, Phys. Rev. D 37, 2743–2754.

Uchiyama, Y., Aharonian, F. A., Tanaka, T., Takahashi, T. and Maeda, Y. (2007), Extremely fast acceleration of cosmic rays in a supernova remnant, Nature 449(7162), 576–578.

Ulrich, K. and Fletcher, A. (2015), Galactic and intergalactic magnetic ﬁelds, Springer International Publishing.

Urban, O., Simionescu, A., Werner, N., Allen, S. W., Ehlert, S., Zhuravleva, I., Morris, R. G., Fabian, A. C., Mantz, A. and Nulsen, P. E. J. (2014), Az- imuthally resolved X-ray spectroscopy to the edge of the Perseus Cluster, MNRAS 437(4), 3939–3961.

Vacca, V., Murgia, M., Govoni, F., Enßlin, T., Oppermann, N., Feretti, L., Gio- vannini, G. and Loi, F. (2018), Magnetic Fields in Galaxy Clusters and in the Large-Scale Structure of the Universe, Galaxies 6, 142.

Vacca, V., Murgia, M., Govoni, F., Feretti, L., Giovannini, G., Orr`u,E. and Bonafede, A. (2010), The intracluster magnetic ﬁeld power spectrum in Abell 665, A&A 514, A71.

Vaillancourt, J. E., Dowell, C. D., Hildebrand, R. H., Kirby, L., Krejny, M. M., Li, H.-B., Novak, G., Houde, M., Shinnaga, H. and Attard, M. (2008), New Results on the Submillimeter Polarization Spectrum of the Orion Molecular Cloud, ApJ 679, L25–L28.

Vallee, J. P. (1980), The rotation measures of radio sources and their data process- ing, A&A 86, 251–253. van der Tak, F. F. S. and van Dishoeck, E. F. (2000), Limits on the cosmic-ray ionization rate toward massive young stars, A&A 358, L79–L82. Bibliography 264 van Dishoeck, E. F. and Black, J. H. (1986), Comprehensive models of diffuse interstellar clouds - Physical conditions and molecular abundances, ApJS 62, 109–145. van Weeren, R. J., de Gasperin, F., Akamatsu, H., Brüggen,M., Feretti, L., Kang, H., Stroe, A. and Zandanel, F. (2019), Diffuse Radio Emission from Galaxy Clus- ters, Space Sci. Rev. 215(1), 16.

Vázquez-Semadeni,E., Banerjee, R., Gómez,G. C., Hennebelle, P., Duffin, D. and Klessen, R. S. (2011), Molecular cloud evolution - IV. Magnetic fields, ambipolar diffusion and the star formation efficiency, MNRAS 414, 2511–2527.

Vazza, F., Brüggen,M., Gheller, C. and Wang, P. (2014), On the amplification of magnetic fields in cosmic filaments and galaxy clusters, MNRAS 445, 3706–3722.

Vazza, F., Br¨uggen,M., Hinz, P. M., Wittor, D., Locatelli, N. and Gheller, C. (2018), Probing the origin of extragalactic magnetic ﬁelds with Fast Radio Bursts, MNRAS 480(3), 3907–3915.

Vedantham, H. K. and Ravi, V. (2019), Faraday conversion and magneto-ionic variations in fast radio bursts, MNRAS 485(1), L78–L82.

Veres, P., Dermer, C. D. and Dhuga, K. S. (2017), Properties of the Intergalactic Magnetic Field Constrained by Gamma-Ray Observations of Gamma-Ray Bursts, ApJ 847(1), 39.

Vernstrom, T., Gaensler, B. M., Brown, S., Lenc, E. and Norris, R. P. (2017), Low-frequency radio constraints on the synchrotron cosmic web, MNRAS 467(4), 4914–4936.

Vernstrom, T., Gaensler, B. M., Rudnick, L. and Andernach, H. (2019), Diﬀerences in Faraday Rotation between Adjacent Extragalactic Radio Sources as a Probe of Cosmic Magnetic Fields, ApJ 878(2), 92.

Virtanen, P., Gommers, R., Oliphant, T. E., Haberland, M., Reddy, T., Courna- peau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S. J., Brett, M., Wilson, J., Millman, K. J., Mayorov, N., Nelson, A. R. J., Jones, E., Kern, R., Larson, E., Carey, C. J., Polat, I.,˙ Feng, Y., Moore, E. W., Vander- Plas, J., Laxalde, D., Perktold, J., Cimrman, R., Henriksen, I., Quintero, E. A., Bibliography 265

Harris, C. R., Archibald, A. M., Ribeiro, A. H., Pedregosa, F., van Mulbregt, P. and SciPy 1. 0 Contributors (2020), SciPy 1.0: fundamental algorithms for scientiﬁc computing in Python, Nature Methods 17, 261–272.

Vogt, C., Dolag, K. and Enßlin, T. A. (2005), PACERMAN- II. Application and statistical characterization of improved RM maps, MNRAS 358(3), 732–741.

Vogt, C. and Enßlin, T. A. (2003), Measuring the cluster magnetic ﬁeld power spectra from Faraday rotation maps of Abell 400, Abell 2634 and Hydra A, A&A 412, 373.

Voit, G. M., Meece, G., Li, Y., O’Shea, B. W., Bryan, G. L. and Donahue, M. (2017), A Global Model for Circumgalactic and Cluster-core Precipitation, ApJ 845(1), 80.

Walker, M. A. (2016), Heating of the Warm Ionized Medium by Low-energy Cosmic Rays, ApJ 818(1), 23.

Wang, J. B., Hobbs, G., Coles, W., Shannon, R. M., Zhu, X. J., Madison, D. R., Kerr, M., Ravi, V., Keith, M. J., Manchester, R. N., Levin, Y., Bailes, M., Bhat, N. D. R., Burke-Spolaor, S., Dai, S., Oslowski, S., van Straten, W., Toomey, L., Wang, N. and Wen, L. (2015), Searching for gravitational wave memory bursts with the Parkes Pulsar Timing Array, MNRAS 446(2), 1657–1671.

Wang, J.-W., Lai, S.-P., Clemens, D. P., Koch, P. M., Eswaraiah, C., Chen, W.- P. and Pandey, A. K. (2020a), Multiwavelength Polarimetry of the Filamentary Cloud IC 5146. II. Magnetic Field Structures, ApJ 888(1), 13.

Wang, J.-W., Lai, S.-P., Eswaraiah, C., Clemens, D. P., Chen, W.-P. and Pandey, A. K. (2017), Multiwavelength Stellar Polarimetry of the Filamentary Cloud IC5146. I. Dust Properties, ApJ 849, 157.

Wang, J.-W., Lai, S.-P., Eswaraiah, C., Pattle, K., Di Francesco, J., Johnstone, D., Koch, P. M., Liu, T., Tamura, M., Furuya, R. S., Onaka, T., Ward-Thompson, D., Soam, A., Kim, K.-T., Lee, C. W., Lee, C.-F., Mairs, S., Arzoumanian, D., Kim, G., Hoang, T., Hwang, J., Liu, S.-Y., Berry, D., Bastien, P., Hasegawa, T., Kwon, W., Qiu, K., Andr´e,P., Aso, Y., Byun, D.-Y., Chen, H.-R., Chen, M. C., Chen, W. P., Ching, T.-C., Cho, J., Choi, M., Chrysostomou, A., Chung, E. J., Bibliography 266

Coudé,S., Doi, Y., Dowell, C. D., Drabek-Maunder, E., Duan, H.-Y., Eyres, S. P. S., Falle, S., Fanciullo, L., Fiege, J., Franzmann, E., Friberg, P., Friesen, R. K., Fuller, G., Gledhill, T., Graves, S. F., Greaves, J. S., Griffin, M. J., Gu, Q., Han, I., Hatchell, J., Hayashi, S. S., Holland, W., Houde, M., Inoue, T., Inutsuka, S.-i., Iwasaki, K., Jeong, I.-G., Kanamori, Y., Kang, J.-h., Kang, M., Kang, S.-j., Kataoka, A., Kawabata, K. S., Kemper, F., Kim, J., Kim, K. H., Kim, M.-R., Kim, S., Kirk, J. M., Kobayashi, M. I. N., Konyves, V., Kwon, J., Lacaille, K. M., Lee, H., Lee, J.-E., Lee, S.-S., Lee, Y.-H., Li, D., Li, D., Li, H.-B., Liu, H.-L., Liu, J., Lyo, A.-R., Matsumura, M., Matthews, B. C., Moriarty-Schieven, G. H., Nagata, T., Nakamura, F., Nakanishi, H., Ohashi, N., Park, G., Parsons, H., Pascale, E., Peretto, N., Pon, A., Pyo, T.-S., Qian, L., Rao, R., Rawlings, M. G., Retter, B., Richer, J., Rigby, A., Robitaille, J.-F., Sadavoy, S., Saito, H., Savini, G., Scaife, A. M. M., Seta, M., Shinnaga, H., Tang, Y.-W., Tomisaka, K., Tsukamoto, Y., van Loo, S., Wang, H., Whitworth, A. P., Yen, H.-W., Yoo, H., Yuan, J., Yun, H.-S., Zenko, T., Zhang, C.-P., Zhang, G., Zhang, Y.-P., Zhou, J. and Zhu, L. (2019), JCMT BISTRO Survey: Magnetic Fields within the Hub- filament Structure in IC 5146, ApJ 876, 42.

Wang, Z.-R., Xi, S.-Q., Liu, R.-Y., Xue, R. and Wang, X.-Y. (2020b), Constraints on the intergalactic magnetic ﬁeld from γ -ray observations of GRB 190114C, Phys. Rev. D 101(8), 083004.

Ward-Thompson, D., Andr´e,P. and Kirk, J. M. (2002), The initial conditions of isolated star formation - V. ISOPHOT imaging and the temperature and energy balance of pre-stellar cores, MNRAS 329(2), 257–276.

Ward-Thompson, D., Kirk, J. M., Crutcher, R. M., Greaves, J. S., Holland, W. S. and Andr´e,P. (2000), First Observations of the Magnetic Field Geometry in Prestellar Cores, ApJ 537(2), L135–L138.

Ward-Thompson, D., Scott, P. F., Hills, R. E. and Andre, P. (1994), A Submillimetre Continuum Survey of Pre Protostellar Cores, MNRAS 268, 276.

Wareing, C. J., Pittard, J. M., Falle, S. A. E. G. and Van Loo, S. (2016), Magneto- hydrodynamical simulation of the formation of clumps and ﬁlaments in quiescent diﬀuse medium by thermal instability, MNRAS 459(2), 1803–1818. Bibliography 267

Waxman, E. (1995), Cosmological Gamma-Ray Bursts and the Highest Energy Cos- mic Rays, Phys. Rev. Lett. 75(3), 386–389.

Webber, W. R. (1998), A New Estimate of the Local Interstellar Energy Density and Ionization Rate of Galactic Cosmic Cosmic Rays, ApJ 506(1), 329–334.

Wentzel, D. G. (1969), The Propagation and Anisotropy of Cosmic Rays. I. Theory for Steady Streaming, ApJ 156, 303.

Wentzel, D. G. (1971), Acceleration and Heating of Interstellar Gas by Cosmic Rays, ApJ 163, 503.

Wentzel, D. G. (1974), Cosmic-ray propagation in the Galaxy - Collective eﬀects, ARA&A 12, 71–96.

West, J., Brown, J.-A., Gaensler, B., Hill, A. S., Irwin, J., Kothes, R., Landecker, T., Robishaw, T., Saﬁ-Harb, S., Stil, J., Van Eck, C. and Wade, G. (2019), Cosmic Magnetism, in Canadian Long Range Plan for Astronomy and Astrophysics White Papers, Vol. 2020, p.54.

Westerhout, G., Seeger, C. L., Brouw, W. N. and Tinbergen, J. (1962), Polarization of the galactic 75-cm radiation, Bull. Astron. Inst. Netherlands 16, 187.

White, T. G., Oliver, M. T., Mabey, P., Kühn-Kauffeldt,M., Bott, A. F. A., Döhl, L. N. K., Bell, A. R., Bingham, R., Clarke, R., Foster, J., Giacinti, G., Graham, P., Heathcote, R., Koenig, M., Kuramitsu, Y., Lamb, D. Q., Meinecke, J., Michel, T., Miniati, F., Notley, M., Reville, B., Ryu, D., Sarkar, S., Sakawa, Y., Selwood, M. P., Squire, J., Scott, R. H. H., Tzeferacos, P., Woolsey, N., Schekochihin, A. A. and Gregori, G. (2019), Supersonic plasma turbulence in the laboratory, Nature Communications 10, 1758.

Whittet, D. C. B., Hough, J. H., Lazarian, A. and Hoang, T. (2008), The Eﬃciency of Grain Alignment in Dense Interstellar Clouds: a Reassessment of Constraints from Near-Infrared Polarization, ApJ 674(1), 304–315.

Whitworth, A. P. and Jaﬀa, S. E. (2018), A simple approach to CO cooling in molecular clouds, A&A 611, A20. Bibliography 268

Whitworth, A. P. and Ward-Thompson, D. (2001), An Empirical Model for Proto- stellar Collapse, ApJ 547(1), 317–322.

Wickramasinghe, D. T. and Ferrario, L. (2000), Magnetism in Isolated and Binary White Dwarfs, PASP 112, 873.

Wickramasinghe, D. T. and Meggitt, S. M. A. (1985), The polarization properties of magnetic accretion columns. III - A grid of uniform temperature and shock front models, MNRAS 214, 605.

Widrow, L. M. (2002), Origin of galactic and extragalactic magnetic ﬁelds, Reviews of Modern Physics 74(3), 775–823.

Widrow, L. M., Ryu, D., Schleicher, D. R. G., Subramanian, K., Tsagas, C. G. and Treumann, R. A. (2012), The First Magnetic Fields, Space Sci. Rev. 166, 37.

Wielebinski, R., Shakeshaft, J. R. and Pauliny-Toth, I. I. K. (1962), A search for a linearly polarized component of the galactic radio emission at 408 Mc/s, The Observatory 82, 158–164.

Wiener, J., Oh, S. P. and Guo, F. (2013a), Cosmic ray streaming in clusters of galaxies, MNRAS 434, 2209–2228.

Wiener, J., Zweibel, E. G. and Oh, S. P. (2013b), Cosmic Ray Heating of the Warm Ionized Medium, ApJ 767, 87.

Wiener, N. (1930), Generalized harmonic analysis, Acta Math. 55, 117–258.

Woltjer, L. (1964), X-Rays and Type I Supernova Remnants., ApJ 140, 1309–1313.

Woosley, S. E. (1993), Gamma-Ray Bursts from Stellar Mass Accretion Disks around Black Holes, ApJ 405, 273.

Wu, K. and Wickramasinghe, D. T. (1990), Three-dimensional structured shocks in AM Herculis-type systems. - I. The angular dependence of intensity and polarization from axisymmetric shocks., MNRAS 246, 686.

Xie, C., van Weeren, R. J., Lovisari, L., Andrade-Santos, F., Botteon, A., Br¨uggen, M., Bulbul, E., Churazov, E., Clarke, T. E., Forman, W. R., Intema, H. T., Jones, C., Kraft, R. P., Lal, D. V., Mroczkowski, T. and Zitrin, A. (2020), The Bibliography 269

discovery of radio halos in the frontier ﬁelds clusters Abell S1063 and Abell 370, A&A 636, A3.

Xu, C. K., Lu, N., Condon, J. J., Dopita, M. and Tuﬀs, R. J. (2003), Physical Conditions and Star Formation Activity in the Intragroup Medium of Stephan’s Quintet, ApJ 595, 665–684.

Xu, H., Govoni, F., Murgia, M., Li, H., Collins, D. C., Norman, M. L., Cen, R., Feretti, L. and Giovannini, G. (2012), Comparisons of Cosmological Magnetohy- drodynamic Galaxy Cluster Simulations to Radio Observations, ApJ 759, 40.

Xu, H., Li, H., Collins, D. C., Li, S. and Norman, M. L. (2010), Evolution and Distribution of Magnetic Fields from Active Galactic Nuclei in Galaxy Clusters. I. The Eﬀect of Injection Energy and Redshift, ApJ 725, 2152–2165.

Xu, H., O’Shea, B. W., Collins, D. C., Norman, M. L., Li, H. and Li, S. (2008), The Biermann Battery in Cosmological MHD Simulations of Population III Star Formation, ApJ 688, L57–L60.

Xu, J. and Han, J.-L. (2014a), A compiled catalog of rotation measures of radio point sources, Research in Astronomy and Astrophysics 14(8), 942–958.

Xu, J. and Han, J. L. (2014b), Redshift evolution of extragalactic rotation measures, MNRAS 442, 3329–3337.

Xu, S. and Lazarian, A. (2017), Magnetic Field Ampliﬁcation in Supernova Rem- nants, ApJ 850(2), 126.

Xu, S. and Zhang, B. (2016), Interpretation of the Structure Function of Rotation Measure in the Interstellar Medium, ApJ 824, 113.

Xu, Y., Kronberg, P. P., Habib, S. and Dufton, Q. W. (2006), A Faraday Rotation Search for Magnetic Fields in Large-scale Structure, ApJ 637, 19.

Yamamoto, S. (2017), Introduction to Astrochemistry: Chemical Evolution from Interstellar Clouds to Star and Planet Formation, Springer Japan.

Yoast-Hull, T. M., Gallagher, J. S. and Zweibel, E. G. (2016), Equipartition and cosmic ray energy densities in central molecular zones of starbursts, MNRAS 457(1), L29–L33. Bibliography 270

Yoon, H. and Cho, J. (2019), Eﬀects of Turbulence Driving and Sonic Mach Number on the Davis-Chandrasekhar-Fermi Method, ApJ 880(2), 137.

Yoshida, N., Omukai, K. and Hernquist, L. (2008), Protostar Formation in the Early Universe, Science 321, 669–.

Yusef-Zadeh, F., Morris, M. and Chance, D. (1984), Large, highly organized radio structures near the galactic centre, Nature 310(5978), 557–561.

Zandanel, F., Pfrommer, C. and Prada, F. (2014), On the physics of radio haloes in galaxy clusters: scaling relations and luminosity functions, MNRAS 438(1), 124– 144.

Zaroubi, S., Jelic, V., de Bruyn, A. G., Boulanger, F., Bracco, A., Kooistra, R., Alves, M. I. R., Brentjens, M. A., Ferriere, K., Ghosh, T., Koopmans, L. V. E., Levrier, F., Miville-Deschenes, M. A., Montier, L., Pandey, V. N. and Soler, J. D. (2015), Galactic interstellar ﬁlaments as probed by LOFAR and Planck, MNRAS 454, L46–L50.

Zatsepin, G. T. and Kuz’min, V. A. (1966), Upper Limit of the Spectrum of Cosmic Rays, Soviet Journal of Experimental and Theoretical Physics Letters 4, 78.

Zeeman, P. (1897), The Eﬀect of Magnetisation on the Nature of Light Emitted by a Substance, Nature 55, 347.

Zhang, C.-P. and Li, G.-X. (2017), Mass-size scaling M r1.67 of massive star- ∼ forming clumps – evidences of turbulence-regulated gravitational collapse, MN- RAS 469(2), 2286–2291.

Zhang, Q., Qiu, K., Girart, J. M., Liu, H. B., Tang, Y.-W., Koch, P. M., Li, Z.-Y., Keto, E., Ho, P. T. P., Rao, R., Lai, S.-P., Ching, T.-C., Frau, P., Chen, H.-H., Li, H.-B., Padovani, M., Bontemps, S., Csengeri, T. and Ju´arez,C. (2014), Magnetic Fields and Massive Star Formation, ApJ 792, 116.

Zheng, Z., Ofek, E. O., Kulkarni, S. R., Neill, J. D. and Juric, M. (2014), Probing the Intergalactic Medium with Fast Radio Bursts, ApJ 797, 71.

Zhuravleva, I., Allen, S. W., Mantz, A. and Werner, N. (2018), Gas Perturbations Bibliography 271

in the Cool Cores of Galaxy Clusters: Eﬀective Equation of State, Velocity Power Spectra, and Turbulent Heating, ApJ 865(1), 53.

Zhuravleva, I., Churazov, E., Schekochihin, A. A., Allen, S. W., Ar´evalo, P., Fabian, A. C., Forman, W. R., Sanders, J. S., Simionescu, A., Sunyaev, R., Vikhlinin, A. and Werner, N. (2014), Turbulent heating in galaxy clusters brightest in X-rays, Nature 515(7525), 85–87.

Zweibel, E. G. (2006), Evolution of magnetic ﬁelds at high redshift, Astronomische Nachrichten 327, 505.

Zweibel, E. G. (2013), The microphysics and macrophysics of cosmic rays, Physics of Plasmas 20(5), 055501.

Zweibel, E. G. and Shull, J. M. (1982), Conﬁnement of cosmic rays in molecular clouds, ApJ 259, 859–868.

Zwicky, F. (1933), Die Rotverschiebung von extragalaktischen Nebeln, Helvetica Physica Acta 6, 110–127.

Zwicky, F. (1937), On the Masses of Nebulae and of Clusters of Nebulae, ApJ 86, 217. 272

Nomenclature

Physical constants

c speed of light in a vacuum 2.99792458 1010 cm s−1 × e electron charge 4.8032068 10−10 esu × g acceleration due to Earth’s gravity 980.665 cm s−2 G gravitational constant 6.67259 10−8 cm3 g−1 s−2 × h Planck constant 6.6260755 10−27 erg s × − ~ reduced Planck constant 1.05457266 10 27 erg s × k Boltzmann constant 1.380658 10−16 erg k−1 B × m electron mass 9.1093897 10−28 g e × m proton mass 1.6726231 10−24 g p × Astronomical units

pc parsec 3.0857 1018 cm × ly light year 9.461 1017 cm × 33 M solar mass 1.9891 10 g × 10 R solar radius 6.960 10 cm × 33 −1 L solar luminosity 3.9 10 erg s ×

Cosmological parameters (Planck 2015)

H Hubble constant (67.8 0.9) km s−1 Mpc−1 0 ± n tilted scalar spectral index 0.968 0.006 s ± Ω matter density parameter 0.308 0.012 m ± σ amplitude of mass ﬂuctuations at 8h−1 Mpc 0.815 0.009 8 ± Nomenclature 273

Greek symbols

α spectral index

αf fine-structure constant β ratio of relativistic speed to the speed of light χ magnetic concentration parameter χ ionisation fraction γ Lorentz factor ∆ change in ∆+, ∆++ Delta resonances δ index for magnetic field diffusion

B energy of magnetic ﬁeld

ν Stokes emission at frequency ν

ph energy of Compton cooling

CR,i CR energy density across all components

CR energy density of CRs

CR,MW energy density of CRs in the Milky Way ISM

I emission coeﬃcient of Stokes intensity I

Q emission coeﬃcient of Stokes intensity Q

U emission coeﬃcient of Stokes intensity U

V emission coeﬃcient of Stokes intensity V Γ(...) Gamma function

ΓHE,i spectral index of HE component for i species

ΓLE,i spectral index of LE component for i species

γe electron Lorentz factor

γi low energy cut-oﬀ of relativistic electrons

γp proton Lorentz factor th γp Lorentz factor of a proton at threshold energy

γA adiabatic index for cosmic rays, value 4/3 η magnetic scaling factor η magnetic diﬀusivity κ absorption coeﬃcient of Stokes intensity I

κ¯π average pion-production inelasticity Nomenclature 274

K5/3 Bessel function of order 5/3 Λ Coulomb factor ln Λ Coulomb logarithm λ wavelength of radiation

λ1 CR resonant scattering length scale parallel to background magnetic ﬁeld

λtd turbulent decay length scale µ mean molecular weight µ˜ cosine of pitch angle ν frequency of radiation

νc critical frequency Ω solid angle

Ωin solid angle of incoming CRs

Ωr energy density of radiation

Ωm energy density of matter

Ωk curvature

ΩΛ energy density of dark energy ϕ angle of polarisation (PA)

ϕ0 intrinsic angle of polarisation ψ angle of the ﬁlament to the sky plane

Πlin degree of linear polarisation

Πcir degree of circular polarisation ρ plasma density

σR standard deviation of RM

σT Thomson cross-section ion σe,x cross-section associated with ionisation process x between electrons and neutral medium ion σp,x cross-section associated with ionisation process x between protons and neutral medium

σpπ total inelastic pp interaction cross section ion σH (Ee; E1) cross section for the production of knock-on electrons of energy Ee

due to an initial CR energy E1 σv rate per particle of momentum exchange between ions and neutral h i particles, averaged over a thermal distribution Nomenclature 275

σex collisional excitation cross section σion ionisation cross section

σ[π±] inclusive cross section for the formation of charged pions

ςR standard deviation of RM across the simulated sky plane τ optical depth

τtd turbulent decay time scale Θ(γ γ ) step function − i Θ(s) weighting factor of electrons contributing to Faraday rotation θ angle between magnetic ﬁeld vector and line-of-sight

θin pitch angle of incoming particle’s velocity and magnetic ﬁeld orientation ψ azimuthal angle on a Poincar´esphere ω angular frequency

ωB electron gyro-frequency

ωc cyclotron frequency

ωL CR gyro-frequency

ωp plasma frequency Υ(s) local fraction of non-thermal electrons υ absorption coeﬃcient of Stokes intensity V ϕ polarisation angle (PA)

ξ0, ξ± multiplicities of neutral & charged pions

ξ(˜µin, µ˜) combined focusing and mirroring eﬀects ζ contrast factor ζH ionisation rate H,min ζLECRs ionisation rate due to the minimum W98 LECR spectrum H,max ζLECRs ionisation rate due to the maximum M02 LECR spectrum H ζHECRs ionisation rate due to HECRs above a GeV ζion diﬀerential direct ionisation rate

Physics notations

A normalisation factor (`) auto-correlation function A a index of cloud density proﬁle Nomenclature 276

a cosmological scale factor

ai area covering factor in ith direction b slope of best-ﬁt power law relation

bC cooling rate via electron-Coulomb interactions

bﬀ cooling rate due to electron Bremssthrahlung (free-free)

bion cooling rate due to interactions with Hydrogen per CR electron

brad Compton and synchrotron cooling rates

be(γe, s) total cooling rate b(E, s) cooling coefficient B magnetic field B magnetic field strength

B0 mean ISM value of magnetic ﬁeld strength

Bkrms root-mean-square of line-of-sight magnetic ﬁeld strength

Bk line-of-sight magnetic ﬁeld strength

Bω Rayleigh-Jeans intensity normalisation factor for relativistic electron energy spectrum C c1, c2, c3 rate coefficients of chemical tracers in molecular clouds D(E, s) diffusion coefficient

D0 effective, empirical normalisation to the diffusion coefficient

Dt turbulent diﬀusivity DD statistic in KS test dispersion measure D d order number E electric ﬁeld E energy

E0 reference energy inn ˜HE,i

Emax maximum energy

Eq energy grid point th Ep threshold kinetic energy for the pp interaction energy a ‘knock-on’ electron contributes towards heating Eh EB binding energy f Faraday rotation coeﬃcient f volume ﬁlling factor Nomenclature 277

fU fraction of CR energy density attributed to the HE component

fH2 molecular Hydrogen fraction

Fe,i electron-ion frictional force [...] Fourier Transform F F (z) Airy integral

FX X-ray ﬂux

gB Gaunt factor g normalisation of binding energy g Faraday conversion coefficient between Stokes intensities Q and V g(Ψ) pitch-angle distribution specific heating rate H h Faraday conversion coefficient between Stokes intensities U and V I Stokes I intensity

Iν speciﬁc intensity

Itot total polarised Stokes intensity

Ilin linear polarised intensity

Icir circular polarised intensity

IX X-ray intensity radio intensity IR ⊥ contribution from the orthogonal magnetic field fluctuations I ib bin index J total angular momentum (λ ) dimensionless parallel fluctuation term J 1 j current j CR (in)flux

K5/3(ξ) modiﬁed Bessel function

κν transfer matrix k wavenumber

kc wavenumber normalisation

kxx rate coeﬃcient for process ID xx

kT rate coeﬃcient related to T process

kP rate coeﬃcient related to P process

kI rate coeﬃcient related to I process Nomenclature 278

kc wavenumber normalisation l correlation length l separation length between PAs orbital angular momentum L L characteristic length scale

Lint integral length scale

LX X-ray luminosity

MJ Jeans mass

mS electron spin quantum number

mL orbital angular momentum quantum number

mJ total angular momentum quantum number m quantum number

mc cloud mass fraction

mπ0 neutral pion rest mass N number N chemical abundance

NH column density

Npair number of unique pairs in a data set of Np individual points

Nbins number of bins normally-distributed ﬂuctuations N n0 normalisation of CR heating rate by Alfv´enwaves

nc density of ﬁlament ridge

np proton number density

ne electron number density

n¯e mean electron number density

nH number density of all gas within the cloud, whether atomic or molecular Hydrogen

nB baryonic number density

ne,th thermal electron number density

ne,nt non-thermal electron number density

nions number density of ions

n˜HE,i normalisation of high-energy (HE) component

n˜LE,i normalisation speciﬁed by the HE spectrum at 1 GeV (at E0) Nomenclature 279

ne,1 number density of primary electrons

ne,2 number density of secondary electrons

nc density of ﬁlament ridge ˜ observable on the lattice grid k Ok P power P pressure

Pc CR pressure

Pµµ FP coefficient Pˆ(k) power spectrum p electron distribution index p probability value p index of density profile p¯ index of magnetic field profile q index of mean magnetic field strength across filaments q absorption coefficient of Stokes intensity Q Q Stokes Q intensity Q injection of CRs/source term e|q,r Q(E, s) source term (CR injection) had Qe injection of CR electron secondaries K Qe knock-on injection term Alfvénheating power QA (s) ionisation heating power QI rotation measure R R ratio of chemical abundances

Rflat characteristic length scale of flat profile or filament width

re classical electron radius r radius

rL CR gyro-radius around magnetic ﬁeld

rL,0 gyro-radius of a 1-GeV CR proton in a 5 µG magnetic ﬁeld

rmax maximum index on spatial grid

rvir virial radius S(E, s) sink term (CR absorption/attenuation) S total spin Nomenclature 280

Sν surface brightness or source function

Sp(γp, s) proton absorption rate by the pp process (`) angular dispersion function of order d Sd sc size of cloud

sr spatial grid point s distance of propagation ∆s step size T temperature

Teﬀ eﬀective temperature

Teq equilibrium temperature t time

td decay time scale

te cooling time scale U Stokes U intensity

UB magnetic energy density u absorption coeﬃcient for Stokes intensity U V Stokes V intensity v absorption coeﬃcient for Stokes intensity V v velocity

vA Alfv´enspeed

vin speed of incoming particle

vk velocity component along magnetic ﬁeld vector

v⊥ velocity component perpendicular to the magnetic ﬁeld vector

xi ionisation fraction

xi,0 normalisation for ionisation fraction x ionisation process

Y solar metallicity Z random number Z magnitude of particle charge | | z redshift Nomenclature 281

Acronyms

AGN active galactic nuclei AIA Atmospheric Imaging Assembly ALMA Atacama Large Millimetre/Submillimetre Array AMR adaptive mesh reﬁnement AR autoregression ATCA Australia Telescope Compact Array Athena Advanced Telescope for High Energy Astrophysics ATNF Australia Telescope National Facility CARMA Combined Array for Research in Millimetre-wave Astronomy CMB cosmic microwave background CDF cumulative distribution function CN cyanide CO carbon monoxide CPRT Cosmological Polarised Radiative Transfer CR cosmic ray CSIRO Commonwealth Scientiﬁc and Industrial Research Organisation CTA Cherenkov Telescope Array DCF Davis-Chandrasekhar-Fermi DM dispersion measure EHT Event Horizon Telescope EM electromagnetic ESA European Space Agency EVN European VLBI Network FAST Five-hundred-metre Aperture Spherical radio Telescope FFT fast Fourier transform FLRW Friedmann-Lemaˆıtre-Robertson-Walker FP Fokker-Planck FRB fast radio burst GADGET GAlaxies with Dark matter and Gas intEracT GBT Green Bank Telescope GCMHD+ Galactic Chemodynamics MagnetoHydroDynamics Nomenclature 282

GK Goldreich-Kylafis GMRT Giant Metrewave Radio Telescope GRB gamma-ray burst HECR high-energy cosmic ray HI hydrogen line in 21 cm HMI Helioseismic and Magnetic Imager HR high resolution ICM intracluster medium IDL Interactive Data Language IGM intergalactic medium IRAM Institut de Radioastronomie Millimétrique ISM interstellar medium JVLA Jansky Very Large Array KS Kolmogorov-Smirnov ΛCDM Lambda–Cold Dark Matter LAPACK Linear Algebra PACKage LAT Large Area Telescope LECR low-energy cosmic ray LMT Large Millimetre Telescope LOFAR Low Frequency Array LOS line-of-sight LSS large-scale structure MC molecular cloud MHD magneto-hydrodynamic MT Mersenne Twister MWA Murchison Widefield Array NASA National Aeronautics and Space Administration NRAO National Radio Astronomy Observatory NVSS NRAO VLA Sky Survey ODE ordinary differential equation OHM hydroxyl Megamasers PDF probability density function PI polarised intensity Nomenclature 283

PRNG pseudo-random number generator PRT polarised radiative transfer QCD quantum-chromo-dynamics RK Runge-Kutta RM rotation measure RMF rotation measure ﬂuctuation rms root mean square ROSAT ROentgen SATellite RT radiative transfer SDC scalar divergence cleaning SF structure function SKA Square Kilometre Array SDO Solar Dynamics Observatory SiO silicon monoxide SKA Square Kilometre Array SMA SubMillimetre Array SNR supernova remnant SPMHD Smoothed Particle MagnetoHydroDynamics SOHO Solar and Heliospheric Observatory VLA Very Large Array VLBA Very Long Baseline Array VLBI Very Long Baseline Interferometry WMAP Wilkinson Microwave Anisotropy Probe WSRT Westerbork Synthesis Radio Telescope

Superscripts/Subscripts

0 initial aniso anisotropic cir circular co co-moving CR cosmic ray e electron Nomenclature 284

eq equilibrium in incoming HE high-energy IR infrared iso isotropic LE low-energy lin linear MW Milky Way max maximum min minimum nt non-thermal obs observed OP optical ord ordered p proton re relativistic reg regular rms root-mean-square td turbulent decay th thermal tot total turb turbulent vir virial x in the x-direction y in the y-direction z in the z-direction xy in the xy plane xz in the xz plane yz in the yz plane parallel k perpendicular ⊥