A study of composition and anisotropy of cosmic rays

using the shape of the air shower front

Guus van Aar

Masters’s Thesis in Physics and Astronomy

Supervisor: Dr. Charles Timmermans

February 2012

GAP-2012-055HEN-472

Acknowledgement

First of all, I’d like to thank my supervisor, Charles Timmermans, for his idea that became thetopic of this thesis and for all his support and guidance while I was working on it.Thanks to my o�ce mates, Harm en Stefan for helping with daily questions and providing inter-esting discussions and advice about my research. Writing the gap-note together on a projectorwhile have a beer was certainly an interesting experience.Thanks to all of my o�ce mates and the rest of the members of the High Energy Physics de-partment for giving me a warm welcome, and the fun we had during the after hours.I’d like to thank the High Energy Physics department for providing the funding that enabledme to present my results to Auger Collaboration, and gave the opportunity to see the PierreAuger Observatory in the wonderful countryside in Argentina. Thanks to Charles, Harm, Johnand Stefan for having a lot of fun and interesting trips during our stay in Argentina.I’d like to thank my parents, Chris and Monique, for financial support during my study andfor keeping to encourage me to go on.Finally, thanks to my girl, Merel, for always being supportive and comforting when things wereless fun. And for always having a special bottle of comfort wine to relax with together after afrustrating day.

Me next to an Auger sd tank.

1

Contents

1 Introduction 4

2 Cosmic rays and air showers 62.1 Cosmic rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Acceleration mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.3 Origin of ultra-high-energy cosmic rays . . . . . . . . . . . . . . . . . . . 82.1.4 Mass composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Air showers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Shower components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Observables of mass composition . . . . . . . . . . . . . . . . . . . . . . . 12

3 The Pierre Auger Observatory 163.1 The Fluorescence detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 The Surface detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.2 Air shower reconstruction with the SD . . . . . . . . . . . . . . . . . . . . 19

4 The radius of curvature as composition sensitive parameter 234.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Zenith correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2.1 SD data cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2.2 Uncertainty due to shower to shower fluctuations . . . . . . . . . . . . . . 244.2.3 The corrected radius of curvature R0 . . . . . . . . . . . . . . . . . . . . 24

4.3 Correlation between R0 and Xmax . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3.1 Correlation coe�cient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3.2 FD data cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3.3 Systematic uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Indication for heavy nuclei from Centaurus A 305.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2 Data selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3 Creating enriched sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.4 Arrival directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.4.1 Two point correlation test . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.4.2 Angular distance to Centaurus A . . . . . . . . . . . . . . . . . . . . . . . 345.4.3 Energy scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.5 Di↵erence between reconstruction algorithms . . . . . . . . . . . . . . . . . . . . 365.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6 Improvements to the reconstruction algorithm 40

2

7 Outlook 42

A Systematic checks 43

References 43

3

Chapter 1

Introduction

In 1939, Pierre Victor Auger carried a few particle counters up to a high altitude laboratory inthe Alps. He used these to measure cosmic rays and found that sometimes all of his countersdetected particles at the same time. When he studied this e↵ect more thoroughly, he saw thatthe frequency of these coincidences decreased when the counters were placed farther apart andwhen he placed his detectors at a lower attitude. From this, he concluded that these coincidencesmust be due to extensive air showers induced by cosmic rays. Auger estimated that the energyof these showers could exceed 1015 eV [1]. After this first discovery of extensive air showersinduced by cosmic rays, many experiments have measured the properties of cosmic rays overmany orders of magnitude in energy. However, the origin and acceleration mechanisms of thehighest energy cosmic rays remains a mystery.

The Pierre Auger Observatory has been build to detect cosmic rays with an energy above3 · 1018 eV. Above this energy, a cuto↵ of the spectrum is predicted due to the gzk-e↵ect. Inaddition, cosmic rays above this energy have so much momentum that their deflection anglesin galactic and extra-galactic magnetic fields become small. Therefore, it is possible that theycould be traced back to their source.

Most cosmic rays with an energy above 1019 eV are either protons or iron nuclei. To studyanisotropies in the arrival directions of cosmic rays, it is important to know which kind ofparticles the cosmic ray primaries are, because their deflection angle is proportional to theircharge. On average, showers produced by proton primaries take longer to develop, so anestimation of the primary type can be made from the longitudinal development of an air shower.This is measured at the Pierre Auger Observatory by fluorescence detectors. Unfortunately,these detectors have the disadvantage that they can only operate during clear and moonlessnights. If measurements of the mass composition of cosmic rays would be possible with thefully e�cient array of particle detectors, a factor ten in the amount of measurements is gained.

More information about the mass composition of cosmic rays is also important for otherstudies. For example, the hadronic interactions at ultra-high-energies could be studied in moredetail. If a source of cosmic rays is identified, it is important to know what kind of particlesit produces in order to study its acceleration mechanisms. Finally, an anisotropy in arrivaldirections could be used to study the galactic and extra-galactic magnetic fields if the chargeof the particles is known.

In the reconstruction of the data of the particle detectors, a curved shower front is usedin order to improve the determination of the arrival direction. In this thesis, the possibility isexplored to use the radius of curvature of the shower front as a parameter that is sensitive tothe longitudinal development of an air shower. First, some background information is givenabout cosmic rays and air showers in Chapter 2. This chapter also contains an interpretation ofthe radius of curvature and the air shower properties that it is sensitive to. Then, in Chapter 3,the Pierre Auger Observatory is described, along with the algorithm with which air showerparameters are reconstructed from the raw data. In Chapter 4, it is shown that the radiusof curvature is sensitive to longitudinal development of an air shower and thus the mass ofa cosmic ray. After that, in Chapter 5 it is shown that there is an anisotropy in the arrival

4

directions of heavy cosmic rays in the direction of the agn Centaurus A. Chapter 6 lists someimprovements that can be made to the reconstruction algorithm. Finally, in Chapter 7, someideas for future studies using the radius of curvature are presented.

5

Chapter 2

Cosmic rays and air showers

2.1 Cosmic rays

2.1.1 Spectrum

Cosmic rays are energetic particles from outer space. They arrive at Earth with energies thatspan twelve orders of magnitude. Besides atomic nuclei of di↵erent mass, cosmic rays can alsobe neutrons, photons or neutrinos. The charged particle cosmic ray energy spectrum is shownin Figure 2.1. The general behaviour is a powerlaw,

f(E) / E

�� (2.1)

however, there are three features where a change of the value of spectral index � has beenmeasured. The first break is called the ‘knee’ at an energy of 4 · 1015 eV. This is believedto be the maximum energy that a supernova remnant can accelerate protons to. Above theknee, the origin of cosmic rays is uncertain but sources are generally believed to be galactic.Around 4 · 1018 eV the spectral index flattens again in a region that is called the ‘ankle’. Thisis believed to be due to the influx of extra-galactic cosmic rays. A cuto↵ of the spectrum isobserved around 6 · 1019 eV, the same energy region where the Greisen, Zatsepin and Kuzminpredict the gzk limit due to interactions with the cosmic microwave background [2].

The highest energy cosmic rays above approximately 1018 eV are called Ultra-High-EnergyCosmic Rays (uhecr) and have a flux in the order of particles per square kilometer per century.This flux is so low that these particles cannot be measured directly, however they cause hugeparticle showers in the atmosphere so they can be detected indirectly.

2.1.2 Acceleration mechanisms

The most popular theory for acceleration of cosmic rays is stochastic particle acceleration, atheory first developed by Enrico Fermi and expanded upon by others. Stanev gives an overviewin [4].

Suppose a relativistic charged particle with energy E1 enters a magnetized cloud movingwith velocity v under an angle ✓1 as shown in Figure 2.2. In the rest frame of the cloud, theenergy of the particle will be

E

01 = �E1(1� � cos ✓1) (2.2)

with � and � ⌘ |v|/c respectively the Lorentz factor and the velocity of the cloud. The particlewill scatter elastically in the magnetic field until it exits the cloud in a direction unrelated to itsinitial direction and with an energy E

02 = E

01. Outside of the cloud, the energy of the particle

in the initial frame isE2 = �E

02(1 + � cos ✓02). (2.3)

Thus, it will have a change in energy by a fraction

⇠ =E2 � E1

E1=

�(cos ✓02 � cos ✓1) + �

2(1� cos ✓1 cos ✓02)

1� �

2(2.4)

6

Figure 2.1: The charged particle cosmic ray spectrum [3].

Figure 2.2: Left: Second order Fermi acceleration in magnetized clouds. Right: First order Fermiacceleration at astrophysical shock fronts.

7

Depending on ✓1 and ✓

02 the particle can either gain or lose energy. For each collision with

the cloud, ✓1 and ✓

02 are random, so on average, there is a net energy gain ⇠ / �

2. This iscalled ‘second order’ Fermi acceleration. With this process, magnetized clouds can accelerateparticles to uhecr energies, but it is not e�cient enough to explain the observed spectral indexof cosmic rays.

This changes when the magnetized area is not a cloud, but an astrophysical shock front.An astrophysical shock is an disturbance that travels much faster than the speed of soundin the medium. This occurs for example in supernovae and in jets flowing out of the centralcore of an active galactic nucleus agn. The shock front moves with velocity u through theinterstellar medium. The matter behind the shock follows with a lower speed v. Equations2.2 and 2.3 now still apply, but the particle will always enter the shock head-on, so cos ✓1 willalways be negative, and cos ✓2 will always be positive. Thus, the particle gains energy everytime it crosses the shock. In this way, ⇠ / � so this ‘first order’ Fermi acceleration process ismuch more e�cient.

2.1.3 Origin of ultra-high-energy cosmic rays

There is no consensus about the source of the highest energy particles above the ankle. Thereare however constraints to the maximum energy Emax that a charged particle can obtain in anastrophysical object. Emax can be estimated by requiring that the gyro-radius of the particleis contained within the acceleration region of the source

✓Emax

EeV

◆= 0.5Z�

✓B

µG

◆✓R

kpc

◆, (2.5)

where Z is the charge of the particle, B the magnetic field strength, R the size of the sourceand � the velocity parameter of the shock front. Thus, BR must have a minimum value foran object to be able to accelerate cosmic rays to a certain energy. This constrains the possiblesources of uhecrs, shown in the Hillas plot in Figure 2.3. Candidates are for example neutronstars, gamma ray bursts (grb) and active galaxies [5].

The deflection of charged particles in galactic and extra-galactic magnetic fields becomessmall for particles at the highest energies, especially for protons. This means that particles canpossibly be traced back to their source location. The Pierre Auger Collaboration has measureda correlation between the arrival directions of cosmic rays with energies above 55 EeV andthe location of 442 nearby agns. This analysis was published in [7] and updated with moredata in [8]. Shown in Figure 2.4 on the left are the arrival directions of the events in the firstpublication (optimization period i, and reference period ii). The right side shows the evolutionof the significance of the signal with more events in period iii. Pdata is the fraction of events thatcorrelate within 3.1� with the locations of the agns, the expected value Piso for an isotropicdistribution is indicated by the dashed line. At the end of period ii, the arrival directions wereanisotropic with a confidence level of about 3�. With more data in period iii, the confidencelevel of this anisotropy remained the same, but the fraction of events that correlates with theagns almost halved.

Note that the locations of the agns are distributed according to the large scale structureof the universe. So while a correlation with the locations of agns was measured, the agnsthemselves do not need to be the actual sources.

2.1.4 Mass composition

Atomic nuclei at ultra-high energies can interact with cosmic microwave background (cmb)or ir/optical/uv backgrounds in the universe. They can lose energy through pair productionand adiabatic losses, but can also lose nucleons by the photodisintegration processes. Themost important process is the Giant Dipole Resonance (gdr), through which a gamma ray canexcite a nucleus temporarily after which it ejects some nucleons. Depending on the nucleus,the photon energy has to be 10 to 20 MeV in the rest frame of the nucleus. Higher energeticgamma rays above 150 MeV can initiate the photopion production [9].

8

Figure 2.3: Hillas plot: size versus the magnetic field strength of astronomical objects which could besources of ultra-high-energy cosmic rays. Objects below the diagonal line cannot accelerate particlesto the indicated energies [6].

Figure 2.4: Left: The celestial sphere in galactic coordinates with the arrival directions of eventsmeasured by the Pierre Auger Observatory above 55 EeV marked with circles of radius 3.1�. The reddots mark the location of the 442 nearest agns and the blue regions indicate larger relative exposure.The white star is the location of Centaurus A, the closest agn to Earth. Right: Degree of correlationof arrival directions of these events with the agn locations. The 1, 2 and 3� uncertainty bands areindicated with the shaded regions. The expected correlation of an isotropic distribution of arrivaldirections is given by the red dashed line.

9

Figure 2.5: Typical lifetime of nuclei propagating trough the cmb, infrared and uv backgrounds due tophotodisintegration. 1016 s corresponds to a distance of 100 Mpc. The solid line is for helium nuclei,the dotted line for carbon, the dashed line for silicon and the dot-dashed line for iron. Photopionproduction, which becomes significant for helium above 1.5 · 1021 eV is not included in this graph [10].

The typical lifetime as function of energy for di↵erent species of cosmic rays is shown inFigure 2.5. The curves have the same shape for each nucleus but are shifted horizontally becausethe nuclei have a di↵erent Lorentz factors at fixed energies. A vertical shift between the curvesis due to the di↵erence in binding energy that has to be added to remove a nucleon from thecore. So around 50 EeV, carbon nuclei start to have gdr interactions with the cmb causing asteep drop in their lifetime. At this energy, iron nuclei cannot have these interactions becausetheir Lorentz factor is lower. This strongly suppresses the flux of light nuclei so the cosmic raycomposition between 50 EeV and 300 EeV is expected to be composed of mainly protons andiron nuclei.

2.2 Air showers

When a cosmic ray enters the atmosphere of the Earth, it will interact with a nucleus in theair. In this interaction, energy is transferred to the nucleus and new particles are created.These secondary particles will also interact with subsequent air nuclei, creating more and moreparticles. This creates a cascade of particles traveling through the atmosphere: an air shower.The air shower is a disc of particles that travels towards the Earth. The front of the showeris curved and the disc gets thicker and less dense farther away from the shower core. Thedevelopment of an air shower depends on the amount of matter it has traversed. This is calledthe slant depth X, measured in gr cm�2. If the atmospheric density ⇢(z) as function of heightz above the Earth is known, then X at height h is given by

X =

Z 1

h

⇢(z)dz. (2.6)

The first interaction of a cosmic ray in the atmosphere is defined to be at depth X1. The depthat which the shower contains the maximum number of particles is called Xmax. An schematicview of an air shower with its di↵erent components in shown in Figure 2.6.

2.2.1 Shower components

An air shower is composed of di↵erent components, characterized by particles that have di↵erentproperties. Discerning between these components can give additional information about the air

10

Figure 2.6: Left: The di↵erent components of an air shower. Right: Schematic view of an air shower.

shower and hopefully the primary particle that initiated it. The di↵erent components are listedbelow.

Hadronic Component

Most uhecrs are atomic nuclei, so most air showers are initiated by a hadron. Hadrons inair showers are mostly created from interactions by other hadrons. Heavy hadrons can beproduced, but they decay or interact quickly and produce pions most of the time. Pions decayinto particles of the muonic and electromagnetic component before they reach Earth. Hence,only a very small fraction of particles that reach Earth are hadrons.

Electromagnetic Component

The electromagnetic component consists of photons, electrons and positrons. These form themain part of an air shower. High energy photons are mainly created from the decay of neutralpions, high energy electrons and positrons are mainly created in the decay-chain of chargedpions. Each high energy electromagnetic particle creates an electromagnetic subshower by pairproduction and bremsstrahlung for as long as there is enough energy to create new particles.Then the electromagnetic component is slowly absorbed by the atmosphere. The depth atwhich there are most electromagnetic particles is defined to be Xmax.

Muonic Component

Muons are mainly produced in the early stages of shower development by the decay of chargedpions. They hardly interact with atoms in the atmosphere, so they travel in a straight line oncecreated. The shower front thus mainly consists of muons because they travel over a shorterpath length between the point of first interaction of the primary and Earth. The number ofmuons in an air shower depends on the type of primary particle. A heavy ion creates a showerwith a bigger muonic component. The muon fraction of a shower thus give information aboutthe type of primary.

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2.2.2 Observables of mass composition

To understand where uhecrs are coming from and how they are accelerated, it is importantto know what kind of particles they are. Di↵erent kinds of cosmic ray primaries are predicted:protons, atomic nuclei with di↵erent masses, neutrons, photons and neutrino’s. All of theseparticles can create air showers that look quite alike, but some properties of the shower areinfluenced by the type of primary and can thus be used to infer the type. The two mostcommonly used are the fraction of muonic particles created, and the longitudinal developmentof the shower.

Unfortunately, uncertainties in hadronic interaction parameters influence the same showerproperties as mass composition. The interaction rate between a cosmic ray and the atmosphereis characterized by the nucleon-air cross section. This determines how likely it is for a highenergy nucleon to interact with an atom in the air, and thus determines the average depth ofthe first interaction. The nucleon-air cross section also applies to heavier cosmic rays, which arenuclei consisting of A nucleons. These can be regarded as a superposition of A nucleons an thushave a cross section that is A times as big. This cross section has been measured at particleaccelerators at much lower energies, and has to be extrapolated over two orders of magnitudein energy in order to apply it to uhecr interactions. Also, air shower simulations predict alower number of muons in air showers than has been measured [11]. This means that we do notyet understand the particle interactions in an air shower.

In addition, the development of air showers is determined by a huge number of stochasticprocesses. Shower observables can show large fluctuations for the same incomming cosmic raydue to these shower to shower fluctuations. Therefore, it is not possible to derive quantitieslike mass composition or the nucleon-air cross section directly from individual events. It is onlymeaningful to compare data to simulations that are created using certain assumptions aboutcomposition and hadronic interaction parameters.

The latest results of the Pierre Auger Observatory [12] include composition analyses basedon muon arrival times, depth of shower maximum and rise time. This thesis looks at the radiusof curvature of the shower front as mass composition sensitive parameter. This section will firstgive an overview of existing composition sensitive parameters and then given an interpretationof the radius of curvature.

Depth of shower maximum

Atmospheric nitrogen molecules emit uv light when excited by air showers. The amount of lightemitted is proportional to the path length of charged particles in an air shower. Thus most lightis emitted when the shower contains its maximum number of charged particles, as is illustratedin Figure 2.7. Fluorescence telescopes can detect this uv light and make a profile of the energydeposited by the shower in the atmosphere. The atmospheric depth at which this depositreaches its maximum is called Xmax. Protons penetrate farther into the atmosphere than heavyprimaries and thus have, on average, a larger value of Xmax. How far these primaries penetrateinto the atmosphere is determined by the proton-air cross section. However, the average valuehXmaxi at a fixed primary energy is related to the mass A of the primaries as hXmaxi / hlnAi.

In addition, the fluctuations of the depth of shower maximum for protonic showers aremuch larger than for iron induced showers so the width of the Xmax distribution rms(Xmax)is also an indication for mass composition. The interpretation of rms(Xmax) however is lessstraight forward in the case of a mixed composition. Xmax is the best understood and mostused composition sensitive parameter for air showers. The rate of change of Xmax per decadeof energy is called the elongation rate. Measurements from the Pierre Auger Observatory ofhXmaxi and rms(Xmax) are shown in Figure 2.8.

Muon fraction

Muons produce the same kind of signal in particle detectors as electrons do. Therefore, theycannot be separated based on their signal in the detector. To measure the amount of muons ina shower, the electromagnetic component of the shower has to be physically removed. This can

12

Figure 2.7: Impression of the development of an air shower as it moves through the atmosphere. Thecurve on the right is proportional to the amount of particles in the shower and to the energy depositedby the shower in the atmosphere at the corresponding depth. Part of this deposited energy is convertedinto fluorescence light that can be detected by the fluorescence telescope on the left.

Figure 2.8: Left: Rate of change of the depth of shower maximum Xmax

with energy, measured by thePierre Auger Observatory [12]. Right: Fluctuations of X

max

as a function of energy. The red and bluelines indicate the predictions by several hadronic interaction models for respectively proton and ironprimaries. Systematic uncertainties are indicated as a band.

13

Figure 2.9: Rate of change of the maximum of the muon production depth Xµ

max

as a function ofenergy, measured by the Pierre Auger Observatory [12]. The red and blue lines indicate the predictionsfrom simulations for respectively proton and iron primaries.

be done by exploiting the stronger penetration power of muons. If the detector is shielded bya large amount of mass, the electromagnetic component will not reach the detector while themuonic component does. This shielding can be done for example by placing a lead cover on thedetector or placing the detector underground.

Muon production depth

Another possibility to select muons in the particle detector signal is to look at strongly inclinedshowers that have traversed a lot of atmospheric mass. The electromagnetic component ofthese showers is almost completely absorbed by the atmosphere, so determination of the muonfraction is hard. However, the time delay between the impact time of the shower core and themuons arriving at Earth can be used to calculate their path length, and thus their productiondepth. The muon production depth distribution is used to find the depth X

µmax where the muon

production reaches a maximum. Note that this is not the equivalent of Xmax for muons sinceX

µmax has maximized muon production instead of muon count. The elongation rate of Xµ

max isshown in Figure 2.9.

Rise time

Another mass composition sensitive observable is the time for the signal of a particle detectorto rise from 10% to 50% of its integrated value. This is called the rise time t1/2 and utilizesthe di↵erence in arrival times between the muonic and the electromagnetic component. Muonscontribute to the early part of the signal, so muon heavy showers (iron) contain a larger fractionof muons in the first 50% and thus have a shorter rise time. Also, deeply penetrating primaries(proton) produce particles over a longer part of their track which leads to a larger spread inpath length and thus a larger rise time.

Radius of curvature

After creation, muons travel in a straight line at the speed of light. Thus, the first muons thatare produced in an air shower form the front of the shower. These first muons are produced overa short length compared to the distance to Earth so they form a front that is approximatelyspherical and expands towards the Earth, illustrated in Figure 2.10. The radius of curvature

14

Figure 2.10: Schematic view of the radius of curvature of an air shower. After the first muons areproduced at the top star, the shower front expands as a sphere which has radius R

c

when the showercore reaches Earth. The bottom star indicates the location X

max

where most particles are produced.These arrive at a later time and increase the thickness of the shower. The left picture illustrates adeeply penetrating proton, the right picture a shallow heavy ion.

is mainly dependent on the longitudinal development of an air shower. Deeply penetratingshowers will have a small radius of curvature, while shallow showers will have a large radius ofcurvature. However, the radius of curvature does not measure the same air shower property asXmax. The depth of shower maximum can be divided into a part X1 before the first interactionand the distance �X that the shower needs in order to develop to its maximum

Xmax = X1 +�X. (2.7)

X1 and �X are independent of each other and the average values should depend on the typeof primary for both.

If we define the amount of atmospheric matter traversed by the shower up to the pointwhere the radius of curvature points to as XR. Then XR has to be somewhere between thepoint of first interaction X1, and the point where most muons are produced, Xµ

max. Figures 2.8and 2.9 show that X

µmax is about 25 % smaller than Xmax. Since XR should be even smaller

than X

µmax it should depend much more on X1 than on �X. If this interpretation is correct, the

radius of curvature could add an independent composition sensitive parameter to the existingarsenal, provided that the accuracy of the measurement is good enough. In Chapter 4, it willbe shown that the radius of curvature is indeed sensitive to the longitudinal development of anair shower.

15

Chapter 3

The Pierre Auger Observatory

The Pierre Auger Observatory is located near the town of Malargue, in the Mendoza provinceof Argentina. It is designed to detect air showers of uhecr’s using two methods. The firstis an array of water Cherenkov radiation detectors placed in the pampa. This array is calledthe Surface detector and can measure a lateral profile of an air shower. The second detectionmethod is a set of optical telescopes that measure the longitudinal profile of an air shower.These optical telescopes together are called the Fluorescence detector. The strength of theobservatory lies in the hybrid operation of these detectors. This allows to estimate the energyof the primary cosmic ray with limited dependence on simulations. An example of an eventobserved by both detectors is shown in Figure 3.1.

3.1 The Fluorescence detector

The Fluorescence Detector (fd) of the Pierre Auger Observatory is designed to observe fluores-cence light emitted by atmospheric nitrogen molecules after they have been excited by passingair showers [13]. The photon production in this process is about 5/MeV of energy depositedin the atmosphere. This is enough to be able to detect and reconstruct air showers from thefluorescence light they produce on clear moonless nights, giving the fd a duty cycle of ⇠ 10%.The fluorescence detector consists of four stations, each containing six telescopes overlookingthe surface detector, as illustrated in Figure 3.2. Each telescope in a station has a field of viewof 30� x 30�, giving a station a total field of view of 180� x 30�. A schematic picture of a singletelescope is shown in Figure 3.3. When light enters the station through the shutter, there is anoptical filter that removes all light except the fluorescence frequencies. Light that passes thefilter is focussed by a spherical mirror into the camera that is an array of 440 hexagonal pmts.Each pmt can be thought of as one pixel of the camera.

An event measured by the fluorescence detector can be reconstructed from fd data only. Theplane created by the shower axis and the fd station (shower-detector plane) can be determinedwith high accuracy. However, the orientation of the shower axis within this plane has a higheruncertainty. Adding the timing information of a single sd station reduces the uncertainty in theshower axis to about 0.5� [13]. When the shower axis is known, a shower profile of the particlecount as function of atmospheric depth is made. This is fitted by the empirical Gaisser-Hillasfunction [14]

fGH(X) = Nmax

✓X �X0

Xmax �X0

◆Xmax�X0�

e

Xmax�X� (3.1)

to determine at which depth Xmax in the atmosphere the maximum amount of particles Nmax

in the shower is reached. In this formula, � is the mean interaction length and X0 is an energydependent shower parameter that, despite its appearance, does not have any physical meaning[15].

Atmospheric conditions have a large impact on the fd measurements and are continuouslymonitored. Air temperature and humidity have an influence on the amount of fluorescence

16

Figure 3.1: An example of an hybrid event measured by two fd stations and the surface detector (id200923507306). The line in the center of the picture indicates the reconstructed shower axis. Thecolored lines show the direction from which the fd stations have observed fluorescence light, with thecolor indicating the timing of the measurement. All colored discs are sd tanks that were hit by theshower, the size and the color indicate total signal and timing.

Figure 3.2: The Pierre Auger Observatory. Each red dot marks a tank of the surface detector. Thegreen lines indicate the field of view of the fluorescence telescopes.

17

Figure 3.3: Schematic view of a fluorescence telescope [16].

photons produced. Clouds can obstruct the field of view of a telescope so at each fd station,an infrared cloud camera continiously monitors the sky. Aerosols can make the atmosphereopaque by scattering the fluorescence light, so profiles of aerosol densities are obtained fromLidar data at each fd station. Next to fluorescence light, particles in air showers also produceCherenkov light. This is emitted in a cone around the shower axis and is generated partly inthe same frequency range as the nitrogen fluorescence. Cherenkov light is also scattered byaerosols and can disturb a measurement even when a shower is not pointed towards a telescope.The amount of Cherenkov light observed by a telescope is calculated using a model and theaerosol profiles and subtracted from the measured signal. All these atmospheric conditions aretaken into account in the fd reconstruction. However, the larger the corrections, the larger theuncertainty they introduce.

3.2 The Surface detector

The Surface Detector (sd) consists of over 1600 water-Cherenkov tanks distributed on an areaof approximately 3000 km2 and placed in a triangular array with a separation of 1500 metersbetween the tanks. The tanks are filled with 12 m3 of purified water and read out using threephoto multiplier tubes (pmts) powered by a solar panel as shown in Figure 3.4. A gps receiveris added to obtain the timing of the signal.

When a charged particle from an air shower travels through the water in a tank, it producesa small flash of Cherenkov light. This light is detected by the pmt’s placed in the tank. Theenergy deposited in Cherenkov radiation by a relativistic charged particle is independent ofits energy, so the total signal that a tank measures only depends on the total track lenth ofcharged particles in a detector. This signal is measured in terms of Vertical Equivalent Muons(vem) which is the average signal that a vertically traveling muon creates in the tank. Theoperation of this detection method is independent of atmospheric conditions, so in contrast tothe fluorescence detector, the surface detector has a duty cycle of 100%.

3.2.1 Trigger

The readout of the Auger surface detector uses a trigger system consisting of five levels. Thefirst level, T1 indicates that a tank was hit by a particle. If there are more particles in atank, the second level T2 triggers to indicate that this tank was possibly near the core of anair shower. The T3 trigger signals that several tanks had a T2 at the same time, and the T4trigger selects only the air shower events from this. The T5 finally, is a quality trigger thatensures that all stations around the hottest station, the station with the highest signal in anevent, are present and functioning at the time. In this way, no crucial information about the

18

Figure 3.4: A sd tank

shower is lost and the reconstruction is expected to be of good quality. However, the T5 triggeris quite restrictive. It turns out that removing one station from the six around the hotteststation hardly a↵ects the quality of the reconstruction if the core location is within a triangle ofoperating stations. This is called the T5-posterior condition because it can only be determinedafter reconstruction of the event [17]. More details about the sd trigger system are given in[18].

3.2.2 Air shower reconstruction with the SD

At the Pierre Auger Observatory, two di↵erent reconstruction algorithms are in use. The Cen-tral Data Acquisition System (cdas) collects the raw sd data, and determines real time if thedata passes the triggers and reconstructs the shower properties. The O↵Line software packageuses a reconstruction algorithm that is able to reconstruct both sd and fd events. Both algo-rithms follow roughly the same procedure. The reconstruction as implemented by O↵Line [19]is described below.

The reconstruction of air shower parameters from raw data of the surface detector is performedin six steps. These are listed below, and described in more detail in the remainder of thissection.

• Stations belonging to the event are selected by estimating the core location and impacttime as the barycenter and bary-time.

• A preliminary shower axis is determined by fitting a plane shower front.

• The lateral particle distribution is fitted in order to obtain a better core location.

• A curved shower front is fitted using the new core location in order to obtain the finalshower axis.

• The lateral particle distribution is fitted again to find the final core location and obtainthe quantity S1000 as a measure for the amount of particles in the shower.

• S1000 is converted into the energy of the shower using the constant intensity cut methodand the calibration using the fluorescence detector.

19

Selection of stations

The data from the triangle of stations that have the largest total signal in an event are usedto define a shower axis based on the start times. The barycenter and bary-time of these threestations are also calculated. In order to determine which tanks belong to the event, the di↵erence�ti between the timing of a station and the expectation from plane shower front is calculated.If a station satisfies the condition -1000 ns < �ti < 2000 ns, it is considered part of the eventand used in the remainder of the reconstruction. If it does not, it is flagged as accidental andconsidered as if it had not triggered.

Plane shower front

Afterwards, data from all stations that are included in the event are used to calculate a newbarycenter which is used as initial core location. The impact time t0 at the barycenter andthe orientation of the shower axis ~a can be estimated by a plane fit to the start times ti andlocations ~xi of tanks in the event. The quantity

�

2 =X

i

(cti � ct0 + ~a · ~xi)2

�

2i

(3.2)

is minimized with the condition that the shower axis vector ~a = (u, v, w) is normalized. Theuncertainty of the locations of the tanks is negligible, so the uncertainty �

2i is the uncertainty

on the station start times, as explained later in the time variance model.

Lateral distribution function

The lateral distribution function (ldf) describes the amount of particles in an air shower atsurface level as function of the distance to the shower axis. It has more or less has the sameshape for each shower and scales with increasing primary energy. In the plane tangent to theshower axis, the particle density is described by

S(r) = S1000 · fLDF (r), (3.3)

where S1000 determines the size of the shower and fLDF (r) is the parameterization of the showershape in such a way that fLDF (1000m) = 1.

The shape of the most suitable ldf depends on the experiment. The atmospheric massabove the experiment, the sample distances and the type of particle detectors used influencethe optimal parameterization. For Auger it was determined [20] that the best description ofthe data is a modified version of the nkg formula [21; 22] in the form of

fLDF (r) =

✓r

r1000

◆� ✓r + r700

r1000 + r700

◆�+�

, (3.4)

with r700 and r1000 reference lengths of 700 m and 1000 m respectively. � and � are twoparameters that optimize the shape of the ldf for an individual shower. These are fixed in thestandard reconstruction and given by a parameterization that depends on S1000 and the zenithangle ✓. The ldf is fitted using a maximum likelihood function that properly takes stationswithout a signal and saturated stations into account, details are given in [19]. The ldf fit isused to determine both the core position and S1000.

Radius of curvature fit

The plane shower front fit was used to make a first estimate of the shower axis. However, theshower front is not a plane, so a better estimation of the shower axis can be made by includingthe curvature of the shower front. Note that at least four stations are needed in an event inorder to be able to fit a radius of curvature, and atleast five stations are needed to obtain anuncertainty on the radius of curvature. If an event is measured in only three stations, in thisstep a new shower axis is determined using a shower front with a fixed radius of curvature.

20

Figure 3.5: Schematic of the spherical shower front development

Suppose the shower starts at time tc at a point(Rc in the atmosphere as shown in Figure

3.5. The timing at location(xi should then satisfy

c(ti � tc) = |(Rc �

(xi|. (3.5)

Point(Rc can be written as

(Rc =

(c+Rc~a with point

(c fixed as the core position obtained from

the ldf fit, ~a the direction of the shower axis, and Rc the radius of curvature of the shower.This can be used to evaluate the timing of a station located ~xi from the shower core as

c(ti � tc) = |Rc~a� ~xi|. (3.6)

Time tc is related to the impact time t0 of the shower core as ctc = ct0 � Rc. Therefore, theradius of curvature be fitted by minimizing

�

2 =X

i

(c(ti � t0) +Rc � |Rc~a� ~xi|)2

c

2�

2ti

(3.7)

with �ti the uncertainty on the timing information of a station. This is given by the timevariance model.

Time variance model

The arrival of particles in an air shower at a sd station is a stochastic process that is assumedto be Poisson distributed [23]. If a station is hit by a shower front that arrives at the stationat true (unknown) time t0, the first particle arrives in the tank with a delay T1. The time Ts

after which the first particle arrives in a station is thus

Ts = t0 + T1. (3.8)

T1 is a random variable that has a mean value ⌧ that depends on the local particle density inthe shower. The variance of Ts is given by the uncertainty in the timing measurement b, andthe variance of T1, V [T1] = ⌧

2. The expectation value ⌧ can be estimated from the arrival timeTk of the k-th particle by ⌧ = Tk/k. Tk and k are obtained from the station data as the totaltime of the signal and the amount of particles that hit the tank. The total amount of particlesn is equal to the signal measured in vem, corrected for the longer path length of a particletraveling with the shower zenith angle through the detector.

The later part of a shower can have a di↵erent particle density than the first part, so inpractice the time interval T50 for the first 50% of the total signal S is used, ⌧ = 2T50/n. An

21

additional factor (n � 1)/(n + 1) needs to be added because Tk is measured from the datathemselves. Another scaling factor a2 is implemented to account for the approximations madein this model [23]. With this, the total variance of Ts becomes

�

2t = a

2

✓2T50

n

◆2n� 1

n+ 1+ b

2 (3.9)

With the extra condition that if n becomes smaller than 2, n is set to 2 to prevent the firstterm to go to zero for stations with a small signal.

A set of Auger stations has two tanks with a separation of 11 m. These give two independentmeasurements of the timing of a station in a shower and are used to determine quantities a andb as a2 = 0.36 and b

2 = 212 ns2 [24].

Energy estimation

The quantity S1000 from the ldf fit is used to estimate the primary energy. For a given energythe value of S1000 decreases with shower zenith angle due to attenuation of the shower particlesin the atmosphere and geometrical e↵ects. Assuming that cosmic rays arrive isotopically, theflux as function of cos2 ✓ should be constant over the Auger energy range. For each bin in cos2 ✓one can determine the value of S1000 such that the amount of events with a larger value of S1000

is constant. This is called the Constant Intensity Cut (cic) and is independent of energy [25].For each shower, a zenith independent reference value of S1000 at 38� is determined using

S38� =S1000

1 + 0.92x� 1.113x2(3.10)

with x = cos2 ✓ � cos2 38�. The translation of S38� to energy is done by calibration with fd

measurements. The latest calibration [26] resulted in

E = 1.68± 0.05 · 1017eV · S38�1.035±0.009

. (3.11)

22

Chapter 4

The radius of curvature ascomposition sensitive parameter

4.1 Introduction

In section 2.2.2, an interpretation is presented of the radius of curvature of the shower front asa parameter that is sensitive to the mass of a cosmic ray primary. However, it is not evident inadvance that the radius of curvature Rc is measured and reconstructed with enough accuracy tobe useful. If Rc is indeed sensitive to composition, it should correlate with other compositionsensitive parameters. The best understood composition sensitive parameter is Xmax so thecorrelation between Xmax and Rc will be examined in this chapter. Even without experimentaluncertainties, this correlation is not expected to be 100% because Xmax measures the depth ofshower maximum, while Rc is more sensitive to the depth of first interaction.

Before Rc can be used as a mass sensitive parameter, it first has to be corrected for itszenith angle dependence. This is done in section 4.2. After that, the correlation between thecorrected Rc and Xmax is tested in section 4.3. The data used in this chapter are all Augerevents measured up to March 2011 and reconstructed with O↵Line v7r2.

4.2 Zenith correction

The radius of curvature of the shower front can be measured, but can not be directly used asmass indicator. The quantity sensitive to the mass of a primary is not the distance from thelocation of first interaction to the impact point of the shower core, but the amount of mattertraversed by the shower in the atmosphere above it. A shower that starts after a certain slantdepth X1 will have a larger radius of curvature when its zenith angle is larger. The path length

of a shower with zenith angle ✓ between a fixed point(Rc (Figure 3.5) in the atmosphere and the

surface of the Earth increases with cos ✓. However, the point where X1 has been traversed foran inclined shower will be higher up in the sky, so the dependence of Rc on cos ✓ gets stronger.Hence, we assume a relation between Rc and ✓ in the from of

Rc(✓) =R0

cos↵ ✓

. (4.1)

This equation can be fitted to the data to obtain the average corrected radius of curvature R0

and power ↵. These parameters may have an energy dependence because cosmic rays at higherenergy interact in a di↵erent part of the atmosphere than those of lower energies. Once ↵(E)has been found, it can be used to eliminate the zenith dependence of Rc by calculating R0 forindividual events.

23

4.2.1 SD data cuts

The radius of curvature is useful as a parameter only if the event is well measured and wellreconstructed. Hence the following quality cuts are used.

Cut Motivation

Full sd reconstruction

T5-posterior triggerThis requires 5 of the tanks around the hottest tank to function,see section 3.2.1.

Energy > 3 EeV The surface array becomes fully e�cient above 3 EeV.

Zenith angle < 60�Above a zenith angle of 60�, a di↵erent reconstruction algorithmis used that shows some biases.

dRc

Rc> 0.015

The distribution of dRc/Rc has some outliers with an uncertaintysmaller than 1.5%. These are removed.

�2LDF

NDFLDF< 3 Ensure a good ldf fit quality.

�2curv

NDFcurv< 8

The radius of curvature fit must have a good quality. This cutremoves approximately the worst reconstructed 10%.

NDFcurv > 1Events with fewer than two degrees of freedom in the curvaturefit show some artifacts in the distribution of Rc and dRc.

4.2.2 Uncertainty due to shower to shower fluctuations

Shower to shower fluctuations create a distribution of the radius of curvature with a certainwidth �S–S . The average value of Rc changes with zenith angle and energy, so �S–S may alsodepend on these quantities. This shower to shower width has to be added as a systematicuncertainty when deducing a relationship for the average Rc on zenith angle.The measured distribution of Rc has an observed width �obs that is the result of a combination ofthe distribution due to shower to shower fluctuations �S–S and the uncertainty on the individualmeasurements of Rc, dRc, so we can approximate

�

2S–S = �

2obs � hdRci2. (4.2)

To determine �S–S as function of zenith angle and energy, the data that pass the cuts aredivided in bins in cos ✓ and logE. �obs is calculated as the standard deviation of the valuesof Rc in a bin. hdRci2 is the mean value of the uncertainty of Rc in a bin. Equation (4.2)is then used to calculate �S–S for each bin. The resulting distribution is fitted with a 2Dpolynomial function, second order in cos ✓ and first order in logE, shown in Figure 4.1, toobtain a continuous estimation of the uncertainty. For each event, the contribution of showerto shower fluctuations to the uncertainty of Rc at the corresponding energy and zenith is addedquadratically to the measured uncertainty.

4.2.3 The corrected radius of curvature R0

The zenith correction for the radius of curvature is performed by fitting parameters R0 and ↵ inequation (4.1) to the data. Because an energy dependence is expected, the data is first dividedin ten bins in logE between 3 EeV and 100 EeV. An unbinned fit is performed in each segmentin energy to obtain ↵. An example of one of the fits is shown in Figure 4.2. The results for ↵and the quality of the fit for each bin are shown in Figure 4.3. The energy dependence of ↵ isthen smoothed by fitting it with a second degree polynomial. This results is

↵(E) = (1.318± 0.056) + (0.11± 0.11) log(E) + (�0.125± 0.055)(log(E))2. (4.3)

The distribution of the corrected radius of curvature as function of energy is shown in Figure4.4.

24

)θcos(0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

log(E/EeV)

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.5

2

2.5

3

3.5

Figure 4.1: Calculated uncertainty on the radius of curvature due to shower to shower fluctuationsindicated by the color scale in km. The black curves are contour lines of a fit using a polyomial that issecond order in cos ✓ and first order in logE.

(deg)θ0 10 20 30 40 50 60

Rad

ius

of c

urva

ture

(m)

5000

10000

15000

20000

25000

30000

Figure 4.2: Determination of power ↵ by means of an unbinned fit with equation (4.1) to the data.Shown is the bin with events with an energy between 24.6 and 34.9 EeV.

EeVElog

0.6 0.8 1 1.2 1.4 1.6 1.8 2

α

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

log(E/EeV)0.6 0.8 1 1.2 1.4 1.6 1.8 2

/Ndof

2 χ

0.8

1

1.2

1.4

1.6

Figure 4.3: left: Fit results of power ↵ in the zenith correction of Rc

as function of energy. Right: Fitquality for each energy bin of the zenith correction.

25

EeVElog

0.6 0.8 1 1.2 1.4 1.6 1.8 (k

m)

0 R

5

6

7

8

9

10

11

12

13

Figure 4.4: The average corrected radius of curvature as function of energy. The error bars indicatethe RMS of the distribution of R

0

for the corresponding bin.

4.3 Correlation between R0 and Xmax

If the corrected radius of curvature R0 is a composition sensitive parameter, there should bea correlation between R0 and the observed depth of shower maximum Xmax. This can beexamined by comparing these parameters for the so-called golden hybrid events. This is a setof events that have been measured with both the sd and one or more fd stations, and can bereconstructed independently from the measurements of both detectors.

4.3.1 Correlation coe�cient

The correlation between two quantities can be quantified by a correlation coe�cient. Thecorrelation coe�cient corr(~x, ~y; ~w) for a set of measurements (~x, ~y) with weights ~w is given by

corr(~x, ~y; ~w) =cov(~x, ~y; ~w)p

cov(~x, ~x; ~w)cov(~y, ~y; ~w). (4.4)

The covariance cov(~x, ~y; ~w) is given by

cov(~x, ~y; ~w) =

Pi(Xi �m(x; ~w))(yi �m(y; ~w))P

i wi(4.5)

and the average m(x; ~w) by

m(x; ~w) =

Pi xiwiPi wi

. (4.6)

The weights will be chosen as w

2i = ( �xi

xi)2 + ( �yi

yi)2 to take the accuracy of the measurements

into account. This correlation coe�cient is only sensitive to linear correlations.

4.3.2 FD data cuts

To obtain an indication of the strength of the correlation between R0 and Xmax, high qualitycoincident data of both fd and sd are needed. Therefore, the following cuts are used in additionto the sd cuts listed in section 4.2.1. Most of these cuts are also used by other authors analyzingfd data [27; 28]. However, here each individual event must be of high quality so the cuts aremore stringent. In addition, the last cut demands that the fd and the sd energy measurementshave to agree with each other.

26

Cut Motivation

Full fd reconstruction

�

2GH/NdofGH < 2.5

The Gaisser-Hillas profile must be a good fit to the observeddata.

Xmax ± 50 gr cm�2 in tele-scope field of view

Ensures that the fitted location of Xmax is correct.

Cherenkov fraction < 50%

dXmax < 40 gr cm�2

dEFD

EFD< 0.2

EFD > 3 EeV Equal to the sd cut to avoid biasses.

✓FD < 60� Equal to the sd cut to avoid biasses.

�2line��2

GH

NdfGH> 1

There must be a su�cient di↵erence in fit quality between alinear fit and a Gaisser-Hillas profile to ensure that the fittedlocation of Xmax is reliable.

Cloud coverage < 20%About half of the events have no Lidar measurement. Theseare not removed to gain statistics.

vaod at 3 km < 0.05

The vertical aerosol optical depth is an indication for theattenuation of the fluorescence light. vaod at 3 km is thevertical integration of the extinction coe�cient from groundlevel to 3 km up.

1

ndf

X

i=sd,fds

(Ei � E)2

�

2Ei

< 5

Events that pass all previous cuts have an independent energyestimation from the sd and fd data.These measurements must be in agreement.

4.3.3 Systematic uncertainties

The weight of an event in the correlation factor is chosen to depend on its relative uncertainty.In this way, the quality of the data is taken into account. The reconstruction algorithm calcu-lates only statistical uncertainties on fitted value of Xmax and Rc. Systematic uncertainties haveto be added by hand. Additional uncertainties on Xmax could be due to the geometrical recon-struction, the atmospheric model and atmospheric conditions. This gives a total contributionto the uncertainty of approximately 23 gr cm�2 [29].

No research has been done on the systematic uncertainty of the radius of curvature andthis is outside the scope of this thesis. However, an estimation of this uncertainty is needed toassign a weight to each event. Therefore, an arbitrary value of 0.08 · R0 is used. This is takento be a relative error so the weight of an event in the correlation factor is not biased by the sizeof R0.

4.3.4 Results

All events that pass the cuts are divided in four equal bins of logE between 3 and 48 EeV. Above48 EeV, only four events pass the cuts so there is not enough statistics to conclude anythingmeaningful about the correlation factor at higher energies. One event can have multiple fd

measurements associated with it. If multiple fd measurements pass the cuts, the weightedaverage of the Xmax measurements is used. This is plotted against R0 in Figure 4.5 and fittedwith a linear function. The obtained correlation factors are shown in Table 4.1.

27

Energy Correlation factor3 < E/EeV < 6 -0.286 < E/EeV < 12 -0.2412 < E/EeV < 24 -0.2724 < E/EeV < 48 -0.61

Table 4.1: Correlation factors between R0

and Xmax

in their respective energy bins.

]-2Xmax [gr cm600 650 700 750 800 850 900 950

[km

]0

R

4

6

8

10

12

14

/ ndf 2χ 289.5 / 321p0 1.811± 18.86 p1 0.002446± -0.01609

/ ndf 2χ 289.5 / 321p0 1.811± 18.86 p1 0.002446± -0.01609

/ ndf 2χ 289.5 / 321p0 1.811± 18.86 p1 0.002446± -0.01609

/ ndf 2χ 289.5 / 321p0 1.811± 18.86 p1 0.002446± -0.01609

3 - 6 EeV

]-2Xmax [gr cm600 650 700 750 800 850 900 950

[km

]0

R

4

6

8

10

12

14

/ ndf 2χ 281.9 / 238p0 4.491± 29.78 p1 0.005973± -0.02968

/ ndf 2χ 281.9 / 238p0 4.491± 29.78 p1 0.005973± -0.02968

/ ndf 2χ 281.9 / 238p0 4.491± 29.78 p1 0.005973± -0.02968

/ ndf 2χ 281.9 / 238p0 4.491± 29.78 p1 0.005973± -0.02968

6 - 12 EeV

]-2Xmax [gr cm600 650 700 750 800 850 900 950

[km

]0

R

4

6

8

10

12

14

/ ndf 2χ 120.7 / 93p0 2.534± 19.3 p1 0.003318± -0.0143

/ ndf 2χ 120.7 / 93p0 2.534± 19.3 p1 0.003318± -0.0143

/ ndf 2χ 120.7 / 93p0 2.534± 19.3 p1 0.003318± -0.0143

/ ndf 2χ 120.7 / 93p0 2.534± 19.3 p1 0.003318± -0.0143

12 - 24 EeV

]-2Xmax [gr cm600 650 700 750 800 850 900 950

[km

]0

R

4

6

8

10

12

14

/ ndf 2χ 9.488 / 23p0 18.89± 56.17 p1 0.02474± -0.06167

/ ndf 2χ 9.488 / 23p0 18.89± 56.17 p1 0.02474± -0.06167

/ ndf 2χ 9.488 / 23p0 18.89± 56.17 p1 0.02474± -0.06167

/ ndf 2χ 9.488 / 23p0 18.89± 56.17 p1 0.02474± -0.06167

24 - 48 EeV

Figure 4.5: Golden hybrid events that pass the cuts are divided in four domains in logE. The data ineach domain is fitted with the function R

0

= p0 + p1 ·Xmax

.

28

4.4 Conclusion

In the interpretation of the radius of curvature as composition sensitive parameter presentedin Section 2.2.2, the shower front is composed of the first muons produced in a shower. Theatmospheric depth that this corresponds to should thus be smaller than the depth of maximummuon production. Because the proton-air cross section increases with energy, the average depthof the first interaction decreases with energy. On the other hand, the depth of maximum muonproduction increases with 40 gr cm�2 per decade of energy and the depth of shower maximum by60 gr cm�2 per decade of energy (according to the simulations in Figures 2.9 and 2.7). Thus, theenergy dependence of a phase in shower development is an indication for the relative depth atwhich it occurs. The measured average value of the corrected radius of curvature increases withenergy in Figure 4.4, so the atmospheric depth that this corresponds to decreases with energy.This is a negative energy dependence, so the phase in shower development that determinesthe radius of curvature should be higher up in the sky than the depth of maximum muonproduction. This means that the interpretation in Section 2.2.2 and the data are consistentwith each other.

The correlation factors indicate that there is a correlation between R0 and Xmax in eachenergy segment. So on average, showers with a high R0 have a low Xmax. Light nuclei produceshowers with on average a higher Xmax, so it is possible to make a distinction between lightand heavy nuclei based on the measured radius of curvature of a shower. Showers with a lowR0 have a higher chance to be the result of a light primary, while showers with a high R0 havea higher chance to be initiated by a heavy primary.

Note that the introduced cuts create a bias in Xmax. The demand that Xmax must be inthe field of view of a telescope creates a bias to including deeply penetrating showers, especiallyat large zenith angles. The influence of this e↵ect on the correlation factors should be smallsince the correlation is determined between parameters from the same event and no averagesbetween events are used.

The correlation factor is a lot higher in the highest energy segment. This can be due todetector e↵ects because higher energy showers are measured with greater accuracy. Especiallythe radius of curvature measurement will benefit from this because the number of degrees offreedom in the curvature fit will increase from a few to tens at high energies. There might alsobe physical e↵ects at work. The rms of the Xmax distribution decreases with energy (Figure2.7). If this decrease of the rms is (partly) due to lower fluctuations of the distance betweenthe depth of first interaction X1 and Xmax (eq. (2.7)), the correlation will become better.

29

Chapter 5

Indication for heavy nuclei fromCentaurus A

An earlier version of the analysis in this chapter has been published as an internal note for thePierre Auger Collaboration [30]. Some parts of the analysis presented here are improved fromthe description in the note.

5.1 Introduction

Results from the Pierre Auger Observatory (Figure 2.7) suggest that the mass composition ofuhecrs is changing between 1018 and 1019.5 EeV from mainly protons to a heavier composition.Cosmic ray primaries most likely consist of protons and iron nuclei at these energies due tointeraction with extra-galactic photon backgrounds. Since the charge of an iron nucleus is26 times as large as the charge of a proton, iron nuclei are deflected much more strongly ingalactic and extra-galactic magnetic fields. So, assuming a mixed composition, it is expectedthat a possible anisotropy in arrival directions of uhecrs is visible in the arrival directionsof proton primaries. If it is possible to separate proton primaries from heavier elements, it isexpected that the power of anisotropy searches increases.

In chapter 4 it was shown that there is a correlation between the depth of shower maximumand the corrected radius of curvature. Thus, the corrected radius of curvature is sensitive to themass of a cosmic ray. Therefore, it is possible to make a division in the data that separates lightfrom heavy primaries based on the corrected radius of curvature. Event by event classification ofthe primary type based on the value of Xmax is not possible because the expected distributionsof Xmax for proton and iron primaries overlap [31]. The radius of curvature is also a probe forthe longitudinal development of a shower and it is also not expected that an event by eventclassification based on the radius of curvature is possible.

This chapter first explains how to create the light or heavy enriched data sets. Surprisingly,an anisotropy is found in the arrival directions of the data set enriched with heavy nuclei. Thisanisotropy is quantified in the later sections.

5.2 Data selection

For this chapter data from the cdas Herald [32] version v4r8 up to april 4th, 2011 are used.Only data that pass the following set of quality cuts are used for the analysis.

• Zenith angle ✓ < 60�

• Estimation and reconstruction compatibles 6= 0

• T5-posterior > 1

30

a b c40% 6.07 1.24 0.4160% 6.96 0.90 0.49

Table 5.1: Parameters of curves that determine the ‘low’ and ‘high’ sets. f(log(E)) = a + b log(E) +c(log(E))2

• Number of sd station > 5, needed to fit a reliable radius of curvature

• cic Energy with latest FD calibration > 3 EeV

• Global fit �2 / ndof < 4

From the selected data, a ‘background’ data set is produced for which in addition is requiredthat

• cic Energy with latest fd calibration < 40 EeV.

The ‘signal’ data set is created from the selected data for which is requested that

• cic Energy with latest fd calibration > 55 EeV.

5.3 Creating enriched sets

Using the same procedure as is described in Section 4.2, the zenith correction for the radius ofcurvature for the Herald data is given by Equation (4.1) with

↵(E) = (1.369± 0.052) + (0.11± 0.11) log(E) + (�0.107± 0.050)(log(E))2. (5.1)

Data that pass the quality cuts are divided into five data sets of equal size based upon theircorrected radius of curvature R0. As R0 is energy dependent, this selection depends on theenergy of the event. The selected events are grouped in bins with a width of 0.025 in log(E).Within each bin, it is determined at which boundary 20% (and 40, 60, 80%) of the events havea R0 lower than this boundary value (see Figure 5.1). In the higher energy bins, the amountof statistics is limited, which causes large bin-to-bin fluctuations in the boundary values. Forcreating the curvature dependent data sets, a second order polynomial is fitted to the 40%and 60% boundary points as a function of log(E) between 3 and 35 EeV. The resulting fitparameters are shown in Table 5.1. These functions are extrapolated to higher energies. Eventswith an energy above 55 EeV and an R0 above the 60% curve form the ‘high’ data set (highradius of curvature), whereas events with the same energies and R0 below the 40% curve fromthe ‘low’ set (low radius of curvature). The ‘low’ data set contains 31 events and the ‘high’data set 28 events. The data with an intermediate value of R0 that is not yet included in a setis discarded to strengthen the enrichment of the sets. A number of systematic checks of theproperties of the events that are included in the ‘low’ and ‘high’ set are performed in AppendixA. These checks show that the radius of curvature is not biased by other shower properties,with the exception that the distributions of the number of sd stations that are hit by the showerare slightly di↵erent for events with a low or a high radius of curvature. It is assumed that thisdoes not strongly influence the conclusions of this chapter. The possible causes of this bias arediscussed in Chapter 6.

5.4 Arrival directions

The arrival directions of the events in the ‘low’ and the ‘high’ data sets are shown in Figure 5.2.The uncertainty on the measured arrival directions at these energies of about 1� is neglectedin this analysis. The ‘low’ set seems more isotropically distributed than the ‘high’ set. In the

31

Figure 5.1: The selected events are grouped per energy bin in five groups of equal number, using R0

todistinguish the groups. The energy dependence (in log(E)) of the boundary-values is fitted up to thevertical bar using second degree polynomials. The fit results are listed in Table 5.1.

latter data set, there seems to be a subset of events clustered around Centaurus A. To testif these distributions deviate from the isotropic expectation, first a two point correlation testis performed to examine if there is clustering of events in the sets. Afterwards, the arrivaldirection of events in the sets are compared with the position of Centaurus A.

5.4.1 Two point correlation test

A two point correlation test is sensitive to a clustering of events and is thus useful to searchfor anisotropies due to point sources. The distribution of angular distances between all pairsof events in the ‘low’ and the ‘high’ data sets are calculated and the agreement of these dis-tributions to the expected distribution is tested. To obtain the expected angular distancedistribution, the average angular distance distribution of 500,000 di↵erent test-sets obtainedfrom the ‘background’ was calculated. Each event in these test-sets is drawn randomly fromthe ‘background’. In this manner, the exposure and other detector e↵ects are properly takeninto account. Figure 5.3 shows the comparison of the angular distance distribution of the ‘high’and ‘low’ data sets to the expected distribution. To test the agreement of these distributionsto the background, a likelihood is calculated assuming a Poisson distribution for the amount ofevents per bin. Note that the amount of events per bin is not independent among the bins be-cause event pairs are considered. The agreement between the observation and the backgroundis then quantified by calculating a p-value using the likelihood as test statistic. A probabilitydensity for the likelihood is made using a toy Monte Carlo simulation. This generates 500,000event sets of equal size as the ‘low’ and the ‘high’ set from the ‘background’ set. The likeli-hood for each random set is calculated to create the probability density distribution. Figure5.4 shows these probability distributions and the likelihood for the observed sets as a verticalline. The p-value of the observed sets is then taken to be the fraction of random sets that havea likelihood to the right of the line. This p-value is interpreted in the following way. If theevents in the test set originate from the same distribution as the background, the p-value is theprobability to observe a set which is less in agreement with the background than the test set.The p-value of the ‘low’ set is 0.68 and the p-value of the ‘high’ set is 0.063.

32

Figure 5.2: Top: Relative exposure of Auger in galactic coordinates and arbitrary units. Bottom:Arrival directions of events in galactic coordinates. The center of Centaurus A is indicated with awhite cross. Left: the ‘low’ set. Right: the ‘high’ set.

Opening angle (deg)0 20 40 60 80 100 120 140 160 180

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Figure 5.3: The average angular di↵erence distributions (dotted) compared to the distribution ofangular di↵erences of all pairs of a set (solid). Left: The ‘low’ set. Right: The ‘high’ set.

33

-LL40 50 60 70 80 90

Freqency

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Figure 5.4: Distribution of log likelihood values of the two point correlation test for 500,000 event setsdrawn randomly from the ‘background’ set. The vertical bars mark the measured values. Left: fordata sets of 31 events and the ‘low’ set. Right: for data sets of 28 events and the ‘high’ set.

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Figure 5.5: Distribution of the angular distance to Centaurus A for the events in two sets (solid line)compared to the expected distribution (dotted). Left: the ‘low’ data set. Right: the ‘high’ data set.

5.4.2 Angular distance to Centaurus A

Figure 5.2 provides a hint for clustering of events around Centaurus A. Therefore, the angulardistance is calculated between the events and the center of Centaurus A (l = -50.5�, b =19.4�). This is shown in Figure 5.5. The distribution of the ‘low’ data set follows the expecteddistribution, whereas the distribution of the ‘high’ set seems to have an excess within 20 degreesof Centaurus A. To quantify the agreement of the ‘high’ and ‘low’ distributions to the expecteddistribution, again a p-value is calculated. This is done in a similar way as described in Section5.4.1 and the results are shown in Figure 5.6. The results are remarkably di↵erent for the ‘low’(p-value 0.31) and the ‘high’ (p-value 0.0062) distribution.

5.4.3 Energy scan

The threshold energy above which the ‘low’ and the ‘high’ set are defined is taken equal to theenergy above which the strongest correlation to the locations of agns were found [7]. In thatanalysis, they obtained this threshold from a scan through energy space optimizing the agn

signal significance. A di↵erent hypothesis is tested here, so the optimal threshold energy mightbe di↵erent. In Figure 5.7 the p-values of the two point correlation test and the anisotropy testtowards Centaurus A are drawn as function of the energy threshold of the two sets. The initialenergy threshold of 55 EeV seems to be optimal for this analysis as well.

34

-LL8 10 12 14 16 18

Frequency

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-LL6 8 10 12 14 16

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Figure 5.6: Distribution of log likelihood values of the two point correlation test for 500,000 event setsdrawn randomly from the ‘background’ set. The vertical bars mark the measured values. Left: fordata sets of 31 events and the ‘low’ set. Right: for data sets of 28 events and the ‘high’ set.

Energy (EeV)45 50 55 60 65

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alue

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Figure 5.7: P-values of the two anisotropy tests as function of the threshold energy above which the‘low’ set (dashed red line) and ‘high’ set (solid black line) are defined. Left: The two point correlationtest. Right: Test for anisotropy in the direction of Centaurus A.

35

Figure 5.8: Arrival directions in galactic coordinates of events reconstructed with O↵Line and an energyabove 50 EeV. The center of Centaurus A us indicated with a white cross. Left: the ‘low’ set. Right:the ‘high’ set. One event in the ‘high’ set is obscured by the cross.

5.5 Di↵erence between reconstruction algorithms

Initially, this analysis was done using data from the cdas Herald. However the O↵Line recon-struction had to be used in the previous chapter to get access to fluorescence detector data. Thecorrelation between Xmax and R0 was found using data reconstructed with O↵Line. It could bepossible that the radius of curvature is handled in a di↵erent way in the cdas reconstruction,which would mean that the ‘high’ set is not necessarily enriched with heavy nuclei for cdas

data. Therefore, analysis of this chapter is repeated with data reconstructed with O↵Line inthis section. The used cuts are slightly di↵erent because the algorithm is di↵erent. Howeverthe same procedure is used as described above.

Using a threshold energy of 55 EeV, the p-value of the two point correlation test is 0.68 forthe ‘low’ set and 0.21 for the ‘high’ set. The test of the angular distance to Centaurus A givesa p-value of 0.74 for the ‘low’ set and 0.23 for the ‘high’ set. A scan of the threshold energyfor both tests is shown in Figure 5.9. These figures show that the anisotropy observed in theHerald data is also present in the O↵Line data but at a lower energy of 50 EeV. The arrivaldirections of events in the ‘low’ and ‘high’ sets above 50 EeV are shown in Figure 5.8.

A comparison of the two reconstruction algorithms is shown in Figure 5.10. All events inthe Herald data that pass the cuts required for the ‘signal’ set are used in this figure. For eachof these events, the di↵erence between the Herald and O↵Line values of the energy and theradius of curvature is compared. The comparison of event energies shows an o↵set of 8%, andan rms around this o↵set of 6%. The average value of radius of curvature is almost identicalbetween the reconstructions but there is a spread of 8%. One event is not shown in the Figure5.10 that has a radius of curvature that is four times as big when reconstructed with O↵Line.A visualisation of the curvature fit with O↵Line of this event is shown in Figure 5.11. Thisevent is rejected by the cuts for the O↵Line data.

5.6 Conclusions

Heavy nuclei in the direction of Centaurus A

High energy events measured by the Pierre Auger Observatory were divided into sets basedon their radius of curvature. Assuming a mixed composition, the set with a high radius ofcurvature is enriched by heavy nuclei and the set with a low radius of curvature is enriched bylight nuclei because the radius of curvature is sensitive to the longitudinal development of anair shower.

There is a clear di↵erence in the distributions of arrival directions for events in the ‘low’ setand events in the ‘high’ set. A two point correlation test did not find any clustering of eventsin the ‘low’ set (p-value 0.68). On the other hand, there is a suggestion of clustering in the

36

Energy (EeV)40 42 44 46 48 50 52 54 56 58 60

P v

alue

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P v

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Figure 5.9: P-values of the two anisotropy tests as function of the threshold energy above which the‘low’ set (dashed red line) and ‘high’ set (solid black line) are defined. This is for events reconstructedwith O↵Line. Left: The two point correlation test. Right: Test for anisotropy in the direction ofCentaurus A.

Entries 75Mean 0.08193RMS 0.06126

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Figure 5.10: Comparison of shower parameters given by Herald and O✏ine for showers with an energyabove 55 EeV according to the Herald. Left: Energy. Right: Radius of curvature.

Figure 5.11: Visualisation of the radius of curvature fit for an event reconstructed with O↵Line. Thegreen line indicates the fitted curvature of the shower front. There is one tank at a large core distancethat was hit early and has a low uncertainty on the timing. This tank completely dominates the fit.Unfortunately, this same figure is not available for the event reconstructed by cdas. (sdid 7161918)

37

‘high’ set (p-value 0.063). This clustering can be attributed to the region around Centaurus Abecause a test of anisotropy around the location of Centaurus A resulted in a p-value of 0.0062.As implied by the two point correlation test, the ‘low’ set does not show an anisotropy in thisdirection with a p-value of 0.31.

Light nuclei are deflected less in magnetic fields. Therefore, if a source producing a mixedcomposition creates an anisotropic signal that is observable on Earth, this should be mostprominent for light nuclei. This is not observed, so this leads to the conclusion that thereis a source of cosmic rays in the direction of Centaurus A that produces above 55 EeV onlyheavy primaries. Unfortunately, it is hard to assign a proper confidence level to this conclusionbecause there was prior knowledge of the data when this analysis was designed.

Reconstruction algorithms

The correlation between the radius of curvature and Xmax was found in data reconstructed withO↵Line. The analysis in this chapter was done with data reconstructed with cdas. To check ifthe found anisotropy is not an aberration of this algorithm, the same analysis was performedwith O↵Line. In Figure 5.9, there is an anisotropy in the direction of Centaurus A for the‘high’ set at 50 EeV. The reconstructed energy of O↵Line is about 8% lower on average thangiven by the Herald, so these energy scales are in agreement with each other. The significanceof the anisotropy in the O↵Line data is about the same as in the Herald data. However theevents included in the sets fluctuate a little between Herald and O↵Line. This di↵erence canbe explained by the fluctuations found in the values of energy and radius of curvature of theevents. It is not clear which algorithm is more accurate. However, for at least one event thatpassed the Herald cuts, the reconstruction by O↵Line was clearly wrong.

Cosmic ray mass composition

Centaurus A is one of the closest of possible uhecr sources to Earth at a distance of approx-imately 3.8 Mpc. This distance is too small to disintegrate cosmic rays of all intermediatenuclear masses between hydrogen and iron (see Figure 2.5). So if Centaurus A is indeed thesource of the observed anisotropy, the mass composition could contain more intermediate nucleithan expected in an isotropic distribution.

The location of Centaurus A is such that it is only visible on the southern hemisphere. IfCentaurus A indeed mainly produces heavy primaries, then this would influence the cosmic raycomposition only on the southern hemisphere. The Telescope Array on the northern hemisphereobserves a mass composition that is in agreement with a proton dominated composition [33].While Auger on the southern hemisphere reported a composition that seems to have a largecomponent of heavy nuclei. The elongation rate for both experiments is shown in Figure 5.12.Heavy cosmic rays from Centaurus A might explain this apparent disagreement.

Source spectrum

The maximum energy that a source can accelerate particles to is proportional to the charge ofthe particle Z (eq. (2.5)). Therefore, if the source in the direction of Centaurus A only producesheavy primaries at 55 EeV, the maximum energy that it can accelerate protons to should belower than that. This can be used to make predictions about the product of the magnetic fieldand the size of the source.

The deflection of cosmic rays in galactic and extra-galactic magnetic fields is proportionalto Z/E. So if heavy primaries are deflected over an angle � at energy E, protons from the samesource should be deflected over the same angle � at an energy E/Z. This fact could be used toverify the results from this analysis. However, this is not yet visible in the scans of the energythresholds in Figure 5.7. For the ‘low’ set, there is no anisotropy down to 45 EeV. The scanfor the ‘high’ set also does not show a clear structure of an end to the spectrum of the source.The p-value is getting a bit larger at higher energies, but this could also be due to the loweramount of events above this threshold. Extending this analysis to lower energies might reveal

38

Figure 5.12: The elongation rate is the change of the average value of Xmax

per decade of Energy. Left:measurements of Auger on the southern hemisphere. Right: measurements of Telescope Array (ta) onthe northern hemisphere.

more about this source. If the resolving power of the radius of curvature is strong enough, thenthe anisotropy should also be present for intermediate masses at intermediate energies.

There are however some di�culties to extend this analysis to lower energies. Assumingmost primaries in the ‘high’ set are iron cores, the energy where an anisotropy in the arrivaldirection of protons should be sought is ⇠ 2 EeV. There is a much larger isotropic backgroundat this energy so more data is necessary to obtain a significant signal. In addition, this energyis below the full acceptance threshold of the Auger surface detector, so events will be detectedin only a few tanks. This will reduce the quality of the reconstructed value of the radius ofcurvature. However, an Infill array has been build in a part of the Auger sd array with a halfthe spacing between the tanks. The threshold energy of the Infill array is approximately 0.3EeV so it should be possible to extend this analysis to lower energies using data from the Infill.

Galactic and extra-galactic magnetic fields

Under the assumption that the path length r of a particle is much larger than the coherencelength lc of the extra galactic magnetic field BeGal, the deflection of charged cosmic rays hasbeen calculated as [34]:

↵eGal = 1.1� · Z✓

E

100EeV

◆�1 ✓r

10Mpc

◆1/2 ✓lc

1Mpc

◆1/2 ✓BeGal

1nG

◆. (5.2)

Assuming a coherence length of 1 Mpc and an average extra galactic magnetic field of 1 nG[35], the deflection of a 55 EeV iron core from Centaurus A is typically ↵eGal = 32�. Theassumption that the coherence length of the field is much larger than the path length of theparticle, is not really valid for particles from Centaurus A. However, these values rule out thepossibility that the anisotropy is due to another source in the direction of Centaurus A that ismuch farther away.

Because the extra galactic magnetic field (egmf) follows the large scale structure, the deflec-tion of crs in the egmf is expected to be smaller than the deflection in the galactic magneticfield (gmf) for nearby sources such as Centaurus A [36]. The strength and direction of the gmfis also not well known, but the deflection in the gmf of a 55 EeV iron nucleus could be esti-mated at approximately 25� [37]. The distribution of opening angles to Centaurus A in Figure5.5 clearly shows that the excess of events lies within 24� of Centaurus A. This is in agreementwith the predicted deflection due to the magnetic fields. With more data, the observed openingangle might be used to obtain better estimations of the galactic and extra galactic magneticfields.

39

Chapter 6

Improvements to thereconstruction algorithm

This is one of the first studies that have been done within the Pierre Auger Collaboration thatused the radius of curvature as a composition sensitive parameter. Therefore, not much hasbeen written about the reliability and the uncertainty of the parameter. This section lists someissues that might improve the reconstruction of the radius of curvature.

Reconstruction chain

The current standard sd reconstruction is optimized to calculate the arrival direction and theenergy of events. The radius of curvature is a tool to do this and therefore the algorithm isnot optimized to reduce biasses in the radius of curvature. This can be concluded from thereconstruction chain. After a first fit of the core location has been done by fitting the ldf, thecurvature is fitted to find the final shower axis. After that, the ldf is fitted again to find a newcore location and to improve the accuracy of the energy estimation. However, this new corelocation is not used to refine the determination of the radius of curvature. Thus, the radius ofcurvature fit is done with a misplaced core location which is visible in some events, for examplein Figure 5.11. This might be solved by doing an extra curvature fit after the second ldf fit.However, ideally, the whole procedure of fitting ldf and the curvature should be done in asingle minimization process.

Time variance model

Another issue is the start time of the signal in an individual tank and its uncertainty given by thetime variance model, described in Section 3.2.2. The model assumes a poissonian distributionof the arrival times of shower particles in a tank. The average time between arrivals is taken asthe total signal time dived by the total amount of particles. This is a good approximation fortanks with a high total number of particles. However, when the total particle count becomes inthe order of one, there should be a large poissonian uncertainty on the total amount of particlesthat hit the tank. This is not taken into account, with the result that the time uncertainty onstations at a large core distance fluctuates heavily. For example, if a tank receives a total oftwo particles that arrive almost simultaneously, the time uncertainty becomes very small andone station at a large core distance can completely dominate the curvature fit. This is whathappens with the event in Figure 5.11. A solution for this problem could be to determine theaverage time between particles ⌧ not only from the station data, but from the particle densitygiven by the ldf while assuming a width of the shower.

Shape of the shower front

Another thing that is worthwhile to investigate is if the parameterization of the shape of theshower front can be optimized. Figure A.3 shows that there is a small positive bias in the

40

number of stations hit by a shower if the radius of curvature is small. There could be twoexplanations for this. The first is that the fitted radius of curvature is correct. The showerswith a high radius of curvature started higher up in the atmosphere, and thus encounteredmore matter while traveling towards Earth. Due to this, more particles are absorbed in theatmosphere than in a shower with a low radius of curvature and the shower is less likely to hitstations at a large core distance.

Another explanation is that the shower front is not spherical at the outer edges of the front.If this is the case, a shower that happens to hit a station at a large core distance, will do thiswith particles that are not the first muons produced in a shower, but with particles that arrivewith a delay compared to the spherical front. In this case, the fitted radius of curvature isnot correct and biased to a lower value if more stations are hit. A better parameterization ofthe shower front might then improve the determination of the depth where the first muons areproduced.

41

Chapter 7

Outlook

This thesis contains one of the first analysis performed within the Pierre Auger collaborationthat uses the radius of curvature as a composition sensitive parameter. It seems to be promisingto use this composition information also in other studies. The most obvious example would beto redo the agn correlation study in [7] while making a distinction between heavy and lightprimaries. This should yield the result that the events that correlate within 3.1� of the locationof the agns are light nuclei, and that the confidence level of the anisotropy increases if heavynuclei are ignored.

One thing that is lacking in the analysis in Chapter 5 is a confidence level of the anisotropyaround Centaurus A for heavy nuclei. The p-value cannot be used as a confidence level becausethere was prior knowledge of the excess of events around Centaurus A. The only way to obtain aproper confidence level is to do the same analysis with an independent data set. Unfortunately,all Auger data that has been recorded up to April 2011 has been used in this analysis, so thereis no independent data set available at the moment. Therefore, it would be wise to make aprescription to do this analysis with future Auger data. It is important to decide to do this upfront, so that there is no prior knowledge of the new data which could be used in designing theanalysis. In this case, the obtained p-value can be interpreted as a proper confidence level.

The accuracy of the study in this thesis can be increased by improving the compositionsensitive parameter that is derived from the fitted radius of curvature. First, this can bedone by improving the reconstruction algorithm according to the suggestions in the previouschapter, second, by using the atmospheric depth XR (see Section 2.2.2) instead of R0. Hence,the zenith correction can be avoided and fluctuations in the atmosphere can be taken intoaccount. Furthermore, XR can be compared to Xmax to gain a better understanding of theshower properties that the radius of curvature is sensitive to and the accuracy with which theradius of curvature is measured.

If XR can be predicted with simulations, it might be possible to determine the mass com-position of cosmic rays using the radius of curvature. Due to the larger exposure of the surfacedetector, the mass composition can then be determined at higher energies than currently pos-sible with the Fluorescence detector.

Finally, one could try to combine the information of multiple composition sensitive param-eters for individual events to try to determine the mass of a cosmic ray on an event by eventbasis. This can be done in a multivariate analysis, where the dependencies of all parameterson each other are taken into account. These algorithms need to be trained on data sets wherethe primary mass is known. Therefore, this requires simulations that correctly reproduce allparameters that are used in the analysis.

42

Appendix A

Systematic checks

In Chapter 5 high energy events are divided into two data sets based on their corrected radiusof curvature R0. The ‘low’ set consist of events above 55 EeV and a low R0, the ‘high’ setconsists of events above 55 EeV and a high R0. The corrected radius of curvature could bebiased by other quantities. In this case the division of events is biased. The distributions ofarrival times are shown in Figure A.1, the distributions zenith and azimuth angles in FigureA.2 and the distributions of energy and number of stations in Figure A.3.

All histograms in this appendix have events with a low R0 drawn with a black line. Theevents with a high R0 are shown in red. In addition, the events from the ‘high’ set that arrivedwithin 24� of Centaurus A are shown in green, so the total ‘high’ set consist of all red and greenevents. The green events are those that create the anisotropy towards Centaurus A. Becausethe ‘low’ and ‘high’ set do not contain a lot of events, it is hard to spot any structure. Therefore,the same histograms are shown for events between 24 and 55 EeV. In this energy regime thestrongest correlation with Xmax was found, so the radius of curvature is expected to be wellmeasured.

43

Time (year)2004 2005 2006 2007 2008 2009 2010 2011

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Figure A.1: Arrival times of events with a low R0

drawn by a black line. Events with a high R0

aredrawn with red. The events marked with green lines are the subset of the ‘high’ set that arrived within24� of Centaurus A. Left: Events with an energy between 24 EeV and 55 EeV. Right: Events above55 EeV. The bottom plots are in gmt, Argentina uses the time zone gmt-3.

44

Azimuth angle (deg)-150 -100 -50 0 50 100 150

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Figure A.2: The zenith and azimuth angles of events with a low R0

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0

are drawn with red. The events marked with green lines are the subset of the ‘high’ set thatarrived within 24� of Centaurus A. Left: Events with an energy between 24 EeV and 55 EeV. Right:Events above 55 EeV.

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Figure A.3: The energy and the number of station hit by a shower for events with a low R0

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are drawn with red. The events marked with green lines are thesubset of the ‘high’ set that arrived within 24� of Centaurus A. Left: Events with an energy between24 EeV and 55 EeV. Right: Events above 55 EeV.

45

Bibliography

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[2] P. Bhattacharjee and G. Sigl, “Origin and propagation of extremely high-energy cosmicrays,” Physics Reports 327 no. 34, (2000) 109 – 247.

[3] J. Cronin, T. K. Gaisser, and S. P. Swordy, “Cosmic rays at the energy frontier.,”Scientific American 276 (1997) 32–37.

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