Cosmic ray shock acceleration in galactic and extragalactic sources

Transcription

1 Faculty of Sciences Academic year Cosmic ray shock acceleration in galactic and extragalactic sources Matthias Vereecken Promotor: Prof. Dr. D. Ryckbosch Supervisor: Dr. A. Meli Submitted in partial fulfillment of the requirements for the degree of Master in Physics and Astronomy

2

3 Acknowledgements The work presented in this thesis would of course not have been possible without the many people offering their advice, guidance and support. First of all I would like to thank my supervisor, Dr. Athina Meli, for her many hours spent guiding and helping me understand, process and apply the theory of cosmic ray acceleration and propagation. It goes without saying that without her dedication, this thesis would not have been half as good. Next, I would like to thank my promotor, Prof. Dr. Dirk Ryckbosch, for guiding me towards this very exciting subject and making it possible to spend my year working on it at the INW. Also, I want to express my gratitude to the other people in the IceCube group, who served as an inspiration for my work in the (bi)weekly meetings. I want to explicitly thank my fellow students Céline, Veronique and Dieter, with whom I was able to share my experiences (and problems) during not only this year, but the previous years as well. Due to them, the stay at the INW was so much more enjoyable. I want to thank my parents and family for supporting me, listening to me and making it possible to devote myself fully to my studies and this thesis in particular. Finally, I end by thanking Bert, for suffering through my (possible) occasional nagging and without whom this year would not have gone nearly as smooth. iii

4

5 Contents List of Figures ix Introduction 1 1 Cosmic rays An introduction Discovering the cosmic rays Measuring the cosmic ray spectrum Detection methods The cosmic ray spectrum Composition Properties of cosmic rays Confinement within the galaxy Energy density and power Sources of acceleration Source properties The Hillas criterion Galactic and extragalactic sources Ultra-high-energy cosmic rays and the end of the spectrum Multi-messenger astrophysics The IceCube Neutrino Observatory IceCube experiment IceTop experiment I Acceleration 25 2 Principles of plasma physics Particles in a plasma Helix orbit Magnetic moment invariance Magnetic mirror Frozen-in magnetic fields Shock waves Non-relativistic shocks Relativistic shocks Laboratory plasma experiments v

6 Contents vi 3 Acceleration mechanism The Fermi mechanism General concept Second order Fermi acceleration First order Fermi acceleration Advanced concepts Diffusive transport Relativistic shocks The role of the magnetic field Reflection Shock-drift acceleration Particle scattering Solving the shock jump conditions Properties of the acceleration process Energy gain Acceleration time scale Injection First adiabatic invariant Modifications Non-linear effects Shock profile Density of accelerated particles Particle shock acceleration simulations Diffusive shock acceleration: large scale versus Monte Carlo simulations Implementation of a Monte Carlo code Test particle approximation Reference frames The shock parameters Injection Diffusion of high energy particles the Monte Carlo method Shock crossing Boundary conditions Discussion Summary: program flowchart Results Simulating and fitting Mild relativistic shocks Parallel shocks Oblique shocks Compression ratio Small angle scattering Time scale Summary Highly relativistic parallel shocks Large angle scattering Hybrid scattering

7 Contents vii 5.4 Highly relativistic oblique shocks Large angle scattering Hybrid scattering Discussion II Propagation Propagation of cosmic rays The transport equation Diffusion-loss equation Convection Adiabatic momentum loss rate Diffusive reacceleration Spallation and decay GALPROP A galactic propagation code Introduction Input Solution to the propagation equation Emission spectra Solar modulation Results Method Results Parameters Cosmic ray spectrum Electron spectrum B/C ratio Be/ 10 Be ratio Gamma rays Conclusion Conclusion 135 A Galprop numerical solution 137

8

9 List of Figures 1.1 An electroscope showing induction [2] Plot of the original measurements by Victor Hess[10] An extensive air shower initiated by a high energy cosmic ray Observed cosmic ray spectrum combined from several experiments[20] Compiled data by Particle Data Group[21] Cosmic ray relative abundance of elements compared to the solar system relative abundances. Data normalised to Si = GCR data for elements heavier than He from ACE/CRIS data, solar system data by Lodders[24]. Figure from ACE news[25] Composition of primary cosmic rays[21] Data on the amplitude and phase (= right ascension of the direction of maximum CR intensity) of the first harmonic of the cosmic ray anisotropy, taken from Erlykin and Wolfendale[30] The Hillas plot, showing the maximum energy up until which particles can be contained in different astrophysical sources [35] Supernova shock wave expanding into the interstellar medium or stellar wind from the precursor star[38] Jet-disk systems as possible cosmic ray sources : microquasar, AGN, GRB[45, 38] Different ways of finding the composition of UHECR s[46] The multi-messenger connection[17] The IceCube detector, figures from the official IceCube website[54] The highest energy neutrino recorded, with an estimated energy of ev. Each blob is the signal from a DOM, the size signifying its energy recorded and the colour giving the tim[54] The IceTop array[54] Helix motion of a positively charged particle in a homogeneous magnetic field and no electric field[56] Helix motion and drift of a proton and an electron in a homogeneous magnetic field and an electric field perpendicular to it[56] An axially symmetric magnetic field with increasing strength along the z-axis[56] An example of a magnetic mirror being used to create a magnetic trap[56] Illustrating magnetic flux freezing[22] Two important reference frames describing the shock The shock rest frame, also called the normal incidence frame ix

10 List of Figures x 2.8 The shock rest frame for a parallel shock Two possible DHT frames describing the shock Fermi s original suggestion for an acceleration mechanism The shock front setup for first order Fermi acceleration Solutions of the compression ratio r, the downstream magnetic field inclination θ u2 and the downstream flow speed deflection angle θ u2 with an upstream magnetic field inclination angle of 5 for different values of the Alfvénic (M A ) and sonic (M S ) Mach number. Figure taken from Summerlin and Baring[63] The shock rest frame for a dynamically unimportant field, where now both the upstream and downstream flow are along the x-axis Comparing the acceleration rate (related to the time scale) with that of ionisation losses gives restrictions on which mechanisms are fast enoughlongair[22] Plot of the velocity profile for different shock thickness parameters for a shock with compression ratio 4. The position coordinate is in units of λ(γ 1 ) and the velocity in units of the upstream flow speed Density distribution of the accelerated particles in function of a normalised distance defined as ξ = ( V shock /κ cos 2 ψ ) x. Figure taken from Gieseler et al.[96] The helix orbit and gyrocentre[100] Describing the scattering of the particle by diffusion of the tip of the momentum vector on a sphere[82]. The original momentum direction is denoted by Ω There is an obvious problem with the original algorithm. The simulation uses a parallel shock with V 1 = 0.1, r = 4 and injects 10 5 particles The problem can be identified by showing the distribution of particles versus position (in log-scale!) and pitch angle. The huge distances crossed by particles with µ 0 causes there overabundance of such particles. The simulation uses a parallel shock with V 1 = 0.1, r = 4 and injects 10 5 particles The correction seems to solve the problem. The simulation uses a parallel shock with V 1 = 0.1, r = 4 and injects 10 5 particles Illustration of the effect of the used correction. The distances crossed by particles with µ 0 are more in line with the other and there is no more overabundance of such particles. The simulation uses a parallel shock with V 1 = 0.1, r = 4 and injects 10 5 particles Schematic of the flow of the Monte Carlo code Energy spectra for mild relativistic parallel shocks. All simulations use r = 4 and inject 10 5 E particles. The energy is expressed in log 10 γ = log 10 m. For comparison the non-relativistic case of V 1 = 0.1 is shown. Observe the low efficiency of the acceleration compared to the other cases Pitch angle distribution of the particles during shock crossing in the shock rest frame for mild relativistic parallel shocks. Shown is the pitch angle in the old frame, before the handling of the shock. In green are the particles crossing from downstream to upstream, in blue the ones from upstream to downstream

11 List of Figures xi 5.3 The average energy of the particles for mild relativistic parallel shocks recorded in the shock frame versus the number of shock crossings, while they are being accelerated Distribution of the total amount of shock crossings a particle underwent for mild relativistic parallel shocks The spatial distribution of a shock near the vicinity of that shock, with the position in units of downstream mean free path. The simulation used a parallel shock with V 1 = 0.1 and injects 1000 particles, which leads to recorded positions already Energy spectra for a non-relativistic and a mild relativistic oblique shock. All simulations use r = 4 and inject 10 5 particles Pitch angle distributions in the DHT frame for a non-relativistic and a mild relativistic oblique shocks at shock crossing, recorded at the moment of scattering. In green are the particles crossing from downstream to upstream, in blue the ones from upstream to downstream Pitch angle distributions for a non-relativistic and a mild relativistic oblique shock at shock crossing, recorded after determining the new pitch angle The average energy of the particles for a non-relativistic and a mild relativistic oblique shock recorded in the shock frame versus the number of shock crossings, while they are being accelerated. Can be compared with the results for parallel shocks in figure Distribution of the total amount of shock crossings a particle underwent for a nonrelativistic and a mild relativistic oblique shock Distribution of the total amount of reflections particles underwent for a non-relativistic and a mild relativistic oblique shocks. The histograms are normalised to 1, so that the ratio can not be compared to the crossing distribution directly The influence of the compression ratio shown for both a parallel and an oblique shock with V 1 = For mild relativistic shocks, small angle and large angle scattering are equivalent. Illustrated for a parallel shock Time scale for the acceleration mechanism for different mild relativistic shocks. Simulation data is taken in shock cycle Summary of the behaviour of the spectral index for mild relativistic shocks. All shocks injected 10 5 particles and had a compression ratio of Energy spectra for parallel relativistic shocks with different shock speeds. All simulations use r = 3 and inject 10 6 particles. The energy is expressed in log 10 γ = log 10 E m The origin of the flat spectra can easily be traced to the stepped spectrum going to extremely high energies, which in reality would not be contained. The spectrum shown is for a parallel shock with Γ 1 = Pitch angle distribution of the particles during shock crossing in the shock rest frame for relativistic parallel shocks with large angle scattering. Shown is the pitch angle in the old frame, before the handling of the shock The average energy of the particles for relativistic parallel shocks recorded in the shock frame versus the number of shock crossings, while they are being accelerated. 106

12 List of Figures xii 5.20 Time scale for the acceleration mechanism for different relativistic parallel shocks with large angle scattering. Simulation data is taken in shock cycle Energy spectra for parallel relativistic shocks with hybrid scattering. All simulations use r = 3 and inject 10 5 particles Pitch angle distribution of the particles during shock crossing in the shock rest frame for relativistic parallel shocks for hybrid scattering. Shown is the pitch angle in the old frame, before the handling of the shock The average energy of the particles for relativistic parallel shocks with hybrid scattering recorded in the shock frame versus the number of shock crossings while they are being accelerated Energy spectra for relativistic oblique shocks with large angle scattering. All simulations use r = 3 and inject 10 6 particles. The power law fits here are purely to give a sense of scale Pitch angle distribution of the particles during shock crossing in the DHT frame for relativistic oblique shocks with large angle scattering. Shown is the pitch angle in the old frame, before the handling of the shock Pitch angle distribution of the particles during shock crossing in the DHT frame for relativistic oblique shocks with large angle scattering. Shown is the pitch angle in the new frame, after the handling of the shock The average energy of the particles for relativistic oblique shocks with large angle scattering recorded in the DHT frame versus the number of shock crossings, while they are being accelerated Energy spectra for relativistic oblique shocks with hybrid scattering. The simulations use r = 3 and inject 10 5 particles Pitch angle distribution of the particles during shock crossing in the DHT frame for relativistic oblique shocks with hybrid scattering Distribution of the total amount of reflections particle underwent during the acceleration process in oblique relativistic shocks with hybrid scattering Distribution of the total amount of meetings a particle underwent during the acceleration process in oblique relativistic shocks with hybrid scattering. Note the log scale! Flux through an infinitesimal element in phase space Individual spectra found from the GALPROP simulation and a fit to the power law part The cosmic ray spectra found from the GALPROP simulation, compared with data from the different experiments quoted in the text The electron spectrum found from the GALPROP simulation, compared with data from the different experiments quoted in the text Ratios of secondary to primary cosmic rays present a check of the basic properties of the simulated propagation. These plots show the results from the GALPROP simulation compared with experimental data quoted in the text The gamma ray spectrum

13 Abstract Cosmic rays have, since their discovery by Victor Hess in 1912, been the subject of much study. The cosmic ray spectrum is an almost perfect power law over about ten decades in energy and thirty decades in flux. In order to study this spectrum and its properties experimentally, many different kinds of experiments have been build, each revealing a new part of the puzzle. Also, many theoretical efforts have gone into explaining the origin of these cosmic rays. Still, after a hundred years, all of their properties are not yet understood. This thesis deals with the origin of cosmic rays as we observe them on Earth. The origin of these highly energetic particles must be found in several astrophysical acceleration sites. At energies below ev, such sources can be identified within the galaxy, the most important example being supernovae. At higher energies, extragalactic sources need to be considered, the primary candidates being active galactic nuclei and gamma ray bursts. The process responsible for creating these high energy particles at these sites is thought to be diffusive shock acceleration, first proposed by Fermi. This acceleration mechanism shall be the subject of study in the first part of the thesis. A Monte Carlo code is developed to simulate the diffusive shock acceleration. This allows a systematic study of the properties and resulting spectrum of the acceleration mechanism. Cosmic rays undergo many different processes during their propagation through the Universe, so that their spectrum is modified along the way. For energies below ev, the cosmic rays are confined within the galaxy. Their propagation will be the subject of the second part of the thesis. With the help of the publicly available code GALPROP, the necessary cosmic ray spectrum at injection by the acceleration sources is found such that experimental results on the cosmic ray, electron and gamma ray spectra are reproduced. 1

14

15 Chapter 1 Cosmic rays An introduction It has long been established that the Earth is bombarded from space by high energy particles we call cosmic rays. The energy spectrum of these particles follows a power law that extends over many orders of magnitude in energy and flux. In this chapter the history[1] and the properties of the cosmic rays will be discussed. 1.1 Discovering the cosmic rays Already in 1785 Coulomb found that electroscopes (devices used to detect electric charge, see figure 1.1) spontaneously discharged, not due to defects but due to the air itself. Using better insulation technology, Faraday confirms this result in In 1879 Crookes measured that the speed of discharge decreased when the air pressure drops, showing that ionised air is directly responsible. In 1898 Pierre and Marie Curie discovered radioactive decay by studying Polonium and Radium. When these radioactive elements were put next to an electroscope they caused discharges in them. Since then electroscopes were used as an instrument to measure the level of radioactivity. These discoveries set the stage for the discovery of cosmic rays and eventually to the exciting field known as astroparticle physics. Figure 1.1: An electroscope showing induction [2] Still, the source of the natural radioactivity that caused the air to be ionised was not known. This ionisation was relatively high: 10 to 20 ions per cubic centimetre of air were created every second. Certainly there was some radiation coming from the Earth and the Sun, but the question was whether there were other sources, and which source was the dominant one. The consensus was 3

16 Chapter 1. Cosmic rays An introduction 4 that the soil was the primary source and that the radiation consisted of γ-rays, since α- and β-rays could be easily shielded from. Experiments however (Rutherford & Cooke[3, 4] and McLennan & Burton[5] 1, ) showed that shielding barely influenced the ionisation rate[7, 8]. In the beginning of the 20 th century many experiments were devised to determine the source of the radiation. Some of them were inconclusive, dominated by experimental errors, or took the wrong conclusion. An example of the latter is an experiment by Elster & Geitel in 1908 who observed a 28% drop when taking a detector to the bottom of a salt mine and concluded that the radiation originates in the earth (in agreement with literature), but that certain environments (water, salt mines) were free from radioactive materials and therefore caused a lower rate. In Wulf measured the ionisation at the surface and at the top of the Eiffel tower, expecting an exponential decrease if the soil was the source. No conclusive evidence was found however. It was Pacini who had a breakthrough by measuring ionisation at different altitudes on mountains, over a lake and over the sea. In 1910 he found that in the hypothesis that the origin of penetrating radiations is only in the soil [...] it is not possible to explain the results obtained. In 1911 he does a measurement under water and finds a significant reduction. He writes in a note called Penetrating radiation at the surface of and in water, in Nuovo Cimento (February 1912): Observations carried out on the sea during the year 1910 led me to conclude that a significant proportion of the pervasive radiation that is found in air had an origin that was independent of the direct action of active substances in the upper layers of the Earth s surface.... [To prove this conclusion] the apparatus... was enclosed in a copper box so that it could be immersed at depth.... Observations were performed with the instrument at the surface, and with the instrument immersed in water, at a depth of 3 m.[9] In order to improve upon Wulf s results, people looked to balloon experiments. The first of these was Gockel in 1909, who ascended to 4000 m and found no decrease in ionisation. After that came Victor Hess. In 1911 he did two balloon experiments (again with electroscopes) and found no variations up to 1300 m. In 1912 he went up to 5200 m and proved that the ionisation, after going through a dip, increases strongly with height (see figure 1.2). The Sun was ruled out as a source by also doing a flight during a near-total eclipse, finding the same results. He concluded that the ionising radiation had an extra-terrestrial origin and called it die Höhenstrahlung. These results were later confirmed by Kolhörster. For his discoveries Hess was awarded with the Nobel prize in In the 1920 s Millikan questioned the existence of the Höhenstrahlung. This doubt was based on new measurements up to m in Texas, where less radiation was measured than by Hess and Kolhöster. However, they did not know about the existence of a geomagnetic effect. In 1926, after new experiments, he changes his opinion and firmly establishes the existence of the cosmic rays. Millikan believed that the cosmic rays were γ-rays coming from the nucleosynthesis of elements like helium and oxygen. In 1927 J. Clay found that the cosmic ray flux varied with latitude, suggesting the particles were influenced by the geomagnetic field and thus charged, not neutral. Also, it was discovered that the cosmic rays are highly penetrative, a property not shared by photons. In 1930, 1 Both Physical Review papers are actually part of Minutes of the Eighteenth Meeting [6] and cannot be found separately.

17 Chapter 1. Cosmic rays An introduction 5 Figure 1.2: Plot of the original measurements by Victor Hess[10]. Bruno Rossi predicted an East-West effect in the cosmic ray intensity[11]: charged particles of a given momentum are deflected by a magnetic field (F = q v B). Particles will, depending on the charge, be deflected towards the earth if coming from East, respectively West and deflected away from it if coming from the other direction. Three independent experiments in confirmed that the cosmic rays intensity was higher coming from the West[12, 13, 14], meaning that the cosmic rays are primarily positive. During these experiments Rossi discovered something called cosmic ray showers [14]. He noticed that when two Geiger counters (a rather new invention) were placed far apart, every now and then there would be a coincidence between the signals. This suggested that a large shower of particles hit the detectors. However, he did not study these in detail so they were later rediscovered by Auger in In the following years and decades it was established by balloon, satellite and surface experiments that the primary cosmic rays (those hitting the Earth atmosphere from outer space) consisted mainly of protons ( 86%) along with α-particles ( 11%), some heavier nuclei ( 1%), electrons ( 2%) and a fraction of antiparticles (e +, p,... ), whereas the secondaries (produced by primaries interacting with the atmosphere) were mainly electrons, photons and muons (which were themselves discovered with the help of cosmic rays). The picture that was formed by all these studies is the following: the Earth is continuously bombarded by particles up to very high energy from outer space. When these cosmic ray particles - primarily protons - interact with the Earth s atmosphere (this first happens at a height of about 15 km), the high energy collisions produce lots of secondaries like other protons, neutrons and pions. These secondaries themselves are also highly energetic and interact again with the atmosphere so that a cascade is formed. Most of the produced particles are unstable and decay (e.g. neutral pions to photon pairs, charged pions to muons). Muons are the most penetrative component and the high energy ones reach the surface in large numbers while the others decay before that. Photons interact with nuclei to produce electrons. Electrons on the other hand can emit photons. The shower thus consists of a hadronic component and an electromagnetic component. Both these shower components continue to grow as they go deeper into the atmosphere

18 Chapter 1. Cosmic rays An introduction 6 until the individual particles start to lack the energy to interact: the shower decreases in size again and eventually dies out. Only part of the shower reaches the Earth s surface, depending on the original energy of the particle. The development of such a shower is shown in figure1.3a of the following section. Since the cosmic ray particles have such high energies that are not otherwise available, they were used extensively in particle physics experiments. Indeed, most of the newly discovered particles before 1950 were done with cosmic ray experiments. Examples of these are the discovery of the positron[15] and the muon[16]. In the 1950 s the accelerator technology had advanced to the point where one could produce highly energetic particles in concentrated and controlled beams, meaning most particle physicists moved away from cosmic rays to the accelerators. The only advantage left for studying cosmic rays was their higher energy, but every time accelerator technology improved, those studies too had to look towards higher energies. In turn the accelerator based experiments were of interest also to the cosmic ray experiments. New knowledge became available on hadronic interactions, which helped to further the understanding of the development of air showers and the detection of particles arriving at the surface. 1.2 Measuring the cosmic ray spectrum Detection methods There are generally two ways of detecting and measuring cosmic rays. The first one is direct detection. This involves putting a detector on a balloon or a satellite and raise it above the height where cosmic rays first interact with the atmosphere. The advantage is obviously that one measures the cosmic rays themselves directly, giving immediate information about the type and energy. The disadvantage however, is the limited energy range for which this can be done. As shown below, the cosmic ray flux drops significantly when going to higher energies. Therefore a bigger detector is needed both to capture (and measure the energy of) the more energetic particle and more importantly to see any particle at all! The limit for direct detection is at about 100 TeV. Examples of this type of experiments are ACE, PAMELA and AMS (space based) and BESS and ATIC (balloon based). The above method is contrasted by the indirect detection methods. Here one measures the particles in the air shower created by cosmic rays (see figure 1.3). After detection of a representative part of the shower, the original particle s properties (type, energy, direction) can be reconstructed. This of course makes measuring the properties of cosmic rays much more difficult. Even now, all the properties of these showers are not known yet, which is the downside to having energies far above what accelerator experiments can achieve. For example, there are many models for hadronic cross sections at the highest energies, each leading to a slightly different reconstruction. The advantage however, is that one effectively uses the entire atmosphere as a giant calorimeter. Thus even very low fluxes are still detectable and the entire energy of the original particle can be contained in the detector. Still, for the highest energies one needs years of data taking to get statistics (or even any data at all), making progress slow. All experiments aiming to measure the high-energy tail of the cosmic ray spectrum use such air showers. Several examples of air shower experiments are

19 Chapter 1. Cosmic rays An introduction 7 Auger, Fly s eye, IceTop, KASKADE and KASCADE-Grande. Auger uses both Cherenkov and fluorescence detectors. IceTop, and its main experiment IceCupe, are described in more detail at the end of this chapter The cosmic ray spectrum The cosmic ray spectrum is shown in figure 1.4, combining data from several experiments. The spectrum is an almost perfect power law extending over many orders of magnitude in flux and in energy. The lower energies are easily reached with accelerators nowadays, but the spectrum goes up to energies far above what is possible with modern technology. Notice, for example, the designed LHC energy in the figure. The low energy deviation from a power law, up to about 1 GeV per nucleon, is not important, since it corresponds to the influence of the Sun in the solar system. The size and shape of this deviation depend on the solar activity, the flux of particles decreasing at moments of high solar activity and vice versa - a phenomenon known as solar modulation. It is easily understood as low energy particles feeling the effects of solar wind and magnetic field, while higher energy particles are not bothered by them. The spectrum also contains meaningful structures however. The most obvious and important ones divide the spectrum into different regimes for which the spectrum follows an unbroken power law. Such a power law is parametrised by dn de = A E x, (1.1) where x is called the spectral index. Starting from 10 9 ev up to about ev the spectral index is about 2.7. The end of this region is a feature called the knee, where the spectrum steepens to an index of about 3.0. Around ev, the second knee, the spectrum steepens again to an index of 3.3. Finally at the ankle, at about ev, the spectrum flattens with an index of 2.7. The knee also roughly corresponds to the division between direct and indirect measurements. As a result the regions ev, and ev and above are mostly studied independently. An important note to make is that there are different ways to picture the cosmic ray spectrum. Figure 1.4 uses flux of particles versus particle energy, the most intuitive one when thinking of cosmic ray energies. Other choices exist however, since certain quantities are more natural than others when looking at different physical phenomena[19]. Flux of particles versus particle energy, for example, is relevant for air showers, since it is the total energy of the particle that is deposited in the shower. Another one is number (or flux) of nuclei per energy per nucleons, which is appropriate for propagation calculation since spallation effects from colliding with interstellar gas approximately conserve the energy per nucleon. Rigidity R = pc Ze (units: energy over charge) is also used instead of the energy, since it is the relevant quantity when the gyroradius of the particle is important, like in acceleration (see section 1.4.2) and propagation through a magnetic field. Another possible representation is number of nucleons per energy per nucleon. This is the relevant quantity when looking at the production of secondaries (antiprotons, muons, pions), since these are produced almost purely in nucleon-nucleon interactions (with only a small effect from being in a nucleus). Another way in which representations of the spectrum may differ is in the choice to multiply out the overall power law from the flux. This is done to allow a more detailed study of the

20 Chapter 1. Cosmic rays An introduction 8 (a) The development of an air shower[17]. (b) Detecting an air shower[18]. Figure 1.3: An extensive air shower initiated by a high energy cosmic ray. structure of the spectrum. Again care must be taken of the units, for the choice of energy also affects the multiplied flux. An example of where this is done is in figure 1.5. In figure 1.5a the structure of the knee and ankle (based on air shower measurements) is more clear. Here it is obvious that the knee is not a sharp boundary, but rather a broad region with lots of substructure that forms the transition between two nicely behaving regimes. Getting the data consistent as in this figure is a highly non-trivial task. Up to a few years ago the different experiments had inconsistent normalisation, which is not a big problem when dealing within one experiment alone. Any general conclusion about structure was made impossible however. An effort was done to revise the normalisation of primary particle energy - reconstructed from the air showers and thus model dependent - with the aim of finding agreement between the experiments, with great success. In 1.5b the end of the cosmic ray spectrum is pictured. The question still remains where the spectrum ends exactly and what the origin is, possibilities for which will be discussed below. This is made difficult by the incredibly low fluxes in this area Composition It was already mentioned that the cosmic ray spectrum consists mostly, but not completely, out of protons. Direct detection gives most accurate results here since simple nuclear detection techniques can be used to identify the particles. Indirect detection requires reconstruction of the shower, which contains no exact information about the primary particle. One can then only try to go find the average mass of initiating particles (as a function of energy) and compare with models. The abundances of elements in galactic cosmic rays (GCR) is compared to the solar system abundances in figure 1.6. The exact energy range in this figure is mostly irrelevant, since most particles are at low energy anyway because of the power law. At first glance, it can be seen that the relative abundances of the different elements are incredibly similar (notice the log scale!). Also the well known odd-even effect from nuclear stability is visible. The important differences however are an underabundance of H, He and an overabundance of elements with masses directly

21 Chapter 1. Cosmic rays An introduction 9 Figure 1.4: Observed cosmic ray spectrum combined from several experiments[20].

22 Chapter 1. Cosmic rays An introduction 10 (a) The all-particle spectrum as a function of E (energy-per-nucleus) from air shower measurements. (b) Expanded view of the highest energy portion of the cosmic-ray spectrum. Figure 1.5: Compiled data by Particle Data Group[21]. under those of Fe and C in the GCR. The first of these differences is not really understood, it could be either due to the fact that H is hard to ionise and thus to inject into the acceleration process, or because of a true composition difference at the source[19, 22]. Another possibility is that H and heavier elements are accelerated at different sources[22, 23]. The second difference can be explained by spallation processes caused by Fe, C and O (primaries) interacting with the interstellar medium, creating particles (secondaries) with mass numbers below the original nuclei. The effect is clearly visible, since the produced particles (Li, Be, B and Sc, Ti, V, Cr, Mn) are essentially non-existent as end products of nucleosynthesis. Through this it is possible to find the amount of matter travelled through on the way from the sources to Earth and, with knowledge of the spallation cross sections, to work back and find the composition at the sources. One important tool in studying the propagation of cosmic rays is the B/C ratio. The reason is that C is a pure primary, while B is a pure secondary species directly produced by C spallation. Their ratio thus gives a very direct measurement of the propagation history and is used as one of the first checks when verifying propagation models. For low energies, the fluxes are high enough that it is possible to measure the spectra of individual species with direct detection. This is shown in figure 1.7. Generally it can be concluded that primary particles have flatter spectra than secondary particles, which is seen in e.g. B compared to C and O[26]. This is easy to understand, since the latter ones are created by spallation interactions of the first ones. The products of this reaction always separately have lower energy, so that there are more low energy, and less high energy particles in the spectrum.

23 Chapter 1. Cosmic rays An introduction 11 Figure 1.6: Cosmic ray relative abundance of elements compared to the solar system relative abundances. Data normalised to Si = GCR data for elements heavier than He from ACE/CRIS data, solar system data by Lodders[24]. Figure from ACE news[25]. Figure 1.7: Composition of primary cosmic rays[21].

24 Chapter 1. Cosmic rays An introduction Properties of cosmic rays Confinement within the galaxy Most of the cosmic rays originate from within the galaxy. The reason for this is that the galactic magnetic field contains particles up to very high energy. The large scale magnetic field of the galaxy can be measured in several ways[27]. The first one is Zeeman splitting of spectral lines. A second one is polarisation at infrared, sub-mm and mm wavebands, since dust grains are oriented preferentially along interstellar magnetic fields, so that their thermal emission has linear polarisation, which is best observed in these bands. Another one is polarisation of starlight, which scatters on the orientated dust grains and thus becomes polarised. Also synchrotron radiation from relativistic electrons can be used to deduce the field strength by assuming equipartition and measuring the emission flux. Lastly, one can look at the Faraday rotation of polarised sources, which is the rotation of the position angle of the electric field vector of linearly polarised radio emission on propagating along the magnetic field direction[22]. Combining results from these different sources, one gets a global magnetic field of about 3 µg. From the galactic magnetic field, one can calculate the radius of curvature for particles of certain energies[28]. In a deflection the centrifugal and Lorentz forces on a particle are equal (assuming v B) mv 2 = ZevB p = ZeρB. (1.2) ρ Filling in units, for protons one gets the easy to use equation ρ[m] = pc/(0.3b) [GeV/T]. (1.3) For protons of ev, this gives ρ = /( ) = 30 kpc. (1.4) For protons of ev, this gives ρ 0.5 pc. (1.5) With a galactic radius of about 15 kpc and a thickness of about 0.3 kpc, one can conclude that particles up to ev are confined within the galaxy, with higher energy particles beginning to leak from the galaxy. Since this is true for every galaxy (at least to the order of magnitude), one can say that the spectrum below ev is coming purely from galactic sources. This gives the first possible explanation for the knee of the cosmic ray spectrum, going as follows. Since starting at this energy particles start to leak from the galaxy, one expects the spectrum to steepen. The knee position would then be different for each element, since the radius is dependent on the charge of the particle. The abundances of the different elements in cosmic rays allows us to reveal part of the history of particles as they travelled towards us. From the ratio of secondaries to primaries in cosmic rays, the mean amount of matter travelled through for most of the cosmic rays is determined to be of the order of 5 10 g/cm 2. With the density in the disk of the galaxy being about 1 proton per

25 Chapter 1. Cosmic rays An introduction 13 cm 3, this gives for the length travelled[19] l = X = 1000 kpc. (1.6) m p ρ One can also look at the ratio of radioactive isotopes versus stable isotopes, compared with theoretically known production ratios. One example of such cosmic ray clocks, as they are called, is Be with a lifetime of years. The production ratio of 9 Be to 10 Be should be about 2:1, whereas the ratio found in cosmic rays is about 10:1. From this the propagation time can be easily estimated. Combining data from Be, Al, Cl, and Mn gives a travel time of about years and an average interstellar hydrogen density of about m 3 [29]. If the particles would come in a straight line towards the Earth from within the galaxy, with a dimension of 1 10 kpc the cosmic rays would travel for only years, clearly incompatible with the above number. Combining the above conclusions, together with the observation that the cosmic rays arrive on Earth very isotropically (see figure 1.8), makes it possible to formulate a model for cosmic ray propagation in the galaxy. The galaxy can be seen as a leaky box, in which the cosmic rays diffuse staying confined for a long time and thus isotropising before at some point escaping into extragalactic space. The high energy particles, by virtue of their bigger radius of curvature, diffuse out of the galaxy faster. This can indeed be observed, again from the composition of cosmic rays. From the energy spectra for the different elements, it can be found that the secondary to primary ratio decreases with energy, meaning the high energy particles must escape quicker (and thus have less time to produce the lower energy secondaries). Figure 1.8: Data on the amplitude and phase (= right ascension of the direction of maximum CR intensity) of the first harmonic of the cosmic ray anisotropy, taken from Erlykin and Wolfendale[30]

26 Chapter 1. Cosmic rays An introduction Energy density and power Simply integrating the measured energy spectrum (and correcting for the solar modulation) gives the total density of cosmic rays. The protons have an energy density of about 0.83 ev/cm 3. Helium and heavier nuclei together contain about 0.27 ev/cm 3. The total energy density of cosmic rays is therefore ɛ CR 1 ev/cm 3. (1.7) This can be compared to the galactic magnetic field with a strength of about 3 µg, giving an energy density of 0.25 ev/cm 3. The two are of comparable magnitude, meaning they are interconnected and both influence each other! Other typical energy densities are that of starlight, at 0.6 ev/cm 3, the cosmic microwave background, at 0.26 ev/cm 3, and the solar wind, at 2.5 kev/cm 3 [31]. From the energy density of the cosmic rays, their residence time in the galaxy and the confinement volume (=the galactic disk), the power necessary to supply the cosmic rays can be calculated. The residence time is taken to be years and the volume can be calculated as πr 2 d π(15 kpc) 2 (200 pc) cm 3. (1.8) The power required is then L CR = V Dɛ CR τ R ergs 1. (1.9) 1.4 Sources of acceleration Source properties The question is then what the source(s) is (are) for the cosmic rays. A first guess could be the Sun, even though Hess already ruled it out as a direct source (see above). However, there is an anticorrelation with the solar activity (see solar modulation above), clearly excluding the Sun. For the still remaining non-believers, a simple calculation shows that the possible sources of energetic particles at the Sun are just not powerful enough[28], accelerating only up to GeV energies. To find a possible source of the highly energetic cosmic rays, one should look for physical environments that are capable of injecting a lot of energy into a single particle, called acceleration and do it efficiently enough to produce lots of cosmic rays. Looking at the 4 fundamental forces, one immediately sees that gravitation and the weak force are far too weak to produce the required amount of energetic particles. The strong force is irrelevant since confinement makes large scale effects impossible. The remaining candidate is the electromagnetic force. Large, strong electric fields due to free or space separated charges are basically non-existent, since the universe seems to be neutral on pretty much any scale 2 and because astrophysical plasmas have such high conductivity (approximately infinite, see also section 2.2) that any static electric field would be immediately short-circuited[22]. Strong electromagnetic waves are a possibility, but do not seem to be able to explain the required energy and flux of cosmic rays. The only remaining possibility is magnetic fields. These fields themselves are incapable of performing work on particles however, since the 2 This can be concluded from the observation that gravity is the dominant effect when looking at the dynamics of the solar system/galaxy/universe even though it is many orders of magnitude weaker than the electric force.

27 Chapter 1. Cosmic rays An introduction 15 force goes as F Magn. = q v B. Thus the required environments need moving magnetic fields, which behave like an electric field (see section 2.3.2). There are several acceleration sites and mechanisms that can be considered, such as the cyclotron mechanism or the collapse of astrophysical objects, which can compress magnetic fields to higher strength or release gravitational energy[28]. The most promising mechanism, however, is Fermi acceleration. The basis of the mechanism is that particles can stochastically gain energy, leading to the automatic formation of a power law spectrum. There are two forms of this mechanism: second order and first order. The first of these was originally proposed by Fermi in 1949[32] and uses the reflection of charged particles by moving magnetic clouds in the galaxy, but is not powerful enough to explain the observations. The latter one uses acceleration in shock environments, which are certain to occur in the many violent phenomena observed such as supernovae. By continuously crossing the shock by frequent scattering, particles are capable of reaching very high energies sufficiently fast. The energy gain is dependent on many factors, one of them being the speed of the shock. The validity of the basic idea of shock acceleration has already been proven by satellite observations in the Earth s bow shock. In this the solar wind collides with the Earth s magnetic field, creating a shock. This shock is accompanied by highly energetic particles. Through the shock acceleration mechanism, Ellison, Reynolds and Jones[33] were able to explain the observed population of particles. The first order Fermi mechanism is still heavily researched today and is also one of the two main subjects of this thesis. A detailed theoretical discussion is given in the third chapter The Hillas criterion In order to accelerate particles to a certain energy, the acceleration site needs to be able to contain particles up to that energy. Using this idea, it is possible to identify the possible sources for cosmic rays. The requirement that particles stay confined during acceleration means that the physical size of the acceleration site must be about the order of the maximum Larmor radius of a charged particle in a magnetic field: L r L = p max ZeB. (1.10) Taking into account the speed of the scattering centres, this is modified to (β = v/c) which immediately gives L r L β, (1.11) E max βzecbl (1.12) which is known as the Hillas criterion[34]. It relates the maximum energy of accelerated particles to the size and magnetic field of the acceleration site, both observationally accessible. This requirement can be put into a plot containing all the known astrophysical objects, the so called Hillas plot, shown in figure 1.9. Note that while there is no guarantee that the particles are able to reach the maximum energy, or that the source can generate enough particles to explain all cosmic ray, it at least helps to easily rule out certain sources as viable candidates. From the Hillas plot one can conclude that there are no known galactic sources able to produce cosmic rays above ev (independent from the fact that the galaxy itself cannot contain such cosmic rays), so for the highest energy cosmic rays, one should search for extragalactic sources.

28 Chapter 1. Cosmic rays An introduction 16 Figure 1.9: The Hillas plot, showing the maximum energy up until which particles can be contained in different astrophysical sources [35] Galactic and extragalactic sources From the Hillas plot some likely candidates for the production of cosmic rays can be found. These will be discussed individually now and will be the basis for the entire discussion of the acceleration mechanism. Supernovae A supernova (SN) is in principle the explosion of a star at the end of its life, where the outer layers of the star are ejected at high speed. The ejecta collide either with the interstellar medium or with the wind generated by the star as it was nearing its death (see figure 1.10), creating powerful shocks where shock acceleration can take place. The speed of the SN shock wave is about 10 4 kms 1. The energy attainable is limited by the finite lifetime of the supernovae. While a supernova only dissolves after about years, most of the acceleration occurs during the undecelerated phase of the evolution[22]. The deceleration starts when the shock has swept up a mass about equal to its own. Consider a supernova with a mass of about 10 M (the original star) which is expanding at a mean velocity of cms 1 into a medium with a density of about 1 proton/cm 3. The lifetime is then about 1000 years. With a magnetic field of about T and the Hillas criterion (equation (1.12)), this gives E max ev (1.13) for protons. A more detailed calculation by Lagage and Cesarsky[36] confirms this result. Heavier particles can, due to E max Z, go to higher energies, meaning that the average mass of cosmic ray particles is predicted to go up at the knee (note that other effects can also produce this effect, such

29 Chapter 1. Cosmic rays An introduction 17 as a leakage depending on the rigidity). The maximum energy can be higher due to turbulence (see section 2.4) or instabilities in young supernova remnants (streaming, Drury and firehose instability, small scale dynamos and downstream the Rightmeyer-Meshkov and charge exchange current-driven instabilities), which compress the magnetic field up to 2 or 3 orders of magnitude higher. Thus, the particles can be accelerated up to the knee at ev. Again the heavier particles can be accelerated to higher energy (maybe up to ev), so that the spectrum would get heavier[23]. This indeed seems to be confirmed by the KASCADE experiment. Even higher energies however, cannot be explained by this scenario. That supernovae can indeed be an excellent candidate, can also be understood from their power output[37]. For example, if there is a type II supernova every 30 years ejecting a mass of 10 M with a velocity of about cms 1, this gives a power output of L SN ergs 1. (1.14) Thus an efficiency of a few percent (still much better than current man-made accelerators) would be enough to supply all the necessary power! (see equation (1.9)) Figure 1.10: Supernova shock wave expanding into the interstellar medium or stellar wind from the precursor star[38]. AGN s and GRB s As described above, higher energies are, as of the current understanding, only reachable by extragalactic phenomena. The two most luminous types of events observed are Active Galactic Nuclei (AGN) and Gamma Ray Burst (GRB), the first of which is a permanent object while the latter is a transient event. Both have been observed to emit powerful gamma rays. Photons cannot be accelerated themselves however, instead they are the product of charged particles that were accelerated to high energy interacting with other particles or photons (see also section 1.5). These two classes of events are thus very likely also a source of highly energetic cosmic rays. A review and model can be found in the paper by Meli, Becker and Quenby[39].

30 Chapter 1. Cosmic rays An introduction 18 An AGN is a supermassive black hole at the centre of a galaxy. Matter can fall into it, releasing a lot of energy in the process, which forms a hot, rotating accretion disk due to the conservation of angular momentum. While this disk itself is a source of high energy particles, it is not the only object of interest. Again, due to the conservation of angular momentum, it is possible for highy energetic gas to be ejected along the rotation axis, forming jets with scales of pc to kpc (see figure 1.11). In these jets, shocks can form, possibly due to matter suddenly falling into the system and being ejected again, so that Fermi shock acceleration can take place. From observation of electron synchrotron radiation, it is found[40, 41] that such shocks have speeds very close to c, with Lorentz boost factors of Γ The assumption is then that protons are accelerated as well. One sees that with a magnetic field of B 10 3 G and a jet radius of about r 1 kpc, the AGN 1021 ev. The typical luminosity of AGN s is Hillas criterion gives a maximal energy of Emax L erg/s, so that the required cosmic ray flux can indeed by delivered by AGN s[42]. A GRB is a very high energy event, observed in gamma rays, with a luminosity that can exceed that of its host galaxy. A GRB is believed to occur when two neutrons stars merge, or a massive star collapses into a black hole. These events would be accompanied by jet formation, where shocks can occur (see figure 1.11). From the observed photon spectra it is found that the jet plasma speeds have Lorentz boost factors in the range 100 < Γ < 1000[43] and energies up to 1021 ev can be reached. From the observed luminosity, the input rate of cosmic rays from GRB s is estimated to be erg/mpc3 /yr, which is the required amount for the UHECR s[44, 40]. Figure 1.11: Jet-disk systems as possible cosmic ray sources : microquasar, AGN, GRB[45, 38] Ultra-high-energy cosmic rays and the end of the spectrum One important property of the cosmic ray spectrum that was not yet discussed concerns the ultra-high-energy cosmic rays (UHECR), which are cosmic rays with an energy above ev. Starting at about ev, a flux suppression is observed, the origin of which is still the subject

31 Chapter 1. Cosmic rays An introduction 19 of intensive study. The current status has been recently summarised by Kampert[46]. There are two plausible candidates for this suppression. The first possible reason for the flux suppression is that the extragalactic accelerators themselves are at their energy limit. The effect is analogous to that in galactic accelerators. Since the maximal energy reachable is proportional to the charge Z, the protons are the first to reach their acceleration limit. The other elements have higher limits, with Fe reaching the highest energy. There are thus separate knees in the spectrum for the different elements, with the Fe-knee being at ev ev, giving the knee a substructure. The result is that at the knee, the composition gets gradually heavier. Towards the ankle (at ev) the proton component becomes dominant again with the p-ankle at ev. If now the flux suppression is due to the limit of the accelerators, the composition is expected to become heavier towards the end of the spectrum. The second possibility is a suppression due to the Greisen-Zatsepin-Kuz min (GZK) cutoff[47, 48] which occurs when high energy protons (> ev) interact with photons in the Cosmic Microwave Background 3 in the following reaction ( -resonance) p + γ +. (1.15) The + can then decay into a proton or neutron and a pion. Since neutrons decay again into protons this mechanism is essentially an energy loss mechanism for protons. The flux of the CMB is so high that the universe becomes opaque for these protons, so that they get a mean free path of about 50 Mpc. This means that the high energy protons from extragalactic accelerators would not be able to reach us at their original energy, but instead get reduced to the threshold 4. This effect is guaranteed to happen, since it relies on very basic physics. The only question is whether the observed suppression is the GZK-cutoff at work, in which case the particle composition would not get heavier.[38] Both these mechanisms (can) give an explanation for the energy of the observed suppression. It is thus important to find the composition of the spectrum at these high energies in order to differentiate between the two scenarios. However, this is highly non-trivial, since at these energies the flux is so low that only huge air shower experiments are capable of studying this. Reconstructing the mass of the original particle which triggered the shower in the atmosphere, is done by observing the shower profile and comparing with numerical models. Since the interactions involved have not been studied in accelerators at these energies yet, this involves extrapolating the known properties at accelerator energies, inducing uncertainties in the models. Recent results are shown in figure At this moment the accelerator scenario seems favoured, but the other has not been ruled out yet. 3 The same effect can of course also happen with higher energy photons in the galaxy, but their flux is too low for this effect to be significant. 4 Due to the low fluxes the hereby produced excess compared to the usual power law would not be observable however

32 Chapter 1. Cosmic rays An introduction 20 (a) Measurement of the energy spectrum by the Pierre Auger Observatory, compared with different cosmic ray composition models[49]. (b) Average mass of the source particle from extended air showers, analysed with the QGSJet01 interaction model[50]. Figure 1.12: Different ways of finding the composition of UHECR s[46]. 1.5 Multi-messenger astrophysics Multi-messenger astrophysics (see Halzen[51], Stamatikos et al.[52] and Becker, Meli and Biermann[53]) is a consequence of the realisation that in order to truly understand high energy astrophysical phenomena in the universe, which are usually also sources of cosmic rays, one needs to combine different observation methods. This is illustrated in figure In AGN s, for example, various high energy particles are created. The accretion disk is so hot that X-rays are created, giving us information about its temperature. From the observed synchrotron spectrum in the radio range, it is known that there must be an accelerated electron population. These electrons can scatter on the photons present and accelerate them in inverse Compton scattering, so that high energy gamma rays are produced, which are indeed observed. It is then also likely that hadrons are accelerated as well, possibly being the main mechanism. In order to understand the production of cosmic rays, it is necessary to know whether AGN s (and other sources) accelerate mainly leptons or hadrons. This cannot be learned from the observed cosmic rays, since ejected hadrons will be affected by extragalactic and galactic magnetic fields, so that they will no longer point back to their source. Electrons suffer heavily from synchrotron radiation and deflection due to magnetic fields. Photons are, although neutral and thus not deflected, affected by all matter and absorbed by the extragalactic background light on their escape and subsequent propagation towards us. There is however also production of neutrinos, which interact only weakly and will thus travel towards us directly, which makes them ideal to gain information about the nature of the acceleration. Such neutrinos would be generated en masse by the following mechanism. High energy protons can collide with other protons or photons, creating for example high energy + s: p + γ +, (1.16) p + p + + p. (1.17)

33 Chapter 1. Cosmic rays An introduction 21 These subsequently decay, producing high energy pions, neutrons and protons: + π + + n. (1.18) π 0 + p The neutrons decay into protons, so that the result is the original protons together with pions. These pions decay as follows (ignoring the difference between neutrinos and antineutrinos for ease of notation) π ± µ ± + ν µ e ± + ν e + ν µ + ν µ, (1.19) π 0 2γ. (1.20) Producing either high energy gamma rays (which can be observed) or high energy neutrinos carrying 20% of the initial particle energy. Of course at these high energies many other, possibly much more complicated interactions can happen 5. Still, the main idea is the same: at the end very high energy neutrinos are produced, which carry information related to the primary particle straight to us. The detection of these very high neutrinos is a new branch in physics, called neutrino astronomy, and is still under development. It is clear that a study of the acceleration mechanism and the observation of such neutrinos complement each other. Figure 1.13: The multi-messenger connection[17]. 1.6 The IceCube Neutrino Observatory IceCube is a neutrino observatory located at the South Pole. Its main goal is (surprisingly) to detect astrophysical neutrinos. The detector consists of two main parts: the actual neutrino detector buried in the ice and a surface experiment (IceTop) that serves as a veto for atmospheric muons and simultaneously measures the cosmic ray spectrum. 5 Actually Kaon production becomes dominant over pion production at high energies.

34 Chapter 1. Cosmic rays An introduction IceCube experiment The main detector consists of 86 strings together carrying Digital Optical Modules (DOM), see figure The working of the detector is based on detecting Cherenkov radiation. As neutrinos pass through the ice, every now and then it is possible that one of them interacts with the ice. Through either charged or neutral current interactions, the neutrino is converted into an electron, muon or tau lepton. Since astrophysical neutrinos have high energies, the resulting lepton goes faster than the speed of light in ice, causing it to emit Cherenkov radiation along a track. This Cherenkov radiation can then be detected by the DOM s, which are essentially just photomultiplier tubes (PMT s) with some electronics on board to locally process the information. The signal from the different DOM s is then transmitted to the top and combined to give one event. The event rate of astrophysical neutrinos is very low and dwarfed by background. The main backgrounds are atmospherical neutrinos and atmospherical muons. The first of these are products of the cosmic ray showers. They produce events that are of very low energy compared to the astrophysical ones however. The second is also a product of cosmic ray showers, more specifically its most penetrating component apart from neutrinos. This background can be reduced significantly by requiring the event track to start in the detector. Also, Icetop serves to detect air showers, so that any event that is also detected by IceTop is automatically uninteresting. Recently the IceCube experiment detected the first extraterrestrial neutrinos[55], for which it was named the 2013 Breakthrough of the Year by the British magazine Physics World. The highest energy event is shown in figure IceTop experiment As mentioned above, Icetop is the surface experiment (shown in figure 1.16) accompanying Ice- Cube, which serves a dual purpose. It consists of an array of tanks filled with ice, each containing two DOM s. Mainly air shower muons will emit Cherenkov radiation in them, which is detected by the DOM s. Any signal in the IceTop array will serve as a veto for the IceCube detector, since for that experiment air showers are considered background. Of course, this also allows the IceTop array to serve as an air shower detector to measure the cosmic ray spectrum at high energies. Therefore the IceTop experiment is by itself already a valuable addition to further our understanding of cosmic rays.

35 Chapter 1. Cosmic rays An introduction (a) A schematic showing the IceCube detector layout. 23 (b) Schematic of the DOM s used in the IceCube detector. Figure 1.14: The IceCube detector, figures from the official IceCube website[54]. Figure 1.15: The highest energy neutrino recorded, with an estimated energy of ev. Each blob is the signal from a DOM, the size signifying its energy recorded and the colour giving the tim[54].

36 Chapter 1. Cosmic rays An introduction Figure 1.16: The IceTop array[54]. 24

37 Part I Acceleration 25

38

39 Chapter 2 Principles of plasma physics From the previous chapter, it is obvious that in order to be able to describe in detail the acceleration mechanism in shock waves, one first needs to understand the plasma and shock physics surrounding this acceleration mechanism. In this chapter some basic principles of plasma theory[56] will be presented. 2.1 Particles in a plasma There are different ways to describe a plasma, such as particle orbit theory, fluid theory, wave theory, MHD theory, and kinetic theory. In this section, particle orbit theory shall be covered. In this, one studies the orbit of a single charged particle in a given electric and magnetic field. This might not seem relevant, since a plasma is highly interacting and correlated, being closer to, and indeed often described similar to, fluids. However, even in usual plasma physics it is informative to study this case, since it is the easiest and because the developed key concepts can help to understand more realistic - and complicated - models. More specifically, the ingredients from this section will be vital to the production of the Monte Carlo simulation. In orbit theory, one looks at the effects of external magnetic and electric fields on the trajectory of a single particle. Radiative effects will be neglected and everything will be first solved using the non-relativistic Lorentz equation. Generally this means solving the following equation: m j r j = e j [E(r, t) + ṙ j B(r, t)] (2.1) for every particle species. checked using Maxwell s equations Of course the applied fields need to be self-consistent, this is to be E = B t (2.2) E B = ɛ 0 µ 0 t + µ 0j (2.3) E = q/ɛ 0 (2.4) B = 0 (2.5) 27

40 Chapter 2. Principles of plasma physics 28 with the current and charge densities j(r, t) = q(r, t) = B e j ṙ j (t)δ(r r j (t)) (2.6) j=1 N e j δ(r r j (t)). (2.7) j=1 Whenever these last densities are significant, statistical or fluid descriptions are usually more appropriate. The following will be done in the test particle approximation, which means that induced fields by the charged particle will be neglected, putting restrictions on the range of physical parameters where this treatment is valid Helix orbit Constant homogeneous magnetic field The simplest case one can describe, which will actually be usefull for the Monte Carlo code, is that of a charged particle in a uniform magnetic field B and with no electric field, E = 0. In these derivations one species will be handled at a time, making the j subscript unnecessary. The z-axis is defined by the the direction of the magnetic field B. Taking the scalar product of the Lorentz equation with ẑ gives z = 0 ż = v = const. (2.8) Also, by taking the scalar product with ṙ m r ṙ = mṙ2 = W = const. (2.9) Combining the above two equations gives the conservation of kinetic energy (as expected since there s only a static magnetic field) W = W + W = const. (2.10) The trajectory is determined by equation (2.8) and the x and y components of (2.1): ẍ = Ωẏ ÿ = Ωẋ. (2.11) with Ω = eb/m. The standard solution method of such equations is to define the complex ζ = x + iy and combine the above equation into one ζ + iω ζ = 0, (2.12) which is easily integrated to give ζ(t) = ζ(0) exp( iωt). (2.13) Defining the initial condition ζ(0) = v exp( iα), this splits into ẋ = v cos(ωt + α) ẏ = v sin(ωt + α). (2.14)

41 Chapter 2. Principles of plasma physics 29 Integrating again gives the particle orbit ( v ) x = sin(ωt + α) + x 0 (2.15) ( Ω v ) y = cos(ωt + α) + y 0 (2.16) Ω z = v t + z 0. (2.17) This orbit is pictured in figure 2.1. It is a superposition of a linear motion along the magnetic field and a rotation in the plane perpendicular to it. The angular dependence of this rotation, φ(t) (Ωt + α), is called the gyro-phase[56] with Ω the gyro-frequency (also called Larmor frequency or cyclotron frequency) and its radius, r L = v / Ω, the Larmor radius. The centre of the circle, r g = (x 0, y 0, v t + z 0 ), is known as the guiding centre. Together these movements generate the known helix motion. The pitch angle θ is defined by and is thus constant along the helix. tan θ = v v (2.18) The importance of the above concepts is that they define the relevant physical scales, which have a natural ordening in most systems. If the physical environment has a scale length L and a characteristic time T, then often (but certainly not always) it is the case that r L /L << 1 and 2π/ΩT << 1. This fact can then be used to ignore the gyration of the particle around the magnetic field and describe instead only the motion of the gyrocentre along the magnetic field. This gyrocentre does not coincede with the location of the particle itself, but can be used to simplify the mathematics, without making big errors. Constant homogeneous magnetic and electric field Now in addition to a constant and homogeneous magnetic field B = (0, 0, B), there is also a constant and homogeneous electric field E = (0, E, E ), where the direction of the component of Figure 2.1: Helix motion of a positively charged particle in a homogeneous magnetic field and no electric field[56].

42 Chapter 2. Principles of plasma physics 30 E orthogonal to B defines the y-axis. The components of the Lorentz equation are then Integrating the last one gives ẍ = Ωẏ (2.19) ÿ = ee m Ωẋ (2.20) z = ee m. (2.21) ż = v + ee t m. (2.22) This equation immediately imposes a restriction on the allowed physical parameters. If left untouched, it gives a limitless increase in speed, so that the non-relativistic approximation would break down. On top of that, the acceleration depends on the charge, meaning positive and negative particles are accelerated in opposite direction. This would inducing arbitrarily large currents and charge separation, giving fluctuating fields ( see the second and third Maxwell equation) contrary to the assumed constant fields. Thus we need to have E = 0 (and rename E = E). Proceeding with the x and y components as before ζ + iω ζ = iee m (2.23) ζ(t) = ζ0) exp( iωt) + v E (1 exp( iωt)) (2.24) with v E = E/B. This gives for x and y separately ẋ = u cos(ωt + α) + v E ẏ = u sin(ωt + α) (2.25) with u and α defined by ζ(0) v E = u exp( iα). (2.26) From the above, the velocity of the guiding centre is v g = (v E, 0, v ). Thus, on top of the motion along the magnetic field, there is a drift across field lines. The drift velocity can be written as v E = (E B)/B 2. Since it is independent of the charge, it does not give rise to currents. Again, the non-relativistic approximation forces us to restrict E << cb. The motion is shown in figure 2.2, for protons and electrons 1. Other cases In the case of non-homogeneous fields, the situation is slightly more complicated. If the nonhomogeneity is small, one can do perturbation theory on the basic motion above and expand the electromagnetic fields around the position of the gyrocentre. Such a perturbation approach was first used by Alfvén and is known as the guiding centre approximation. In this, the fast Larmor motion is averaged out, retaining only the slower guiding centre motion. The most general inhomogeneous field is characterised by the nine components of B i x j. Of these, the gradient and curvature terms give rise to a drift. The remaining terms, describing divergence terms and shear, do not give rise to drifts. 1 The protons and electrons drift equally quick since the smaller electron drift is compensated by the faster gyro-frequancy

43 Chapter 2. Principles of plasma physics 31 Figure 2.2: Helix motion and drift of a proton and an electron in a homogeneous magnetic field and an electric field perpendicular to it[56]. If going to the relativistic case, one needs to modify the Lorentz equation by changing m to γm so that the results would be essentially the same. In the next section the non-relativistic approximation will be continued (since it is the standard derivation), but its results will remain valid in the relativistic case (as can be shown be explicitly rederiving the steps), see for example chapter 7 in Longair[22] Magnetic moment invariance The rotation of the charged particle corresponds to a microscopic current I L, which induces a magnetic field opposed to the applied magnetic field due to Ampère s law. With this microcurrent, there is associated a magnetic moment µ B given by µ B = πr 2 LI L B/ B = πr 2 L ( ) eω B 2π B, (2.27) since the current is just the charge divided by the Larmor period. Using r L = v Ω, this becomes µ B = W B. (2.28) B2 The magnetic moment will be used extensively in the simulation since it is under certain conditions approximately conserved, as will be shown below. First, the case of magnetic fields varying slowly in time ( Ḃ / B << Ω ), but not in space, will be considered. No longer will W and W be separately conserved, but there will be a counterpart that is (approximately). A time-dependent, axial magnetic field will induce an azimuthal electric field. The derivations above are no longer valid, so that v and v are no longer conserved. Indeed, taking the scalar product of the Lorentz equation with v gives ( ) d 1 dt 2 mv2 = e E v. (2.29) In one Larmor orbit, the energy changes by δ( 1 2 mv2 ) = E dr = e ( E) ds (2.30)

44 Chapter 2. Principles of plasma physics 32 with the surface integral being enclosed by the particle orbit. equations, this becomes Recognising one of Maxwell s δ( 1 B 2 mv2 ) = e ds. (2.31) t Approximating this integral, due to the slowly varying fields, gives δ( 1 2 mv2 ) πr2 L e Ḃ = mv2 2 2π Ḃ Ω B (2.32) independent of the sign of the charge, since the surface element is orientated: e > 0 has B ds < 0 and vice versa. Recognising the Larmor period 2π/ Ω, this can be rewritten as δb δw = W B (2.33) or ( ) W δ = δ (µ B ) = 0. B (2.34) In other words, the magnetic moment is approximately conserved. Such an approximate constant of motion is known as an adiabatic invariant in Hamiltonian mechanics. It is dependent on the fact that the global trajectory changes slowly compared to the fast periodic rotation of the particle around the gyrocentre. There can also be other adiabatic invariants corresponding to different periodic motions. Therefore this one is also known as the first adiabatic invariant. The invariance can also be proven for spatially varying fields for which the divergence terms are non-zero. The considered configuration, an axially symmetric magnetic field with increasing strength along the z-axis, is shown in figure 2.3. The divergence of B, which of course disappears, can be written in cylindrical coordinates as B = 1 r r (rb r) + 1 B θ r θ + B z = 0. (2.35) z The B θ component is equal to 0 for an axially symmetric field. Integrating gives rb r = r 0 r B z dr. (2.36) z Over one Larmor orbit the field is approximately constant, so that the B-term can be taken out of the integral, giving, together with B r << B z, B r (r L ) r L B z 2 z r L B 2 z. (2.37) Substituting this into the z-component of the Lorentz equation gives immediately m dv dt = 1 2 e r B Lv z = W B This can be rewritten as B z = µ B B z. (2.38) ( ) d 1 B dt 2 mv2 = µ B v z = µ db B dt. (2.39) The time derivative of the orthogonal energy component can be rewritten as ( ) d 1 dt 2 mv2 = d dt (W ) = d dt (µ BB). (2.40) Adding the last two equations and using total energy conservation gives dµ B = 0, (2.41) dt finding again, by approximation, magnetic moment conservation.

45 Chapter 2. Principles of plasma physics 33 Figure 2.3: An axially symmetric magnetic field with increasing strength along the z-axis[56] Magnetic mirror The conservation of magnetic moment in a spatially inhomogeneous field has some interesting consequences. An illustration of this is the magnetic mirror. Consider the setup shown in figure 2.4. As the particle is propagating, the magnetic field strength B increases, so that W should increase as well to keep the magnetic moment constant. Obviously this needs to be compensated by a decrease in W to keep the energy constant. It is then possible that W becomes 0 while B is still rising, so that the particle will then undergo reflection. Figure 2.4: An example of a magnetic mirror being used to create a magnetic trap[56]. The conservation of the magnetic moment µ B v /v = tan θ v = sin θ together give = W /B and the definition of the pitch angle W B = sin2 θw B = const sin2 θ B This can be rewritten, by defining the constant B R, as = const. (2.42) sin θ = (B/B R ) 1/2. (2.43) If the particle can penetrate the field up to the maximum before being reflected, then B R = B M, since the pitch angle will only be 0 when B = B M. This means that particles with a pitch angle sin θ = (B/B R ) 1/2 > (B/B M ) 1/2 will be reflected before reaching the maximum magnetic field while those with sin θ (B/B M ) 1/2 will not be reflected.

46 Chapter 2. Principles of plasma physics Frozen-in magnetic fields It is easily proven that in a plasma, the conductivity is very high[22]. In many cases it is formally put equal to infinity. Below it will be shown that in a plasma with infinite conductivity, any movement of the plasma will be followed by the magnetic field, as if it were frozen into it. This will then be the case for astrophysical plasmas. The treatment below follows the one given by Longair[22] which is based on the book by Ratcliffe[57]. We start by considering a current loop in a plasma (see figure 2.5a). Changing the magnetic flux through the loop induces an electromotive force E = dφ dt. (2.44) The magnetic flux φ is the sum of flux from the external currents, φ ex, and the induced flux in the circuit φ i. The induced flux is found through the inductance L φ i = Li. (2.45) Any change in the external currents causes an electromotive force in the circuit and the resulting current can be found by combining the above equations When the resistance is zero, this gives L di dt + Ri = dφ ex dt. (2.46) In other words L di dt = dφ ex dt = dφ ex dt. (2.47) φ i + φ ex = constant (2.48) Now figure 2.5b will be used to study what happens when the circuit changes shape. If the vertical conductor moves a distance dx at a velocity v, there will be an induced electric field E = v B = vb 1. There is an induced electromotive force in the conductor, with l the distance between the parallel wires, E = El = vb 1 l. (2.49) (a) A current loop in a plasma. (b) A superconductor setup to illustrate the change in size of a current loop. Figure 2.5: Illustrating magnetic flux freezing[22].

47 Chapter 2. Principles of plasma physics 35 From equation (2.44) it is found that this is accompanied with a magnetic flux dφ = (vb 1 l) dt (2.50) opposite to B 1. Also, because the loop has increased in size, there is an increase in flux dφ = B 1 ldx = (vb 1 l) dt (2.51) in the same direction as B 1. Adding these two effects, there is no change in flux. Generally this result can be formulated as B ds = constant. (2.52) S So, saying that the flux is constant is equivalent to saying that the magnetic field lines follow the plasma, so that the statement about frozen-in fields is now proven. 2.3 Shock waves The Fermi mechanism that will be described in the next chapter makes use of the properties of astrophysical shocks propagating in a plasma medium at rest. These shocks are in general described by magnetohydrodynamics. The key ingredients for treating these astrophysical shocks, such as the different reference frames used and the equations relating the media on both sides of the shock, will be developed in the following sections Non-relativistic shocks Hydrodynamical shocks Usually, any disturbance in a fluid (for example an aircraft) causes a wave to propagate at the speed of sound in the medium. This means that the information of the disturbance is transported to the rest of the fluid, so that the fluid can react to it timely and appropriately. This causes any change in macroscopic quantities, such as pressure and temperature, to happen adiabatically. When the source of the disturbance propagates faster than the sound speed, it can overtake the sound wave. The fluid can no longer react adiabatically and undergoes a sudden change, a shock. The shock thus forms a transition region between the already shocked fluid and the fluid which has received no information about the disturbance yet[56]. As hinted to in the introduction, it will prove useful to describe the shock in different reference frames. The description above corresponds to the frame most natural to an external observer. It is always possible to choose a frame where the shock plane coincides with the yz-plane and the propagation direction of the shock is the negative x-direction, by defining the first one to be the case and then transforming along the yz axis until the latter is true 2. This specific choice will henceforth be referred to as the lab frame (see figure 2.6a). A second very important frame is the shock frame, where the shock is at rest. It is found by transforming with the shock speed along 2 More accurately, there is actually a second possibility where the direction of motion is along the shock front. In normal hydrodynamics these shocks have propagation speed equal to zero however and they are called slipstreams instead[58]. In MHD such shocks are possible.

48 Chapter 2. Principles of plasma physics 36 the propagation direction of the shock. The medium that was initially at rest, will now be going towards the shock, in the positive x-direction, with the original shock speed. The already shocked medium, which was pulled along with the shock with a speed slower than the shock speed (as will be shown below) will, after the transformation, be going away from the shock, also in the positive x-direction (see figure 2.6b). This latter statement is actually a non-trivial one. It is found that in normal hydrodynamics this is indeed the case, since there is no physical mechanism that is capable of generating a new component in the shocked fluid. The unshocked region is called upstream and the shocked one downstream. There are two additional frames, the fluid rest frames, where the medium at one side of the shock is at rest. The transition region of a shock is very difficult to describe in detail. Dissipative effects are needed to be able to change the state of the fluid, adding considerable complexity to the equations. Luckily this will not be needed in the following, it is adequate to relate the macroscopic quantities on both sides of the shock (regardless of the exact shock structure)[59]. To find these we need to start from the 1D Euler equations for a fluid, given by ρ t = (ρv ), x (2.53) t (ρv ) = x (ρv 2 + P ), (2.54) t (ρe) = x (ρv (e V 2 + p/ρ)), (2.55) where V is the flow speed, ρ is the density, P is the pressure and e is the internal energy. The first of these is simply the continuity equation, stating that a change in density should be accompanied by an opposite change in flux. The second one is the momentum equation, stating that a change of momentum is due to the bulk momentum flux and a random momentum flux, which is actually described as pressure. The third one is the energy equation, where the first two terms on the right hand side are, together, the total energy (internal energy ρe and fluid velocity energy) and the third term is due to the random movement in the fluid. The last equation can be rewritten, assuming the following equation of state for an ideal gas p = (γ 1)ρe, (2.56) where γ = C P C V is the ratio of specific heats. Demanding now conservation of mass, momentum and energy, this gives ρv = 0, (2.57) x x (ρv 2 + P ) = 0, (2.58) ( 1 x 2 ρv 2 + γ ) γ 1 V P = 0 (2.59) These equations need to hold everywhere in the plasma, in particular the conservation of mass, momentum and energy are still valid across the shock. Given their form, they can be trivially integrated from to any value of x. Doing this for both upstream and downstream gives the

49 Chapter 2. Principles of plasma physics 37 y y Upstream Upstream V SH x V 1 V 2 x Downstream Downstream (a) The lab frame. (b) The shock rest frame. Figure 2.6: Two important reference frames describing the shock. following relations: [ρv ] 2 1 = 0, (2.60) [ρ 1 V 1 V + P ] 2 1 = 0 (2.61) [ 1 ρ 1 V 1 2 V 2 + γ ] 2 γ 1 V P = 0, (2.62) where [O] 2 1 = O 2 O 1 and the first equation was used to replace any ρv with its constant value ρ 1 V 1 everywhere. These relations are called the Rankine-Hugoniot jump conditions and they make it possible to solve the downstream conditions, given the upstream conditions. By rewriting the first equation, one finds the compression ratio Further introducing the upstream Mach number M 2 1 = V 1 2 Vsound,1 2 the set of equations can be solved, giving[60, 59] 1 r = V 1 V 2. (2.63) = ρ 1V1 2, (2.64) γp 1 r = (γ + 1)M2 1 (γ 1)M (2.65) and P 2 = P 1 [1 + γm 2 1 ( 1 1 )] r (2.66) 2γM 2 1 (γ 1) = P 1. (2.67) γ + 1

50 Chapter 2. Principles of plasma physics 38 In case of a simple sound wave, meaning M 1 = 1, this gives r = 1 and P 2 = P 1 i.e. there is no shock. It can be found that M 1 < 1, which gives r < 1 and P 2 < P 1, is thermodynamically ruled out because it would give a decreasing entropy. The more interesting case is that for M 1. Plugging this in, it is found that the pressure ratio can increase without limit. The compression ratio however reaches a constant value r = γ+1 γ 1. In the case of a monatomic, non-relativistic gas, which has γ = 5 3, one finds the compression ratio r = 4. (2.68) MHD approach to astrophysical plasmas Now we turn to astrophysical plasmas, meaning we need to add magnetic fields to the picture. These magnetic fields can have arbitrary orientations with respect to the shock normal. When the magnetic field is parallel to the shock normal (ψ = 0), the shock is called parallel. When the magnetic field is perpendicular to the shock normal, the shock is called perpendicular. Lastly, for all other orientations, the shock is called oblique. In the last two cases the magnetic field will be able to transfer momentum perpendicular to the shock propagation direction, so that it will no longer be possible to find a shock frame where both V 1 and V 2 are along the x-axis. The inclusion of the magnetic fields allows for transverse modes of propagation, instead of only longitudinal as in hydrodynamical shocks[56]. Consider for example the transverse movement of the plasma. Since the magnetic field is frozen into the plasma, it will be dragged along, causing a wave. There are thus three modes of propagation, two transversal and one longitudinal, meaning three degrees of freedom. These degrees of freedom regroup into three propagation velocities, namely slow, intermediate and fast, the first and last of these causing shocks (see also Kirk and Duffy[61]). Another consequence of magnetic field lines is that it is now possible for the shock to be much thinner than the collisional mean free path [56]. These shocks are called collisionless shocks and they cannot technically be MHD shocks. A correct treatment requires the use of kinetic theory 3 which explicitly includes non-linear effects. The starting point of such a treatment is still the fluid approximation however. For the acceleration mechanism to work, a collisionless shock is needed. In order to make a simple treatment possible, the fluid equations are assumed to be valid to good approximation. The derivation of shock jump conditions can then be done analogously to the one above, again circumventing the shock structure itself, if the shock is described as a discontinuity. As said above, the inclusion of a magnetic field necessitates considering not just the x-components of the equation, but other coordinates as well. Luckily the addition of Maxwell s equations will allow the set of equations to be complete. Observe the system again in the shock rest frame2.7. In a steady state, all variables are time-independent. From Maxwell s equation (2.3) we have ( µ 0 j = 0, db z dx, db ) y. (2.69) dx On both sides of the shock, Ohm s law is valid E 1 = j 1 σ 1 V 1 B 1 E 2 = j 2 σ 2 V 2 B 2. (2.70) 3 In kinetic theory, instead of describing the plasma with fluid equations, where the individual properties of the particles are forgotten, the plasma is described by the evolution of the distribution function, which keeps the information about the individual velocities of the particles.

51 Chapter 2. Principles of plasma physics 39 Finally, from Maxwell s equations (2.5) and (2.2) we have db x = 0, dx (2.71) de y = 0, dx (2.72) de z = 0. dx (2.73) Integrating the last three equations across the shock and using the two before them, we find [B x ] 2 1 = 0, (2.74) [V x B y V y B x ] 2 1 = 0, (2.75) Now the mass, momentum and energy relations need to be used. [V x B z V z B x ] 2 1 = 0. (2.76) The effect of the magnetic field is basically adding an extra isotropic pressure B 2 /(8π) and a tension B 2 /(4π) along the field direction. Integrating the five conservation relations, one gets in a convenient notation the following jump conditions, with ˆn the direction of the shock normal (along the negative x-direction) [ρv ˆn] 2 1 = 0, (2.77) [ ρv(v ˆn) + (P + B 2 /8π)ˆn (B ˆn)B/4π ] 2 = 0, (2.78) 1 [ ( 1 V ˆn 2 ρv 2 + γ ) ] 2 γ 1 P + B2 (B ˆn)(B V) = 0, (2.79) 4π 4π [B ˆn] 2 1 = 0, (2.80) [ˆn (V B)] 2 1 = 0. (2.81) 1 Two important remarks need to be made here. Firstly, it can be found that in case of a parallel shock, with both magnetic fiels along the x-axis, the magnetic field drops out of the equations and they reduce to the ordinary Rankine-Hugoniot equations. The magnetic field thus serves no dynamical purpose in parallel shocks so that these shocks are equivalent to hydrodynamical y B 2 Upstream ψ V 2 2,S V 1 ψ 1,S x Downstream B 1 Figure 2.7: The shock rest frame, also called the normal incidence frame.

52 Chapter 2. Principles of plasma physics 40 ones 4. Secondly, the magnetic field will make the shock weaker compared to its hydrodynamic equivalent since the flow energy is now converted to both heat and magnetic energy in the transition region[56]. A more rigorous, fully relativistic, discussion of how to deal with the magnetic field will be left for the next section Relativistic shocks In this section the shock jump conditions and relevant reference frames for relativistic magnetohydrodynamic shocks will be rederived, following the work of De Hoffmann and Teller[58] done in For simplicity, only plane shocks will be considered. The jump conditions will be valid under the approximation of infinite conductivity.. The derivation is for general orientations of the magnetic field with respect to the shock normal and is fully relativistic. The electric field vanishes in the plasma rest frames, due to the approximately infinite conductance (as explained in subsections and used in section 2.2) and the lack of a global motion (see below). Also, the dielectric displacement D vanishes in this system. Further assuming H = B, the only quantities that need to be considered are E and B. Parallel shocks This is the simple case of a plane shock with a magnetic field parallel to the shock normal. Since this means the plasma flow is along the magnetic field, the hydrodynamic motion is not influenced by this field. The shock thus behaves as in normal hydrodynamics and the discription is very similar to the discussion above. In what follows, use will be made of the different reference frames, so that quantities will depend on the frame considered. The plasma rest frames are denoted by O i, where i = 1, 2 means upstream or downstream side respectively. The shock rest frame (shown in figure 2.8) will be denoted by O i,s, where the i means whether the quantity is from the upstream or downstream side. This frame does contain an electric field different from zero. As in normal hydrodynamics, the plasma flow speeds V 1 and V 2 both lie simply among the x-axis, since the symmetry of the system does not allow the plasma flow to be deflected. Thus in the plasma frames, all field components vanish except for B i,x. The Lorentz transformations for electromagnetic fields are given by: With this, the shock frame components become E = E (2.82) B = B (2.83) E = γ (E + V B) (2.84) ( B = γ B 1 ) c 2 V E (2.85) E i,s,x = 0 B i,s,x = B i,s,z E i,s,y = 0 B i,s,y = 0 E i,s,z = 0 B i,s,z = 0, (2.86) 4 Actually the magnetic field will serve to support Alfvén waves that will be important to the acceleration mechanism.

53 Chapter 2. Principles of plasma physics 41 y Upstream Downstream V 1 V 2 B 1 B 2 x Figure 2.8: The shock rest frame for a parallel shock. so that the electric field vanishes in the shock frame as well. The magnetic fields on either side are related to each other by using B = 0 across the shock discontinuity and applying the divergence theorem B 1,S,x = B 2,S,x. (2.87) To solve the shock jump conditions, the energy-momentum tensor for the plasma is needed (a generalisation of the derivation in the previous section). The plasma will again be viewed as a simple fluid. Define the relativistic energy density (E i is the kinetic energy per particle) as ρ i c 2 = n i (m 0,i c 2 + E i ), (2.88) and p i as the pressure in the plasma frame. For the simple system under consideration, the total energy-momentum tensor, which is the sum of the energy-momentum tensor for a fluid and the electromagnetic energy-momentum tensor, has the following non-zero components in the plasma frame T i,xx = p i B2 i,x 8π T i,yy = T i,zz = p i B2 i,x 8π Transforming to the shock system, this becomes T = ρ i V 2 +p i B2 i,x 1 V 2 /c 2 (2.89) (2.90) T i,tt = ρ i + B2 i,x 8πc 2 (2.91) 8π 0 0 [V ρ i +(V/c 2 )p ] 1 V 2 /c 2 0 p i + B2 i,x 8π p i + B2 i,x 8π 0 [V ρ i +(V/c 2 )p i ] 1 V 2 /c ρ i +(V 2 /c 4 )p i 1 V 2 /c 2 + B2 i,x 8πc 2, (2.92)

54 Chapter 2. Principles of plasma physics 42 which holds for both regions 1 and 2. From the equation ν T νµ = 0 (2.93) and the fact that there are no dependencies on coordinates other than x, it follows immediately that T xµ / x = 0, i.e. those components are independent of x. Integrating across the discontinuity, it is found that the xx and xt components of T in the two regions are equal. Together with using B = 0 now in the shock frame (similar to equation (2.87)), this gives immediately (ρ 1 V p 1 )/(1 V 2 1 /c 2 ) = (ρ 2 V p 2 )/(1 V 2 2 /c 2 ) (2.94) (ρ 1 V 1 + V 1 c 2 p 1)/(1 V 1 2 c 2 ) = (ρ 2V 2 + V 2 c 2 p 2)/(1 V 2 2 ). (2.95) c2 Equating the fluxes of particles on either side of the shock, which expresses the conservation of particles, gives n 1,S v 1,S = n 2,S v 2,S, (2.96) or in the plasma frames Γ 1 n 1 v 1 = Γ 2 n 1 v 1, (2.97) This means that in relativistic shocks, the density can increase without limit, even though the velocities always stay finite. The compression ratio r = V 1 /V 2 is thus the relevant parameter for these shocks[59]. These last 3 equations reduce to the Rankine-Hugoniot equations in the non-relativistic limit. Oblique shocks The second configuration that needs to be considered is that of a magnetic field not parallel to the shock normal (but not perpendicular either). Start from the upstream plasma frame, where there is no electric field. Again it is always possible to find a system where the shock normal and V 1 are along the x-axis. It will be shown that this will no longer be the case for v 2 because of the presence of the oblique magnetic field. The y-axis is chosen such that B 1 lies in the xy-plane. Thus we have B 1,x 0, B 1,y 0, B 1,z = 0. (2.98) Now go to a special shock system (labelled by HT ), where the shock is at rest, by transforming along the magnetic field lines. It is then the case that B 1,HT,x B 1,HT,y = V 1,HT,x V 1,HT,y = φ 1. (2.99) Since V 1,HT and B 1,HT are parallel, there is also no electric field in this system. This easily follows from Maxwell s equations. Consider two frames, one is the rest frame of a magnet, the other the rest frame of a conductor (or a plasma). The magnetic field in the plasma frame B will be related to that in the magnet frame B: Thus the Maxwell-Faraday equation B (x, t) = B(x + vt). (2.100) E = B t (2.101)

55 Chapter 2. Principles of plasma physics 43 can be written as E = (v )B = (B v) v( B) = (v B) (2.102) giving E = v B. (2.103) Thus a plasma flow with a magnetic field would generally give rise to an electric field. V 1 B 1 however, this is not the case in this special shock frame Since E 1,HT = 0. (2.104) Because the transformation was along the magnetic field lines, V 1,z = B 1,z = 0 is still valid and B 1,HT,x = B 1,x 0, B 1,HT,y = B 1,y 0. (2.105) Now the fields in the downstream region need to be determined. Using B = 0 across the shock gives B 1,HT,x = B 2,HT,x, (2.106) just as in the parallel case. In the downstream plasma frame there is no electric field, so E 2 B 2 = 0. This last expression is Lorentz invariant, so that is also valid in the special shock frame E 2 B 2 = 0, so that the electric and magnetic field are perpendicular. Since B (both upstream and downstream) doesn t change in time, we also have E = 0 at the shock front. It was already proven that E 1,HT = 0, meaning E 2,HT is perpendicular to the shock front. However, since B 2,HT,x 0, the electric field cannot be perpendicular to both the shock front and the magnetic field. Thus it is also needed that E 2,HT = 0. In order to not generate an electric field, it necessary that v 2 B 2, i.e. B 2,HT,x = V 2,HT,x = φ 2 B 2,HT,y V 2,HT,y (2.107) B 2,HT,z = V 2,HT,z = φ 3. B 2,HT,y V 2,HT,y (2.108) Note that there has been no reason up until now why there should not be a non-zero z-component to v 2,HT or B 2,HT. It is necessary to look at the energy-momentum tensor to further relate the quantities on both sides. Transforming from the upstream plasma frame to the new frame with a velocity V (with B i,ht = B i from equation (2.105)) valid on both sides of the shock. T i,ht,xx = p i + (V 2 x /c 2 )µ 2 (p i + c 2 ρ i ) + (B 2 i /8π) (B 2 x/4π) (2.109) T i,ht,xy = (V x V y /c 2 )µ 2 (p i + c 2 ρ i ) (B i,x B i,y /4π) (2.110) T i,ht,xz = (V x V z /c 2 )µ 2 (p i + c 2 ρ i ) (B i,x B i,z /4π) (2.111) T i,ht,xt = ( V x /c 2 )µ 2 (p i + c 2 ρ i ) (2.112) Since in the upstream frame everything happens in the xyplane by choice, T 1,HT,xz = 0 automatically. Because of equation (2.93), it then follows that also T 2,HT,xz = 0.

56 Chapter 2. Principles of plasma physics 44 There are two sets of solutions. The first one is when taking T i,ht,xy 0. By comparing equations (2.110) and (2.111) with the help of (2.108), it is obvious that T i,ht,xz = V z V y T i,ht,xy. (2.113) It was already shown that T 2,HT,xz = 0. From the last equation and the assumption that T i,ht xy 0, it follows that V 2,z = 0 is needed and thus also B 2,z = 0. This is the most natural case: everything takes place in the xy-plane. The second set is when T i,ht,xy = 0. It can be shown that there is then a degenerate set of solution rotated along the shock normal. In that way out-of-pain solutions are possible while still respecting the symmetry of the system. The De Hoffmann-Teller frame The frame discussed above is the so-called de Hoffmann-Teller (DHT) frame, since they were the first ones to realise the usefulness of this frame[58]. In this frame, the shock is at rest and the plasma flow speed is parallel to the magnetic field lines so that there are no electric fields. Because the shock is oblique, both the flow speed and the magnetic field lines are refracted at the shock front. In normal hydrodynamics such a configuration is not possible since the shock is only capable of transferring momentum perpendicular to the front. The refracted magnetic field lines make the necessary momentum transfer possible. This frame is interesting to work in both for theoretical derivations and for simulations, mainly because it is a shock frame that does not have any electric fields. Note that it was silently assumed that a transformation to such a frame is possible. This is not always the case. In order to understand why, it is useful to consider what frame exactly is transformed to. The frame where the shock is standing still and V B is the one where the intersection points of the magnetic field with the shock front are standing still. We are thus transforming to the frame of these intersection points. If the shock is sufficiently fast and the magnetic field sufficiently oblique however, these intersection points can go faster than light. Such a shock is called a superluminal shock. The trick of the De Hoffmann-Teller frame can only be used in subluminal shocks. Below, a criterion for subluminal shocks will be derived. Actually, the De Hoffmann-Teller frame is not unique, there is an entire set of solutions that corresponds to it. The two important ones are the following. There is the one described above, reached by transforming along the magnetic field lines. The second one is reached by doing the transformation in two steps. first a transformation from the plasma frame to the normal incidence frame (see figure 2.7 from before), the shock frame where V 1 is parallel to the shock normal, followed by a transformation along the shock front to a frame where the intersections are at rest. In this case equation (2.105) is no longer valid. The difference between the two is explained nicely by Kirk and Heavens[62] and Summerlin and Baring[63] and is illustrated in figure 2.9. In the first way of transforming, due to relativistic aberration, the shock front is rotated, so that it is no longer coinciding with the yz-frame. In the second way this is not the case, since the shock front is in both steps either completely perpendicular or completely parallel to the boost velocity. The latter transformation method is therefore the preferred way of reaching the De Hoffmann-Teller frame.

57 Chapter 2. Principles of plasma physics 45 y B 2,HT B 2,HT Upstream Upstream ψ 2,HT V 2,HT ψ 2,HT V 2,HT ψ 1,HT x ψ 1,HT x V 1,HT Downstream V 1,HT Downstream B 1,HT B 1,HT (a) The DHT frame in the original derivation. (b) The DHT frame that is easiest to work with. Figure 2.9: Two possible DHT frames describing the shock. Lorentz transformation of the shock parameters By using the second way of transforming, the only new transformation is that from the shock frame to the De Hoffmann-Teller frame, which is along the shock front. Following now the derivation by Summerlin and Baring[63], the different angles and speed between the two frames can easily be related. Doing a transformation along the y-axis with a speed β t changes the components of a general velocity vector as V x,ht = In the upstream frame, there is no z-component, so that we have V x,s Γ t (1 + V t V y,s ), (2.114) V y,ht = V y,s + V t 1 + V t V y,s. (2.115) V 1,x,HT = V 1,x,S Γ t, (2.116) V 1,y,HT = V t. (2.117) The Lorentz transformation of the magnetic field (equations (2.83) and (2.85)) become The ratio of these equations give B x,ht = B x,s Γ t, (2.118) B y,ht = B y,s. (2.119) tan ψ i,ht = Γ t tan ψ i,s i = {1, 2} (2.120) for the angle of the magnetic field with respect to the shock normal in both the upstream and downstream frame. Using the above equation, the fact that V i,ht B i,ht and taking the ratio of the upstream expressions for equations (2.114) and (2.115), we find V t = V 1,x,S tan ψ 1,S. (2.121)

58 Chapter 2. Principles of plasma physics 46 The transformation to the De Hoffmann-Teller frame is thus possible only if V 1,x,S tan ψ 1,S < 1. (2.122) Repeating now the same idea for the transformation between the shock frame and the fluid frame gives tan ψ i,s = Γ i,s tan ψ i. (2.123) where Γ 1,S = 1/ 1 β1,x,s 2 and Γ 2,S = 1/ 1 β2,x,s 2 β2 2,y,S. The above equations relate all the necessary quantities in different frames, starting from the parameters in one frame. 2.4 Laboratory plasma experiments The details of (relativistic) plasmas have long been not fully understood, being only available through indirect astronomical observations or complex plasma simulations. Recently, however, such plasmas have become experimentally accessible through plasma acceleration[64]. Plasma acceleration is a technique for accelerating charged particles, such as electrons, positrons and ions, using an electric field associated with electron plasma wave or other high-gradient plasma structures (e.g shock or sheath fields). The plasma acceleration structures can be created either using ultra-short laser pulses or energetic particle beams that are matched to the plasma parameters. These techniques offer a way to build high performance particle accelerators of much smaller size than conventional devices. The basic concepts of plasma acceleration and its possibilities were originally conceived by Tajima and Dawson[65]. Plasma accelerators hold immense promise for innovation with a wealth of applications in astrophysics and plasma physics. The use of high-power lasers to create hot plasmas is common to laboratories around the world. The advantage of lasers is that very high powers can be delivered, with focussed intensities onto target in excess of Wcm 2. To get a sense of perspective, note that this intensity is more than 20 orders of magnitude higher than what delivered by bright sunlight. The electric field in such a beam is more than Vm 1. This is enough to rip electrons from atoms directly. When laser light is absorbed by a solid, the matter is evaporated and heated into a plasma with a temperature that can easily be in excess of 10 8 K. The sort of lasers range from the very large systems, with nanosecond pulses delivering kilojoules of energy, to smaller systems with ultra-fast pulses of less than s which can be focussed to high intensities. There are several important physical phenomena associated with laser-matter interactions, such as the following. It is possible to generate plasmas that can be used as models of astrophysical systems. Strong shock waves can be droven in excess of 100 million atmospheres pressure. Beams of electrons or protons with > MeV energies can be generated. It is possible to generate ultri-high magnetic fields in excess of 1000 T. A plasma can be pumped to generate X-ray lasers operating in the soft X-ray regime. At the moment there are many experimental laser plasma facilities around the world, such as: the laser plasma accelerator at Lawrence Berkeley National Laboratory, the Texas Petawatt laser facility (University of Texas at Austin) where electrons can be accelerated mono-energetically to 2 GeV over about 2 cm! There is also the centre for Plasma Physics of Queen s University Belfast

59 Chapter 2. Principles of plasma physics 47 and the Central Laser Facility at the Rutherford Appleton Laboratory in UK, the Laboratoire pour l Utilisation des Lasers Intenses in Palaiseau in France and the the PALS system in Prague, Czech Republic, and others. Very recently a team of scientists (Meinecke et al.[66]) managed to recreate a supernova explosion in the laboratory, using the Vulcan laser facility at the Rutherford Appleton Laboratory UK. In this impressive experiment three laser beams were focused onto a carbon rod target, not much thicker than a strand of hair, in a low density gas-filled chamber. The enormous amount of heat generated more than a few million degrees Celsius by the laser caused the rod to explode creating a blast that expanded out through the low density gas. In the experiment the dense gas clumps or gas clouds that surround an exploding star were simulated, by introducing a plastic grid to disturb the shock front. A series of lab experiments with this set-up, and relevant follow-up simulations, demonstrated that as the blast of the explosion passes through the grid, it becomes irregular and turbulent just like the images from Cassiopeia A supernova. Among other, it was found that the magnetic field is higher with the grid than without it. Since higher magnetic fields imply a more efficient generation of radio and X-ray photons, this result confirms that the idea that supernova explosions expand into uniformly distributed interstellar material is not always correct, and it is consistent with both observations and numerical models of a shock wave passing through a clumpy medium.

60

61 Chapter 3 Acceleration mechanism In this chapter diffusive shock acceleration of charged particles in astrophysical shocks, based on the original proposal by Fermi[32] in 1949, will be explained. The first section will deal with the basic principle behind the mechanism. Subsequent sections will treat in more detail the more advanced concepts developed in the last few decades. As was found in the last chapter, a plane shock can be described in the xy-plane only (ignoring the degenerate case mentioned). The approximation of a plane shock is certainly permissible in this case, given the size of the astrophysical sources. The accelerators considered could be both leptonic if they are primarily accelerating electrons, or hadronic, if they are mainly accelerating protons and heavier nuclei. Obviously, to generate cosmic rays, the hadronic kind would be favoured. Determining in what category the different accelerators belong can only be done by combining different observation methods (see the Multimessenger astrophysics of section 1.5). For the purpose of this chapter, the kind of particle is mostly irrelevant however. In fact, all of the acceleration theory is, in principle, valid for all charged particles. The type of particle will determine which effects are relevant and this is mostly important for loss effects. In the context of this thesis, however, we will mostly be concerned with the hadronic type. To start, the acceleration will be studied in the test-particle approach. This means that the particles considered will feel the effect of the electromagnetic fields, but will not themselves contribute to them. 3.1 The Fermi mechanism General concept There are three primary properties the sources of cosmic rays are required to have. Firstly, they should have sufficient power output to provide the density of cosmic rays. Secondly, their maximum energy attainable must be high enough to explain the already observed particles. Both of these were discussed in the first chapter. Finally, the spectrum should have the form of a power law. The Fermi mechanism described below generates such a power law very naturally, meaning 49

62 Chapter 3. Acceleration mechanism 50 it is an attractive acceleration candidate to consider and is indeed the primary focus of current research. Following the argument by C. Grupen[28], we will show the mechanism which generates a power law distribution. The main idea of the Fermi mechanism is that particles undergo repeated acceleration cycles, each having an energy gain which is proportional to the current energy This means that after n cycles, a particle would have an energy of Rewriting to the number of cycles needed to get to a certain energy E = ξe. (3.1) E = E 0 (1 + ξ) n. (3.2) n = ln(e/e 0) ln(1 + ξ). (3.3) After each cycle, the particle has an escape probability P esc = 1 P. Thus the number of particles after n cycles is N = N 0 P n. (3.4) Combining these last two equations gives for the energy distribution ( ) ( ) ( N E ln P ln = n ln P = ln N 0 E 0 ln(1 + ξ) = s ln E0 E ( ) ln 1 1 P esc s = ln P/ ln(1 + ξ) = P esc ln(1 + ξ) ξ ) (3.5) (3.6) dn(e) de E (1+s) (3.7) If one can find a system with such repeated accelerations, where the energy gain is proportional to the current energy, then a power law spectrum is guaranteed. The exact power law index still depends on the details of the mechanism and the time scale on which the power law distribution is formed has not been considered yet (see section 3.4.2). In what follows, the two main mechanisms to exhibit such behaviour are described in more detail. Note that the above conditions basically express that the system is scale-free and it can be shown that this automatically leads to a power law (see the iconic paper by Newman[67]) Second order Fermi acceleration This is the mechanism originally considered in 1949 by Fermi[32]. The idea is that charged particles get accelerated as they propagate through the galaxy by encountering moving dense interstellar plasma clouds containing magnetic fields. The approximate speed of such clouds is of the order of 30 km s 1 (root mean square radial velocity of 15 km s 1 from observation[68], multiplied with root of 3 for the velocity). When a particle encounters such a cloud, it is scattered by it. This scattering is not due to particle-particle interactions however, since such clouds have low density and are thus collisionless plasmas. If the particle were to suffer particle-particle scatterings, it

63 Chapter 3. Acceleration mechanism 51 would immediately thermalise, so that a build-up of energy cannot occur. This is now not the case, instead is scattered off the frozen-in magnetic field irregularities in the clouds (or in some versions, off magnetic mirrors), so that it does not lose its energy to the cloud. The particle can approach the cloud in two ways, either the particle and the cloud move towards each other in a head-on collision, or the particle can catch up to the cloud (since the particle speed will approach c, while the cloud moves slowly) in an overtaking collision. To picture what happens, consider a one dimensional set-up (only movement in the x-direction) and suppose that the particle is always reflected by the cloud (see figure 3.1a). In a head on collision the particle will increase its energy, extracting it from the cloud (which has approximately infinite mass, meaning the energy loss can be neglected) while an overtaking collision leads to a decrease. In general this simple reasoning is not always true however (see below). The derivation below to calculate the mean energy gain in one encounter follows the one by Gaisser[19]. The system is described in the frame of the cloud in 2 dimensions 1 as shown in figure 3.1b. The scattering is due to magnetic field irregularities, which can only change the direction of the particle and perform no work on it. Thus in the cloud frame, in which the irregularities are at rest (since they are frozen into the cloud plamsa), there is no energy change during scattering the scattering is purely elastic. For a sufficiently relativistic particle, so that E pc, a simple Lorentz transformation on the momentum four-vector of the charged particle from the lab frame gives the particle energy as seen from the gas cloud E 1 = γe 1 (1 β cos θ 1 ), (3.8) 1 with γ =, β = V/c and V the speed of the gas cloud (primed quantities are in the 1 V 2 /c2 cloud rest frame). When the particle escapes with an energy E 2, it still has the same energy E 1. Transforming the escaping particle back to the original lab frame gives Combining the above two equations E 2 = γe 2(1 + β cos θ 2). (3.9) E E 1 = 1 β cos θ 1 + β cos θ 2 β2 cos θ 1 θ 2 1 β 2 1. (3.10) Now to get the mean energy change, the above equation needs to be averaged over the incoming and outgoing directions. For gas clouds the outgoing angle is uniformly distributed in the cloud frame, since there is no preferred direction at all, giving the distribution (see also section 4.2.5) dn dθ 2 sin θ 2 dn d cos θ 2 = const, 1 cos θ 2 1. (3.11) This means cos θ 2 = 0 so that the fractional energy change becomes E 2 E 1 = 1 β cos θ 1 1 β 2 1. (3.12) 1 The third dimension would not change anything, you can just add the extra angle φ around the horizontal axis and integrate it out.

64 Chapter 3. Acceleration mechanism 52 (a) Simple picture of a particle encountering magnetised clouds[28]. (b) Defining quantities used in deriving energy gain[19]. Figure 3.1: Fermi s original suggestion for an acceleration mechanism. To average over cos θ 1, the probability for a collision with a cloud needs to be calculated. The relative velocity between the cloud and the particle (with v c) is c V cos θ 1. The probability for collision of an isotropically distributed set of particles with a cloud is directly proportional to this relative velocity (seen by comparing the path of different particles for a given time interval). There is no angular dependence because the cloud is a blob (meaning it is a single point if seen from large enough distance), so that if a particle crosses there is no preferential direction. This gives then the normalised distribution so that cos θ 1 = 1 V 3 c dn = c V cos θ 1, 1 cos θ 1 1, (3.13) d cos θ 1 2c and the fractional energy change becomes ξ = E 1,2 E 1 For a non-relativistic cloud speed, β << 1, this can be approximated to = β2 1. (3.14) 1 β2 ξ 4 3 β2. (3.15) Physically, the source of the energy gain is the cloud as a whole, just like in elastic scattering in classical mechanics. Because of its squared dependence on the (small) β, the original Fermi mechanism is called second order. This means acceleration goes slow: only after many collisions can there be a significant net gain. The reason for this is that in some collisions the particle energy decreases. Also it can be seen from the general formula (3.10) that particles colliding head-on can still sometimes lose energy, if they leave the cloud out the back, so that the general statement head-on collisions result in a gain, while overtaking collision result in a loss is indeed not always correct. From formula (3.6), the spectral index can be expressed as s = P esc ξ = 1 ξ T cycle T esc. (3.16) Fermi originally considered the galactic disc as acceleration region, with an escape time of the order 10 7 years. The acceleration rate is the rate of collisions with the clouds. The cosmic rays go

65 Chapter 3. Acceleration mechanism 53 at a speed c and the clouds have a density ρ c and cross section σ c, so that T cycle 1/(cρ c σ c ). This gives for the spectral index 1 s 4. (3.17) 3 β2 cρ c σ c T esc There is thus no unique spectral index, moreover the result is usually a large number[69]. There are however a few problems with this mechanism. The first one is that the acceleration is incredibly slow. Not only is the gain small due to the squared dependence on the cloud speed, but on top of that the physical parameters are suboptimal. The small cloud speed quoted above gives V/c 10 4, giving small average gain per collision. Furthermore, the mean free path of cosmic rays in the interstellar medium is about 0.1 pc, giving only a few collisions per year. Together this means that the particles gain energy very slowly over time. Secondly, in order for the mechanism to work efficiently, the particles need to be quite energetic already. Low energy particles are heavily bothered by loss effects (primarily ionisation) however. Getting the acceleration even started is thus very difficult. This is known as the injection problem and is very hard to solve. Lastly, there is the problem that this mechanism does not predict the exponent of the power law. Many separate occurrences of this mechanism could give varying exponents, contrary to the very robust and uniform power law seen over many orders of magnitude. Therefore this model is no longer seen as the primary mechanism for the spectrum of cosmic rays. However, it still has uses for cosmic ray propagation models, under the name of diffusive reacceleration, where this mechanism slightly alters the cosmic ray spectrum during the propagation through the galaxy First order Fermi acceleration The first reason for the slow working of the above mechanism was that there were also encounters that lost energy instead of gaining. The difference between the two corresponds roughly (but not exactly) to the separation between head-on and overtaking collisions. It would be much more advantageous if there were only head-on collisions. It was first realised independently by Krymskii[70], Axford, Leer and Skadron[71], Bell[72, 73] and Blandford and Ostriker[74] in the years that such a system can be realised in the case of a shock front propagating through the interstellar medium, which renewed the interest in the Fermi mechanism. Shocks occurs, for example, when a star explodes in a supernova event, causing a wave of ejecta which to sweep through the local medium at a speed higher than the speed of sound of that medium. These astrophysical environments consist of plasmas with magnetic field lines and magnetic field irregularities that allow for collisionless scattering, so that again no energy is lost due to thermalisation. The derivation of the spectral index will follow that by Longair[22], which is based on the original work by Bell[72]. Like in the previous chapter, we consider a plane shock wave which forms a discontinuity between the upstream and downstream plasmas. These conditions imply that the gyroradius of the particle is very large compared to the actual shock thickness, but small compared to the radius of curvature of the shock front (which is related to the containment criterion). Also the magnetic field in the plasma will not be explicitly considered here yet, but the general results are not altered by this.

66 Chapter 3. Acceleration mechanism 54 As in sections and 2.3 it will be useful to define different reference frame to easily and accurately describe the physics involved. In the upper left panel of figure 3.2a the shock is shown in the lab frame (note that this is the only figure in this entire thesis in which the shock propagates to the right) and in the upper right panel the shock rest frame is shown. As found in section 2.3.1, the downstream flow speed is 1/r of the upstream flow speed. The last two frames, at the bottom of the figure, are the plasma frames where the plasma medium is at rest. Using simple Lorentz transformations from the shock rest frames, it can be seen that both the upstream and downstream plasma see the other one coming towards itself with the same speed ( 3 4 U in the case r = 4). This feature is essential. For example, assume one has a high energy particle somewhere upstream (lower left panel of figure 3.2a). This particle is scattered isotropically in the upstream plasma medium. At some point it inevitably crosses the shock, since the shock is propagating towards the upstream medium and the particle undergoes random scattering events so that its effective speed is reduced to the Alfvén speed of the plasma (see the original paper by Bell[72]), which is the equivalent for magnetohydrodynamic waves of the sound speed in normal hydrodynamics. The particle will then undergo a head-on collision with the downstream plasma. In the downstream plasma (lower right panel) the particle again scatters isotropically. Either the particle never catches up to the shock again, at which point it is said to escape, or it does catch up and undergoes a new head-on collision, this time with the upstream plasma. The particle has now done one complete cycle in which it only undergoes head-on collisions and the process can start all over again. We will now show that there is indeed an energy gain proportional to the current energy. The discussion above uses the same concepts as the second order Fermi mechanism. In particular, the derivation up to equation (3.10) is still valid. The only difference occurs in the averaging over angles. From now on the convention as in figure 3.2b will be used, meaning upstream is to the left. The angles of incoming and outgoing particles are defined similar to figure 3.1b and primed quantities are again used for the cloud frame while the unprimed are used for the lab frame. The distribution of outgoing angles is determined by the projection of an isotropic flux onto a plane. Consider a flux tube making an angle θ 2 with the surface normal and having an area A. If the particle density is n the fraction of particles going at this same angle to the surface normal hitting (a) Different frames describing a shock front[22]. (b) Defining quantities used in deriving energy gain[19]. Figure 3.2: The shock front setup for first order Fermi acceleration

67 Chapter 3. Acceleration mechanism 55 the surface in a time dt is given by flux = n A cdt cos θ 2. (3.18) A dt This has to be multiplied by the chance that a particle makes this angle. For a uniform particle distribution this goes as sin θ 2. The normalised distribution thus becomes dn d cos θ 2 = 2 cos θ 2, 0 cos θ 2 1. (3.19) The cosine is restricted between 0 and 1 since outgoing particles only cross the shock if they are going to the left. Thus cos θ 2 = 2/3 and the fractional energy gain averaged over the outgoing angle is given by E 2 E 1 = 1 β cos θ β 2 3 β2 cos θ 1 1 β 2 1 (3.20) For the incoming particle the distribution is again the same (isotropic flux projected on a plane), but now the cosine is restricted to 1 cos θ 1 0, since particles need to go to the right to cross the shock. This means that cos θ 1 = 2/3, giving for the averaged energy gain ξ = E 1,2 E 1 = β β2 1 β 2 1. (3.21) For non-relativistic shock speeds this gives ξ 4 3 β = 4 3 u 1 u 2. (3.22) c To find the spectral index, the escape probability is needed. This probability is the ratio of the rate of particles encountering the shock to the rate of particles being swept away downstream. The first number is found by projecting (= cos θ) the isotropic cosmic ray flux (cρ CR /4π) onto the plane shock front and then integrating for all particles going towards the shock 1 0 2π d cos θ dφ cρ CR 0 4π cos θ = cρ CR 4. (3.23) The rate of convection downstream is just given by the density times the speed with which the downstream plasma flows away from the shock (ρ CR u 2 ), giving for the escape probability P esc = ρ CRu 2 cρ CR /4 = 4u 2 c. (3.24) Thus, combining equations (3.22) and (3.24) gives the spectral index s = P esc ξ = 3 u 1 /u 2 1. (3.25) Going back to the example in figure 3.2a where the compression ratio was 4, this gives a spectrum dn(e) de E 2. (3.26) In other words, the first order Fermi mechanism naturally gives a value of the spectral index close to the one observed. The exact value is very robust, only mildly dependent on the precise

68 Chapter 3. Acceleration mechanism 56 conditions in the acceleration environment, meaning a variety of sources could produce roughly the same spectrum. Since the first order mechanism is also much faster, not only due to the gain being β instead of β 2. The mechanism takes place in a shock so that, compared to the moving clouds in the galaxy, the encounter rate is higher (since the system is compact) and the shock speed gives a higher β. For example, the speed of shocks in supernovae is of the order 10 4 kms 1, much higher than the sound and Alfvén speed of the interstellar medium ( 10 kms 1 ). Such as with all acceleration mechanisms however, the injection problem is still present. Note that in the above calculations, the energy change was found purely by transforming between the relevant reference frames. It is this trick that will be used intensively in the simulation to almost magically accelerate the particles. 3.2 Advanced concepts Obviously the above derivation is rather basic and more advanced techniques exist. While it is not the goal to explain all of these in detail, it is instructive to go over the main ideas. Apart from being relevant results themselves, these theoretical efforts are often the inspiration for simulating the acceleration process. Essentially there exist two approaches to finding the spectrum. In the first one, by Bell[72], one considers the behaviour of one individual particle and then take averages, like was done above. The second approach, which follows the papers by Krymskii[70], Axford, Leer and Skadron[71] and Blandford and Ostriker[74], uses a macro-approach where one considers the distribution function of the particles. Many good reviews on the theoretical efforts in the last decades exist[75][76][59][61] Diffusive transport The main idea of the first order mechanism described above is that the accelerated particles diffuse on each side of the shock, allowing for head-on collisions required on shock crossing, leading to the alternative name diffusive shock acceleration. Therefore, to theoretically model the acceleration process, one can try to solve the distribution function f(x, p, t) on each side of the shock from the transport equation for diffusive transport (or actually more appropriately called the Fokker-Planck equation, since we are concerned with individual particles and not flows) and then connect the distributions at the shock front. Most generally the equation is[75] f t + V f = (κ f) Vp f p + 1 p 2 p ( p 2 D f p ), (3.27) with V the plasma flow speed. The second term on the left hand side represents advection by the scattering centres which are moved by the background motion. The first term on the right describes ordinary diffusion, with generally κ the anisotropic diffusion tensor. The second term on the right represents adiabatic changes as the large scale motion of the medium is diverging or converging, where Liouville s theorem states that such a change should be matched by, respectively, an equal convergence or divergence in momentum space. The last term represents the movement of the scattering centres relative to the background and therefore describes second order acceleration.

69 Chapter 3. Acceleration mechanism 57 Obviously some approximations are necessary. It was already reasoned that second order Fermi acceleration is not very powerful compared to first order Fermi acceleration, so it is usually 2 ignored (meaning the scattering centres are considered to be stationary). The adiabatic change is a minor effect as well. Lastly, following the reasoning by Blandford and Ostriker[74] and Kirk and Schneider[78], in the simplest case the diffusion can be described by pitch angle diffusion only[76], where the diffusion coefficient is independent of the direction. For the one-dimensional geometry considered above, the Fokker-Planck equation then reduces to f t + V f = µ D µµ(p)(1 µ 2 ) f µ = ( ) f t coll (3.28) where now D µµ, the pitch angle diffusion coefficient, is the only part left of the diffusion tensor. The right hand side is a collision or scattering operator, theoretically defined by the diffusion coefficient and is the primary source of differences between various approaches. Finally to solve this, one searches for a stationary solution, meaning f t = 0. One then expands the distribution function in Legendre polynomials. Assuming isotropy of the distribution function, meaning only the lowest order polynomials are kept, then allows to easily solve the transport equation on both sides and match the solution at the shock front Relativistic shocks In the above, we were concerned with non-relativistic flows. When relativistic shock fronts are considered, higher order effects become important. The most important one is that the distribution function becomes highly anisotropic. The reason for this is simple: since the shock speed becomes relativistic, an initial isotropic distribution of particles in the plasma frames undergoes relativistic beaming when Lorentz transforming to the shock frame. This means that, as one considers higher and higher shock speeds, more and more particles will have a velocity directed along the shock normal. This obviously implies that there will be more purely head-on collisions, increasing the energy gain. Physically this just means that the upstream particles are caught by the shock as soon as they go towards it. Relativistic shocks were first considered by Peacock[79] in the individual particle approach and by Kirk and Schneider[78] in the distribution function approach. In the latter, the transport equation was modified to[78, 61] Γ 1,2 (1 + v 1,2 µ) f t + Γ 1,2(v 1,2 + µ) f x = µ D f µµ µ, (3.29) with µ the angle of the particle with respect to the flow and Γ the Lorentz boost factor of the plasma flow speed. The above equation is found by transforming the variables t and x from the plasma frame, where the Fokker-Planck equation is most easily found, to the shock frame where the upstream and downstream distributions can be easily matched. Again, the equation can be solved by expanding it in functions, for this the Q J -method as developed by Kirk and Schneider[78]. 2 The effect it has on the spectrum is, for example, studied by Virtanen and Vainio[77].

70 Chapter 3. Acceleration mechanism 58 There are semi-analytic solutions to this problem, which take into account correctly with the anisotropy of the particles. This comes down to solving the equation up to higher order, which is an arduous task. In practice, one needs to turn to numerical solutions to correctly deal with the complications seen in relativistic plasma flows[80]. Different approaches are available and will be discussed in the next chapter. 3.3 The role of the magnetic field The above derivations are all valid for parallel shocks, where the effect of the magnetic field can be neglected (as explained in section 2.3.1). At the moment of shock crossing the particle angle with respect to the shock normal is in that case equal to the pitch angle. Therefore, the inclusion of an oblique magnetic field will create complications. This has two immediate consequences. The first one was discussed in the previous chapter: the plasma flow will be refracted at the shock front. This influences the Lorentz transformations used to calculate the energy gain. The second consequence is that since the particles follow the magnetic field lines, they essentially see not the speed of the shock but instead chase (or are chased by) the intersection of the shock front with the magnetic field lines. Since this intersection goes faster than the shock itself, it can be expected that such shocks will be better at accelerating particles to high energies (giving flatter spectra and/or faster acceleration rates). Of course this effect will be most relevant for non-relativistic shocks, where the difference can be considerable. In the following, the magnetic fields will lie purely in the xy-plane, as defined at the end of the previous chapter. The total magnetic field consists of two components. There is the mean magnetic field which defines the geometry (through its inclination angle with the shock normal) and there are magnetic field irregularities which cause the scattering events. As mentioned in the previous chapter, there is a distinction between subluminal and superluminal shocks. It will turn out that these two shocks behave very different in the acceleration mechanism Reflection The oblique magnetic field allows for a new phenomenon: reflection at the shock front. In the parallel case, where the magnetic field was along the shock normal everywhere, the magnetic field had equal strength on both sides on the shock (see equation (2.87)). In the oblique case, the component parallel to the shock normal is still conserved across the shock (equation (2.106)), but the other component is changed. This means that, since one component needs to be equal, the two fields can no longer be equal in strength (if the geometry is indeed restricted to the xy-plane). Since the downstream flow speed is lower than the upstream flow speed, the downstream plasma density is higher (formally explained with the compression ratio defined by the Rankine-Hugoniot conditions). Since the magnetic field is frozen in, it will also be compressed, i.e. the downstream magnetic field will be stronger. From the treatment in section it follows that at the shock, certain particles will no longer be able to cross from upstream to downstream depending on their pitch angle they see a magnetic mirror. Since magnetic field is constant on both sides of the shock and changes discontinuously at the shock front, the pitch angle after reflection will be equal and opposite to the original one. This reflection was extensively studied by Hudson[81].

71 Chapter 3. Acceleration mechanism 59 This possibility of reflection causes the energy spectrum to be flatter as the magnetic field becomes more oblique, as observed and explained by Ostrowski[82]. When a particle reflects at the shock, it still suffers a head-on collision, meaning it gains energy. However, since particles upstream cannot escape the shock, at some point they will inevitably be swept up again. This means that it is likely that there are more collisions before the particle escapes from the downstream frame. Moreover, as the field becomes more oblique, the velocity component away from the shock will become small (since the particles still need to follow field lines). Thus after a reflection the particle will very quickly encounter the shock again, before it had a chance to (significantly) change its pitch angle and the particle will be reflected again. The repeated reflections will cause the gain to be large, causing flatter spectra (more particles with high energies). Moreover, due to the quickly repeating reflections the time scale of the acceleration can be much shorter, up to a few order of magnitude[83]. While the individual gains in (highly) oblique shocks are indeed huge, there is also a downside. It was found by Ellison, Baring and Jones[84] in non-relativistic flows that, as the magnetic field becomes more oblique, particles are less efficiently injected from the thermal population into the acceleration mechanism in the first place (see below for a discussion of injection). One can therefore not expect a huge flux of particles from such accelerators. Only in the case that there is a lot of turbulence can there be enough injection. There is one possible saviour for this mechanism however. If there is a population of particles, already accelerated to high enough energies (e.g. in the stellar wind or from different magnetic field inclinations at other positions) then the highly oblique shocks can still be used Shock-drift acceleration An important consequence of an oblique magnetic field is that there is now a v B electric field present, as proven in chapter 2. This electric field can lead to something called shockdrift acceleration, where particles are accelerated in a single shock crossing by drifting along the magnetic field. This mechanism was considered in a paper by Begelman and Kirk[85] where they model the acceleration in superluminal shocks at hot spots in extragalactic sources. The work proposed that in these radio sources, the shock is superluminal, in order to explain some otherwise contradicting observations. This, however, has as a consequence that normal diffusive shock acceleration becomes (in its simplest form) impossible, since the intersections of the magnetic field and the shock go (by definition) at a speed larger than c. The shock-drift acceleration can then be used to accelerate the particles instead by letting the particles cross the shock due to the electric field drift, which does not follow the magnetic field lines. It has since been found that the superluminal shock-drift mechanism is inefficient in accelerating particles up to the highest energies (> ev) and that only subluminal shocks are capable of generating the ultra-highenergy cosmic rays[86, 39]. Even in subluminal shocks, the shock-drift acceleration is a fundamental part of the Fermi acceleration mechanism[59, 84], since an electric field appears in both the lab frame and normal incidence frame. Including this drift in the description of shock crossing would normally complicate the treatment (both theoretically and numerically), but it can be efficiently dealt with. In

72 Chapter 3. Acceleration mechanism 60 subluminal shocks the electric field vanishes in the De Hoffmann-Teller frame. Thus, by describing shock crossing in this frame, which is also a shock rest frame, the shock-drift acceleration is automatically included in the treatment without the accompanying complications Particle scattering Up until now only the mean magnetic field was discussed. However it is obvious that in the violent phenomena considered (like supernovae, AGN s and GRB s) there should be considerable turbulence present. Moreover, magnetic irregularities are required for the Fermi mechanism to work in the first place, since the particles need to be scattered without losing energy. While in some papers the cause of the irregularities is left unspecified, it is attributed to the Alfvén waves created by the high energy particles going faster than the Alfvén speed in the[72, 74]. The Alfvén waves serve as a sort of glue between the plasma and the energetic particles[59]. The presence of turbulence allows for a new type of movement, not along the mean magnetic fields, called cross-field diffusion. For subluminal shock, this effect is an unnecessary complication, but for superluminal it is an essential part of the physics, since there particles diffusing along the magnetic field lines have great difficulty in crossing the shock from downstream to upstream. A discussion relating the turbulence to cross field diffusion was given by [87][87]. Some of the work in the field explicitly treat this turbulence when trying to solve the accelerated particle spectrum, usually by adding a number of finite amplitude perturbations on top of the magnetic field and numerically integrating the particle trajectories[88, 82, 89]. Different approaches for the perturbations exist; a short review is given by Kirk and Duffy[61]. One of the findings is that the perturbations serve to increase the mean magnetic field in a parallel shock, so that the particles see an oblique effective field at shock crossing, allowing for reflection so that the energy gain increases. The above method is quite involved however, and a simpler solution exists. Particles are allowed to randomly scatter with a certain frequency without explicitly treating the perturbations, but with the understanding that the underlying reason for scattering is a certain turbulence spectrum (see for example Meli and Quenby[86]) Solving the shock jump conditions An important part of solving the acceleration spectrum is correctly dealing with the plasma conditions under which the particles are accelerated. It is necessary to find correct values of the upstream and downstream plasma flow speeds and magnetic field inclination angles. This is done by finding appropriate solutions to the Rankine-Hugoniot shock jump conditions. The solution of the jump conditions was discussed by several authors. Kennel and Coroniti[90, 91] solve them explicitly to construct a full MHD-model of the Crab nebula. Jones and Ellison[59] solve the jump conditions to find a relation between the Mach number of the shock, its compression ratio and the spectral index of the resulting particle spectrum. Heavens and Drury[92] provide solutions for different populations. A more recent solution can be found in the paper by Summerlin and Baring[63]. The first part of it is devoted to developing accurate solutions to the most general

73 Chapter 3. Acceleration mechanism 61 conditions, which then serve as input for independent simulations of the actual particle acceleration process. Their results are shown in figure 3.3. In the simulation developed for this thesis, the approach used follows the papers by Begelman and Kirk[85] and Kirk and Heavens[62] where the limit of a strong shock is considered. It is in these shocks that the most efficient accelerators are expected. In such shocks the magnetic field is dynamically unimportant and, since it is the magnetic field that makes it possible to refract the downstream plasma flow, such a refraction can now be neglected in the shock rest frame. That this approximation is valid can be appreciated from the downstream deflection shown in figure 3.3. In the shock frame both the upstream and downstream flow velocities are then along the shock normal, as shown in figure 3.4. The ratio of the magnetic field strengths in the upstream and downstream region will be calculated for the above case, following the review by Kirk and Duffy[61]. From the Lorentz transformations of the electromagnetic fields, it can be seen that a transformation from the shock frame to the plasma frame will not change the x-component of the magnetic field (see equation (2.83)). was proven in the last chapter that in the shock frame the x-component of the magnetic field is conserved across the shock (see equation (2.106)). Therefore B 1 cos ψ 1 = B 2 cos ψ 2 (3.30) where B 1 and ψ 1 are the magnetic field strength and angle with respect to the shock normal in the upstream region as measured in the upstream plasma frame (and analogous is for the index 2). Equation (2.75) can generally be rewritten as [B µ u ν u µ B ν ]l µ = 0 (3.31) with l µ the shock normal and where we defined the four-vector B µ = u ν () with F µν the dual electromagnetic tensor. Using the Minkowski product of this jump condition with itself gives Γ 2 1B 2 1(V 2 1 cos 2 ψ 1 ) = Γ 2 2B 2 2(V 2 2 cos 2 ψ 2 ) (3.32) 1 with Γ 1,2 =. Combining this result with equation (3.30) then results in 1 V1,2 2 /c2 It B 2 B 1 = ( r 2 Γ 2 1(r 2 1)(cos 2 ψ 1 v 2 1) ) 1/2 (3.33) with the compression ratio r = v 1 v Properties of the acceleration process Energy gain Most of the important qualitative behaviour was already explained in the appropriate sections above and will not be repeated here. There is one remaining remark that needs to be made however. The mean energy gain was indeed already shown for non-relativistic shocks, but it is also possible to say something meaningful about relativistic shocks, following the reasoning by

74 Chapter 3. Acceleration mechanism 62 Figure 3.3: Solutions of the compression ratio r, the downstream magnetic field inclination θ u2 and the downstream flow speed deflection angle θ u2 with an upstream magnetic field inclination angle of 5 for different values of the Alfvénic (M A ) and sonic (M S ) Mach number. Figure taken from Summerlin and Baring[63]. y B 2 Upstream V 1 V 2 ψ 1,S ψ 2,S x Downstream B 1 Figure 3.4: The shock rest frame for a dynamically unimportant field, where now both the upstream and downstream flow are along the x-axis.

75 Chapter 3. Acceleration mechanism 63 Gallant and Achterberg[93] on parallel relativistic shocks. It is of course assumed that the same principles that worked for non-relativistic shocks are still valid for relativistic shocks. The energy gain is again easiest to calculate with the help of Lorentz transformations. Consider a cycle that starts and ends in the upstream frame (unprimed quantities), the energy gain is given by E f E i = Γ 2 r(1 β r µ d )(1 + β r µ u) (3.34) where primed quantities are measured in the downstream frame, β r is the relative velocity between the upstream and downstream frame with Γ r the associated Lorentz factor, µ is the angle of a crossing particle with respect to the shock normal and u (d) signifies crossing to the upstream (downstream) frame. The same equation holds for cycles starting in the downstream frame. The second term is of the order unity, since to cross from downstream to upstream, it is needed that 1 µ u β s with β s the shock speed in the downstream frame. If now µ d is assumed to be isotropically distributed, then the first term is also of order unity, so that the typical energy gains would be E f Γ 2 E s. (3.35) i This is correct in the first cycle, before the shock is able to disturb the medium and one indeed expects an isotropic injection population upstream. After this however, the distribution becomes highly anisotropic, so that the above reasoning is invalid. It can be shown that for further crossings, the gain becomes of order unity Acceleration time scale It has already been mentioned that the acceleration time is a very important characteristic of an acceleration mechanism, since it determines whether the maximum energy calculated from the Hillas criterion can actually be reached. There are two time scales which are important in the picture of cosmic ray acceleration. Firstly, there is the lifetime of the event. Taking for example a supernova, the shock only exists for a limited amount of time, so that if the acceleration takes too long, the particles can never reach high energies. Secondly, there is the problem of loss effects. Electrons, for example, lose energy mainly due to ionisation (at low energy) and synchrotron radiation (at high energy). If the acceleration rate is too low, the electron would never be able to increase its energy (see figure 3.5). For protons this is less of a problem, due to their significantly higher mass. There are of course theoretical derivations of the acceleration time scale. The easiest one is from a microscopic viewpoint as given by Drury[75], following the original derivation by Lagage and Cesarsky[94] for non-relativistic shocks. The flux of particles crossing from downstream to upstream, assuming an isotropic flux (which is correct in all frames for non-relativistic shocks), is easily calculated to be 1 0 µvn dµ = nv (3.36) with n the density of particles, v the speed of such a particle and µ = cos θ with θ the incoming angle. From the steady state solution of the diffusion equation on one side, the total number of upstream particles is 0 n exp(v 1 x/κ 1 ) dx = κ 1n V 1 (3.37)

76 Chapter 3. Acceleration mechanism 64 Figure 3.5: Comparing the acceleration rate (related to the time scale) with that of ionisation losses gives restrictions on which mechanisms are fast enoughlongair[22]. where κ 1 is the diffusion coefficient on the upstream side and V 1 is the upstream plasma flow speed. Combining these last two equations gives a mean residence time of ( ) κ1 n (nv ) 1 4κ 1 = 4 V 1 v V 1 (3.38) for the upstream particles. For the downstream particles the situation is slightly more complicated, since they can also escape. What is needed is the mean residence time of the particles that return to the shock front. Consider a flow of particles inserted at a position x 0 > 0, with an absorbing boundary at the origin. The diffusion equation n t = V n x = n κ 2 x 2 + Qδ(x x 0) (3.39) n(0) = 0 n( ) < (3.40) with Q the source term, has the solution Q ( ( V exp V x ) ) ( ) κ 1 exp V x 0 κ n = Q ( ( V 1 exp V x 0 )) κ 0 x x 0. (3.41) x 0 x < The flux to the origin is then given by κ n ( x = Q exp V x ) 0. (3.42) κ The return probability for a downstream particle at the position x 0 is then P ret = exp ( V x ) 0 κ. Since we are concerned with a steady state, the number of particles returning from downstream to the shock is then calculated by integrating this chance over the particle distribution downstream 0 P ret (x)n dx = κ 2n V 2. (3.43) The amount of particles going downstream is in this approximation equal to the amount of particles going upstream, giving a mean residence time for the downstream particles of 4κ 2 V 2 v. Adding the downstream and upstream residence times gives the time of one full acceleration cycle, which is

77 Chapter 3. Acceleration mechanism 65 the time for a particle to enter the downstream region, return to the upstream region and entering downstream again t = 4 v ( κ1 + κ ) 2. (3.44) V 1 V 2 From equation (3.1) and the result of the first section in equation (3.22), the momentum gain per cycle is p = 4 V 1 V 2 p. (3.45) 3 v The typical time scale of the acceleration is then given by t acc = p t ( p = 3 κ1 + κ ) 2 (3.46) V 1 V 2 V 1 V 2 which can be easily compared with for example the time scale of loss effects. For oblique shocks the above equation needs to be modified. The solution is given by Lieu and Quenby[95] in the De Hoffmann-Teller frame 3 t acc = V 1 cos ψ 1 V 2 cos ψ 2 ( κ1 V 1 cosψ 1 + κ 2 V 2 cos ψ 2 ) (3.47) with κ i the spatial diffusion coefficient on the upstream or downstream side. The diffusion coefficient can be decomposed into κ = κ cos 2 ψ + κ sin 2 ψ, where κ (κ ) is the component parallel (perpendicular) to the magnetic field. This second term describes cross-field diffusion Injection In order for the acceleration mechanism to work, high energy particles need to be already present. It was mentioned in section that this is a non-trivial problem. The obvious first guess would be that the thermal population, which has a high energy tail, provides enough of these seed particles to inject into the acceleration. Ellison, Reynolds and Jones[33] developed a Monte Carlo code that was able to simulate the acceleration process in both the non-relativistic and relativistic regime. When injecting thermal particles into the acceleration process, the resulting spectrum was found to agree with spacecraft observations of the energetic particles at the Earth s bow shock. In the paper by Summerlin and Baring[63] the injection problem was also briefly illustrated for mildly relativistic shocks in the context of shock-drift acceleration. They found that more turbulent environments were able to inject the particles more efficiently, but the resulting spectrum was also steeper. In the review by Jones and Ellison it was concluded that the acceleration process is not a peculiarity, but actually an inherent part of (supercritical) shocks in collisionless media. In fact, it is the dissipation mechanism of choice of such shocks. They present the following reasoning, comparing shocks in collisional and collisionless gases. The essence of a shock is the conversion of an upstream flow into the downstream flow, which thus needs to be accompanied by a conversion of energy. In a collisional gas the energy from one degree of freedom, the upstream flow, can be transformed to many degrees of freedom (order ) through particle-particle interactions. In a collisional gas such a transformation is not possible and the energy is given, through the complicated plasma interactions, to the limited number of plasma modes, whose energy becomes very

78 Chapter 3. Acceleration mechanism 66 large. The particles then try to come into equilibrium with these modes, giving a few high energy particles. Unlike with collisional shocks, some of these particles will be able to recross the shock, allowing for a series of repeated energy gain. The result is a few high energy particles which escape the system before any equilibrium is achieved. Still the exact mechanism is not fully understood yet, certainly not for highly relativistic shocks. A possibility is that the plasma already contains pre-accelerated particles from previous phenomena that were able to accelerate particles from the thermal population. For example pulsars existing within supernova remnants[19]. When exploring the acceleration mechanism, one is usually not bothered by this problem and just assumes that there are ample seed particles present. To fully describe injection, large-scale plasma simulations are the most appropriate First adiabatic invariant There are many different methods of handling shock crossing, both in theoretical and in numerical treatments. One of those is the assumption of magnetic moment conservation. In the previous chapter (section 2.1.2) it was shown that the magnetic moment is an adiabatic invariant, meaning that in the limit of fields varying slowly in space and time it is approximately conserved. It is obvious that at the shock front, the magnetic field is not slowly varying. The approximation is still used however, because of its attractiveness, both physically (it seems natural) and mathematically (it is very simple to use). Of course the validity of this approximation needs to be checked and this was done by several authors. Hudson[81] theoretically studied the problem of reflection and transmission in relation to the magnetic moment. Lieu and Quenby[95] numerically investigated the acceleration process for subluminal shocks by explicitly following the particle trajectory (the helix orbit), handling shock crossing by matching the orbits in upstream and downstream frame. They then compared the magnetic moment before and after shock crossing. Both of these authors found that the adiabatic invariant is conserved almost exactly only for non-relativistic near-perpendicular shocks and for all pitches except the very small ones. For relativistic shocks, the conservation essentially breaks down[85, 61]. However, the approximation can still be used safely for (nearly) parallel shocks. The reason is that although there is a spread on the value of the magnetic moment after transmission or reflection, the mean is approximately equal to the original value. On average the conservation of magnetic moment is thus a valid approximation[62, 62, 96]. Recently Summerlin and Baring[63] used their own simulation to check the conservation by comparing to the paper by Kirk and Heavens[62] and found that the deviation is small. 3.5 Modifications From the formulation of the acceleration set-up, it is obvious that a few simplifications were present. The most notable ones are the test-particle approximation and the discontinuity of the shock.

79 Chapter 3. Acceleration mechanism Non-linear effects Since the particles that partake in the acceleration process obtain large energies, it can be expected that they will contain a significant part of the total energy of the shock. This has a noticeable effect on the dynamics of the shock and alter its structure[75]. Since this effect is shown to be important for non-relativistic shocks, it is likely also the case in relativistic shocks[33]. The backreaction of the high energy particles influences the shock in several ways. The streaming of a significant amount of high energy particles would cause plasma instabilities to form, which add to the turbulence. The added energy density and pressure and changed ratio of specific heat will serve to modify the Rankine-Hugoniot conditions[59]. These complications will not be considered any further however. For a fully self-consistent treatment, large-scale plasma simulations are ideal Shock profile In the above it was always assumed that the particle gyroradius was larger than the shock thickness, so the accelerated particles cannot resolve the shock structure. In reality the plasma must be continuous, thus there exists something called a shock profile. accelerated particles will also serve to smooth out the shock. The back-reaction of the Instead of describing the shock by having two regions with fixed velocity (or density, since these are related), the shock can then be described with a function v(x). One possible parametrisation of this function was proposed by Schneider and Kirk and recently discussed in detail by Virtanen and Vainio[77][98] and is given by V (x) = V 1 V 1 V 2 2 ( [ ]) x 1 + tanh W λ(γ 1 ) (3.48) where W is the shock thickness parameter and λ(γ 1 ) is the particle mean free path at an energy (Lorentz factor) equal to that of the upstream flow. A nearly steplike shock can be reproduced by the value W = 0.01 as is shown in figure Density of accelerated particles It is also possible to look at the density of accelerated particles as a function of position. Again this problem can be solved both analytically and numerically. Generally, some kind of jump is found at the shock front. However, care has to be taken when trying to find the exact nature of this jump numerically. Different results are found depending on the spatial resolution of the code and the use of the magnetic moment conservation approximation. Gieseler et al.[96] approached the problem with special attention for these problems and found analytically that when the particle distribution at the shock front is isotropic, the particle density needs to be continuous at the shock front irrespective of the obliquity of the shock. When the obliquity induces anisotropy however, a jump is permitted. Their results are shown in figure 3.7.

80 Chapter 3. Acceleration mechanism 68 Figure 3.6: Plot of the velocity profile for different shock thickness parameters for a shock with compression ratio 4. The position coordinate is in units of λ(γ 1 ) and the velocity in units of the upstream flow speed. Figure 3.7: Density distribution of the accelerated particles in function of a normalised distance defined as ξ = ( V shock /κ cos 2 ψ ) x. Figure taken from Gieseler et al.[96].

81 Chapter 4 Particle shock acceleration simulations In the last chapter it became obvious that describing diffusive shock acceleration analytically in the relativistic regime is difficult. The acceleration mechanism requires that anisotropy, for example, is treated in detail. For that reason, one needs to turn to simulations in order to further understand the physics behind the acceleration mechanism and its properties. In this thesis a code was written using the Monte Carlo method, and its implementation shall be described in this chapter. 4.1 Diffusive shock acceleration: large scale versus Monte Carlo simulations There are different approaches to simulate diffusive shock acceleration, each having its own advantages and disadvantages. They can be roughly subdivided into two categories: large scale plasma simulations or Monte Carlo simulations. An extensive overview of the two mechanisms can be found in a review by Jones and Ellison[59]. Large scale plasma simulations try to solve the movement of particles in the plasma. There are several approaches to this. The first is the hydrodynamical approach, where the plasma is treated as a fluid, described by fluid equations. Such a method is however incapable of accurately determining the accelerated population since it explicitly ignores the details of individual particles. The second is with full particle simulations. The main idea of those is to start from Maxwell s equations and a particle population to solve the plasma as a whole, studying simultaneously the shock structure formation, thermal particle population, injection into the acceleration process, the high energy particles and any non-linear effects. The obvious advantage to this is that it is the most complete and correct simulation one can do, using no assumptions. Of course this comes at a price. Such simulations are not only horribly complicated, they are also incredibly slow. The huge amount of particles necessary and their interactions make it impossible to follow the process for a long time. In the third approach one tries to solve this by using a hybrid method. Part of the population is described as a fluid (usually the electrons) while the rest is described individually. Most of the time, the particle-in-cell-method is used, where individual macro-particles (which 69

82 Chapter 4. Particle shock acceleration simulations 70 are representative of a set of physical particles) are followed exactly, but densities and currents are calculated on a grid. Even with these improvements however, such simulations still struggle to find the accelerated particle population. The power law spectrum suppresses the high energy particles, so that an excessive amount of CPU time would be needed to reach those energies. Monte Carlo codes follow a completely different approach. The plasma itself is no longer described in detail, instead its global properties, such as flow speed, magnetic field inclination and compression ratio, are fixed by solving the Rankine-Hugoniot equations for an MHD shock separately. High energy particles are then allowed to scatter in this plasma. The exact details of scattering events are no longer considered, instead only its statistical properties are kept. The diffusion of a particle through the plasma is then modelled by a random walk, where each event consists of the stochastic generation of a scattering length and angle. These are determined by generating pseudo-random numbers from distributions that equal those that would be found if the particle interactions were solved exactly. By using this approach, the Monte Carlo method captures the relevant physics, with the complexity being hidden in the distributions. While at first sight it would seem that this method is inferior to the large scale plasma simulations, this is not the case. In fact the two complement each other. Monte Carlo codes have the advantage that no computer resources are wasted on irrelevant details. These codes are then capable of implementing complex events, for example found in earlier theoretical studies or PIC simulations, at very little cost. The result is that very large amounts of high energy particles can be simulated directly, so that the power law spectrum can be studied over many orders of magnitude in energy; something large scale plasma simulations are incapable of doing. 4.2 Implementation of a Monte Carlo code This section will describe the specifics of the code written and applied for this thesis. The program follows the Monte Carlo approach explained above. It describes a plane shock with shock speeds that can vary from v = 0.01c to very high Γ s, since the code is fully relativistic. The magnetic field can take any orientation, as long as the shock stays subluminal. Most of the required ingredients are already mentioned in the previous chapters, they need only be brought together here Test particle approximation The code follows the test particle approximation, simulating idealised particles that feel only the effects of the externally defined fields and have themselves no influence on the electromagnetic fields or back-reaction on the shock structure. As usually in numerical simulations, the units are chosen such that the calculations are easy. In this case the units used are m = 1, c = 1 and distances are expressed in number of mean free paths (see below). From these last two, the unit of time immediately follows. The particle energy is then equal to its Lorentz factor γ (not to be confused with the plasma flow boost factors Γ i ) since E = mγc 2. (4.1) The advantage of this description is that one simulation can be rescaled to any kind of particle. For example, a particle with γ = 1.5 can be either a proton of 1.5 GeV or a 4 He nucleus of 6 GeV.

83 Chapter 4. Particle shock acceleration simulations 71 This works not as well for electrons, since those have significant loss effects which are not included in this simulation Reference frames As was remarked at the end of chapter 2, it is possible to describe the shock in reference frames such that all quantities lie in the xy-plane only (except for a degenerate case that is not treated). The shock will propagate in the direction of the negative x-axis, so that the upstream side is on the negative x-axis and the downstream side on the positive. Even though all reference frames were already mentioned before, they will be summarised again here. The Lorentz transformations for the particle between the frames shall be given in section Lab frame The lab frame is the frame one intuitively thinks about when describing a shock wave, for example in a supernova. The shock front is moving at a certain speed through the interstellar medium at rest. It turns out that this frame is not very useful when doing calculations, so that it will not be considered any further. Shock frame The shock frame, shown in figure 3.4 is the frame where the shock is at rest, coinciding with the y-axis and the upstream velocity is parallel to the shock normal, along the x-axis. The upstream side is on the negative x-axis. Any quantity described in this frame carries an index S and an index i where i = 1 means quantities on the upstream side and i = 2 means quantities on the downstream side. This frame is used to define the input quantities. Quantities in all the other frames are found by Lorentz transformations which were given in section In the approach used in this code, where the magnetic field is considered to be weak, also the downstream speed is along the x-axis, as explained in section Plasma rest frames The plasma frames are the rest frames of the upstream and downstream plasma. As was explained in section 2.3.2, there is no electric field in these frames. Due to this, the scattering is most naturally described in this frame, as the scattering here is elastic and isotropic. De Hoffmann-Teller frame The De Hoffmann-Teller frame (DHT), shown in figure 2.9b, is the shock frame where the electric field vanishes, due to V i B i (i = 1, 2). As was discussed in section this frame is not unique, but the easiest one is reached by first transforming to the shock frame and then to the DHT frame. It is the frame that will be able to describe shock crossing the easiest, since it connects both plasmas (due to being a shock frame) without interference from any electric field. As was explained in section this automatically includes shock-drift acceleration that is observed in any other frame. Of course for a parallel shock this frame coincides with the shock frame.

84 Chapter 4. Particle shock acceleration simulations The shock parameters The way to handle the plasma and the shock it contains in the code is rather straightforward. The geometry and magnitude of all the fields are fixed beforehand and cannot be changed during the simulation. The input file determines the following parameters: the upstream flow speed V 1 (which is the shock speed in the lab frame), the compression ratio r and the upstream magnetic field inclination in the shock frame ψ 1,S. From these, all other parameters can be determined using the solved Rankine-Hugoniot jump conditions. The downstream flow speed is simply V 2 = V 1 /r (4.2) and is parallel to the x-axis (see above). The downstream magnetic field is found from combining equation (2.106), written as with equation (3.33) B 1,S cos ψ 1,S = B 2,S cos ψ 2,S (4.3) B 2,S B 1,S = ( r 2 Γ 2 1(r 2 1)(cos 2 ψ 1,S V 2 1 ) ) 1/2 to find cos ψ 2,S. The shock is then completely determined in the shock frame and from there it can be transformed to any other frame, as done in section (4.4) Injection Particles are injected far upstream at a single, high energy (to avoid influence from a pre-existing distribution) with a random pitch angle oriented towards the shock. The particles will then almost certainly meet the shock at some point. This would not be true for downstream injection, since particles there have a chance to escape the acceleration process. The injection energy is chosen automatically by the program through the formula γ start = Γ 1. (4.5) It was checked that this gives the particles a sufficiently high starting energy. By injecting the particles at high energy, they are almost assured to partake in the acceleration mechanism. The injection problem is thus circumvented and left as a separate problem (see also section 3.4.3, along with a physical justification there) Diffusion of high energy particles the Monte Carlo method The guiding centre approximation A charged particle in a plasma propagates in a helix orbit which follows the magnetic field lines. While some authors choose to follow this trajectory exactly (see for example Ostrowski[88]), this code uses the guiding centre approximation. A nice review of this approximation can be found in the book by Parks[99]. The particle is, for most of the simulation, replaced by a fictitious particle located at the gyrocentre of the helix orbit and the gyrophase is ignored. The pitch angle θ of the particle is kept however, since it determines the speed of the gyrocentre v gyr = v part cos θ c cos θ (4.6)

85 Chapter 4. Particle shock acceleration simulations 73 and is an important parameter to handle shock crossing. Figure 4.1 illustrates the actual particle and its gyrocentre following the magnetic field with inclination ψ. The particle gyrocentre will henceforth be referred to as the particle, from the context no confusion with the actual charged particle should arise. Figure 4.1: The helix orbit and gyrocentre[100]. Random walk The particle spends most of its time in the upstream or downstream plasma frames, where it propagates along the magnetic field lines in between the scattering events caused by the magnetic field irregularities. While it is propagating, the pitch angle is kept constant. Essentially, the gyrocentre performs a random walk[101], where the travelled distance is a randomly distributed value characterised by the mean free path λ. The way to simulate such a random walk was originally developed by Cashwell and Everett[102]. The distribution of scattering lengths is a simple exponential, since the principle is the same as radioactive decay, given by P scatter (l) = 1 ( λ exp l ) λ so that < l >= λ. It is then necessary to generate a random scattering length that obeys this distribution. This is by now a standard problem in physics. One of the ways to do this, the most efficient in this case, is the following. Equate the above distribution to another one by the following definition (4.7) P (l) dl g(r) dr. (4.8) If one demands then that the variable R is uniformly distributed, so that g(r) = 1, the above equation can be integrated to R = F (l) l = F 1 (R) (4.9) where F (l) is the cumulative distribution of l. By drawing a random number R from a uniform distributions, it is thus possible to find a random scattering length obeying the right distribution using the inverted cumulative distribution. In this case the cumulative distribution is easily found analytically F (l) = l 0 ( 1 λ exp l ) ( dl = 1 exp l ). (4.10) λ λ

86 Chapter 4. Particle shock acceleration simulations 74 Equating this to R gives l = λ ln(1 R). (4.11) Since 1 x is also uniformly distributed between 0 and 1, it is just as simple to use l = λ ln R. (4.12) After the scattering length has been determined, the particle position is advanced by moving this distance along the magnetic field lines. The time is also advanced, the increment being determined by the scattering length and the speed of the gyrocentre, which is related to the pitch angle θ through l t = (4.13) v cos θ where v is the particle speed. Although this is very close to c, it was chosen to use the exact formula v = 1 1 γ 2 (4.14) in order to be able to use the program for non-relativistic shocks without the need to use highly relativistic particles. The mean free path of the particle is no constant however. Instead it is generally a function of momentum, here chosen as[33, 84, 103] λ(p) = λ 0 p = λ 0 γ 2 1. (4.15) This choice makes sense physically, since it just expresses that higher energy particles scatter further. A motivation for this choice can be found in Summerlin and Baring[63] where they considered the connection of the mean free path with turbulence. As already mentioned above, distances are expressed in mean free paths. In practice this goes as follows. The program has a separately determined λ 0 for the upstream and downstream regions. The smallest of these (the downstream one) is taken as a definition for the unit distance and the other one is a multiple of this. Physically it makes sense that these regions have different mean free paths since the compression of the downstream plasma might increase the amount of scattering. This free choice makes no difference in the current program however, since every movement is proportional to the mean free path so upstream and downstream positions can be expressed in units of mean free path separately. Random number generators for uniform distributions are widely available, although care needs to be taken that the generator has a high enough quality (that the numbers are random enough ). In this case the Mersenne Twister generator provided by the C++11 library was used, which passes a lot of the usual tests (contrary to the standard C++ generators, which are known to be poor). Scattering operator At the scattering events, the charged particle deviates from the original helix orbit in what is called pitch angle diffusion. The goal is then to find a way to model the scattering operator introduced

87 Chapter 4. Particle shock acceleration simulations 75 in as accurately as possible. There are two main approaches to this: large angle scattering and small angle scattering. Large angle scattering is the simplest of the two approaches. The principle is that each scattering event completely randomises the pitch angle of the particle. This may technically not be true, since the particle changes its pitch angle slowly over several scatterings. For non-relativistic shocks however, these individual scatterings need not be considered and only the global behaviour needs to be studied. The physical reason for this is that the shock is slow compared to the high energy particles, so that they would meet relatively rarely. Determining a new pitch angle in this approach is relatively straightforward and the method was first described by Ellison, Reynolds and Jones[33]. The scattering of the particle s momentum vector can be described by the diffusion of its direction on a sphere, shown in figure 4.2. Since in the plasma rest frame there is no preferred direction, one needs to generate a random Ω that is distributed uniformly over the sphere. Finding this distribution is a standard problem in physics. A uniformly distributed set of points on a sphere has an equal amount of points in each solid angle element dω, so that the following distribution is derived for the polar angle (with a range between 0 and π) and phase (with a range between π and π) P (µ) dµ = 1 dµ, (4.16) 2 P (φ) dφ = 1 2π dφ (4.17) were θ is the pitch angle, µ = cos θ (the term pitch angle will be used for both θ and µ as the difference is clear) and φ is the phase. Using the cumulative distributions, the direction can then be generated with uniformly distributed variables R i between 0 and 1: µ = 2R 1 1, (4.18) φ = π(2r 2 1). (4.19) Using these equations to generate deflection angles δθ and φ, the new pitch angle µ is then found with the cosine rule from spherical trigonometry from the old pitch angle µ by µ = µ cos δθ + 1 µ 2 sin δθ cos φ. (4.20) As mentioned above, for relativistic shocks the large angle scattering scenario is expected to break down. Instead it may make more sense to describe the particle as slowly diffusing in pitch angle, slightly changing its pitch angle in each scattering event. Gallant and Achterberg[93] showed that this is indeed necessary and particles can only slightly change their pitch angle in each scattering. They found the following expression for the deflection angle δθ 1 Γ (4.21) with Γ the Lorentz factor of the shock speed. From this condition, it can already be inferred that for non-relativistic shocks the difference between the two schemes disappears. The physical reason for this small deflection angle is that after crossing the shock from downstream to upstream, the

88 Chapter 4. Particle shock acceleration simulations 76 Figure 4.2: Describing the scattering of the particle by diffusion of the tip of the momentum vector on a sphere[82]. The original momentum direction is denoted by Ω. particle is quickly caught again by the shock, since the shock speed is very close to c. The particle therefore does not have the chance to undergo many scatterings, so that the individual small angle deflections need to be modelled. In the simulation this is achieved by doing two modifications to the above scheme. Firstly, the δθ is restricted to the range from 0 to δθ max, instead of allowing it to encompass the entire range from 0 to π, as given by Ostrowski[82]. The distribution of polar angles µ is essentially still the same, differing only in normalisation P (µ) = 1 1 cos δθ max dµ. (4.22) Using again the cumulative distribution, the deflection angles can be generated by µ = 1 (1 cos δθ max )R 1, (4.23) φ = π(2r 2 1). (4.24) The second adjustment that needs to be made here, concerns the scattering length. The algorithm derived above is actually only valid for the large angle scattering case. When small angle scattering is considered, the restricted deflection needs to be taken into account. The treatment is inspired from theoretical considerations taking into account turbulence (see for example Meli[104]). The only equation needed for this implementation is δθ max = 6δt t c (4.25) with δt the time between pitch angle scatterings and t c = l v the collision time associated with the scattering length derived in the section above. This equation thus basically reduces the above scattering behaviour back to the individual pitch angle scattering events. In the code, δθ max is automatically determined by the following formula δθ max = θ F actor Γ 1, (4.26) with θ F actor an input parameter to make it possible to loosen the restriction on the deflection angle and Γ 1 is the upstream Lorentz factor. One unfortunate consequence of the change to small angle scattering is that the particle travels less far between two scattering events than in the large angle scattering case and that it thus takes more simulation time for the particle to completely

89 Chapter 4. Particle shock acceleration simulations 77 change its direction. This method is therefore significantly more CPU intensive. This is especially noticeable in the downstream frame, where a new encounter with the shock is not guaranteed and the particles need to catch up to the shock. In fact, for highly relativistic shocks very few to none of the simulated particles are capable of reaching the shock again in (what is for the purpose of this thesis considered) a reasonable simulation time. (See also the discussion in section 4.2.7) The above two options correspond to the standard treatments, where either large angle scattering or small angle scattering is implemented for the complete simulation. The derivation of Gallant and Achterberg actually concerns only upstream scattering however. By implementing small angle scattering in all frames, valuable CPU time is wasted. Since this is not necessary a third, hybrid method was developed for this thesis. In the upstream plasma, the particles still undergo small angle scattering, to accurately describe the interaction with the relativistic shock. In the downstream frame however, the particles are allowed to undergo large angle scattering. Using this method, it is possible to capture the same physics as the full small angle scattering approach with an efficiency resembling that of the large angle scattering approach. Another effect was not yet discussed: cross-field diffusion. When the particle scatters it is also possible that it moves perpendicular to the magnetic field lines. In the simulation this would be modelled by shifting the position of the gyrocentre to a different magnetic field line (parallel to the original). In subluminal shocks however, including this effect is not important. For superluminal shocks it is essential however, since it allows a particle following a field line to more easily cross the shock again after it has crossed into the downstream region. This can be seen by observing the intersections of the magnetic field lines with the shock, which in the case of superluminal shocks go at a speed higher than c. By diffusing purely along the magnetic field lines, the gyrocentre would need to catch up to this intersection, which is impossible. The only other way for the particle to recross the shock is by its extended helix orbit itself, which can still intersect with the shock a few times after the gyrocentre has passed the shock front. Since this case is not studied in this simulation however, cross-field diffusion can be ignored Shock crossing Most of the time the particle is scattering in either the upstream or the downstream plasma. Sometimes it of course happens to cross the shock, so that the magic of the acceleration mechanism can happen. Just like in the basic proof in section 3.1.3, the energy gain will automatically appear by doing Lorentz transformation between the different relevant reference frames. In this section shock crossing shall be described for a general, oblique shock with the understanding that the situation simplifies considerably for parallel shocks. Consider a particle that is scattering in the upstream plasma. At a certain point the scattering algorithm will cause the particle to go to the other side of the shock, meaning that its x-coordinate is positive in the shock frame. At this moment the simulation is paused and a new subroutine takes over. Since the scattering algorithm automatically moves the particle, the particle has already propagated into the downstream frame without a proper handling of shock crossing, therefore the particle is, in this code, first rewound back to the shock (both its position and time coordinates are reduced). The guiding centre approximation is then left, returning to the helix orbit by generating

90 Chapter 4. Particle shock acceleration simulations 78 a random gyrophase φ. The momentum vector of the actual physical particle is then where v is the particle speed, which is close to c. p x = γv cos θ cos ψ 1 γv sin θ cos φ sin ψ 1, (4.27) p y = γv cos θ sin ψ 1 + γv sin θ cos φ cos ψ 1, (4.28) p z = γv sin θ sin φ. (4.29) This formula can be easily found by first considering the momentum vector of the particle in the helix, projecting out the z-component, the component along the magnetic field and the component orthogonal to the magnetic field and the projecting these last two onto the x- and y-axis. A two-stage Lorentz boost is then initiated, transforming first along the x-axis to the shock frame and then along the y-axis to the DHT frame. In this frame, the electric field is equal to zero so that the magnetic moment is to a good approximation conserved. First it needs to be checked if the particle can pass to the downstream frame, since the magnetic field there is stronger and reflection might occur. For downstream to upstream crossing, the magnetic field diminishes so that reflection can not occur. Using the results from section 2.1.3, the critical pitch angle is defined through sin θ crit = B 1,DHT B 2,DHT (4.30) since there only two possible values of the magnetic field. From equation (2.106), which is valid in any frame, we find B 1,DHT B 2,DHT = cos ψ 2,HT cos ψ 1,HT. (4.31) This means that any particle with a pitch angle smaller than θ crit in the DHT frame will pass through the shock, the others will be reflected by it. Reflection is described simply by performing the substitution µ µ (4.32) in the DHT frame. If the particle passes through, the adiabatic invariant is used to determine its pitch angle in the new magnetic field. Indeed, the conservation of the adiabatic invariant p2 B section 2.1.2) can be rewritten to (see 1 µ 2 B 1,DHT = 1 µ 2 B 2,DHT (4.33) by using that the magnitude of the momentum is conserved as well in the absence of an electric field (see for example Gieseler et al.[96]). Using the definition of the critical angle we then find the new pitch angle as µ = ( ) µ µ 2 µ 2 crit µ 1 µ 2 crit (4.34) where the new pitch angle needs to have the same sign as the old one for the particle to keep moving in the same direction. The derivation is completely analogous for downstream to upstream scattering. Using the formulas given by Terasawa[100] it is then also possible to change the

91 Chapter 4. Particle shock acceleration simulations 79 gyrophase accordingly with cos φ = 1 cos 2 θ cos φ sin θ, (4.35) sin φ = cos(ψ 2,DHT ψ 1,DHT ) 1 cos 2 θ sin φ sin θ (4.36) sin(±(ψ 2,DHT ψ 1,DHT )) cos θ (4.37) where an error in the second formula has been corrected and the sign in the second formula is decided by the frame it is in after the shock crossing, where the plus sign is the downstream frame. The particle can then be transformed back to the shock frame and into the downstream frame. The guiding centre approximation is picked up again and the gyrophase is forgotten. What is left to do, is determining a new position downstream, away from the shock. This is done by the usual scattering algorithm, but without determining a new angle. After this the subroutine exits and the scattering algorithm is resumed. Note that the rewinding and rescattering of the particle is an approach specific to this code. In the literature using the same general principles for Monte Carlo simulations (guiding centre, adiabatic invariant), no reference to such steps was found. Instead it appears that most authors just continue scattering from the new position immediately. This method was also checked in the simulation. There is a slight difference in the spectra between the two approaches. Due to the approximations used in the description of the shock (mainly it being discontinuous) and the use of the adiabatic invariant it is hard to say which approach is the better one. Modifications The combination of the random walk of the gyrocentre with pitch angle diffusion from the helix orbit has an unwanted consequence. The problem will be illustrated for the large angle scattering case. If particles have a pitch angle close to π/2, then the speed of the gyrocentre is close to 0. By construction, however, the particle is allowed to propagate freely for the entire scattering length that was randomly drawn, independent of this angle. This means that the particle needs to spend a lot of time in this state. The result is that more particles have a pitch angle around π/2 than there are particles at other angles. The first problem with this is that the particle distribution in the plasma frame is then no longer isotropic, so that it will not be isotropic in the shock frame even for non-relativistic shocks, as is illustrated in figure 4.3a. Since scattering is described in the plasma frame, this also means that the particle is co-moving with the plasma for a long time, such that it can cross a large distance in the shock frame. The result is that more particles will cross the shock at such pitch angles. Since the acceleration mechanism is at its strongest for truly head-on collisions and the distribution now favours particles that are basically at rest in the plasma frame, this means that the energy gain is severely limited, leading to very steep spectra (significantly steeper than those predicted by theory) as shown in figure 4.3b. The way to identify that this really is the problem, is by plotting the 2D histogram in figure 4.4. Figure 4.4a shows the cosine of the pitch angle and x-coordinate of the particles before the first shock crossing. Figure 4.4b shows the downstream position determined by the scattering algorithm, so before the correct handling of the shock crossing. It can be seen that there are indeed a lot of particles with µ 0, as was already shown in figure 4.3a and that they cross far larger distances than any other particle.

92 Chapter 4. Particle shock acceleration simulations 80 (a) The pitch angle distribution in the shock rest frame. There is a significant overabundance of particles at µ 0 (notice the log-scale). (b) The resulting spectrum is much steeper than theory predicts. Figure 4.3: There is an obvious problem with the original algorithm. The simulation uses a parallel shock with V 1 = 0.1, r = 4 and injects 10 5 particles. (a) Distribution showing the position of particles before scattering across the shock. (b) Distribution showing the position of particles after scattering across the shock. Here the distance crossed is obviously much larger for particles with µ 0. Figure 4.4: The problem can be identified by showing the distribution of particles versus position (in log-scale!) and pitch angle. The huge distances crossed by particles with µ 0 causes there overabundance of such particles. The simulation uses a parallel shock with V 1 = 0.1, r = 4 and injects 10 5 particles.

93 Chapter 4. Particle shock acceleration simulations 81 There is no easy way to solve this problem. The first idea is be to make the mean free path proportional to µ. This however results in a pitch angle distribution that is not at all isotropic, which goes against the basic premise of the treatment. The second idea is then to apply the correction factor for the mean free path µ only to the range µ < 0.1. This overcompensates for most of the pitch angles however (except for those with µ very close to 0), meaning those particles cross far smaller distances than the other pitch angles, so that a significant dip is produced in the µ-distribution. It turns out that a simple solution is to have a correction factor equal to µ in the range 0 < µ < 0.1. This reduces most of the scattering lengths in the shock frame to sizes more in line with the other particles and avoids the dip produced in the previous correction, so that the pitch angle distribution is mostly isotropic. Figures 4.5 and 4.6 show again the 2D histograms for the corrected case. The result is that spectra are flatter and the non-relativistic spectral indices agree with theoretical solutions. The problem faced here is inherent to the approach used to model scattering and needs to be handled. The usual approach is to completely exclude a range of pitch angles close to 0. The reason for choosing not to do this here is that it artificially enhances the shock acceleration, since the weaker gains are eliminated. Both approaches are obviously not ideal, but the existence of different solution methods can only be seen as an enrichment Boundary conditions At some point the simulation of a particle needs to end and this moment is determined by boundary conditions. There are two sources of such conditions: physical and numerical. There are two conditions in the physical category. Firstly, the particle might at some point escape the shock downstream (see section 3.1.3). In the simulation it is of course not possible to follow the particle for an infinite amount of time to see if it ever returns. Instead, it needs to be guessed whether there is a reasonable chance of returning. This can be done through rather intricate methods, such as employed by Ellison, Baring and Jones[84], where the exact probability to return is calculated and returning particles are followed explicitly in reversed time. In this thesis, a more simple approach is used however. A downstream boundary is placed at a certain high number of mean free paths downstream λ(p)(for example 100), where the scaling mean free path needs to be used since highly energetic particles can return to the shock from greater distances. The particles that reach this boundary have no significant chance to ever return to the shock and are set to leave the system. The validity of this approach is empirically checked: if the spectrum no longer changes when increasing the distance, then it is far enough away. The second physical reason why particles leave the acceleration mechanism is when their gyroradius has increased to the point where the particles are no longer contained by the accelerator source. This is artificially imposed on the particles by allowing the simulation to run up to a certain maximal energy determined from the input file. For most of the simulations, this energy boundary is chosen very high in order to easily see which energies can realistically be achieved by the mechanism, even though statistically every energy can be reached as long as enough particles are simulated. The second source of boundary conditions is for computational reasons. There is one condition in this category. If a particle is scattering an excessive amount of times (defined by the input

94 Chapter 4. Particle shock acceleration simulations 82 (a) The bump in the pitch angle distribution has disappeared and the distribution is close to isotropic. (b) The resulting spectrum is more in line with theoretical predictions. Figure 4.5: The correction seems to solve the problem. The simulation uses a parallel shock with V 1 = 0.1, r = 4 and injects 10 5 particles. (a) Distribution showing the position of particles before scattering across the shock. (b) Distribution showing the position of particles after scattering across the shock. Figure 4.6: Illustration of the effect of the used correction. The distances crossed by particles with µ 0 are more in line with the other and there is no more overabundance of such particles. The simulation uses a parallel shock with V 1 = 0.1, r = 4 and injects 10 5 particles.

95 Chapter 4. Particle shock acceleration simulations 83 file, here put to ), it is removed from the simulation so that the simulation cannot go on indefinitely. Under normal circumstances, this boundary condition should not be triggered. In the case of small angle scattering however, a significant fraction of simulated particles exits with this condition. The reason is that, because of the small increments in pitch angle, the particles have a hard time returning to the shock. Unless the downstream boundary boundary is very close to the shock, it is highly likely that the particle spends a long time somewhere in the downstream plasma without doing anything interesting. This is the reason for developing the hybrid model. 4.3 Discussion While the developed code includes many of the important physical effects, the code is of course, like any other numerical simulation, not a perfect simulating tool. It is necessary to briefly review what the code is not capable of in order to be able to correctly apply the simulations and interpret the results. The first point is the geometry used. Obviously, in a real shock different inclinations are present, due to the curving of the magnetic field lines. At large scales however the approximately the same inclination will be present for a relatively long time. Moreover, it is actually advantageous to study the effects of different inclinations separately. Spectra from different inclinations can then be superimposed to get an average spectrum as in the paper by Meli, Becker and Quenby[39], or to produce a spectrum with breaks as in the model by Biermann et al.[23]. The above point raises a more general concern, namely that the description of the shock and the plasma background used seems simplistic. Of course more detailed methods exist, explicitly taking into account shock structure, turbulence or the thermodynamic properties of the plasma (see for example the theoretical studies by Heavens and Drury[92]). Again, the approach used here does not have to be a disadvantage however. Since the input parameters are quite general, it is possible to use any kind of shock parameters without needing an explicit treatment of the plasma. The parameters can also be based on independent solutions of the plasma, as was indeed the approach used by Summerlin and Baring[63]. They also based the shock acceleration simulation on a certain number of simple parameters, such as the compression ratio (which is in the same range as the ones used in this simulation) see also section Another point is the lack of feedback in the system, which can weaken the shock or limit its lifetime. The only alternative however is to increase the complexity of the simulation, at the cost of significant computing resources. Such approaches would then not be able to reach the huge number of particles that can be simulated by the currently developed code. Therefore other simulations which do include this, are very much complementary to the code here. There are other possible areas of critique. The validity of the adiabatic invariant approximation was discussed in section Another one is the shock transition region, which is taken to be discontinuous. The change to a smooth shock was discussed in section 3.5.2, it is however a minor effect (since collisionless plasmas have very thin shocks anyway) that could only be implemented at the expense of significant computer resources. Energy loss was not included, but this is not

96 Chapter 4. Particle shock acceleration simulations 84 really relevant here, since the focus was on protons and heavier nuclei anyway. For electrons the energy losses are significant and a completely different approach is necessary. We conclude therefore, that the disadvantages of the Monte Carlo code developed here are either defensible or can be worked around. It should be repeated however that the Monte Carlo code of course has many advantages, lying in its simplicity to simulate lots of particles and their multidimensional phase space very fast and its ability to include many more different physical effects rather easily. 4.4 Summary: program flowchart To conclude this chapter, an overview shall be given of the complete program flow. The program follows the steps illustrated in figure 4.7. Particles are injected into the system and followed throughout their lifetime separately. At various moments in the program, information on the behaviour of the particles is saved to output files. This includes for example the pitch angle and energy gain in each shock crossing, the time taken to perform the entire acceleration process, the final energy and the boundary condition triggered.

97 Chapter 4. Particle shock acceleration simulations 85 Start Input Calculate all frame quantities Inject Particle upstream Transform to upstream frame Start crossing algorithm Transform to shock frame Transform to dh-t frame Scatter particle Did particle cross shock? yes Can particle cross? yes no Reflect particle: µ µ no Transform µ: adiabatic invariant Transform to shock frame no Does particle exit? Transform to new plasma frame yes Transform back to shock frame for exit quantities Write out all data of particle Output no Have done all particles? yes Stop Figure 4.7: Schematic of the flow of the Monte Carlo code.

98

99 Chapter 5 Results This chapter will finally deal with the actual simulation of the shock acceleration mechanism. The discussion is split up in two parts: shocks that are considered mild relativistic and shocks that are highly relativistic. The first part can easily be compared to theoretical results and should help to verify that the simulations are valid and identify some trends already. The second part can accurately be described by simulations only and will exhibit some interesting properties. 5.1 Simulating and fitting The simulation was first studied extensively by scanning the parameter range manually, so that any issues in the code could be identified and the important physical effects isolated. The power law spectral index is determined by fitting a linear function in log-log space with the least squares method provided by the numerical package SciPy[105]. Specialists in the field of power laws will gladly point out that this method is not powerful enough, see for example the paper by Newman[67]. The motivation for deciding to stay with this method is the following. The more complicated methods are mainly used to fit power laws to distributions that only start to show power law behaviour near the middle or end of their range and that suffer from lack of statistics in the tails. For mild relativistic shocks however, the power law is an amazingly good fit and statistics is not a problem. For highly relativistic shocks on the other hand the power law is impossible to fit, certainly for shocks with large angle scattering. Even the most intricate of statistical method would produce a meaningless fit. The simple least squares method is thus more than adequate for this case. The SciPy algorithms also provide an error estimate which is not powerful (or even correct) enough for a thorough statistical analysis. However, the error estimate on any fit is certainly far below the errors due to the approximations used, so that it is used anyway (as is standard practice). In order to present here a systematic study of the shock acceleration, it was necessary to be able to simulate a lot of different parameter values. For this purpose a script was developed that runs the simulation from a set of given parameters and automatically plots the energy spectrum, pitch angle distribution and many others. The power law fit is also automatically done. As will be seen in the plots, there is an injection problem present in this kind of simulations (see for example also Ellison, Reynolds and Jones[33]). To fit a power law then, a fitting range must be decided. It was found that the following algorithm gives satisfying results. The fit starts from the fourth 87

100 Chapter 5. Results 88 consecutive dropping bin and ends at the second bin with zero counts. If this criterion cannot be satisfied or the fitting range does not contain at least 10 bins, the fit fails. In this case it might be possible to manually determine a fitting range or, more likely, a power law cannot be fitted at all. 5.2 Mild relativistic shocks First the mild relativistic shocks will be treated. This encompasses the range up until V 1 = 0.9 (this last value borders on being considered highly relativistic), where again everything is in units c = 1 like explained in chapter 4. For most of the simulations only one value of the compression ratio shall be used, namely r = 4. The reason for this choice can be found in sections and The effect of different compression ratios shall be studied separately Parallel shocks First parallel shocks shall be considered for different shock speeds. The case V 1 = 0.1 can be considered as non-relativistic and should be comparable to the theory developed in section All other shocks can be considered as mild relativistic ones. Energy spectrum The differential energy spectrum, determined by recording the particle energy in the shock frame when it exits the simulation, is shown in figure 5.1 for different shock speeds, along with a power law fit. It is immediately obvious that the spectra are almost perfect power laws. For the slowest shocks the power law behaviour starts immediately, while for faster shocks the simulation suffers from a slight injection problem. This is however an artefact of the simulation and should not be considered physical, since injection is not treated correctly anyway. For low shock speeds the spectra are a bit steeper than the canonically predicted spectral index of 2. For higher shock speeds, as the shock becomes more relativistic, there is a clear trend towards flatter spectra. Note that the results from section 3.1.3, which considers strictly non-relativistic shocks, do not show such a behaviour, since there the spectral index is dependent only on the compression ratio, even though the energy gain per cycle found there is proportional to the speed of the shock. Thus even in mild relativistic shocks, there are additional effects that are important. This deviation from the simple theoretical result is found in all treatments, see for example the review by Kirk and Duffy[61]. An additional consequence of these flatter spectra is of course that higher energies can more easily be reached. The simulations indeed exhibit this behaviour, so that the acceleration mechanism does not seem to lose efficiency for more energetic particles. Pitch angle distribution The cause for the flattening of the energy spectra can be found in the pitch angle distribution. In the simulation the particle pitch angle in the shock frame during shock crossing is recorded, both the value at the moment of scattering and the new value calculated with the adiabatic invariant (see section 4.2.6). The distribution of this first one is shown in figure 5.2 for different shock speeds, since the latter one doesn t give any new insight. For the lowest shock speeds the

101 Chapter 5. Results 89 (a) Shock speed V 1 = 0.1 (b) Shock speed V 1 = 0.3 (c) Shock speed V 1 = 0.5 (d) Shock speed V 1 = 0.7 (e) Shock speed V 1 = 0.9 Figure 5.1: Energy spectra for mild relativistic parallel shocks. All simulations use r = 4 and inject 10 5 particles. The energy is expressed in log 10 γ = log 10 E m. For comparison the non-relativistic case of V 1 = 0.1 is shown. Observe the low efficiency of the acceleration compared to the other cases.

102 Chapter 5. Results 90 Lorentz transformation between the plasma frame and shock frame reduce to simple Galilean transformation. The distribution for these shocks is thus the same one as in the plasma frame and is indeed isotropic (see also equation (4.16)). Due to the significantly larger amount of crossings for the extremely mild relativistic case (see the next section), such shocks have more statistics for the pitch angle distribution so that there are less fluctuations visible. When going to higher shock speeds, the pitch angle distributions become deformed. The reason for this is relativistic beaming: boosting an initially isotropic pitch angle distribution results in a distribution where all angles are along the boost direction. The result is that in the shock frame, there are more completely head-on collisions, so that the average energy gain is increased. The downstream distribution stays isotropic, since its boost speed there is always much lower. Average energy gain per cycle Another quantity to study is the energy gain per shock cycle, where a cycle consists of a particle starting in the downstream frame, going to the upstream frame and returning to the downstream frame. Every time the particle is in the downstream plasma (meaning after 1, 3, 5,... shock crossings), its energy is recorded in the shock frame. The average energy of a particle versus its number of crossings already done is then shown in figure 5.3. Note that this means that all particles contribute to the first bin, while only a few will reach the last bin. It is immediately obvious that shocks with low and high speeds show huge differences. The slowest shocks have barely any gain per cycle, but some particles perform a huge number of cycles, so that they reach high energies eventually. High speed shocks on the other hand have a huge gain per cycle, but the particles can only perform a few crossings before leaving the system. This already implies that although high energies are reachable even in slow shocks, it will take considerably longer to do this. Shock crossing distribution The crossing distribution shows the information that was hidden in the plots from the previous section. Figure 5.4 shows the distribution of total crossings a particle underwent in the acceleration process, recorded at exit from the simulation. It shows the expected behaviour that most particles cross only a few times and some of them can perform more crossings, with an exponential fall-off due to the chance to escape, which is the basic premise of the Fermi mechanism. Spatial distribution While it may not give any new insights, it is still interesting to check the spatial distribution of the accelerated particles. This is recorded by saving the position of each particle as it is scattered. Such a distribution is shown in figure 5.5. The actual range goes far beyond the values shown, with the same behaviour. The result can be compared with figure 3.7.

103 Chapter 5. Results 91 (a) Shock speed V 1 = 0.1 (b) Shock speed V 1 = 0.3 (c) Shock speed V 1 = 0.5 (d) Shock speed V 1 = 0.7 (e) Shock speed V 1 = 0.9 Figure 5.2: Pitch angle distribution of the particles during shock crossing in the shock rest frame for mild relativistic parallel shocks. Shown is the pitch angle in the old frame, before the handling of the shock. In green are the particles crossing from downstream to upstream, in blue the ones from upstream to downstream.

104 Chapter 5. Results 92 (a) Shock speed V 1 = 0.1 (b) Shock speed V 1 = 0.3 (c) Shock speed V 1 = 0.5 (d) Shock speed V 1 = 0.7 (e) Shock speed V 1 = 0.9 Figure 5.3: The average energy of the particles for mild relativistic parallel shocks recorded in the shock frame versus the number of shock crossings, while they are being accelerated.

105 Chapter 5. Results 93 (a) Shock speed V 1 = 0.1 (b) Shock speed V 1 = 0.3 (c) Shock speed V 1 = 0.5 (d) Shock speed V 1 = 0.7 (e) Shock speed V 1 = 0.9 Figure 5.4: Distribution of the total amount of shock crossings a particle underwent for mild relativistic parallel shocks.

106 Chapter 5. Results 94 Figure 5.5: The spatial distribution of a shock near the vicinity of that shock, with the position in units of downstream mean free path. The simulation used a parallel shock with V 1 = 0.1 and injects 1000 particles, which leads to recorded positions already Oblique shocks Now the effect of changing inclination of the magnetic field shall be studied for two selected shock speeds, namely V 1 = 0.1 and V 1 = 0.7. The first one (a non-relativistic) is studied for the inclination angles ψ 1,S = 40, 60 and the latter one (a mild relativistic) for ψ 1,S = 20, 40. The slow shock only reaches superluminality when the shock is near perpendicular, while the latter one already at an inclination angle a bit above 45 (see formula (2.122)), which is the reason for the different choice of angles. Just like in the parallel case, there is no difference between the large angle and small angle scattering cases. Energy spectrum The differential energy spectra for the selected shock parameters are shown in figure 5.6. The effect of the magnetic field is quite dramatic. The spectrum becomes very flat even for V 1 = 0.1 if the shock is sufficiently oblique. The physical reason for this was already explained in section 3.3: since the particles follow the magnetic field lines, they chase the intersections of the magnetic field lines with the shock and see a relativistic shock. Pitch angle distribution Two representative pitch angle distributions are shown in figures 5.7 and 5.8, where the first one is as before and the second one shows the distribution of the new pitch angle determined after shock crossing (formulas (4.32) and (4.34)). These angles are now recorded in the DHT frame, since shock crossing is handled there for oblique shocks. The plots in figure 5.7 show that the particles approaching the shock still have the same properties as in the parallel case. The reason for showing the new angle in figure 5.8 is to illustrate that reflection does indeed occur. The blue histograms shows particles that tried to cross into the downstream region, but as the shock gets more oblique, more of them are reflected back. The difference between the reflected distributions of the two shock speeds is due to the original

107 Chapter 5. Results 95 (a) Shock speed V 1 = 0.1, inclination ψ 1,S = 0 (b) Shock speed V 1 = 0.7, inclination ψ 1,S = 0 (c) Shock speed V 1 = 0.1, inclination ψ 1,S = 40 (d) Shock speed V 1 = 0.7, inclination ψ 1,S = 20 (e) Shock speed V 1 = 0.1, inclination ψ 1,S = 60 (f) Shock speed V 1 = 0.7, inclination ψ 1,S = 40 Figure 5.6: Energy spectra for a non-relativistic and a mild relativistic oblique shock. All simulations use r = 4 and inject 10 5 particles.

108 Chapter 5. Results 96 pitch angle distributions that were different as well. One of them was isotropic, while the other one was beamed, which is then also the case for the reflected distribution (since reflection is just the substitution µ µ). Notice also that the green and blue histogram still complement each other perfectly in figure 5.8. The reason for this is that formula (4.34) used to determine the new pitch angle is completely reversible. This means that particles going from downstream to upstream can only end up in a narrow pitch angle range, since only those would also be able to cross from upstream to the downstream (the direction is ignored, since the adiabatic invariant is only sensitive to µ 2 ). The remaining range encompasses those particles with µ that would reflect if going towards the shock and indeed reflected particles going away from the shock end up only there. Average energy gain per cycle The average energy in each cycle for the most oblique case considered is shown in figure 5.9. It can be seen that the gain in each cycle is larger than in the parallel case. The reason for this extra gain is twofold. Firstly, the particles see the magnetic field intersections at a speed that is higher than the shock speed. Secondly, there is the presence of reflected particles. These particles already gain energy in the usual manner, since a reflection is still a head-on collision. Moreover, since these particles immediately return upstream, they need to encounter the shock again seeing as escape is only possible downstream. Since a reflected particle is in this simulation not counted as a crossed particles, the result is that the gains in each cycle are larger on average. Another feature of these plots is that there are less shock crossings on it. This is caused by the downstream particles having a harder time returning to the shock. This is also the reason for the fluctuations at the end of the plot, where there is less statistics. Shock crossing distribution The reduction in amount of crossings as the shock gets faster seen in the parallel case still occurs. The same thing now happens as the shock gets more oblique. This effect is shown in figure 5.10, which can be compared with the parallel case. Reflection distribution The reflection distribution, which of course has no equivalent in the parallel case, is shown in figure The expectation that particles are reflected more often as the shock gets more oblique is indeed observed for V 1 = 0.7. For the non-relativistic shock this is not the case however (in fact, when going to even lower speeds, there are actually less reflections). The reason is that in this case particles have too hard a time returning to the upstream frame, so that it is less likely that a particle has a large amount of reflections Compression ratio The influence the compression ratio has on the spectral index is shown in figure 5.12 for a shock of V 1 = 0.5, both in the parallel and in the oblique case (with ψ 1,S = 40 ). The behaviour is very smooth so that it is easy to predict the value of the spectral index for a range of compression

109 Chapter 5. Results 97 (a) Shock speed V 1 = 0.1, inclination ψ 1,S = 40 (b) Shock speed V 1 = 0.7, inclination ψ 1,S = 20 Figure 5.7: Pitch angle distributions in the DHT frame for a non-relativistic and a mild relativistic oblique shocks at shock crossing, recorded at the moment of scattering. In green are the particles crossing from downstream to upstream, in blue the ones from upstream to downstream. (a) Shock speed V 1 = 0.1, inclination ψ 1,S = 40 (b) Shock speed V 1 = 0.7, inclination ψ 1,S = 20 Figure 5.8: Pitch angle distributions for a non-relativistic and a mild relativistic oblique shock at shock crossing, recorded after determining the new pitch angle.

110 Chapter 5. Results 98 (a) Shock speed V 1 = 0.1, inclination ψ 1,S = 60 (b) Shock speed V 1 = 0.7, inclination ψ 1,S = 40 Figure 5.9: The average energy of the particles for a non-relativistic and a mild relativistic oblique shock recorded in the shock frame versus the number of shock crossings, while they are being accelerated. Can be compared with the results for parallel shocks in figure 5.3 (a) Shock speed V 1 = 0.1, inclination ψ 1,S = 60 (b) Shock speed V 1 = 0.7, inclination ψ 1,S = 40 Figure 5.10: Distribution of the total amount of shock crossings a particle underwent for a non-relativistic and a mild relativistic oblique shock.

111 Chapter 5. Results 99 (a) Shock speed V 1 = 0.1, inclination ψ 1,S = 40 (b) Shock speed V 1 = 0.7, inclination ψ 1,S = 20 (c) Shock speed V 1 = 0.1, inclination ψ 1,S = 60 (d) Shock speed V 1 = 0.7, inclination ψ 1,S = 40 Figure 5.11: Distribution of the total amount of reflections particles underwent for a non-relativistic and a mild relativistic oblique shocks. The histograms are normalised to 1, so that the ratio can not be compared to the crossing distribution directly.

112 Chapter 5. Results 100 ratios. Still, from the discussions in sections and it follows that there is only a limited range of viable compression ratios Small angle scattering For mild relativistic shocks large angle and small angle scattering are equivalent, since δθ max for such shocks is equal to π, which causes the same behaviour as large angle scattering. This is illustrated by the comparison in figure 5.13 for parallel shocks, but is equally valid for oblique shocks. All other recorded quantities behave the same as well Time scale The time scale of the acceleration is found by using the definition in section t acc = t E/E. (5.1) This formula gives a measure of time independent of the current energy of the particle, since these influence the energy gain and cycle time. Using this formula, it is possible to compare the theoretically predicted time scale derived for non-relativistic shocks with the one found in the simulations. The simulation records, for a certain shock cycle, the cycle time, energy gain and energy at the start of the cycle for each particle. The average of the cycle time t from the simulations can then be compared with that predicted by theory with the help of the above equation, using the recorded energies 1. This comparison is done in figure 5.14, where the recording was done in shock cycle 3. The left plot shows the absolute time recorded and predicted, from which it can be seen that theory and simulation follow the same trend. There is a clear difference however, illustrated by the plot on the right. This plot shows the ratio of simulated time to the theoretical time. There is clearly faster acceleration for more oblique and/or faster shocks. The same typical values were found by Lieu et al.[108]. The deviations seen in the trends (kept in for completeness) are a consequence of a lack of statistics and do not represent any new physics Summary To summarise, the spectral indices of mild relativistic shocks with different speeds and different magnetic field inclinations are shown in figure The previously observed trends are indeed visible. From this plot the spectral indices for intermediate values can be guessed. The error bars on this plot are barely visible. It should be repeated that these error bars are only statistical ones and that the errors due to the made approximations would be much larger. 1 Other methods exist, based on the time dependence of the cut-off of the power law spectrum, see Ostrowski and Bednarz[106] and Bednarz and Ostrowski[107]. This method can not be used here however.

113 Chapter 5. Results 101 Figure 5.12: The influence of the compression ratio shown for both a parallel and an oblique shock with V 1 = 0.5. (a) Shock speed V 1 = 0.5, large angle scattering (b) Shock speed V 1 = 0.5, small angle scattering with θ F actor = 5 Figure 5.13: For mild relativistic shocks, small angle and large angle scattering are equivalent. Illustrated for a parallel shock.

114 Chapter 5. Results 102 (a) The average time of cycle 3 from simulation and from theory, for different shocks. The two follow the same behaviour. (b) The ratio of the simulated time and the theorecial time. Shocks with higher speed and magnetic field inclination are more efficient than non-relativistic theory predicts. Figure 5.14: Time scale for the acceleration mechanism for different mild relativistic shocks. Simulation data is taken in shock cycle 3. Figure 5.15: Summary of the behaviour of the spectral index for mild relativistic shocks. injected 10 5 particles and had a compression ratio of 4. All shocks

115 Chapter 5. Results Highly relativistic parallel shocks Now highly relativistic shocks shall be considered (starting with parallel ones), meaning the shock speeds are now expressed in terms of Lorentz boost factors Γ 1. The case V 1 = 0.9 will be repeated here, since it can also be considered relativistic with Γ The other chosen shock Lorentz factors are Γ 1 = 10, 50, 100, 300, The first two of this set correspond to values applicable to AGN s, the latter three to GRB s. For relativistic values a compression ratio of 3 is more appropriate. It will be found that the behaviour of relativistic shocks is significantly more complicated than the mild relativistic ones. The crossing distribution shall not be shown any more, since it exhibits the same trend as before Large angle scattering First the large angle scattering model shall be considered as a continuation of the mild relativistic shocks. The simulations in this case inject 10 6 particles for increased statistics. Energy spectrum The energy spectra are shown in figure 5.16, with the energy range limited to log γ = 14. For protons this corresponds to an energy of ev, which is significantly above the end of the cosmic ray spectrum. An immediate feature that stands out is the development of a plateaued spectrum. The shock with the highest Γ 1 even shows a flat spectrum for most of the range, followed by a quick drop. To show that this behaviour is physical and not just an injection effect, the plot is repeated up until log γ = 34 in figure While the energy range itself is unrealistic, it shows that the very flat spectrum is just a result of the plateaued behaviour extending over a much larger energy range. It is of course no longer meaningful to define a spectral index for such spectra. Pitch angle distribution The pitch angle distribution continues to be beamed, such that at the largest Lorentz factors, basically all particles upstream cross the shock along its normal as shown in figure The downstream particles still cross the shock isotropically, since their boost speed is only V 2 1 3, which can be compared with a similar boost for the upstream pitch angle distribution in figure 5.2. Average energy gain per cycle These plots can be found in figure Only two cases are shown now to illustrate that the same trend from the mild relativistic parallel case continues. In the fastest of these shocks it can clearly be observed that the first cycle has a huge gain, while the subsequent cycles each have lower gain than the one before. This is a different behaviour than the mild relativistic one, where each cycle was equally powerful. It is in these plots that the reason for the plateaued spectrum can be found. There are only a few acceleration cycles possible at best, but each one of them is so powerful that the different cycles become distinguishable in the energy spectrum.

116 Chapter 5. Results 104 (a) Shock speed V 1 = 0.9 (b) Shock speed Γ 1 = 10 (c) Shock speed Γ 1 = 50 (d) Shock speed Γ 1 = 100 (e) Shock speed Γ 1 = 300 (f) Shock speed Γ 1 = 1000 Figure 5.16: Energy spectra for parallel relativistic shocks with different shock speeds. All simulations use r = 3 and inject 10 6 particles. The energy is expressed in log 10 γ = log 10 E m.

117 Chapter 5. Results 105 Figure 5.17: The origin of the flat spectra can easily be traced to the stepped spectrum going to extremely high energies, which in reality would not be contained. The spectrum shown is for a parallel shock with Γ 1 = (a) Shock speed Γ 1 = 10 (b) Shock speed Γ 1 = 1000 Figure 5.18: Pitch angle distribution of the particles during shock crossing in the shock rest frame for relativistic parallel shocks with large angle scattering. Shown is the pitch angle in the old frame, before the handling of the shock.

118 Chapter 5. Results 106 (a) Shock speed Γ 1 = 10 (b) Shock speed Γ 1 = 1000 Figure 5.19: The average energy of the particles for relativistic parallel shocks recorded in the shock frame versus the number of shock crossings, while they are being accelerated. Time scale Again the time scale can be studied; the results are shown in figure The actual mechanism is more efficient than non-relativistic theory predicts and the ratio of the two continues to decrease. Notice however that these plots are in log-log scale, so that the decrease is now linear with Γ 1 and the relativistic shocks are obviously much more efficient that non-relativistic theory would give Hybrid scattering This section studies the effect of hybrid scattering, which means small angle scattering upstream and large angle scattering downstream, on parallel shocks. The computing cost of these simulations is still rather high, even if they are far more efficient than in the small angle scattering case. Therefore, these simulations inject less particles (10 5 instead of 10 6 ) and only a few parameters will be studied. This last fact then also makes it rather unexciting to consider summary plots of the spectral index or time scale. For the latter one, it should come as no surprise that the ratio of simulated to predicted times is again decreasing with shock speed. Energy spectrum The energy spectrum is shown in figure In the large angle scattering case both of these developed a plateaued spectrum, but this structure is now completely gone so that the spectra can again be characterised by spectral indices. The spectral indices found here are about the same values as found in the mild relativistic parallel shocks towards the higher V 1 (see figure 5.15). The energy range reached by the particles is also smaller than in the large angle scattering case and this difference can not be explained by the decreased amount of particles simulated alone. Pitch angle distribution The pitch angle distributions for the two fastest shocks in the set are shown in figure The distribution is now obviously less beamed. The reason for this is the small angle scattering that

119 Chapter 5. Results 107 (a) The average time of cycle 3 from simulation and from theory, for different shocks. The two follow the same behaviour. (b) The ratio of the simulated time and the theorecial time. Figure 5.20: Time scale for the acceleration mechanism for different relativistic parallel shocks with large angle scattering. Simulation data is taken in shock cycle 3. (a) Shock speed Γ 1 = 10 (b) Shock speed Γ 1 = 50 Figure 5.21: Energy spectra for parallel relativistic shocks with hybrid scattering. All simulations use r = 3 and inject 10 5 particles.

120 Chapter 5. Results 108 particles that came from downstream to upstream undergo. The particles can then change their pitch angle only slowly and they stay longer in the vicinity of the shock. The result is that particles caught up by the shock might have pitch angles that have only barely been able to turn around, so that even shock crossings with µ closer to 0 are possible. In large angle scattering, on the contrary, particles were able to leave the vicinity of the shock more easily (due to longer scattering lengths) and the distribution can isotropise in only one scattering, so that the beamed isotropic distribution approaches the shock. Average energy gain per cycle The gain, shown in figure 5.23, is very similar to the one in the large angle scattering case. The individual gains are much lower however, due to the decreased anisotropy in the upstream pitch angle distribution on shock crossing. This explains the smaller energy range achieved, since the amount of cycles possible shows no significant change. This last fact is understandable for both pure small angle scattering and hybrid scattering, since the particles in the downstream flow are not chased by the shock so that a change in the scattering mechanism does not influence the chance to escape. 5.4 Highly relativistic oblique shocks Lastly, highly relativistic oblique shocks shall be studied. These simulations use the same speed and number of particles as the parallel case, but use inclination angles ψ 1,S = 20, Large angle scattering Energy spectrum The energy spectra are shown in figure As shocks start to get more oblique, their spectrum gets more plateaued. This is again expected, since the particles see a faster shock on the magnetic field lines. For the case Γ 1 = 10 a fit of a power law is drawn. No attempt is made to claim these spectra are power laws, instead the fits are there to help give a sense of scale. For Γ 1 = 300 this is impossible, since the spectra have become flat over (almost) their entire (physically acceptable) energy range. Pitch angle distribution The pitch angle distributions are shown in figure The beaming effect is again present. A new, dramatic effect occurs however: there are no crossings from downstream to upstream visible anymore. None of the simulated particles were able to return to the upstream plasma after shock crossing, due to the inclination of the magnetic field (even with Γ 1 = 1000 in parallel shocks the downstream particles were always able to return). This also means that it is not possible to calculate the time scale for such shocks with the current definition, since the method is based on particles doing a full acceleration cycle. Reflection however still occurs (see figure 5.26), which enables particles to have multiple shock encounters so that the stochastic generation of a power law-like spectrum is still possible.

121 Chapter 5. Results 109 (a) Shock speed Γ 1 = 10 (b) Shock speed Γ 1 = 50 Figure 5.22: Pitch angle distribution of the particles during shock crossing in the shock rest frame for relativistic parallel shocks for hybrid scattering. Shown is the pitch angle in the old frame, before the handling of the shock. (a) Shock speed Γ 1 = 10 (b) Shock speed Γ 1 = 50 Figure 5.23: The average energy of the particles for relativistic parallel shocks with hybrid scattering recorded in the shock frame versus the number of shock crossings while they are being accelerated.

122 Chapter 5. Results 110 (a) Shock speed Γ 1 = 10, inclination ψ 1,S = 0 (b) Shock speed Γ 1 = 300, inclination ψ 1,S = 0 (c) Shock speed Γ 1 = 10, inclination ψ 1,S = 20 (d) Shock speed Γ 1 = 300, inclination ψ 1,S = 20 (e) Shock speed Γ 1 = 10, inclination ψ 1,S = 40 (f) Shock speed Γ 1 = 300, inclination ψ 1,S = 40 Figure 5.24: Energy spectra for relativistic oblique shocks with large angle scattering. All simulations use r = 3 and inject 10 6 particles. The power law fits here are purely to give a sense of scale.

123 Chapter 5. Results 111 (a) Shock speed Γ 1 = 10, inclination ψ 1,S = 20 (b) Shock speed Γ 1 = 300, inclination ψ 1,S = 20 Figure 5.25: Pitch angle distribution of the particles during shock crossing in the DHT frame for relativistic oblique shocks with large angle scattering. Shown is the pitch angle in the old frame, before the handling of the shock. (a) Shock speed Γ 1 = 10, inclination ψ 1,S = 20 (b) Shock speed Γ 1 = 300, inclination ψ 1,S = 20 Figure 5.26: Pitch angle distribution of the particles during shock crossing in the DHT frame for relativistic oblique shocks with large angle scattering. Shown is the pitch angle in the new frame, after the handling of the shock.

124 Chapter 5. Results 112 Average energy gain per cycle The energy gains in each cycle are shown in figure Since for the inclined case no particles return to the upstream plasma, all energy is gained in one single shock crossing. The plots become meaningless then, showing only the mean energy after reaching the downstream frame Hybrid scattering Large angle scattering is now replaced by hybrid scattering. The spectra again become smoothed out, as shown in figure The pitch angle distribution, shown in figure 5.29, is again less beamed than its large angle scattering equivalent, with the same reason as in the parallel case but now for the reflected particles. The reflected distribution, part of the right panel of the same figure, also has this property. Again there are no more particles returning to the upstream plasma, so that there are no complete acceleration cycles. There is however a new interesting effect. Whereas in the oblique relativistic case with large angle scattering, the distribution of the number of reflections follows the same behaviour as the mild relativistic case (which is why it was not shown), this is no longer the case here. The reflection distribution is shown in figure Particles now prefer to undergo a large amount of reflections. The reason for this is similar to the one given in section As particles are reflected, they can only slowly change their pitch angle, so that it is very likely that they are reflected again. To put into context how many particles were reflected, the meeting distribution is shown in figure 5.31 on a log scale. If a particle crosses the shock, it meets the shock once, whereas if the particle reflects the shock, the code considers it as two meetings (since the particle comes back upstream). This latter method of counting is only a matter of definition however. Due to the different nature of the crossing and reflection distribution, the two are easily distinguishable in the meeting distribution (which was not the case for all previous cases). 5.5 Discussion In the numerical simulations of mild relativistic shocks, an almost perfect power law was always observed. The basic, non-relativistic theory in section predicts that the spectral index would depend only on the compression ratio. In mild relativistic shocks, it was seen here that the spectral index indeed has values around the canonical value of 2, but it varies considerably with both the shock speed and magnetic field inclination, on top of the dependence on compression ratio. Higher shock speeds or higher compression ratios result in flatter spectra. As the shock speed is increased, the upstream distribution in pitch angles gets beamed, leading to increased gains. In general the spectra of oblique shocks are flatter, due to the particles seeing the intersection of the shock with the magnetic field lines only. Due to the inclined magnetic field and the resulting compression, the particles can also undergo reflection, leading to increased energy gains for a complete shock cycle. For the extreme case of highly relativistic shocks, the situation is even more complicated. When using large angle scattering, the spectrum starts to develop plateaued shapes as the Lorentz boost factor is increased. These plateaus are the result of the individual shock cycles becoming so

125 Chapter 5. Results 113 (a) Shock speed Γ 1 = 10, inclination ψ 1,S = 20 (b) Shock speed Γ 1 = 300, inclination ψ 1,S = 20 Figure 5.27: The average energy of the particles for relativistic oblique shocks with large angle scattering recorded in the DHT frame versus the number of shock crossings, while they are being accelerated. (a) Shock speed Γ 1 = 10, inclination ψ 1,S = 20 (b) Shock speed Γ 1 = 10, inclination ψ 1,S = 40 Figure 5.28: Energy spectra for relativistic oblique shocks with hybrid scattering. The simulations use r = 3 and inject 10 5 particles.

126 Chapter 5. Results 114 (a) Shock speed Γ 1 = 10, inclination ψ 1,S = 20, old pitch angle. (b) Shock speed Γ 1 = 10, inclination ψ 1,S = 20, new pitch angle. Figure 5.29: Pitch angle distribution of the particles during shock crossing in the DHT frame for relativistic oblique shocks with hybrid scattering. (a) Shock speed Γ 1 = 10, inclination ψ 1,S = 20 (b) Shock speed Γ 1 = 50, inclination ψ 1,S = 20, new pitch angle. Figure 5.30: Distribution of the total amount of reflections particle underwent during the acceleration process in oblique relativistic shocks with hybrid scattering.

127 Chapter 5. Results 115 (a) Shock speed Γ 1 = 10, inclination ψ 1,S = 20 (b) Shock speed Γ 1 = 50, inclination ψ 1,S = 20, new pitch angle. Figure 5.31: Distribution of the total amount of meetings a particle underwent during the acceleration process in oblique relativistic shocks with hybrid scattering. Note the log scale! potent, due to the very strong beaming of the pitch angle, that they can be distinguished in the spectrum. At a Lorentz boost factor of 1000, corresponding to highly relativistic GRB s, the spectrum is flat over most of its range (in log-log scale). This behaviour was also reported by Meli and Quenby[103, 86], Summerlin and Baring[63] and other authors. Such flat spectra were indeed observed in certain GRB s, see for example the paper by Meli, Becker and Quenby[109] and references therein. This validates the physics observed in the simulations here. Increasing the obliquity again increases the plateaued behaviour, since the particles see a faster shock. A new effect is that downstream particles have great difficulty in returning to the upstream plasma, to the point that no particles returned at all in a simulation with 10 6 particles. The spectrum is thus created in a single crossing. The stochastic behaviour leading to the power law is now generated by the possibility to reflect many times. In the case of small angle scattering, investigated with the help of the the newly developed hybrid scattering, the spectrum smooths out so that an almost perfect power law is once again found. The spectral indices thus found are again steeper than the mild to high relativistic ones in the large scattering case. This same trend was found by Baring[111]. The beaming effect is reduced due to the small angle scattering allowing only little change in pitch angle immediately after returning upstream. In the oblique case the same effect happens after reflection. Moreover, due to this, it was observed that such reflections become more frequent. In conclusion, it becomes apparent that highly relativistic shocks no longer adhere to a universal power law. Instead, the spectrum and acceleration properties are highly dependent on the speed of the shock, the magnetic field inclination and the scattering operator. This same conclusion was found in different works, such as by Niemiec and Ostrowski[110], Meli, Becker and Quenby[39], Baring[111] and Summerlin and Baring[63]