Deep Learning Benadering voor Directionele Reconstructie van Hoog-Energetische Kosmische Stralen met de IceTop Detector

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Deep Learning Approach for

Directional Reconstruction of

High-Energy Cosmic Rays with the

IceTop Detector

Ian B

AUWENS

Promotor:

Prof. Dr. Dirk R

YCKBOSCH

Supervisor:

Stef V

ERPOEST

A thesis submitted in partial fulfillment of the requirements

for the degree of

M

ASTER OF

S

CIENCE IN

P

HYSICS AND

A

STRONOMY

Department of Physics and Astronomy

Academic year 2019-2020

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iii

Acknowledgements

First and foremost, I would like to thank Prof. Dr. Ryckbosch for the oppor-tunity of doing an internship at Imperial College London, allowing me to experience scientific research abroad during the summer break. In this pe-riod, I learned immensely about various topics in physics, which resulted in increasing my interest in experimental particle physics even more! Your theory lectures in physics are definitely among the most legendary and mem-orable ones.

I would also like to thank my supervisor Stef Verpoest, who was always ready to give plenty of helpful advice and feedback. Furthermore, I am very grateful for your explanations of the physics behind many processes.

In addition, I would like to thank Dr. Alessio Porcelli for giving helpful feed-back from time to time.

I would also like to thank my fellow students as well, for the many lunch breaks (and occasional ping-pong sessions), before the COVID-19 quaran-tine lockdown was in effect and the many philosophical discussions we have had over the past five years of our studies.

Last but not least, I would like to thank my brother Robin, for his everlast-ing technical support and computer wizardry, together with my parents and friends, on whom I could always rely for support.

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v

Summary

The study of cosmic rays provides a method to detect matter from both inside and outside our galaxy, the Milky Way. These cosmic rays are mainly rela-tivistic nuclei (proton dominated) and have a flux governed by a power-law behaviour. Much is still unknown about these cosmic rays, e.g. their origin and acceleration to their highest energies (around 1021 eV, much higher com-pared to the energies reached by particle accelerators on Earth).

One way to measure these cosmic rays is by making use of an indirect method: investigating their interactions in the atmosphere. When these highly ener-getic cosmic rays enter the Earth’s atmosphere, they will collide with the air particles and disperse their energy in the form of the production of many energetic secondary particles. These new particles will then continue on col-liding with even more air particles and thus continuing the air shower. The footprint of these secondary particles that reach the detectors can then be used to analyze and study the primary cosmic ray that initiated the air shower.

One of such detectors built for this purpose is the IceTop array, an integral part of the IceCube Neutrino Observatory located at Antarctica.

In this thesis, a deep learning approach has been investigated for the direc-tional reconstruction of primary cosmic rays, detected by their air showers with the IceTop detector.

We found that there is a possibility of utilizing these deep learning tech-niques, especially for the very high energy cosmic rays.

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Samenvatting

Het onderzoek naar kosmische stralen levert ons een methode om materie afkomstig van zowel binnen als buiten ons sterrenstelsel, de Melkweg, te detecteren. Deze kosmische stralen zijn voornamelijk relativistische kernen (gedomineerd door protonen) en hun flux is bepaald door een machtsfunc-tie. Er zijn nog veel onzekerheden omtrent deze kosmische stralen, voor-namelijk gerelateerd met hun oorsprong en versnelling naar hun extreem hoge energieën (rond 1021 eV, een waarde véél groter dan de energieën die we kunnen bereiken door deeltjesversnellers op aarde).

Een mogelijke manier om deze kosmische stralen te detecteren is door ge-bruik te maken van een indirecte detectiemethode: door hun interactie in de atmosfeer te gebruiken. Wanneer de hoogenergetische kosmische stralen binnentreden in de atmosfeer van de aarde, zullen ze namelijk botsen met haar luchtdeeltjes en hun energie verdelen door vele energetische secundaire deeltjes te produceren. Deze secundaire deeltjes zullen dan op hun beurt verder botsen met andere luchtdeeltjes en de air shower verder in stand houden. De voetafdruk van deze secundaire deeltjes die de detectoren bereiken kan dan gebruikt worden voor de analyse en studie van de primaire kosmische stralen die de air shower geïnitialiseerd hebben.

Eén van dergelijke detectoren gebouwd voor dit doel is de IceTop detector, een onderdeel van de IceCube Neutrino Observatory, gesitueerd op Antarctica. In deze thesis werd de bruikbaarheid van een deep learning benadering on-derzocht voor de directionele reconstructie van primaire kosmische stralen door hun air showers, gedecteerd door de IceTop detector, te gebruiken. We concluderen dat de mogelijkheid bestaat voor dergelijke deep learning technieken, in het bijzonder voor de heel energetische kosmische stralen.

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Science Outreach

In context of promoting the research performed at the IceCube Neutrino Ob-servatory, I made an introductory video (in Dutch) where a short presenta-tion of the aurora, cosmic rays and the IceCube Neutrino Observatory itself is given. This video, titled "Het noorderlicht, kosmische stralen en IceCube", was shared by the "Vereniging Voor Natuurkunde" (Physics Association) and can be found here1:

https://vvn.ugent.be/blog/het-noorderlicht-kosmische-stralen-en-icecube/.

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List of Figures

1.1 Rate of ionization in function of altitude . . . 2

1.2 Flux of cosmic ray anti-matter . . . 3

1.3 Primary cosmic and solar elemental abundances . . . 4

1.4 Scaled energy spectrum of primary cosmic rays . . . 5

1.5 Fermi acceleration . . . 13

1.6 Hillas plot . . . 14

1.7 Primary cosmic ray flux per nuclei . . . 17

1.8 Anti-correlation of sunspots and amount of CR . . . 18

1.9 Cosmic microwave background spectrum . . . 20

1.10 Energy losses of high energy proton . . . 22

1.11 Particle populations of cosmic ray energy spectrum . . . 23

2.1 Geometry of incident cosmic ray . . . 26

2.2 Electromagnetic air shower development . . . 28

2.3 Extensive air shower development . . . 29

2.4 Air shower development via Heitler-Matthews model . . . 32

2.5 Longitudinal profile of air showers . . . 34

2.6 Lateral profile of air showers . . . 36

3.1 Illustration of Cherenkov effect . . . 41

3.2 Schematic of a DOM . . . 42

3.3 Schematic of the PMT present in a DOM . . . 43

3.4 Schematic of the data flow in the primary IceCube online sys-tems . . . 46

3.5 Illustration of the IceCube Neutrino Observatory . . . 48

3.6 Illustration of the layout of IceTop . . . 50

3.7 Illustration of an IceTop tank . . . 51

4.1 Illustration of different loss functions . . . 57

4.2 Example of a feedforward neural network . . . 61

4.3 Examples of activation functions . . . 63

4.4 Examples of underfitting and overfitting . . . 66

4.5 Underfitting and overfitting example . . . 67

4.6 Example of a 2D convolution procedure . . . 69

4.7 Sparse connectivity . . . 71

4.8 Example of convolutional neural network . . . 72

5.1 Illustration of air shower . . . 74

5.2 Illustration of air shower front . . . 77

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5.4 Example of a Laputop reconstructed event . . . 79 6.1 Hexagonal IceTop and orthonormal grid . . . 83 6.2 Illustration of hexagonal to orthonormal grid transformation . 83 6.3 IceCube standard coordinate system . . . 85 6.4 Unweighted primary cosmic ray zenith and azimuth

distribu-tions . . . 85 6.5 Unweighted primary cosmic ray energy spectrum . . . 87 6.6 Charge and time distribution . . . 88 6.7 Weighted primary cosmic ray zenith and azimuth distributions 91 6.8 Weighted log energy distribution of primary cosmic rays . . . 91 7.1 Typical structure of used convolutional neural networks. . . . 94 7.2 Accuracy of network good proton events . . . 96 7.3 Histogram of reconstructed angles for good proton events . . 97 7.4 2D histogram of angles for good proton events . . . 98 7.5 Profile plot of reconstructed angles in function of logarithm

energy for good proton events . . . 98 7.6 Profile plot in function of energy of Laputop for good proton

events . . . 99 7.7 Angular resolution of the network for good proton events . . . 100 7.8 Accuracy of network uncontained proton events . . . 102 7.9 Histogram of reconstructed angles for uncontained proton events103 7.10 2D histogram of angles for uncontained proton events . . . 104 7.11 Profile plot of reconstructed angles in function of logarithm

energy for uncontained proton events . . . 104 7.12 Profile plot in function of energy of Laputop for uncontained

proton events . . . 105 7.13 Angular resolution of the network for uncontained proton events105 7.14 Accuracy of network uncut proton events . . . 108 7.15 Histogram of reconstructed angles for uncut events . . . 109 7.16 2D histogram of angles for uncut proton events . . . 110 7.17 Profile plot of reconstructed angles in function of logarithm

energy for uncut proton events . . . 110 7.18 Profile plot in function of energy of Laputop for uncut proton

events . . . 111 7.19 Angular resolution of the network for uncut proton events . . 111 7.20 Zoomed in angular resolution of the network for uncut proton

events . . . 112 7.21 Accuracy of network good all nuclei events . . . 114 7.22 Histogram of reconstructed angles for good all nuclei events . 115 7.23 2D histogram of angles for good all nuclei events . . . 116 7.24 Profile plot of reconstructed angles in function of logarithm

energy for good all nuclei events . . . 116 7.25 Profile plot in function of energy of Laputop for good all nuclei

events . . . 117 7.26 Angular resolution of the network for good all nuclei events . . 117

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xvii 7.27 Zoomed in angular resolution of the network for good all

nu-clei events . . . 118 7.28 Zoomed in angular resolution of the network for good all

nu-clei events . . . 118 7.29 Accuracy of network uncontained all nuclei events . . . 121 7.30 Histogram of reconstructed angles for uncontained all nuclei

events . . . 122 7.31 2D histogram of angles for uncontained all nuclei events . . . 123 7.32 Profile plot of reconstructed angles in function of logarithm

energy for uncontained all nuclei events . . . 123 7.33 Profile plot in function of energy of Laputop for uncontained

all nuclei events . . . 124 7.34 Angular resolution of the network for uncontained all nuclei

events . . . 124 7.35 Zoom of angular resolution of the network for uncontained all

nuclei events . . . 125 7.36 Angular resolution of the network for uncontained all nuclei

events, with Laputop results . . . 125 7.37 Accuracy of network uncut all nuclei events . . . 128 7.38 Histogram of reconstructed angles for uncut all nuclei events 129 7.39 2D histogram of angles for uncut all nuclei events . . . 130 7.40 Profile plot of reconstructed angles in function of logarithm

energy for uncut all nuclei events . . . 130 7.41 Profile plot in function of energy of Laputop for uncut all

nu-clei events . . . 131 7.42 Angular resolution of the network for uncut all nuclei events . 131 7.43 Zoomed in angular resolution of the network for uncut all

nu-clei events . . . 132 7.44 Zoomed in angular resolution of the network for uncut all

nu-clei events . . . 132 A.1 Accuracy of network weighted good all nuclei events . . . 141 A.2 Histogram of reconstructed angles for weighted good all

nu-clei events . . . 142 A.3 2D histogram of network weighted good all nuclei events . . . 143 A.4 Profile plot in function of energy of the network for weighted

good all nuclei events . . . 143 A.5 Profile plot in function of energy of Laputop for weighted good

all nuclei events . . . 144 A.6 Angular resolution of the network for weighted good all nuclei

events . . . 144 A.7 Zoomed in angular resolution of the network for weighted

good all nuclei events . . . 145 A.8 Angular resolution of the network for weighted good all

nu-clei events . . . 145 A.9 Accuracy of network weighted uncontained all nuclei events . 146 A.10 Histogram of reconstructed angles for weighted uncontained

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A.11 2D histogram of network weighted uncontained all nuclei events148 A.12 Profile plot in function of energy of the network for weighted

uncontained all nuclei events . . . 148 A.13 Profile plot in function of energy of Laputop for weighted

un-contained all nuclei events . . . 149 A.14 Angular resolution of the network for weighted uncontained

all nuclei events . . . 149 A.15 Zoom of angular resolution of the network for weighted

un-contained all nuclei events . . . 150 A.16 Angular resolution of the network for weighted uncontained

all nuclei events, with Laputop results . . . 150 A.17 Accuracy of network weighted uncut all nuclei events . . . 151 A.18 Histogram of reconstructed angles for weighted uncut all

nu-clei events . . . 152 A.19 2D histogram of network weighted uncut all nuclei events . . 153 A.20 Profile plot in function of energy of the network for weighted

uncut all nuclei events . . . 153 A.21 Profile plot in function of energy of Laputop for weighted

un-cut all nuclei events . . . 154 A.22 Angular resolution of the network for weighted uncut all

nu-clei events . . . 154 A.23 Zoomed in angular resolution of the network for weighted

un-cut all nuclei events . . . 155 A.24 Angular resolution of the network for weighted uncut all

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List of Tables

6.1 Amount of nucleons of generated nuclei. . . 86

6.2 Amount of generated primary cosmic ray events. . . 89

7.1 Tested CNNs of good proton events . . . 95

7.2 Optimized CNNs for good proton events . . . 96

7.3 Tested CNNs of uncontained proton events . . . 101

7.4 Optimized CNNs of uncontained proton events . . . 101

7.5 Tested CNNs of uncut proton events . . . 107

7.6 Optimized CNNs for uncut proton events . . . 107

7.7 Tested CNNs of good all nuclei events . . . 113

7.8 Tested CNNs of uncontained all nuclei events . . . 120

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1

Chapter 1

Cosmic Rays

The field of astroparticle physics is a relatively young domain in physics and thus has many unanswered questions. Two of these questions are: "Where do cosmic rays come from?" and "How can they be accelerated to very high energies?".

The universe is filled with high energy particles which propagate through the near vacuum of space. They originate from various sources within the universe: the low energy cosmic rays mainly originate from the Sun, particles with a higher energy typically come from sources beyond our solar system and the highest energy sources are even extragalactic [1]. When cosmic rays hit the outer layer of the Earth’s atmo-sphere, they are called primary cosmic rays. This term is used to distinguish primary incident cosmic rays from the secondary particles which are being produced by inter-actions of these particles with the Earth’s atmosphere. In this chapter, we will focus on the details of these primary cosmic rays.

1.1 Discovery of Cosmic Rays

In 1912, Victor Hess discovered during a balloon flight that the amount of ionizing radiation increased with altitude, which can be seen in Figure1.1. Additionally, he also performed measurements during both day and night and found no particular difference during both periods. This implied that the radiation was not coming from the Earth itself nor from the Sun [2]. In the period between the 1930s and 1950s, when the particle accelerators on Earth were limited in energy compared to the accelerators today, the cosmic rays functioned as ’particle collider experiments’ (with the atmosphere) and subsequently led to the discovery of new subatomic particles [3].

In 1932, Carl David Anderson discovered the existence of positrons (e+)1 in cosmic rays. Four years later, he again discovered another particle: the muon (µ−), also from cosmic ray measurements. In 1947, he discovered the exis-tence of the charged pions (π±) and around 1947-1950 the existence of the strange particles.

During the mid 1930’s, the discovery of extensive air showers (see Chapter

2), consisting of secondary cosmic rays originating from the collision of the high energy primary cosmic rays with the atmospheric air nuclei, occurred.

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FIGURE1.1: Rate of ionization in function of altitude, by Hess (1912) (left) and by Kolhörster (1913, 1914) (right) [4].

This result was found independently by Bruno Rossi and Pierre Auger [5,6]. In this thesis, the term rays in cosmic rays refers to all relativistic, charged particles, in contrast to another convention which refers to all relativistic par-ticles (i.e. including uncharged parpar-ticles, e.g. (anti-)neutrinos or high energy

γ-rays) [7].

1.2 Composition

The composition of charged primary cosmic rays consists mainly of protons (86%), together with alpha particles (11%), electrons (2%) and a small com-ponent of heavy charged nuclei up to a nuclear charge number Z = 92 (1%). Additional types of particles have also been found in cosmic rays, namely positrons and antiprotons, which most likely find their origin in secondary particle productions of cosmic rays with interstellar gas (see Figure1.2).

In Figure1.3, a comparison of the composition of primary cosmic ray nuclei with the solar-system abundances of elements is shown, where we can see that both distributions are very similar. Both the cosmic and solar elemental abundances show an odd-even staggering effect in Z. This is related to the observation that neutrons are more tightly bound when the amount of neu-trons (N = A - Z) is even compared to the situation where N is odd (due to a nuclear pairing interaction) [9]. This implies that nuclei with an even A

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1.3. Energy Spectrum 3

FIGURE1.2: Flux of positrons, scaled with E3(left) and ratio of

antiproton - proton flux, rescaled by 104(right), both measured by AMS-02 [8].

and Z are more strongly bound compared to odd A and/or Z, making them favoured in stellar thermonuclear reactions. The abundances for C, N, Fe and O have similar values for both primary cosmic rays and solar elements, which leads to the assumption that many of these elements originate from the Sun.

However, we can clearly see differences for elements with Z between 3 and 5, corresponding to respectively Li, Be and B. For these elements, we can see that the primary cosmic ray abundances are greater than the solar abun-dances. This larger abundance for primary cosmic rays is explained by spal-lation of C and O nuclei during their propagation through interstellar hydro-gen, producing lighter nuclei. Similarly, for nuclei just below Fe (Z < 26) (i.e. Sc, Ti, V, Mn) we have the same spallation argument, but now of abundant Fe and Ni nuclei instead. The low solar presence of the light elements with Z between 3 and 5 in Figure1.3is due to these elements having a low Coulomb barrier and being weakly bound, making them swiftly absorbed in nuclear reactions taking place in stellar cores.

These results for the composition of primary cosmic rays, which is energy dependent, have been determined for the low energy region by use of direct measuring techniques. Since the intensity decreases with an increasing en-ergy (see Section1.3), the possibility to directly observe high energy (around 100 TeV) compositions of charged primary cosmic rays becomes more diffi-cult and thus also the determination of the composition (see Section1.4).

1.3 Energy Spectrum

As mentioned in the previous section, the intensity of primary cosmic rays decreases with increasing particle energy. This trend can be seen in Figure

1.4, where data of various experiments have been combined.

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FIGURE1.3: Composition of solar elemental abundances com-pared with the abundances of primary cosmic ray nuclei (mea-sured by the Cosmic Ray Isotope Spectrometer on the ACE spacecraft). All the abundances have been normalized to a

rel-ative abundance of 103_{for Si [}₁₀_].

factor E2, in order to facilitate in recognizing certain structures in the figure. A very noticeable characteristic is that the energy spectrum follows a power-law, which ranges over an enormous energy range (around 12 orders of mag-nitudes!). This power-law behaviour is very reminiscent of non-thermal be-haviour, since the energies span over many orders of magnitude, showing no sign of any characteristic - temperature - scale. This implies that these particles are not produced - with these high energies - by ordinary thermal emission mechanisms [11].

From the previous we can formulate the energy spectrum of the cosmic rays as follows, defined with a spectral index (γ):

N(E)dE∝ E−γ_dE, _(1.1)

where N(E) is the amount of particles within an energy range [E, E+dE]. This equation is often expressed in differential form:

dN dE ∝ E

−(γ+1) _(1.2)

or redefined in differential flux form (amount of particles per energy per area per solid angle and per time interval):

dΦ(E) dE = dN(E) dE dA dΩ dt nucleons GeV m2sr s . (1.3)

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1.3. Energy Spectrum 5

FIGURE1.4: Energy spectrum of all primary cosmic ray parti-cles combined, scaled by a factor E2[12].

In Figure 1.4, a ’bump’ in the curve can be observed, around 5×1015 eV, which is called the knee of the spectrum. Around 4×1018, we can see a sim-ilar bump which is referred to as the ankle of the spectrum. These areas are called the transition regions, where the knee marks a reduction of the in-tensity of particles originating from galactic cosmic accelerators and the an-kle the appearance of extragalactic particles. At around 1020 eV, there is an abrupt cut-off, where there are barely any particles observed. From this, we can identify 3 different energy regions in this spectrum, each with their own characteristics and underlying physics. This also implies different values for the spectral index of the power law in different energy regions [13,14].

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• Low energy cosmic rays (LECR): E.1010eV:

The flux in this energy region originates mainly from solar origin and is thus greatly affected by effects of the Earth’s magnetic field and the solar wind. More information on the latter process can be found in Section1.5.3.

• High energy cosmic rays (HECR): 1010 eV.E.1015 eV (≈knee): The spectral index (γ) corresponding to this energy interval is approxi-mately 1.7, for which most of the particles originate within our galaxy. The flux of primary cosmic rays is approximately 1 particle per m2per second. Around the knee region, this flux decreases to approximately 1 particle per m2per year.

• Ultra-high energy cosmic rays (UHECR): 1015 eV.E:

The spectrum between these energies steepens, with a spectral index

γ≈2.1. Most of the particles find their origin from extragalactic sources.

In the energy range around the ankle, the flux decreases even more to approximately 1 particle per km2per year.

For energies higher than 1018eV, the spectrum appears to flatten slightly to γ ≈1.6 and abruptly falls off around 1020eV. A possible physical ex-planation for this observation can be found in Section1.6.2.

1.4 Detection of Cosmic Rays

As previously mentioned, the primary cosmic ray flux decreases with an in-creasing energy, spanning over many orders of magnitudes. Using only a single detection method for this enormous energy range would be too good to be true.

For the low energy region, one can resort to direct observation methods at high altitudes, near the top of the atmosphere. The most often used methods for this are satellites and balloon-borne experiments. They have the advan-tages of being very accurate and being able to distinguish between matter and anti-matter (via spectrometers relying on magnets). The problems with this technique is that they have to be small scale (to be transported in satel-lites or balloons) and that eventually the statistics at higher energies runs out, so that we have no direct detection above 1 PeV. For these energy ranges we have to resort to different measuring techniques. Examples of direct detec-tion include the Alpha Magnetic Spectrometer (AMS) located on the Interna-tional Space Station (ISS) [15,16] and the CREAM experiment (Cosmic Ray Energetics and Mass) [17].

This brings us to the ground based experiments, which have large collection areas in order to probe high energy cosmic rays from energies starting at 100 TeV. Since a cosmic ray with an energy of 1020 eV only occurs once per km2 per century, we can either wait very long or build a massive detector. The disadvantage of this technique is that they have to rely on indirect measure-ments, where an analysis of the shape of an air shower - created by cosmic

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1.5. Galactic Cosmic Rays 7 rays interacting in our atmosphere and producing many secondary particles - is used to identify the primary cosmic ray. Examples of these detectors are the Pierre Auger Observatory [18] (in Argentina), High Resolution Fly’s Eye (HiRes) Experiment (in Utah, USA) [19], Telescope Array Project (TA) (in Utah, USA) [20] and IceTop (in Antarctica) [21].

In this thesis, we will focus on a large scale, ground based detector called Ice-Top. Its energy sensitivity for detection of cosmic rays is indicated in Figure

1.4, ranging from a primary cosmic ray energy of approximately 100 TeV up to a few EeV [13,14,22,23].

1.5 Galactic Cosmic Rays

Galactic cosmic rays originate from within our galaxy, the Milky Way. They propagate through the interstellar medium, which has an average interaction depth2of X = 5 g/cm2, before reaching the Earth’s atmosphere. Inside the disk of a galaxy, the number density is around ndisk ≈ 1 proton/cm3, so that

the corresponding distance travelled by the cosmic rays is given by:

l = X

mpndisk

=3×1024 ≈1 Mpc3,

which is a very large distance compared to the average half-thickness of the disk of a galaxy (h≈0.1 kpc).

By balancing the centrifugal and Lorentz force, we can find an estimate of the momentum of a particle with charge Z (given that the particles velocity is perpendicular to the magnetic field,~v ⊥ ~B) which is contained within a galaxy:

p= ZeBr.

By assuming a galactic magnetic field of B = 10−10 T4 and a bending ra-dius5 of 5 pc - from where particles are able to escape - we get a maximum momentum for which protons and electrons are contained within the galaxy of:

pmax =4.6×106GeV/c.

Since heavier nuclei will have a Z > 1, their maximum momentum for con-tainment will be higher, so that they are more contained within the galaxy compared to e.g. protons, thus providing a possible explanation for the ap-pearance of a second knee observed in the energy spectrum [13,22].

2_{Interaction depth for the ISM is defined as X} ₌ _n

ISMl, where nISM the interstellar

medium density and l the distance travelled by cosmic rays.

3_{1 parsec is 3.1}_×₁₀13_{km, or 3.26 light-years.}

4_{Around 10}5_{times weaker than Earth’s magnetic field.} 5_{Also called the gyroradius/Larmor radius, r.}

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1.5.1 Acceleration of Galactic Cosmic Rays

One of the fundamental questions in cosmic ray physics is the acceleration of cosmic rays, especially to the highest observed energies (∼1020 eV). It is cur-rently unknown whether these accelerations occur in large scale structures in the galaxy or near discrete, point-sources. This is however, known for the low-energy component in the solar system, since these are produced by in-terplanetary shock waves and other shocks associated with the solar wind (to keV and MeV energies) and by solar flares (to GeV energies).

As an example, we can estimate the power requirement to accelerate cos-mic rays in our galaxy, knowing that the thickness of the disk is D ≈ 0.3 kpc, with a radius R ≈ 15 kpc, with an average age of cosmic rays in the galaxy of τ≈3×106years [14], for an average cosmic ray energy density of n_E,CR ≈ 1 eV/cm3, so that the total power requirement to accelerate cosmic rays in our galaxy:

LCR =

nE,CRπR2D

τ ≈10

60 _{eV yr}−1_.

For a type II supernova ejecting 10 M into the interstellar medium with a

velocity v ≈ 107m/s, which occurs in a galaxy around 3 times per century, we have:

LSN ≈1062 eV yr−1.

Which suggests that supernovae remnants (SNR) in galaxies may be the source for acceleration, since only a few percent of the average power output by su-pernovae in a typical galaxy is enough for the power requirement for galac-tic cosmic rays. Even though there are large uncertainties in these values, it seems possible that that supernova blast waves can indeed accelerate the galactic cosmic rays, up to a certain energy.

Fermi Acceleration

The mechanism behind Fermi acceleration is based on transfer of kinetic en-ergy from large magnetized plasma to cosmic ray particles, to achieve the high non-thermal energy spectrum of cosmic ray protons and nuclei. If a particle with an initial energy E0 gains ∆E = ηE amount of energy per

ac-celeration cycle, its final energy En after n amount of cycles can be written

as:

En =E0(1+η)n. (1.4)

Rewriting the previous to find the amount of cycles required to reach a cer-tain energy En:

n =ln En E0

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1.5. Galactic Cosmic Rays 9 Defining the escape probability of the cosmic ray particle from the accelerat-ing region per cycle as Pesc, so that the probability of remaining in the

accel-erating region after n cycles is(1−Pesc)n.

The fraction of particles accelerated to an energy greater or equal to E is thus given by6: N(≥ E)∝

∑

m≥n (1−Pesc)m = (1−Pesc)n Pesc ∝ 1 Pesc E E0 −γ , (1.6)

with (knowing that Pescand η <1)7 8:

γ = −ln(1−Pesc) ln(1+η) ≈ Pesc η (1.7) and thus dN(≥ E) dE = constant × E E0 −(γ+1) . (1.8)

So that the Fermi mechanism (both first and second order, see next sections) results in a power-law energy spectrum (with spectral index γ), which is also observed experimentally.

From this, we can rewrite the probabilities to time periods: the characteristic time required for one acceleration cycle (Tcycle) and the characteristic time

it takes to escape the acceleration region (Tesc ∝ 1/Pesc). Their ratio is the

escape probability per cycle from the acceleration region. This means that cosmic ray has accelerated, after a time t, to an energy:

E ≤E0(1+η)t/Tcycle. (1.9)

So that we find that the high energy cosmic rays will require more time to be accelerated, compared to the cosmic rays with lower energies. Furthermore, if the accelerating source has a finite lifetime, it will only be able to acceler-ate particles to a certain maximum energy, putting a limit on the cosmic ray energies.

Second Order Fermi Acceleration

In this model, a relativistic cosmic ray with an initial energy Ei elastically

’scatters’ (collisionless, otherwise it would lose energy) on a magnetic gas cloud, moving at a velocity vgas c, resulting in the cosmic ray gaining

energy. This can be seen in the left panel of Figure1.5. After some time, the average motion of the cosmic ray will be equal to that of the gas cloud. The initial energy of the cosmic ray measured in our reference frame and in the

6_Using_∑∞

k=0xk = 1−x1 for|x| <1.

7_{Using ln}₍₁₊_x_{) ≈}_x₋_x2_/2₊_x3_/3_{+ O(}_x4₎_for_|_x_{| <}_1. 8_{We only retain the first order terms here.}

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gas cloud frame9are connected by a Lorentz transformation E0_i =γEi(1−βcos θi) where β=vgas/c and γ =1/

q

1−β2.

And the final energy of the cosmic ray (noting that it scatters elastically E_i0 =

E0_f):

Ef =γE0f(1+βcos θ0f)

=γE_i0(1+βcos θ0_f)

=γ2Ei(1−βcos θi)(1+βcos θ0_f).

So that the relative energy gained:

η = ∆E

Ei

= Ef −Ei

Ei

=γ2(1−βcos θi)(1+βcos θ0f) −1. (1.10)

The direction of the cosmic ray escaping the cloud is randomized, so that after averaging

<cos θ0_f >=0.

The probability of collision between the relativistic cosmic ray and the non-relativistic gas cloud is proportional to their relative velocity (with a factor 1/2 for normalization): dN d cos θi = 1 2(1−βcos θi), where <cos θi >= R1

−1cos θi(1−βcos θi)d cos θi

R1

−1(1−βcos θi)d cos θi

= −β

3. So that the average fractional energy gain η (using that β1):

η = ∆E Ei =Dγ2(1−βcos θi)(1+βcos θ0f) E −1 = 4β 2_/3 1−β2 ≈ 4 3β 2_.

Since the average energy gain is quadratic in the velocity of the gas cloud, this model is referred to as the second order Fermi acceleration.

The problem with this method is that the random velocities of gas clouds are relatively small (β≈10−3), so the acceleration is very small, and that we have neglected collisional energy losses between particles in the gas cloud, which decrease the energy. Additionally, this explanation yields a too soft power

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1.5. Galactic Cosmic Rays 11 law exponent (in Equation1.7, Pesc ≈10−7, so that the spectral index γ∼10,

compared to the realistic value of 1.7). Therefore, we need to find a second method of acceleration, which brings us to the first order Fermi acceleration, commonly referred to as shock wave acceleration.

First Order Fermi (Shock) Acceleration

In this situation, we have a large, plane shock front moving with a velocity

~

−u1through plasma. The shocked gas will then move away from this shock,

with a velocityu~2relative to it (with|~u2| < |~u1|). Viewed from the lab frame,

the gas behind the shock will thus move with a velocity v~gas = − ~u1+ ~u2.

This can be seen in the right panel of Figure1.5.

The β in Equation1.10 can still be used, for the velocity of the shocked gas (’downstream’) relative to the unshocked gas (’upstream’).

The outgoing distribution of particle directions is now not averaged to 0, since there is an asymmetry present in this model, because particles in the upstream will re-enter the shock and those in the downstream will escape. So that the distribution is that of a uniform, isotropic flux projected onto a plane.

The outgoing angle: dN d cos θ0_f =2 cos θ 0 f for 0 ≤cos θ 0 f ≤1

and with (realizing that dN/d cos θ0_f ∝ cos θ0_f)

<cos θ0_f > = R1 0 cos θ 0 fd cos θdN 0_fd cos θ 0 f R1 0 dN d cos θ0_fd cos θ 0 f = R1 0 cos θ 2 f0d cos θ0f R1 0 cos θ 0 fd cos θ0f = 2 3. For the incoming angle, we have:

dN d cos θi

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and with (realizing that dN/d cos θi ∝ cos θi) <cos θi > = R0 −1cos θi dN d cos θid cos θi R0 −1 dN d cos θid cos θi = R0 −1cos θ2id cos θi R0 −1cos θid cos θi = −2 3.

Plugging these results in Equation1.10and averaging out:

η = ∆E Ei = 1+4β/3+4β 2_/9 1−β2 −1≈ 4 3β= 4 3 u1−u2 c .

Analogously, explaining why this is referred to as the first order Fermi accel-eration.

The incoming flux of cosmic rays is given by the projection of an isotropic flux onto the plane of the shock front (with nCR the number density of

cos-mic rays being accelerated): Φin= Z 1 0 d cos θ Z 2π 0 dφ cnCR(E) 4π cos θ= cnCR 4 .

The outgoing flux of cosmic rays, escaping downstream from the shock front, is given byΦout =nCRu2, so that the escape probability (defined as the ratio

between the outgoing and incoming fluxes) is given by: Pesc = Φ_Φout

in

= 4u2

c .

Recalling the equation for the spectral index for acceleration (Equation1.7):

γ = Pesc

η =

3 u1/u2−1

.

Using a value of u1/u2 = 4 results in a spectral index γ = 1, which agrees

well with the observed value (around 1.7) for the energy spectrum of cosmic rays. The fact that the following energy spectrum is close to the observed spectrum implies that supernovae are most likely the acceleration events for high energy cosmic rays [13,24,25].

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1.5. Galactic Cosmic Rays 13

FIGURE1.5: Illustration of the second (left) and first (right)

or-der Fermi acceleration [26].

1.5.2 Hillas Plot

The maximum energy that can be gained from shock accelerations is given by Emax ≤Zβs R kpc B µG 109GeV, (1.11)

with B the magnetic field strength, R the radius of the expanding shockwave (i.e. acceleration region) and βsthe velocity of the shockwave. In 1984, Hillas

concluded that in order for cosmic ray acceleration, the acceleration region needs to be at least twice the bending radius of the particle [27]. He summa-rized these conclusions in a plot, nowadays called the Hillas plot (see Figure

1.6), to show which objects are capable of accelerating cosmic rays to cer-tain high energies. In case of relativistic shockwaves (βs ≈1), many sources

are capable to accelerate protons up to 1020 eV. If we are dealing with non-relativistic shockwaves (βs 1), however, the amount of possible candidates

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FIGURE1.6: Illustration of the Hillas plot, showing the possible acceleration sources in function of the magnetic field strength B

and radius of acceleration region R [24].

1.5.3 Propagation of Galactic Cosmic Rays

Diffusion

The transport of cosmic rays in a galaxy (e.g. our Milky Way) can be de-scribed by the diffusion equation. The conservation of particles can be ex-pressed by:

∂n

∂t + ~∇ · ~J =0

where n is the number density of cosmic ray particles and~j the particle flow. Combining the previous equation with Ficks’ law for an isotropic medium (with D the diffusion coefficient, which is often a tensor):

~_J _{= −}_D₍_r_)~_∇_n₍_{r, t}₎_,

which results in following equation for a stationary medium:

∂n ∂t − ~∇ ·

h

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1.5. Galactic Cosmic Rays 15 Accounting for motion relative to a velocity field~v(r):

∂

∂t −→

∂

∂t + ~v(r) · ~∇

and adding additional terms to account for physical sinks and sources for propagation of a given nuclei i in the interstellar medium (ISM):

∂ni ∂t + ~v(r) · ~∇ni− ~∇ · h D(r)~∇ni(r, t) i =Qi(E, r, t) − βcρ λi + 1 γτi ni(E, r, t) − ∂ ∂E(βni(E, r, t)) + βcρ mp _k

∑

_≥_i Z ∞ E dE 0dσi,k(E0, E) dE nk(E 0 , r, t). (1.12) The source term Qi(E, r, t) reflects the creation rate of energetic particles of

type i, distributed in the disk of the galaxy. Other terms that arise in this equation are respectively: for the loss rate of nuclei by collisions and decay (where λi is the interaction length, in g/cm2, and τi the mean lifetime in the

restframe of particle i), for continuous energy losses of particle i (mainly syn-chrotron radiation and redshift energy losses due to the expanding Universe) and for a cascade term, to account for particles from nucleonic cascades and for nuclear fragmentation/spallation processes by particles k on i.

Leaky Box Model

In this model the assumption is made that cosmic rays propagate freely in a containment volume (e.g. disk of galaxy), having a constant escape prob-ability per time, τesc−1 c/h (with h the half-thickness of the disk and c the

speed of light). The name Leaky box illustrates the similarity to a closed of box, where confined, relativistic, particles have a small probability of escap-ing whenever they reach the boundary of a box, and thus sometimes leak out of it. When we replace the diffusion and convection terms in Equation1.12, with the characteristic escape time for particles ni/τesc, and assume that ni

has reached (steady-state) equilibrium, so that ∂ni/∂t =0 (and omitting the

continuous energy losses): ni(E) τesc(E) =Qi(E) − βcρ λi + 1 γτi ni(E) + βcρ mp _k

∑

_≥_i Z ∞ E dE 0dσi,k(E0, E) dE nk(E 0₎ . Applying this model to a proton cosmic ray, we can substantially simplify previous equation. A proton is a very stable particle with a decay time much

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longer than the age of the universe [28]. Additionally, for a primary pro-ton (for which creation from fragmentation from heavier nuclei can be ne-glected), the previous equation can be simplified to:

np(E)

τesc(E)

=Qp(E) − βcρ

λp np

(E).

Solving for npresults in

np =

Qp(E)τesc(E)

1+λesc(E)

λp

where λesc(E) = βcρτesc(E).

For protons the interaction length λp is large compared to the escape length

λesc(λp =55 g/cm2 λesc), so that:

np(E) ≈ Qp(E)τesc(E)∝ Qp(E)E−δ.

If at high energies the observed proton spectrum is measured to be np(E) ∝

E−(γ+1)_{, then this implies that}

Qp(E) ∝ E−(γ+1−δ) ≈E−2.1.

For heavier nuclei, such as Fe, the interaction length λFe is much smaller

(λFe = 2.6 g/cm2), so for low energies many particles will heavily interact

before escaping, which can be seen in its flatter low-energy spectrum and gradual steepening (where λesc <λFe), compared to He and protons. On the

high energy side, we expect quasi parallel lines in the spectrum, for all nuclei. Figure1.7 shows the primary cosmic ray flux for various different particles, ranging from protons to iron nuclei [7,13].

Solar Modulation

As mentioned previously, the low energy cosmic rays are modulated by the magnetic field of the Earth and by the solar wind. The latter is a flow of plasma, containing low-energy protons and electrons, which are ejected from the Sun. Every 11 years, the poles of the Sun change polarity, resulting in variations in the solar wind. This is called the solar sunspot cycle and is characterised by a change in the amount, and surface area, of sunspots. If the solar wind happens to hit the Earth (called solar flare), it is possible to see the Aurora Borealis/Australis (northern and southern lights), caused by excita-tions of air molecules by low-energy charged particles, which are trapped in the Earth’s magnetic field.

During periods of maximum solar activity, the amount of incoming cosmic rays is reduced compared to solar minimum (i.e. anti-correlation between cosmic ray intensity and solar activity), which can be seen in Figure1.8. This

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1.6. Extragalactic Cosmic Rays 17

FIGURE1.7: Primary cosmic ray flux of various nuclei in

func-tion of its kinetic energy. Upper right panel shows the ratio of H/He at constant rigidity10[29].

process is called solar modulation, and can be explained by the suppres-sion of galactic cosmic rays reaching the inner solar system because of the outward-flowing solar wind [13,14,22,30].

1.6 Extragalactic Cosmic Rays

1.6.1 Acceleration of Extragalactic Cosmic Rays

Shockwave acceleration via supernovae remnants are typically able to accel-erate cosmic rays to an energy Emax,SN ≈100Z TeV. This means that for

ener-gies higher than this, we require a different acceleration source. From Figure

1.6, we can see that possible candidates for these extragalactic cosmic rays are: active galactic nuclei (AGN), neutron stars (NS) and gamma ray bursts (GRB). These sources have large enough acceleration regions and sufficiently large magnetic field strengths. [24,25].

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FIGURE 1.8: Anti-correlation of amount of solar activity (amount of sunspots) and count rate of cosmic rays [31].

1.6.2 Propagation of Extragalactic Cosmic Rays

The high energy primary protons will interact with galactic and extragalac-tic magneextragalac-tic fields, and possibly the Earth’s magneextragalac-tic field. This means that only the most energetic protons (E1018eV) can be used for particle astron-omy (since they have a high rigidity and thus small magnetic deflection) [32].

GZK Suppression

At energies higher than 5 ×1019 eV the flux of primary cosmic rays is ob-served to be highly suppressed, to only one event per km2 per century. A partial explanation can be given by the so called ’GZK suppression’, named after Greisen [33], Zatsepin and Kuzmin [34]. In 1966, they proposed that very high energetic protons are suppressed by interactions with the cosmic microwave background (CMB) photons, which are a blackbody radiation remnant of the Big Bang. Very high energetic protons are able to collide with these CMB photons to produce pions (via Delta resonances), so that the uni-verse is opaque to these highly energetic particles:

p+γ −→∆+ −→ p+π0

−→ n+π+.

From these considerations, we can calculate the threshold energy for photo-pion production, via conservation of four-momentum (where the proton has

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1.6. Extragalactic Cosmic Rays 19 a mass M, an initial momentum~ppand energy Ep11and the microwave

pho-ton a momentump~γand energy Eγ = |~pγ|, in natural units). Since the square of four-momenta is Lorentz invariant, we have that:

(pp+pγ)2= (p 0

p+pπ)2.

Setting the energy of the proton equal to the threshold energy (Ep =Ethr) for

production of another proton and a pion (at rest) and noting that the kinetic energy of the proton is much higher than its rest mass (Ep M), so that

(Ep ≈ | ~pp|):

M2+2(EpEγ− | ~pp|| ~pγ|cos θ) ≥ M2+2Mmπ+m2π 2EthrEγ(1−cos θ) ≥2Mmπ+m2π.

With θ the angle between the proton and photon directions and mπ the mass of the neutral pion:

Ethr ≥

2Mmπ+m2π 2Eγ(1−cos θ)

. (1.13)

The photon number density in the frequency range [ν, ν+dν] for a universe filled with blackbody radiation of temperature T is given by:

dnγ =

8πν2dν c3

1 ehν/kT₋₁,

with the last term the Bose-Einstein distribution function, since photons are vector bosons (i.e. spin 1 particles). Integrating this over the photon energies yields 412 photons per cm3in current cosmological epoch (redshift z=0):

nγ = Z ∞ 0 dnγ = Z ∞ 0 8πν2dν c3 1 ehν/kT₋₁ =412 T3 2.725 cm−3.

The most important interaction is with the aforementioned cosmic microwave background but also the radio, optical and infrared background are impor-tant targets photon fields [35].

The CMB energy spectrum, which can be seen in Figure1.9, is a blackbody distribution with a temperature T = (2.72548±0.00057) K, so that the photon energy Eγin Equation1.13:

Eγ =kT =2.35×10 −4

eV. (1.14)

11_{The variables after the interaction are denoted with ’.} 12_{This plot has been made with data provided from [}₃₆_].

Note that the errorbars are included but are not visible due to them being too small.

MJy/sr stands for Mega Jansky per steradians, where 1 Jy is equal to 10−26W/(m2Hz), a unit for the spectral flux density.

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FIGURE 1.9: Cosmic microwave background spectrum

mea-sured by FIRAS on the COBE satellite, with in red the data points and in blue its blackbody fit12.

In the situation where we have a proton colliding with a CMB photon with an energy Eγ = ykT, we have that the proton energy threshold is given by (for a head-on collision, where we have maximum energy transfer cos θ = −1, with y=5):

Ethr =6×1019eV.

So that protons with an energy higher than Ethr are not expected to be

de-tected, since they scatter with the CMB (in this situation, a proton generally loses about 15 to 20% of its initial energy per interaction, until it is below Ethr.)

The cross-section for proton and photon interactions near the threshold en-ergy is σ ≈ 2.8×10−28 cm2, so that the collision mean free path length,

λ=1/(nγσ), is 2.8 Mpc for cosmic microwave photons. Cosmic microwave background photons with an energy greater than Eγ =5kT constitute around 10% of the distribution (high energy tail of the distribution), so that their mean free path length would be around 50 Mpc. This implies that protons with an energy Ep > Ethr, originating from beyond the local galactic

super-cluster (relatively small distance compared to extragalactic distances), would be attenuated by interactions with the CMB.

The GZK mechanism, which has been specified here, is only applicable for highly energetic protons, and not for heavy nuclei, for which the dominant energy loss mechanism would be photonuclear disintegration instead of pion

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1.6. Extragalactic Cosmic Rays 21 production by interactions with the CMB (they have a higher threshold en-ergy for pion production since their masses are higher). The AUGER collab-oration observed the highest energy primary cosmic rays to be heavy nuclei instead of protons [37], indicating that the GZK mechanism holds for pro-tons.

Inverse Compton Scattering

The energetic protons are also able to lose energy by inverse Compton scat-tering on blackbody photons, where a highly-energetic particle13 scatters with a relatively low-energetic photon, resulting in a highly energetic pho-ton. This interaction, however, doesn’t require a minimum threshold energy - and thus occurs at every proton energy:

p+γ−→ p+γ.

The cross-section for this scattering process varies with the inverse square of the center-of-mass energy (σ ∝ 1/s), but is small compared to the resonant pion production, so that its effect to the energetic proton spectrum is very minimal.

Electron-Positron Pair Creation

Next to photopion production and inverse Compton scattering, a third possi-ble process by which energetic protons can lose energy is by electromagnetic production of electron-positron (e+e−) pairs, also called Bethe-Heitler pair production. In this interaction, an e+e− pair is produced by energetic pro-tons impacting on the cosmic microwave background:

p+γ−→ p+e++e−.

The center-of-mass energy for this interaction is given by sthr = (mp+2me)2,

so that the energy threshold is given by Ethr ≈ 6×1017 eV. This energy

threshold is lower compared to the threshold for pion production and there-fore is its effect on the energy loss (and propagation) of highly energetic pro-tons negligible, except for proton energies between 3×1018and 4.8×1019eV. The average energy loss of a cosmic ray proton to produce an electron-positron pair is approximately 0.1%, which is small compared to the energy loss of a pion production event (15 to 20%). Which can be attributed to the mass of an electron being around 130 times smaller than that of a pion.

Redshift Losses

The most dominant energy loss contributions below 3×1018 eV are redshift losses. For proton energies>1018 eV, this effect is approximately constant.

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Since the average energy losses for pion producing interactions are the high-est overall, this implies that this is the limiting process for propagation of high-energetic cosmic ray protons.

The collection of various loss mechanisms can be seen in Figure1.10 [13,14,

22].

FIGURE1.10: Different energy loss contributions of a high

en-ergy proton (magenta: redshift losses, green: pair production, blue: pion production/GZK) [38].

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1.7. Population 23

1.7 Population

One of the goals of astroparticle experiments, ranging over broad ranges of energy, is to determine whether different underlying particle populations dominate in certain energy regions, and, if this is the case, to measure the locations (and types) of their sources.

In Figure1.11, we can see a possible model of the populations which are asso-ciated with galactic supernova remnants (population 1), a hypothetical high energy galactic source (population 2) and the highest energy particles with extragalactic origins (population 3). The composition of the particles around the knee region changes from a light to a heavy component, regardless of whether the knee reflects the maximum energy of their source or whether it reflects a rigidity-dependent change in propagation. Furthermore, the energy spectrum of particles above 3×1018 eV shows no sign of anisotropy, which can be expected for sources within the galactic plane, and are thus assumed to be from extragalactic origin. This transition from galactic to extragalactic cosmic rays is reflected in the hardening of the energy spectrum around the ankle [13].

FIGURE1.11: Illustration of different underlying particle pop-ulations of the cosmic ray energy spectrum (1: galactic super-nova remnants, 2: galactic high energy source, 3: highest

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25

Chapter 2

Air Shower Production

Once the primary cosmic rays reach the Earth and subsequently enter its atmo-sphere, they suddenly stumble upon many particles with which they can interact. The particles contained in the atmosphere will then act as targets for the high en-ergetic primary cosmic rays, resulting in deep-inelastic scattering, destroying them and producing new particles. These secondary, high energy particles can then also produce new particles, generating a cascade of more and more particles, until they have been absorbed and their energy has decreased to the point that new particle cre-ation is no longer possible. This crecre-ation of new particles in the atmosphere is called an air shower.

2.1 Earth’s Atmosphere

Before introducing the general features of air showers, we will first start with certain characteristics of the atmosphere of the Earth, required for further explanations. A useful parameter for cosmic ray interactions with the atmo-sphere is the vertical atmospheric depth (also called column density), defined as the altitude integrated density above a certain height h,

X(h) = Z ∞ h ρ(h 0₎ dh0 (2.1) and is expressed in g/cm2.

The relation between the distance l, measured from the the surface of the Earth to the cosmic ray, and the vertical altitude h is given by

h=l cos θ+1 2 l2 R⊕sin 2 θ,

with R⊕ the radius of the Earth ( R⊕ ≈ 6400 km). In the following

deriva-tions, we will neglect the curvature of the Earth1 (valid for zenith angles

θ <65◦), so that:

h ≈l cos θ. (2.2)

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h l X_slant X X = 0 h = 0 θ

FIGURE2.1: Illustration showing the geometry of an incident cosmic ray.

An illustration of this situation can be seen in Figure2.1. The density change with height, required to calculate X, can be found by assuming the atmo-sphere to be in hydrostatic equilibrium and locally isothermal (constant tem-perature).

The balance of forces in the atmosphere can be described as: P A− (P+dP) A =ρAg dh,

with P the pressure at a certain height, A the surface area, g the gravitational constant and dh the change in height. So that

dP= −ρg dh.

And using the ideal gas law,

P= ρRT

M ,

with R the ideal gas constant (R=8.31 J/(K mol)), T the temperature and M the molar mass of the air molecules. Combing both the previous equations,

dP

P = −

Mg RT dh. So that (with the isothermal condition):

P =P0e−h/H,

with H= RT/(Mg)the scale height and P0the pressure at ground level. Any

changes in pressure are due to changes in density (because of the isothermal condition), so that

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2.2. Electromagnetic Cascades 27 So that finally (via Equation2.1):

X(h) =

Z ∞

h ρ0e

−h0/H _dh0

= X0e−h/H with X0 =ρ0 H.

The vertical atmospheric depth of the atmosphere at sea level is approxi-mately X(h =0) ≈1000 g/cm2corresponding to an atmospheric pressure of around 1000 hPa.

The value for the scale height, H, in this ideal model with T = 273 K and M = 29 g/mol is approximately 8 km, reaching the troposphere. In reality, however, the atmosphere is not in isothermal balance, since its temperature changes with increasing altitude [13,24].

Inspired by Equation2.2, we can define the slant depth Xslantas

Xslant(h) =

X(h)

cos θ.

2.2 Electromagnetic Cascades

When a highly energetic electron or photon collides with molecules at the upper part of the atmosphere, an electromagnetic cascade will be initiated. In this cascade, pair production of e+e− and bremsstrahlung are the dom-inant processes from which the cascade develops. The shower energy con-tinues until all energy is dissipated by ionization of the atmosphere by the electrons and positrons present in the cascade. Considering only highly en-ergetic particles, we can neglect collisional energy losses and Compton scat-tering during the cascade development. Furthermore, pair production and bremsstrahlung require the presence of a nearby Coulomb field of an atomic nucleus, so that the processes will be screened by their bound electrons. The cascade equations for describing an electromagnetic cascade were formu-lated by Hans Bethe and Walter Heitler in 1934 [40].

2.2.1 Heitler Model for Electromagnetic Air Showers

In 1944, Heitler constructed a simple toy model for the basic aspects of the de-velopment of electromagnetic (EM) air showers [41]. In this model, an incom-ing electron interacts with the atmosphere after a certain distance, producincom-ing bremsstrahlung photons, which in turn, will undergo pair-production them-selves (provided the photon energy is sufficient, i.e. Eγ ≥2me):

e± →e±+γ γ →e++e−.

This process continues, resulting in a cascading shower of particles, which can be seen in Figure2.2.

Statistically, after every step of travelled splitting length, the energy of a par-ticle is half that of the previous parpar-ticle. Assuming that the cross-sections

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e

−

e

−

γ

e

−

e

−

e

−

γ e

−

e

−

γ γ

n = 1

n = 2

n = 3

e

+

e

+

e

+

γ

FIGURE2.2: Illustration of the development of an

electromag-netic cascade, initiated here by an electron.

of the bremsstrahlung and pair-production processes are equal and constant throughout the shower, and neglecting any additional energy losses, the split-ting length of the photon, electron and positron in the atmosphere is related to the radiation length, X0= 37 g/cm2, via

λ=X0ln 2.

After n amount of splitting lengths (steps), we have a distance of X =n λ =

n X0ln 2 where the amount of particles of the cascade doubles, giving

follow-ing equation for the amount of particles: N(X) = eX/X0

or rewritten in amount of steps n:

N(X) =2n, with a remaining energy per particle of

E(X) = E0

N(X) = E0 e

−X/X0_.

This branching and multiplication process continues until the energy has de-creased to a certain critical energy, Ec. This is the characteristic energy where

ionization losses dominate and after which particles will only lose energy, be absorbed or decay. The critical energy for EM showers in the atmosphere is Ec ≈87 MeV.

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2.3. Extensive Air Showers 29 The maximum number of particles, at this critical energy, is thus

N(Xmax) = 2nc = E0 Ec with nc = ln _E 0 Ec ln 2 . Which corresponds to an atmospheric depth of:

X_maxEM =ncλ=X0ln E0

Ec

.

Although the Heitler model is a toy-model to simulate electromagnetic show-ers, it is a very good approximation to illustrate and explain its features. Most notable, the final total number of electrons, positrons and photons N(Xmax)

is proportional to the initial energy E0and the depth of the maximum shower

Xmax is proportional to ln(E0). [13,14,41,42]. p π− π+ p π0 γ γ e+ e− π0 _π− π+ p n µ − νµ νµ µ− e− µe µν D0 µ− νµ K+ µ+ π+ π− νµ γ γ γ γ e− e+ e+ γ e+ e− e− γ e− e− γ e− e+ e+ e− γ e− e−

Electromagnetic Component Hadronic Component Muon and Neutrino Component

p

π− n

FIGURE2.3: Illustration of the development of an extensive air shower, initiated by a high energy cosmic ray.

2.3 Extensive Air Showers

In Section 2.2.1, we discussed EM cascades, initiated by photons and elec-trons. Extensive air showers (EAS), on the other hand are initiated by high energy cosmic ray nuclei and contain high-energy hadrons. These showers

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essentially have three particle components: the electromagnetic, muonic and hadronic part. An illustration of an extensive air shower can be seen in Figure

2.3. The electromagnetic part is maintained primarily by the decay of neutral pions and eta particles to photons. The hadronic cascade is maintained by the nucleons and high-energy hadrons, while the muonic part is generated by the decay of lower energetic charged pions and kaons.

Whenever a primary cosmic ray hits an air molecule, an extensive air shower is initiated, where mostly pions are produced. The mean free path length for nuclear interactions in the atmosphere is around λint ≈ 100 g/cm2 for

a proton. The total atmospheric depth is approximately X = 1030 g/cm2, so that pions are mainly created in the stratosphere. The mean lifetime of charged pions is τ =26 ns, corresponding to a mean free path length before decaying of ddecay =γcτ. At energies for the pion around∼1 GeV, practically

all charged pions will decay in flight and barely interact with the atmosphere. At higher energies, nuclear interactions start playing an important role for charged pions. The decay channels of charged pions are:

π+ →µ++νµ π− →µ−+νµ

and for the neutral pion, with a lifetime of 8.4×10−8ns:

π0 →γ+γ.

Where these produced photons will then initiate the electromagnetic com-ponent of the shower, mainly in the upper atmosphere since the absorption length is short compared to the total atmospheric depth. The electrons and photons produced in this component is also called the soft component, since they are easily absorbed.

If the produced muon, from the charged pions, has a too low energy (.1 GeV), it will also decay in flight (since its mean lifetime is τ = 2200 ns, corre-sponding to a mean free path length before decaying of ddecay=6.6 km):

µ+ →e++νe+νµ µ− →e−+νe+νµ.

If the muons have an energy greater than 3 GeV, they can overcome the ion-ization energy loss (≈2 MeV cm2/g) and reach the surface of the Earth with-out decaying. Even higher energetic muons are called the hard component of cosmic radiation, since they are able to penetrate far underground the sur-face.

The main difference between an electromagnetic shower - initiated by a high-energy e−or γ - and an extensive air shower - initiated by a high-energy pro-ton or nucleus - is that in the latter a hadronic cascade develops throughout the atmosphere, with a longitudinal scale (see Section2.3.2) larger compared

Deep Learning Benadering voor Directionele Reconstructie van Hoog-Energetische Kosmische Stralen met de IceTop Detector

Deep Learning Approach for

Directional Reconstruction of

High-Energy Cosmic Rays with the

IceTop Detector

Ian B

Promotor:

Prof. Dr. Dirk R

Supervisor:

Stef V

A thesis submitted in partial fulfillment of the requirements

for the degree of

M

S

P

A

Department of Physics and Astronomy

Academic year 2019-2020

Acknowledgements

Summary

Samenvatting

Science Outreach

Contents

List of Figures

List of Tables

Chapter 1

Cosmic Rays

1.1

Discovery of Cosmic Rays

1.2

Composition

1.3

Energy Spectrum

1.4

Detection of Cosmic Rays

1.5

Galactic Cosmic Rays

1.5.1

Acceleration of Galactic Cosmic Rays

∑

1.5.2

Hillas Plot

1.5.3

Propagation of Galactic Cosmic Rays

∑

∑

1.6

Extragalactic Cosmic Rays

1.6.1

Acceleration of Extragalactic Cosmic Rays

1.6.2

Propagation of Extragalactic Cosmic Rays

1.7

Population

Chapter 2

Air Shower Production

2.1

Earth’s Atmosphere

2.2

Electromagnetic Cascades

2.2.1

Heitler Model for Electromagnetic Air Showers

e

e

γ

e

e

e

γ e

e

γ γ

n = 1

n = 2

n = 3

e

e

e

γ

2.3

Extensive Air Showers