3D Reconstruction of Extensive Air Showers at thePierre Auger Observatory

Miguel Figueiredo Vaz Pato

Dissertação para obtenção do Grau de Mestre em

Engenharia Física Tecnológica

Júri

Presidente: Professor João Seixas

Orientador: Professor Mário Pimenta

Vogal: Doutora Soa Andringa

Julho 2007

i

Acknowledgements

I am thankful to my supervisor Mário Pimenta mostly for the much I have learned with him and the full

support given during the development of the thesis. And to Soa Andringa, that accompanied at close

distance the whole work and made possible many aspects therein.

The thesis was entirely developed in Lisbon at Laboratório de Instrumentação e Física Experimental

de Partículas (LIP), to which I am very grateful for the exceptional research conditions oered. I would

also like to thank to LIP members Catarina, Bernardo, Patrícia, Pedro Assis and Ruben for numerous

fruitful discussions and help in technical issues. I address special thanks to André and Sara as well.

Last but not the least, I am grateful to my parents, brother and sister and my friends for putting up

with me! És szeretnèk különösen köszönetet mondani Gabriellànak.

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Resumo

É apresentada uma síntese acerca de raios cósmicos de energia ultra elevada e dos respectivos desaos

de detecção, bem como o actual estado da arte das observações neste particular domínio. O Observatório

Pierre Auger é descrito de forma detalhada e, em seguida, propõe-se um método tridimensional original

para reconstruir cascatas atmosféricas extensas a partir de dados de uorescência e de natureza híbrida.

Este novo método utiliza o tempo de amostragem ao nível dos telescópios como terceira dimensão espacial,

produzindo uma imagem 3D da cascata atmosférica. Reconstrói-se então o perl da cascata considerando

a óptica detalhada dos telescópios e a luz de uorescência e de erenkov (directa e difusa). Os resultados

das reconstruções da geometria e do perl são vericados através de simulação de cascatas iniciadas por

protões e, nalmente, usam-se dados reais para efectuar a medição de pers laterais.

Palavras-chave: raios cósmicos de energia ultra elevada, cascatas atmosféricas extensas, Observatório

Pierre Auger, luz de uorescência, radiação de erenkov, perl lateral da cacasta

Abstract

A brief review on ultra high energy cosmic rays and the associated detection challenges is drawn

together with the present status of observations in the eld. The Pierre Auger Observatory is described

in a detailed manner and, afterwards, an original three dimensional procedure to reconstruct extensive

air showers from uorescence and hybrid data is proposed. This new method uses the sampling time at

the telescopes as a third dimension in space producing a 3D image of an air shower. The shower prole

is then reconstructed by considering the detailed optics of the telescopes and uorescence and (direct

and scattered) erenkov light. The results from both geometry and prole reconstructions are checked

through proton simulation and, nally, real data is used to perform the measurement of shower lateral

proles.

Keywords: ultra high energy cosmic rays, extensive air showers, Pierre Auger Observatory, uores-

cence light, erenkov light, shower lateral prole

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v

Contents

1 Introduction 1

2 UHECR within the cosmic ray eld 3

2.1 Energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Theoretical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.2 Acceleration and production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Relevance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 UHECR detection 9

3.1 Extensive air showers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 Electromagnetic showers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.2 Hadronic showers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Indirect detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.1 Ground arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.2 Light detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 Present status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 The Pierre Auger Observatory 21

4.1 Southern site description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1.1 Surface Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1.2 Fluorescence Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1.3 Laser facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1.4 Atmospheric monitoring devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2 Event reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2.1 Surface Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2.2 Fluorescence Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2.3 Hybrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Future steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.3.1 Southern site enhancements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.3.2 The northern site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 The 3D FD reconstruction 38

5.1 Geometry reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.1.1 The 3D method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.1.2 Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40vi

5.2 Prole reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2.1 The 3D shower prole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2.2 Light at diaphragm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2.3 Spot and mercedes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2.4 Expected and observed signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6 Lateral prole measurements 57

6.1 Systematic study of the 3D reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.2 Lateral sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7 Conclusion and prospects 65

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List of Figures

2.1 Energy spectrum of cosmic rays above 108 eV [2]. . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Degradation of proton energy during propagation through the universe [2]. . . . . . . . . . 6

3.1 The Heitler's model in electromagnetic cascades (a) and in hadronic showers (b) [8]. . . . 11

3.2 A ctious event as recorded by a ground array [1]. . . . . . . . . . . . . . . . . . . . . . . 12

3.3 The uorescence spectrum of the atmospheric nitrogen [12]. . . . . . . . . . . . . . . . . . 14

3.4 An air shower as seen by a uorescence detector [1]. . . . . . . . . . . . . . . . . . . . . . 16

3.5 Percentage of missing energy for dierent energies and primaries as calculated in Monte

Carlo simulations. Open circles represent proton showers, open squares He nuclei, lled

circles CNO and lled squares Fe [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.6 The energy spectrum of cosmic rays with the conicting results from AGASA and HiRes in

the UHE range. The actual ux is multiplied by E2.5 to ease the visualisation of spectrum

features [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.7 The energy spectrum as measured by HiRes (a) and the PAO (b). . . . . . . . . . . . . . 18

3.8 The transition to a light composition trend that occurs in the ankle region [21]. The

simulation results from proton and iron showers are indicated by the upper and lower

parallel lines respectively, while the full circles represent the experimental data. . . . . . . 19

4.1 The Pierre Auger Observatory southern site in Argentina [24]. The dots represent SD

tanks, while the lines show the eld of view of the FD telescopes. As of 31st March 2007,

1215 tanks and all 24 telescopes were fully installed and operational [39]. . . . . . . . . . . 22

4.2 A water erenkov tank of the Surface Detector [24]. . . . . . . . . . . . . . . . . . . . . . 22

4.3 The Schmidt telescope used in the FD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.4 The (β, α) coordinates and their relation to spherical coordinates (θ, φ). In this particular

case, φt = 90 (adapted from [45]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.5 The dimensions of an FD pixel and the mercedes structures in (β, α) coordinates. . . . . . 26

4.6 An entire FD camera with its 440 pixels and mercedes stars. The telescope axis centre is

signaled with a full circle and the origin with an open one. . . . . . . . . . . . . . . . . . . 27

4.7 The spot on θ ∈ [0, 5] and φ = 2 as previewed by KG simulation [47]. The plot

compares the incident direction of each photon (θin, φin) with the direction in the focal

surface (θfs, φfs). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.8 The position of the laser facilities (CLF and XLF) in the southern site. Other atmospheric

monitoring devices are also signaled [50]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.9 The Central Laser Facility (on the right) and the Celeste SD tank (on the left) [40]. . . . 29

4.10 The FD geometry reconstruction setup (adapted from [63]). . . . . . . . . . . . . . . . . . 31

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4.11 Timing t to equation (4.6) of the monocular event SD 2521005 (FD 2/1066/35) recorded

on 2006/08/01 [58]. In this case, Rp ' 5.3 km, χ0 ' 65.5 and T0 ' 24800 ns. . . . . . . . 32

4.12 The light prole at the diaphragm as a function of time for the event SD 2521005 (FD

2/1066/35) recorded on 2006/08/01 [58]. The actual quantity represented is the light ux

that crossed the diaphragm. The several components of direct and scattered light are

represented as well (see text). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.13 The reconstructed longitudinal prole and the Gaisser-Hillas t [44]. . . . . . . . . . . . . 33

4.14 Comparison of monocular and hybrid reconstructions using laser shots [64]. On the left

the dierence between the reconstructed Rp and the actual one is plotted, while on the

right the same dierence for χ0 is presented. . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.15 Calibration between S38 and the energy reconstructed by the uorescence technique [25]. 36

5.1 The 3D geometry setup and event visualisation. . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2 The distribution of dCP−eye and log10EKG in the data collected from January 2006 until

September 2006 with KG and 3D prole reconstructions and χ3D0 ≥ 45. The red and blue

lines indicate the border of the empirical cut (5.3) with d∗CP−eye = 5 km and d∗CP−eye = 10

km, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.3 Comparison between the Rp, χ0 and T0 values as reconstructed by the standard and 3D

approaches. Data collected from January 2006 until September 2006 with KG and 3D

prole reconstructions, χ3D0 ≥ 45 and passing cut (5.3) with d∗CP−eye = 10 km was used

to produce the plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.4 Distributions of rmin, rmed and rmax in (a) and their dependence on χ0 in (b). The data

set used is the same as in gure 5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.5 3D visualisation of event SD 2553671 (FD 2/1081/2704). In this case, Rp ' 9.7 km and

χ0 ' 33.1. Note the dierence between the reconstructed volumes here and those of the

event presented in gure 5.1(b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.6 Dependence of rmax on Rp for the same data set as in gure 5.3 but passing the quality

cut χ0 ≥ π2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.7 The Ixx/Iyy and Izz/r2med distributions for the same data set as in gure 5.3 but requiring

rmed 6= 0 and at least one Iii 6= 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.8 The distributions of√Ixx + Iyy − Izz (a) and

√Izz (b) for the same data set as in gure

5.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.9 The geometric setup used in the 3D prole reconstruction. . . . . . . . . . . . . . . . . . . 46

5.10 Gaussian t to the Nγ,ik distribution for Nγ,ik < 0. All data collected throughout July

2006 was used to produce the plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.11 In (a) and (b), the dependence of the observed (grey shaded area) and expected (red line)

signals on XNP and RCP are presented for the 1018.5 eV simulated event SD 78 (from

job 0). The direct and Rayleigh scattered erenkov expected fractions are signaled by the

green and blue lines respectively. This event presents Rp ' 7.6 km and χ0 ' 90.8. The

behaviour of the ratio e/o with XNP (c) and RCP (d), the χik distribution (e) and the

dependence of the mean lnP1

(Nγ,ik, Nγ,ik

)on RCP (f) are also shown for the same event. 55

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5.12 Observed (grey shaded area) and expected (red line) signals for the 1018.5 eV simulated

event SD 14 (from job 0). As in gure 5.11, the green and blue lines represent direct

and scattered erenkov contributions at the telescopes respectively. This event presents

Rp ' 1.6 km and χ0 ' 139.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.1 In (a) and (b), the dependence of the observed (grey shaded area) and expected (red line)

signals on XNP (a) and RCP (b) for 30 1018.5 eV simulated events. The direct and Rayleigh

scattered erenkov expected fractions are signaled by the green and blue lines respectively.

Dashed lines refer to expected signals calculated with the KG parameters, while solid lines

represent the use of the simulated parameters. The value of RM was xed to 9.6 gcm−2 to

produce these plots. The behaviour of the ratio e/o with XNP (c) and RCP (d), the χikdistribution (e) and the χ2/Ndf values per event (f) are also shown for the same simulation

set. Notice that in plot (d), for RCP & 25 gcm−2, the quantity of detected volumes is low

and thus there are signicant uctuations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.2 The behaviour of the estimators χ2 (RM ) /Ndf (in black) and lnL (RM ) (in red) with the

eective parameter RM for the close simulated event SD 78 (from job 0) (a) and the distant

one SD 10 (from job 0) (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.3 In (a) and (b), the dependence of the observed (grey shaded area) and expected (red

line) signals on XNP and RCP are presented for the 15 events from the Lecce-L'Aquilla

simulation and passing the cut (5.3) with d∗CP−eye = 10 km. The direct and Rayleigh

scattered erenkov expected fractions are signaled by the green and blue lines respectively.

The behaviour of the ratio e/o with XNP (c) and RCP (d), the χik distribution (e) and

the dependence of the mean lnP1

(Nγ,ik, Nγ,ik

)on RCP (f) are also shown for the same

simulation sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.4 In (a) and (b), the dependence of the observed (grey shaded area) and expected (red line)

signals on XNP (a) and RCP (b) for 50 events collected in June/July 2006 and passing the

cut (5.3) with d∗CP−eye = 5 km. The direct erenkov expected fraction is signaled by the

green line, while Rayleigh and Mie scattered erenkov components are represented in blue

and magenta respectively. The value of RM was xed to 9.6 gcm−2 to produce these plots.

The behaviour of the ratio e/o with XNP (c) and RCP (d), the χik distribution (e) and the

dependence of the mean lnP1

(Nγ,ik, Nγ,ik

)on RCP (f) are also shown for the same data

set. Notice that in plot (c) the ratio e/o grows for XNP & 1150 gcm−2, possibly because

the Mie light fraction (5.24) is over estimated in this region − recall that the component

shown in magenta is only a mean one. Besides, the statistics in that region is small and

consequently large uctuations are expected. . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.5 In (a) and (b), the dependence of the observed (grey shaded area) and expected (red line)

signals on XNP (a) and RCP (b) for 50 events collected in July 2006 and passing the cut

(5.3) with d∗CP−eye = 10 km. The direct erenkov expected fraction is signaled by the

green line, while Rayleigh and Mie scattered erenkov components are represented in blue

and magenta respectively. The value of RM was xed to 9.6 gcm−2 to produce these plots.

The behaviour of the ratio e/o with XNP (c) and RCP (d), the χik distribution (e) and

the dependence of the mean lnP1

(Nγ,ik, Nγ,ik

)on RCP (f) are also shown for the same

data set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

x

6.6 The behaviour of the estimators χ2 (RM ) /Ndf (in black) and lnL (RM ) (in red) with the

eective parameter RM for two real showers recorded in July 2006. Plot (a) corresponds

to event SD 2425381 with dCP−eye ' 4.7 km, while (b) corresponds to event SD 2425226

with dCP−eye ' 7.5 km. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

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Abbreviations

ADC Analog to Digital Converter

AGASA Akeno Giant Air Shower Array

AMIGA Auger Muons and Inll for the Ground Array

AMS Alpha Magnetic Spectrometer

APF Aerosol Phase Function Monitors

CDAS Central Data Acquisition System

CLF Central Laser Facility

CMB cosmic microwave background

CORSIKA COsmic Ray SImulations for KAscade

EAS extensive air shower

FD Fluorescence Detector

GPS Global Positioning System

GZK Greisen-Zatsepin-Kuzmin

HAM Horizontal Attenuation Monitors

HEAT High Elevation Auger Telescopes

HiRes High Resolution Fly's Eye

IACT imaging atmospheric erenkov telescope

ICRC International Cosmic Ray Conference

KG Karlsruhe group

LDF lateral distribution function

LIDAR LIght Detection And Ranging

MAGIC Major Atmospheric Gamma-ray Imaging erenkov (telescope)

NKG Nishimura-Kamata-Greisen

PAO Pierre Auger Observatory

PMT photomultiplier tubes

SD Surface Detector

SDP shower detector plane

UHE ultra high energy

UHECR ultra high energy cosmic rays

UV ultraviolet

VEM vertical equivalent muon

VHE very high energy

XLF second Central Laser Facility

xii

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Chapter 1

Introduction

Since its discovery in 1912 by Victor Hess, a great deal was understood about the phenomenon of cosmic

rays. For several decades the eld led the research on high energy particle physics in a time when

particle accelerators underwent an early phase of development. Nowadays, the area of cosmic rays is

split into several dierent branches dened according to the energy range pursued. The lower energy

range is reasonably well documented and understood, although there are still some problems to attack.

But, by far, the ultra high energy cosmic ray (UHECR) eld has remained over the years the hardest

to study. The reason behind this diculty is essentially two-fold. On the one hand, the ux of ultra

high energy cosmic rays hitting the atmosphere of the Earth is extremely low and, so, large detection

areas are required. On the other hand, the extensive air showers (EAS), formed while UHECRs cross

the atmosphere, call for detectors that present several technical challenges. Hence, there are still many

open issues in the UHE range: the energy spectrum, the primary composition, the arrival directions, the

sources and the propagation throughout the universe.

Even though several experiments were of extreme importance to the research in UHECRs, the Pierre

Auger Observatory (PAO) is expected to help in the solution of some of the above mentioned puzzling

questions. Indeed, by the end of this year (2007), when fully operational, the Observatory will become, by

far, the largest cosmic ray experiment ever and is supposed to benet from the use of a hybrid technique

that combines the two most successful detection types in the eld. The PAO comprises both a Surface

Detector (SD), that samples the shower lateral prole at the ground, and a Fluorescence Detector (FD),

that records the light emitted during shower development. Moreover, as proved by earlier experiments,

the control of systematics in a ground-based observatory is vital and, thus, the Pierre Auger Observatory

is equipped with several complementary systems to monitor the atmosphere and estimate systematic

uncertainties.

The research in UHECRs undergoes today a very exciting period since the quantity and quality

of the experimental data gathered until now are on the verge of allowing thorough tests of theoretical

models. And the theoretical relevance of this eld spans dierent branches, including particle physics,

astrophysics, cosmology and fundamental physics.

The present work introduces a three dimensional method for the reconstruction of extensive air showers

using the Fluorescence Detector at the Pierre Auger Observatory. As an application, the measurement of

shower lateral proles is performed in simulation and data. Part of the work developed during the thesis is

subject of a poster accepted for presentation at the 30th International Cosmic Ray Conference (Mérida,

México) in July 2007 and of an oral presentation to be held at the 6th New Worlds in Astroparticle

1

Physics (Faro, Portugal) in September 2007. The structure of the thesis is organised as follows. Chapter

2 contains a brief review on the most important UHECR features. The indirect detection techniques are

then explained in chapter 3 where a short summary of the present observations is presented as well. The

fourth chapter is dedicated to a description of the Pierre Auger Observatory with special emphasis on the

Fluorescence Detector. In chapter 5 the 3D reconstruction procedure is proposed and chapter 6 follows

with the measurement of shower lateral proles in both simulation and data. Finally, chapter 7 draws

the main conclusions and the future prospects of the work developed.

2

Chapter 2

UHECR within the cosmic ray eld

According to a broad denition, cosmic rays are energetic particles of extraterrestrial origin that hit the

top of the atmosphere of the Earth. They include many kinds of particles, both charged and neutral,

namely protons, atomic nuclei, electrons, antiprotons, positrons, photons and neutrinos [1, 2].

At low energies (. 1014 eV), the relative composition of cosmic rays is well known: they are mostly

protons and there are signicant quantities of He, C, N, O, Si and Fe nuclei [3]; minimal amounts of

nuclei heavier than Fe are also present [2]. The Sun is the main source of low energy cosmic rays as it is

the most signicant astrophysical object in the neighbourhood of our planet. Indeed, the abundances of

elements found in this energy range follow the solar composition except in some cases where spallation

prevents the elements from arriving at Earth.

In the intermediate range 1014 eV . E . 1018 eV , the composition still remains subject of contro-

versy, while the sources of these cosmic rays are believed to be the sun and supernova remnants in our

galaxy.

At ultra high energies, which may be understood as E & 1018 eV although there is not a strict

denition, the panorama is quite dierent. There are both technical diculties in the detection of

UHECR and theoretical challenges to explain their existence. Up to now the composition of this kind of

cosmic rays is poorly known and there are no identied sources. It is, therefore, an interesting and open

area of physics, with tight bounds to high energy particle physics and to astrophysics as well.

2.1 Energy spectrum

The energy of the cosmic rays detected up to now spans 15 orders of magnitude: from 106 eV to 1020 eV

[2]. The spectrum for energies above 108 eV is shown in gure 2.1 and it exhibits 33 orders of magnitude

in ux. In fact, while cosmic rays of 1011 eV are seen at a rate of 1 m−2s−1, those of 1018 eV only hit

Earth at 1 km−2yr−1 ' 3 · 10−14 m−2s−1. This outstanding range both in energy and in ux is simply

due to the wide variety of cosmic ray sources, from the Sun to extragalactic objects.

The spectrum is well tted to a simple power law [2]:

dN

dE∝ E−γs (2.1)

where γs is called the spectral index. Overall, γs is close to 3, but there are two major deviations: the

so-called knee at 5 · 1015 − 5 · 1016 eV and the ankle at about 1018 eV.

For E . 1015eV, the spectral index is approximately 2.7. This region includes cosmic rays coming

from the Sun and other galactic sources such as supernova remnants. Then, at the knee region there is a3

Figure 2.1: Energy spectrum of cosmic rays above 108 eV [2].

gradual increase of γs up to 3.0; the ux presents now a quicker decrease with energy and this behaviour

remains until the ankle. Besides, as energy increases, the lighter components gradually disappear, because

they are the rst not to be conned by the galactic magnetic eld. This is due to their greater magnetic

rigidity E|q| . Indeed, a particle of charge q and rest mass m under a magnetic eld B perpendicular to the

particle velocity v feels a radial force given by F = |q|vB = γmv2

r , where r is the radius of the trajectory.

Since p = γmv and E2 = p2c2 +m2c4,

r =√E2 −m2c4

|q|Bc' E

|q|Bc(2.2)

The approximation is valid at knee energies for all known particles because E mc2. Thus, lighter cosmic

rays are usually less charged and r is greater − they are less bounded to the galaxy by its magnetic eld.

At the ankle, there is a decrease of the spectral index back again to 2.7 − the spectrum attens.

Although there is no consensus about this feature, the ankle is believed to correspond to a transition

from galactic to extragalactic sources: at these energies not even the heavier nuclei (such as Fe) are

conned to the galaxy 1 . And this energy range originates so high magnetic rigidities that the particles

roughly point back to their sources. Then, if cosmic rays of E & 1018 eV are mainly from galactic origin,

1The transition implies a decrease in the galactic component rather than its disappearance, since galactic ultrahigh energy cosmic rays may still hit Earth although less bounded to the galaxy.

4

there should exist anisotropy to the galactic plane from events above the ankle. As for ultra high energy

composition, protons are feasible candidates because heavier nuclei are more likely to interact or lose

energy while crossing the interstellar space. Nevertheless, this is still an open question.

Finally, at energies beyond 1019 eV there is no sucient statistics to recognise any feature in the

spectrum [2]. However, some refer a third irregularity − the toe − even though its cause is not yet clear.

2.2 Theoretical problems

The ultra high energy regime gives rise to two main theoretical dilemmas. On the one hand, it is

challenging to explain how particles travel cosmological distances through interstellar space and arrive at

our planet with such impressive energies − this is the propagation dilemma. On the other hand, there is

the question of production and acceleration of particles up to multi-joule energies.

2.2.1 Propagation

The rst requirement one should impose on UHECR candidates is that they are stable and present

minor losses of energy while propagating. Among known particles, natural possibilities are then protons,

electrons, photons, neutrinos and atomic nuclei.

These particles, except for neutrinos, must undergo a cuto mechanism that is based on simple,

well-established results from particle physics. The rst intervenient of this mechanism is the cosmic

microwave background (CMB), discovered by A. Penzias and R. Wilson in 1966. It consists of almost

isotropic radiation with an energy spectrum very similar to that of a black body at TCMB ' 2.73 K,

presenting a mean wavelength < λCMB >' 1.96 mm which corresponds to a mean energy < ECMB >=hc

<λCMB>' 6.34 ·10−4 eV [1]. Because of its isotropy, the CMB should be scattered uniformly around the

universe so that travelling particles are in contact with it. In this way, little after the discovery of the

CMB, K. Greisen and (independently) G. Zatsepin and V. Kuzmin [4, 5] used well-known results from

special relativity to predict that suciently energetic protons interact with the CMB and lose part of

the initial energy. They found a minimum initial energy of the proton above which the interaction may

occur − the so-called GZK cuto.

Let us analyse, for instance, the process pγCMB → pπ0, where < ECMB > is the supposed energy

of the cosmic photon 2 . The threshold condition for this reaction to occur states that the nal proton

and the pion must be at rest in the center of mass reference system, that is, s = (mp +mπ0)2. Since

pCMB =< ECMB > and pp = βpEp (βp is the initial proton velocity in c units), one also computes the

center of mass energy as

s = (pCMB + pp)µ (pCMB + pp)µ = (< ECMB > +Ep)

2 − (~pCMB + ~pp)2

= m2p + 2 < ECMB > Ep (1− βpcosθ) (2.3)

where the calculation was performed in the laboratory reference system and θ is the angle between the

initial proton and the CMB photon. The GZK cuto is found by restricting (2.3) to the threshold

condition:

EGZKp =m2π0 + 2mpmπ0

2 < ECMB > (1− βpcosθ)(2.4)

2The cosmic microwave background radiation presents an energy spectrum spanning all positive energies, butit is fairly concentrated over its mean value, < ECMB >. So, it is reasonable to consider the referred process withγCMB of energy < ECMB >.

5

Considering frontal collisions (θ = π) and UHE protons (βp ' 1), EGZKp ' 1.07 · 1020 eV. As seen

above, the center of mass energy is√s = mp + mπ0 ' 1073 MeV, which is inside the energy range

of standard particle accelerators. Consequently, the total cross section σ(pγ → pπ0

)can be obtained

experimentally and, nally, the mean free path of the proton (the distance it travels before interacting

with a photon from CMB radiation) is given by Lp =[nCMB · σ

(pγ → pπ0

)]−1, nCMB being the CMB

photon density. Using nCMB ' 411 cm−3 [1] and σ(pγ → pπ0

)' 200 µb [4], one gets Lp ' 1.22 · 1023

m' 3.95 Mpc (1 pc' 3.26 light-years). On average terms, after this distance the proton interacts and

loses 0.13 of its initial energy [4]. It continues interacting on 3.95Mpc intervals until its energy falls below

the GZK cuto, as gure 2.2 shows. The main conclusion from gure 2.2 is that, independently of the

source energy, the proton cannot travel more than ∼ 100 Mpc [2] with post-GZK energies; in other words,

the universe is opaque to protons of energies greater than the GZK cuto. Therefore, protons arriving

at Earth with post-GZK energies must come from a source situated less than ∼ 100 Mpc away.

Figure 2.2: Degradation of proton energy during propagation through the universe [2].

Of course, the GZK mechanism is of statistical nature. It is possible that a proton from a source

several Gpc away hit the atmosphere with post-GZK energies, but that constitutes an extremely unlikely

scenario. The GZK eect foresees, in fact, a sharp decrease, rather than an uncontinuous cut, on the

cosmic ray spectrum for extremely high energies.

For the process pγCMB → pπ0, analysed in the previous paragraphs, one should consider as well the

∆+(1232MeV ) resonance, pγCMB → ∆+ → pπ0, which yields EGZKp ' 2.51 · 1020 eV. Moreover, there

are other reactions to take into account when studying the proton GZK cuto, such as pγCMB → nπ+

and pγCMB → e+e−p, with EGZKp ' 1.12 ·1020eV and EGZKp ' 7.57 ·1017eV, respectively. However, their

cross sections are low in comparison to that of pγCMB → pπ0 and so the latter is the most important

process in the proton case. Notice that the above referred reactions are all possible with photons other

than those of CMB − the importance of the CMB steams from being uniformly spread around the

universe and not locally clustered as other kinds of electromagnetic radiation.

6

A GZK-like mechanism also applies to photons(γγCMB → e+e−, E

′GZK′

γ ' 4.12 · 1014 eV)and to

nuclei whose nucleons are lost by spallation on CMB [6]. As for neutrinos, they do not couple directly

to photons according to the standard model since they are neutral and then, in principle, they do not

interact with CMB photons. There are higher order diagrams for the dispersion νγ → νγ but they have

low amplitudes due to the several vertices present. But neutrinos may suer a GZK-like eect due to the

existing cosmic neutrino background formed in the expansion of the universe. In fact, extremely energetic

neutrinos can interact with the just mentioned relic background that, considering all three neutrino

avours, presents a mean density nνr = 3 · 311nCMB ' 336 cm−3, a temperature Tνr =

(411

)1/3TCMB '

1.95 K and a mean energy < Eνr >∼ 10−4 eV [7]. Considering the annihilation ννr → Z0, one gets a

GZK cuto several orders of magnitude above the proton GZK cuto: E′GZK′

ν ' 2.08 · 1025 eV.

The GZK paradox consists in the fact that several events with energies above 1020 eV were detected

and that there are no candidate near sources. Indeed, a source of such extreme energies should be easily

visible to astronomers, that cannot nd a candidate source of multi-joule cosmic rays within ∼ 100 Mpc

of the Earth. A rst solution to this paradox is to consider that post-GZK cosmic rays are mainly

neutrinos, although it is not commonly accepted that neutrinos are responsible for such a signicant

quantity of events. Moreover, neutrinos cannot be accelerated by known physical processes. Thus, they

must come from the decay of particles with even higher energies, which leads to the next challenge −that of understanding how particles can reach ultra high energies.

2.2.2 Acceleration and production

Two dierent types of models attempt to explain the existence of ultra high energy particles: bottom-up

scenarios, which deal with acceleration processes, and top-down models, that involve the decay of super

massive particles.

As for bottom-up scenarios, particle acceleration is achieved through electromagnetic processes, which

must follow a primitive requirement: the particle, of charge q, must be conned to the acceleration site

of typical dimension R and average magnetic eld B. Equivalently, Emax ' |q|BcR [1, 3], as in equation

(2.2). Above this energy the particle abandons the site and cannot be further accelerated. Typical

processes of acquiring energy are synchrotron-like ones, where particles are accelerated by electric or

variable magnetic elds and their trajectories bent by magnetic elds. This phenomenon may occur in

astrophysical sites, but there are signicant energy losses to take into account [2]. Besides, there exist

other more complex acceleration mechanisms that may take place in a large number of astronomical

sources. These sources can accelerate particles up to a certain maximum energy according to their

BR product; however, it seems extremely dicult to nd cosmological objects able to hit 1020 eV even

disregarding the GZK feature.

The top-down models look at the ultra high energy question through a dierent angle. These models

are based on super massive particles (of masses up to ∼ 1025 eV [6]) presumably originated in the early

universe and that managed to survive until now. The decay of such particles would produce several mesons

and hadrons; charged mesons yield neutrinos (π+(π−) → l(l)νl(νl)) while neutral mesons yield photons

(π0 → γγ). Therefore, top-down models usually foresee large uxes of ultra high energy neutrinos and

photons rather than protons and nuclei. This feature is quite dierent from the previsions of bottom-up

scenarios and it is an useful feature to test dierent models.

7

2.3 Relevance

The research in the UHECR eld is relevant to several areas of modern physics, namely particle physics,

astrophysics, cosmology and fundamental physics.

In the particle physics eld, the study of UHECR makes possible the study of reactions at high√s.

Indeed, considering in the laboratory reference system a particle of mass m1 and energy E1 hitting a

target particle of mass m2 at rest, s = (p1 + p2)µ (p1 + p2)µ = m21 + m2

2 + 2m2E1. If the two particles

are protons and the moving one has an ultra high energy of E1 ∼ 1019 eV, then√s ∼ 137 TeV, while in

the Large Hadron Collider (LHC) the same collision will be possible up to√s ∼ 14 TeV or, equivalently,

E1 ∼ 1017 eV. Of course, the main setback in cosmic ray physics is that the uxes and energies observed

cannot be controlled as in a particle accelerator.

As for astrophysics, since UHECRs point back to their origins, important sources can be further

studied and acceleration mechanisms understood. New astronomy channels may be opened soon and

there is also the possibility of performing cosmology studies as UHECR are potential messengers of

earlier stages of the universe.

Finally, a wide variety of fundamental physics models, such as SUSY, extradimensions and Lorentz

violation, may be tested using UHECR data.

8

Chapter 3

UHECR detection

The detection of cosmic rays with energies below 1014 eV may be direct [2] since the ux is high at that

energy range. Indeed, as pointed out in section 2.1, cosmic rays of 1011 eV arrive at a rate of approximately

1 m−2s−1, which means that small detectors are enough to gather signicant statistics. However, at low

energy, the atmosphere makes ground-based experiments inadequate and therefore balloon or satellite-like

detectors are required. A modern example of this kind of detectors is the Alpha Magnetic Spectrometer

2 (AMS2), which is planned to be installed in the International Space Station by 2009 and that will be

able to measure the energy and the direction of charged cosmic rays and photons.

At energies above 1014 eV there is a fortunate combination of two factors that calls for an indirect

detection. On the one hand, the low ux (roughly 10 m−2day−1 for E > 1014 eV [2]) demands for great

exposure areas that cannot be easily launched above the top of the atmosphere. On the other hand,

cosmic rays of this energy range hit the atmosphere and originate the so-called extensive air showers

(EAS). These showers may be detected by ground-based experiments whose main challenges are the

reconstruction of energy, nature and direction of the primary particle, the original cosmic ray.

3.1 Extensive air showers

High energy cosmic rays (E & 1014 eV) interact with atmospheric nuclei typically on the top of the

atmosphere and originate large cascades of dierent kinds of particles − the extensive air showers. In this

process the atmosphere works as a calorimeter: it gradually absorbs the energy of the created particles.

But it is much more complicated than a lead calorimeter, because of the quite variable density and loads

of meteorological phenomena to take into consideration. Note as well that some kinds of cosmic rays,

like neutrinos, have such a small interaction cross section that may not interact in the atmosphere and

go undetected.

The dynamical features of an EAS are best parameterised by the slant depth X measured in gcm−2:

X(~r0, ~r) =∫ ~r

~r0

ρ (~r′) d~r′ (3.1)

where ρ is the density of the atmosphere and ~r0 is the rst contact point of the cosmic ray with the

atmosphere. Usually X is computed as a function of the altitude h of the point ~r and the zenith angle θ

of the trajectory using the approximation

X '∫∞hρ(h)dhcosθ

≡ Xv

cosθ(3.2)

which is valid for almost vertical showers. The slant depth is more convenient to use than a simple

distance because of the variable atmospheric density. As a reference, a vertical shower crosses a slant9

depth of approximately 1000 gcm−2 until it reaches sea level, while a skimming one traverses up to 36000

gcm−2.

3.1.1 Electromagnetic showers

Electromagnetic showers are the ones originated by photons, electrons or positrons and are fed mainly

by two electromagnetic processes: bremsstrahlung (e± → e±γ), where the electron (or positron) is

deaccelerated in the presence of electromagnetic elds and radiates, and pair production (γ → e+e−),

where a photon of energy higher than 2me ' 1022 KeV creates a pair electron-positron in the presence

of matter (the atmospheric nuclei) for momentum conservation. These processes create a chain reaction

that evolves until the electrons and positrons start to lose their energy essentially by collisions rather

than by bremsstrahlung radiation. From that moment on, the creation of new generations of particles

stops and the cascade declines.

The development of electromagnetic showers is fairly described by Heitler's model, illustrated in gure

3.1. This very simple but meaningful model denes several splitting levels separated by a length d; at

each level the reactions γ → e+e− and e± → e±γ are assumed to occur with equal distribution of energy

and in an elastic manner [8]. The length d is given by d = Xlln2 (Xl ' 37 gcm−2 being the radiation

length of the electron in the air) since that is precisely the length after which the electron has lost half

of its initial energy − in fact, Ee(X) = Ee(X0)e−(X−X0)/Xl . Therefore, according to the model, at a

certain slant depth

Xj = jXlln2 (3.3)

there are N = 2j particles in the cascade (electrons, positrons and photons) all of which with an energy

E0/2j , where E0 is the energy of the primary particle. This behaviour ends when the shower hits its

maximum or, in other words, when E0/2j falls below ε0, the critical energy 1 . In this way, one may

calculate the primary energy if one knows the number of particles Nmax at shower maximum Xmax:

E0 = ε0Nmax (3.4)

Moreover, (3.3) and (3.4) together yield

Xmax = jmaxXlln2 = log2NmaxXlln2 = ln

(E0

ε0

)Xl (3.5)

Equations (3.4) and (3.5) contain the two most important previsions of Heitler's model: (a) the

number of particles at shower maximum is proportional to the primary energy and (b) the slant depth of

shower maximum presents a logarithmic increase with the primary energy. Nevertheless, the model tends

to overestimate the number of electrons in the shower [8]. For further information in electromagnetic

shower development detailed simulation is needed.

3.1.2 Hadronic showers

Hadronic showers are originated by the interaction of a primary nucleon or nucleus with atmospheric

nuclei. Part of the primary energy is converted by strong interaction in secondary mesons [1] − almost

entirely charged and neutral pions and some kaons as well. The remaining energy goes to secondary

nucleons that will initiate other showers. The pions produced during the cascade may either decay or

interact depending on their energy. The decay length is given by Ldecay = cγτ = Eτmc (in m), while the

1Energy at which the electron energy loss by bremsstrahlung and by ionization are equal. Following [12, 13],the electron critical energy in air is ε0 ' 81 MeV.

10

(a) (b)

Figure 3.1: The Heitler's model in electromagnetic cascades (a) and in hadronic showers (b) [8].

interaction length for pions in air is Xπint ' 120 gcm−2 for Eπ < 1014 eV [1, 8]. The determination of

the dominant process for a specic energy is performed comparing Xπint to Ldecay, both in gcm−2 units.

Anyway, due to their very short mean life, π0 essentially decays: π0 → γγ. The resulting photons start

sequential electromagnetic subshowers, feeding the electromagnetic component of the cascade. As for

charged pions, their mean life is suciently large to allow that some, low energy ones, decay and the

others, high energy ones, interact. Decaying charged pions originate mainly muons and muon neutrinos

[9] (π+(π−) → µ+(µ−)νµ(νµ)) giving rise to the muonic component. Finally, the charged pions that

interact with atmospheric nuclei originate 13 π0 and 2

3 π± [1], continuing the hadronic component. As

the shower evolves, the energy of charged pions decreases and more and more of them decay instead of

interacting. That is why in a hadronic shower there are more electrons, positrons, photons, neutrinos and

muons than hadrons. Specically, for each hadron in the cascade there are in mean terms ∼ 102 muons

and neutrinos and ∼ 104 electrons, positrons and photons [10].

As in the electromagnetic case, the hadronic shower hits a maximum, characterised by Xmax and

Nmax, when the charged pions energy falls below the energy at which decay and interaction occur with

equal probability. Although these showers are complex, a modied Heitler's model may be applied using

Xπint instead of Xl [8]. Again, shower simulation codes, such as CORSIKA (COsmic Ray SImulations for

KAscade), are needed to study hadronic shower development with more detail.

The distinction between proton (hadronic) and photon (electromagnetic) initiated showers of the same

primary energy is achieved considering that proton cascades present Xmax lower than photon ones.

3.2 Indirect detection

Indirect detection of UHECR is possible through the measurement of EAS features. Two dierent kinds

of detection techniques are usually implemented: ground arrays, that sample the transverse prole of the

cascade at the surface level, and light detectors, that record the light emitted during the development of

the EAS.

3.2.1 Ground arrays

Ground arrays are sets of individual particle detectors scattered around Earth surface and that can

cover considerably large eective areas. The detectors may be electromagnetic (sensitive to photons and

electrons), muonic or hadronic − usually scintillators and/or water erenkov tanks are used [11]. This

11

technique presents a threshold energy of approximately 1014 eV since only high energy cosmic rays are

able to originate air showers large enough to detect at the ground. Moreover, the target energy range

of a ground array experiment determines the optimal altitude of the site: the shower maximum should

occur just above the array in order to measure greater quantity of particles with less uctuations [1].

Therefore, EAS of primary energies of 1014−1015 eV are best recorded at high altitudes, while ultra high

energy cascades should be detected at lower altitudes − otherwise, the shower maximum is missed and

the reconstruction becomes dicult.

As an EAS crosses the experiment site, each detector measures a particle density as exemplied in

gure 3.2; to convert this information into the transverse prole of the cascade its direction is needed. At

rst approximation, the air shower is a at disk propagating at the speed of light as a plane front and,

thus, the hit time and the position of the individual detectors are used to calculate this direction. In

practice, however, one must take into account a more realistic propagation shape, which is curve rather

than at. Given the direction, the measured densities may be tted to a transverse prole − whose

specic parameterisation is highly inuenced by the experiment conguration − and the shower core

is determined. Afterwards, the detectors that deviate from the tted prole may be excluded and the

process of determination of shower direction and shower core is repeated iteratively until it converges.

Figure 3.2: A ctious event as recorded by a ground array [1].

In what refers to primary energy reconstruction, one usually turns to shower simulation that demon-

strates that the particle density at a certain distance from the shower axis is roughly proportional to the

primary energy and that this behaviour is almost independent from the primary mass number. So, the

measured particle density at 500 − 1000 m from the axis yields the primary energy estimate through a

constant of proportionality given by Monte Carlo calculations.

Finally, the nature of cosmic rays is not straightforward to determine, although a hint on the primary's

mass number is given by the relative abundance of the muon component on the EAS: the bigger this

relative abundance, the heavier is the primary.

Some setbacks are inherent to the ground array detection technique. For instance, it is not possible to

perform direct studies on the longitudinal development of the shower. Besides, the energy reconstruction

is based on Monte Carlo simulations, which assume that the primary nature is known and use extrapolated

cross sections from low energy calorimeter measurements. Another disadvantage is the non-reliability in

the determination of the cosmic ray nature. However, ground arrays are not strongly inuenced by

meteorological, atmospheric nor luminosity conditions and, thus, can make measurements continuously

12

presenting a duty cycle of virtually 100%. Adequate rst estimates on primary energy and on shower

direction are also advantages of the ground arrays [2].

One of the most signicant ground array experiments was the Akeno Giant Air Shower Array (AGASA),

situated in Japan. The site covered a total area of approximately 100 km2 with 111 surface detectors

of 2.2 m2 each separated by ∼ 1 km and 27 underground muon detectors [14]. AGASA studied most

features of UHECR, in particular the far end of the cosmic ray spectrum.

3.2.2 Light detectors

The development of air showers through the atmosphere gives rise to the emission of two kinds of elec-

tromagnetic radiation that may be collected by optical devices on the ground: erenkov and uorescence

light. The production and attenuation of these types of light are controlled by the atmosphere, which is

an integral part of ground-based light detectors. Therefore, in the next paragraphs relevant atmospheric

properties are detailed to some extent, before describing erenkov and uorescence detection techniques.

Light production and attenuation in the atmosphere

As stressed above, an extensive air shower crossing the atmosphere emits both uorescence and erenkov

light. The uorescence depends only slightly on atmospheric characteristics, but most shower parame-

ters are functions of the atmospheric temperature, pressure and/or density. These variables have well-

understood behaviours with height and are adequately described by models such as the US standard

atmosphere.

In this section, the production of erenkov and uorescence light is briey described as well as the

most inuent scattering processes, Rayleigh and Mie, that are responsible for both the existence of

scattered erenkov light and the attenuation of electromagnetic radiation from its emission point until

the detector.

erenkov light erenkov radiation is emitted by charged particles while travelling in a medium of

refraction index n at a speed v greater than the phase velocity of light in the medium, cn . It is emitted on

a cone along the propagation direction with maximum opening angle θ = arcos(c/nv

). In an extensive

air shower most charged secondary particles travel almost at the speed of light in vacuum c and so, as

nair ' 1.0003 [1], θ ' 1.4. Although the mentioned value for nair corresponds to the refraction index of

the air at sea level, it is useful in getting a rst approximation on the erenkov opening angle. However,

due to the charged particles transverse momentum, erenkov light is seen at the ground with an angle

up to ∼ 20 with respect to shower axis in the case of UHE protons [12].

Atmospheric nitrogen uorescence Another remarkable process occurs as an EAS develops in the

atmosphere: the shower charged particles excite N2 molecules and N+2 ions that afterwards return to

their ground states by isotropic emission of electromagnetic radiation in the ultraviolet and visible bands.

This phenomenon is called scintillation [21], but the above described radiation is known in the cosmic ray

eld as uorescence light even though uorescence is the process in which an atom or molecule absorbs

photon(s) of certain wavelength λ and re-emits photon(s) of λ′ > λ. When the charged particles of an air

shower excite the atmospheric nitrogen (N2 and N+2 ), two competing phenomena may occur: either the

excited molecules collide with other atmospheric components such as the oxygen or they emit uorescence

light, in the 300-400 nm band. In this wavelength range, N2 and N+2 are indeed the main contributors to

13

uorescence production and their spectrum, shown in gure 3.3, presents three strong spectral lines: at

337 nm, 357 nm and 391 nm [12].

Figure 3.3: The uorescence spectrum of the atmospheric nitrogen [12].

But the nitrogen uorescence spectrum is not enough to reconstruct the shower longitudinal prole:

one needs to relate the photons produced with the charged particles in cascade development. This relation

is accomplished by the uorescence yield yfγ that is approximately 4 γ/e/m in the 300-400 nm band

[12, 20]. It has a mild dependence on atmospheric density and temperature and it may be parameterised

through the energy deposited by the shower along its longitudinal development.

Rayleigh scattering When an electromagnetic beam crosses a gas, it is scattered o by the gas'

molecules and/or atoms − if the size of these scattering centres is small in comparison with the beam

wavelength λ, then Rayleigh scattering takes place presenting a cross section proportional to λ−4. As-

suming dNγ photons crossing the atmosphere along the path dl, their extinction rate due to Rayleigh

scattering is given by [16]:dNγdl

= −ρNγXR

(400nmλ

)4

(3.6)

being XR = 2974 gcm−2 the characteristic Rayleigh pathlength [18]. The radiation is preferentially

deected in the forward and backward directions according to the angular distribution [17]:

d2NγdldΩ

=3

16π(1 + cos2θ

) ∣∣∣∣dNγdl∣∣∣∣ (3.7)

where θ is the angle between the original beam direction and the scattered one.

Since dX = ρdl (confer (3.1)), the integration of (3.6) results in Nγ (X2) = Nγ (X1) e−|X1−X2|XR

( 400nmλ )4

,

where 1 and 2 refer to the initial and nal points of the trajectory, respectively. Or in terms of a

transmission coecient:

TR = e− |X1−X2|

XR( 400nm

λ )4

(3.8)

Mie scattering The Mie scattering occurs when the scattering centres have sizes similar to the wave-

length of the radiation and this is, for instance, the case of UV light crossing the aerosol content of the

atmosphere. Aerosols are natural and human-made particles with various sizes and dierent vertical den-

sity proles, namely low altitude clouds, pollutants and dust. The Mie cross section is not as dependent

14

on the wavelength λ as the Rayleigh one is and its extinction rate is [16]

dNγdl' −Nγe

− hhM

lM (λ)(3.9)

where h is the height and hM and lM are characteristic distances of aerosol distributions. The angular

dependence, which is strongly peaked forward, is given by [16]:

d2NγdldΩ

' aM · e−θθM

∣∣∣∣dNγdl∣∣∣∣ (3.10)

with θM = 26.7. The parameter aM is forced to 0.891 so that∫ 2π

0

∫ π0aM · e−

θθM sinθdθdφ = 1.

The transmission coecient is obtained integrating (3.9):

TM = e− 1lM (λ)

∣∣∣∣∫ 21 e− hhM dl

∣∣∣∣ (3.11)

Following [16], hM ' 1.2 km and lM (λ = 400 nm) ' 14 km. With the approximation dl ' dh/cosθ′ (θ′

being the angle between the beam trajectory and the vertical), (3.11) yields:

TM = e− hMlM (λ)cosθ′

∣∣∣∣∣e− h1hM −e

− h2hM

∣∣∣∣∣ (3.12)

The Mie attenuation is specially important in polluted sites and within the rst ∼4 km of atmosphere.

Besides, the values given in this section for aM , θM , hM and lM must be understood as mean ones,

because the aerosol content of the atmosphere is highly variable in space and time. So, a large cosmic

ray experiment must have several regular atmospheric surveys in order to update such parameters in a

daily basis.

erenkov technique

Optical detectors may collect erenkov radiation directly if the shower approximately points at them

− otherwise, only Rayleigh and Mie scattered light can be observed. The main advantage of measuring

direct erenkov light is the substantial photon density that is greater than the electron density at the

ground and proportional to the number of charged particles created during shower development. Besides,

the analysis of the lateral distribution of this kind of radiation may be used in the estimation of the

shower maximum and primary energy.

A modern example of an imaging atmospheric erenkov telescope (IACT) is the Major Atmospheric

Gamma-ray Imaging erenkov (MAGIC) device, settled in the island of La Palma, Canarias, Spain and

taking data since late 2004 [19]. The rst telescope includes a 236 m2 mirror with 17 m of diameter and a

camera of 576 photomultiplier tubes (PMT) in order to image the erenkov light emitted during shower

development. A second telescope is now under construction. Rather than focusing on UHECR, MAGIC

is designed to study very high energy (VHE) gamma rays (1011 eV . E . 1014 eV ) that, being neutral,

suer no magnetic deection and enable the study of astrophysical sources.

Fluorescence technique

Since uorescence light is isotropic, it can be detected far away from the emission point [1] as long as

it is abundant enough − the threshold energy for uorescence detection is situated around 1017 eV. The

detectors usually consist of large mirrors that concentrate the collected light in a PMT camera whose

task is to divide the image into pixels, as shown in gure 3.4, and record the photon densities and arrival

times of each pixel. Therefore, this technique allows the imaging of the shower longitudinal prole.15

Figure 3.4: An air shower as seen by a uorescence detector [1].

Before, however, one has to proceed to the geometry reconstruction. Firstly, the positions of the

activated pixels are used to nd the shower-detector plane (SDP), that contains both the detector and

the shower axis. Then, if there is only one uorescence detector, the exact geometry of the axis within

the SDP is tted with the help of the pixels' arrival times. In the case of stereoscopic observations, i.e.

more than one uorescence detector triggered, the existing SDPs may be intersected to determine the

shower axis with better precision. Another way to improve the timing t necessary to single uorescence

detectors is to use the information given by a ground array that has seen the event.

Once the event geometry is xed, several factors must be taken into account to derive the shower

longitudinal prole from the raw signal received by the detector. To begin with, direct and scattered

(Rayleigh and Mie) erenkov light is calculated using the geometry and subtracted from the original

signal in order to obtain the uorescence fraction. This fraction must be corrected according to the solid

angle seen by the device because uorescence light is isotropic. Then, Rayleigh and Mie attenuation

undergone by photons while propagating from the emission point to the detector is considered − to

minimise this attenuation uorescence-based experiments are usually settled in non-polluted, dry places.

Besides, regular monitoring of atmospheric conditions (aerosol and cloud measurements in special) is

essential to compute attenuation in a precise manner. Lastly, using the uorescence yield, the number

of electrons in the shower as a function of the slant depth, Ne(X), this is the longitudinal prole, is

reconstructed, allowing the determination of Xmax and Nmax.

In what regards energy reconstruction, the total energy from the cascade electromagnetic component

is proportional to∫∞

0Ne(X)dX, being the constant of proportionality given by the electron mean loss.

But to nd the primary energy one has to estimate the so-called missing energy − the energy taken by

hadrons, muons and neutrinos, that are invisible to uorescence techniques. Although the calculation of

the missing energy is model-dependent, Monte Carlo simulations show that it represents less than 10%

at primary energies greater than 1018 eV. Figure 3.5 sketches the evolution of this undetected energy

both with primary nature and primary energy. The heavier the primary, the more important is the

missing energy. Indeed, at the same primary energy heavier nuclei present smaller energy per nucleon

and consequently the charged pions and other mesons produced are more likely to decay than interact

[1], thus feeding the muonic component which is almost invisible to uorescence detectors. Moreover,

as primary energy increases, the missing energy declines because the interaction of secondary mesons is

more probable and that feeds the electromagnetic, detectable component of the cascade. To sum up, at

16

ultra high energies the electromagnetic energy is a good estimate on the primary energy (within < 10%)

and that is why uorescence detectors are highly ecient for UHECR showers.

Figure 3.5: Percentage of missing energy for dierent energies and primaries as calculated in Monte Carlosimulations. Open circles represent proton showers, open squares He nuclei, lled circles CNO and lledsquares Fe [1].

Besides the high eciency just mentioned, the main advantage of uorescence techniques is the ability

to study the longitudinal prole which allows the calculation of Xmax. Nevertheless, this kind of detection

is extremely sensitive to atmospheric and meteorological conditions and may only be used in moonless

nights − about 10% of duty cycle. Moreover, one relies on simulations to estimate the missing energy

even though this may be much greater in showers initiated by exotic particles. Finally, as in the ground

arrays case, it is not possible to identify the primary nature on a event-by-event basis [2] since several

features of a shower, such as Xmax, present important statistical uctuations. Therefore, only a 'mean'

primary nature may be determined.

A successful example of a uorescence-based experiment was the Fly's Eye (then, High Resolution

Fly's Eye, HiRes), settled in the USA and operated from 1981 until 1993. Originally, it consisted of one

uorescence eye with 67 spherical mirrors of ∼ 1.6 m diameter corresponding to 880 PMTs with a 25 ns

time resolution. Each PMT covered a 5x5 eld of view in hexagonal-shaped pixels in order to maximise

light collection and achieve full coverage of the night sky [21]. In 1986 a second eye, with 36 mirrors

and situated 3.4 km away from the rst one, was installed allowing stereoscopic observations. Fly's Eye

is responsible for the knowledge of several features of UHE physics and for the detection of the most

energetic phenomenon ever recorded − 3.2 · 1020 eV.

3.3 Present status

There are currently three main topics in ultra high energy cosmic ray physics that still remain unanswered:

the energy spectrum, the composition and the arrival directions. This section will briey describe the

present status of observations at this energy range gathering data from dierent experiments, including

the Pierre Auger Observatory. The PAO comprises both a Surface Detector (SD) and a Fluorescence

Detector (FD), allowing the use of a hybrid technique that combines the two detection types described

in 3.2.

Over the last years, the energy spectrum at the highest energies has been thoroughly studied by

two major experiments, AGASA and HiRes. These groups present conicting results. For instance,

while HiRes sees the ankle at ∼ 3 · 1018 eV [21], AGASA seems to locate this structure at about 1019

17

Figure 3.6: The energy spectrum of cosmic rays with the conicting results from AGASA and HiRes inthe UHE range. The actual ux is multiplied by E2.5 to ease the visualisation of spectrum features [22].

(a) The ratio between the observed ux using HiResdata and the expected one if there were no GZK cuto[23]. There exists a clear break at about 1019.8 eV.

(b) The energy spectrum as measured by the PAOwith data prior to 2007 [30]. The three spectra pre-sented correspond to dierent sets of data: SD events,hybrid events and inclined showers (of zenith anglesgreater than 60).

Figure 3.7: The energy spectrum as measured by HiRes (a) and the PAO (b).

eV [1]. Figure 3.6 shows the cosmic ray spectrum as measured by several experiments. The actual

quantity represented is the ux times E2.5 to better distinguish spectrum features: little errors in energy

determination correspond to large errors in the yy axis. In the rst place, it is clear that AGASA predicts

a greater ux than HiRes in the whole UHE range. The dierence corresponds to 30− 40% of systematic

error in energy reconstruction [1, 3] which may be corrected if one assumes that AGASA overestimates

the event energy by ∼ 20% and HiRes underestimates it by ∼ 20%. These values are in agreement

with the expected systematic uncertainties for the two experiments [3]. Anyhow, AGASA results seem

to deny the existence of a GZK feature, whereas HiRes data is consistent with a GZK cuto at 1019.8

eV as represented in gure 3.7(a). The lastest Pierre Auger Observatory spectrum dates from 2007

[27, 28, 29, 30] and is shown gure 3.7(b). It seems to indicate a cuto above 1019.6 eV [30], altough there

are still signicant statistical and systematic errors. In what refers to events with energies above 1020 eV,

the AGASA group claims 11 events, but HiRes in single eye mode, with great exposure, detected only

one. As of 2007, the maximum primary energy recorded at the PAO was 1020.25 eV [30].

From this analysis it is obvious that the UHE spectrum may be further studied only with improved

statistics and a better control of systematic errors specially in energy reconstruction. The PAO is believed

to solve the statistics problem with the expected 60 events per year with energies above 1020 eV [3];18

besides, it has the necessary equipment to perform detailed monitoring of systematics.

The cosmic ray composition at the highest energies is still an open topic as well. In the knee region,

between 5 · 1015 eV and 1016 eV, several experimental groups conrm a shift to a heavier composition

[1]. On the contrary, the ankle signals the transition to a lighter cosmic ray trend, as pointed out in

section 2.1. On the one hand, uorescence-based experiments, such as HiRes, may study the composition

through the evolution ofXmax with the primary energy E0. Lighter cosmic rays are more likely to interact

deeper in the atmosphere (presenting a greater Xmax) than heavier ones. However, hadronic interaction

models predict a slope dXmaxdlog10E0

, the so-called elongation rate, almost independent from primary nature:

50-60 gcm−2decade−1 [1, 21]. As the elongation rate measured by HiRes is somewhat larger than these

values, one concludes that the composition at the ankle is becoming lighter and that UHECR are more

consistent with protons than with iron nuclei − see gure 3.8. On the other hand, ground arrays may

infer the composition through the relation between muonic and electromagnetic components: heavier

primaries initiate showers with more muons than those present in cascades originated by lighter cosmic

rays. AGASA does not conrm nor contradict a transition to a lighter component above the ankle [1].

Figure 3.8: The transition to a light composition trend that occurs in the ankle region [21]. The simulationresults from proton and iron showers are indicated by the upper and lower parallel lines respectively, whilethe full circles represent the experimental data.

Another important composition study is that of knowing the fraction of γ-initiated showers at ultra

high energies. Indeed, as stressed in section 2.2.2, such a study tests top-down models, since these

preview a signicant ux of ultra high energy photons. Two ground array experiments, Haverah Park

and AGASA, estimated an upper limit to the fraction of γ-initiated showers above 1019 eV obtaining 48%

and 28%, respectively [32]. As of 2007, the Pierre Auger Observatory data points at 13% [34]; this result

is determined by comparison between the mean values of some ground parameters observed in hybrid

data and the corresponding values from simulated photon showers. Using observed and simulated Xmax

also in hybrid data, the limit is set on 16% [35].

The distribution of arrival directions is the third hot topic in UHECR observations. As explained

in chapter 2, UHE charged particles have great magnetic rigidity and are deected up to a few degrees

from their initial directions, usually less than the angular resolution of the experiments designed to this

19

energy range [3]. Therefore, the distribution of arrival directions is meaningful in the study of cosmic

ray sources. As in the spectrum case, AGASA and HiRes present contradictory results. While HiRes

is compatible with an isotropic distribution, AGASA recorded 47 events with energy above 4 · 1019 eV

of which three doublets and one triplet [1]. Each set of events dier less than 2.5 in arrival direction.

Again, the statistics is still very low, but AGASA indicates a certain degree of anisotropy even if the

clusters directions do not seem to point at signicant astronomical sites. The rst PAO indications on

this subject [36, 37], dated from 2005, are consistent with isotropy. New results are expected within one

year.

20

Chapter 4

The Pierre Auger Observatory

The Pierre Auger Observatory 1 was proposed to the scientic community in 1992 by Alan Watson and

James Cronin in order to study the ultra high energy range with unprecedent statistics − about 60 events

per year with energies above 1020 eV [3]. The Observatory consists of two dierent sites: one in each

hemisphere, both situated at mid-latitudes. This geographical choice allows an uniform, full sky coverage

that is essential to anisotropy studies [11]. The northern site will be located in Lamar, Colorado, USA,

but still undergoes an early phase prior to construction. The southern site is situated on the Pampa

Amarilla near the Andes − Malargüe, Province of Mendonza, Argentina − at about 1400 m above sea

level. It is now in an advanced stage of construction and, when fully operational (by the end of 2007), it

will enclose a total area of 3000 km2. The whole present work refers uniquely to the southern site.

The innovative character of the PAO steams from the use of a hybrid technique. Indeed, the southern

site, represented in gure 4.1, comprises a ground array of 1600 water erenkov tanks (the so-called

Surface Detector, SD) and a set of 24 telescopes placed into four uorescence eyes (the Fluorescence

Detector, FD), combining the two most successful UHECR detection techniques, described in section 3.2.

Therefore, the PAO is able to study longitudinal and lateral proles of cosmic ray showers presenting the

advantages of both detection types. In addition, an improved geometrical reconstruction is achieved for

hybrid events − those simultaneously seen by the SD and the FD. Moreover, these events may be used to

perform a calibration in energy between both techniques. Such a calibration frees the PAO from Monte

Carlo simulations (vastly used in ground array experiments) and may explain why the results obtained

so far with ground arrays and uorescence detectors, such as AGASA and HiRes respectively, do not

coincide. However, the hybrid detection requires the synchronisation between the Surface Detector and

the Fluorescence Detector, which is a sensitive task.

Finally, the control of systematics, as pointed out in 3.3, plays a decisive role in cosmic ray data

analysis. The Pierre Auger Observatory is, thus, equipped with several complementary systems to monitor

the most signicant atmospheric eects and estimate accurately the systematic uncertainties, unlike most

earlier experiments.1Pierre Auger was the French scientist responsible for the concept of ground arrays: he detected particles

arriving at the ground in coincidence and separated by several meters. In 1939, Auger also proved the existenceof 1015 eV cosmic rays.

21

Figure 4.1: The Pierre Auger Observatory southern site in Argentina [24]. The dots represent SD tanks,while the lines show the eld of view of the FD telescopes. As of 31st March 2007, 1215 tanks and all 24telescopes were fully installed and operational [39].

4.1 Southern site description

4.1.1 Surface Detector

The Surface Detector is an 1600 water erenkov tanks array in an 1.5 km spacing triangular grid covering

an area of roughly 3000 km2. The grid spacing was chosen to achieve full eciency (ve triggered tanks)

at 1019 eV, even though 3 · 1018 is enough to achieve it for non-horizontal showers [24, 11]. As of 31st

March 2007, 1297 tanks were positioned in the eld, 1272 lled with water and 1215 with electronics

installed [39].

Figure 4.2: A water erenkov tank of the Surface Detector [24].

Each water erenkov tank − see gure 4.2 − is a 10 m2 base, 1.5 m tall plastic cylinder lled with 12

m3 of puried water up to an 1.2 m height [24, 41]. The inner surface of the tank presents high reectivity

to maximise the collection of erenkov light, which is done by three 9 inch PMTs located on the top. The

electronics apparatus is powered by two 12 V batteries charged by two solar panels. The connection to

the Central Data Acquisition System (CDAS) is set up through a communication antenna and the timing22

information is obtained from a GPS unit. The whole tank was designed to be highly robust in order to

face the extreme environment of the Pampa: dust, sand, rain, hail, snow, salinity, humidity, wind and

day-night temperature variations of 20C.

When an EAS crosses the SD, charged shower particles travel in the water of the tanks at a speed

greater than the phase velocity of light in the medium, thus emitting erenkov light. Photons are

not charged but, when crossing a depth in water of ρwater∆l ' 1 gcm−3 · 1.2 m = 120 gcm−2 which

corresponds to ∼ 3.2 radiation lengths (see section 3.1.1), almost all convert into relativistic electron-

positron pairs that radiate erenkov light. This light is collected by the three PMTs and the signal

is analysed in terms of vertical equivalent muons (VEM), i.e. the average charge deposited by a high

energy vertical down-going muon. In this way, only the neutrino part (and eventually an unknown

neutral, weakly interacting fraction) of the shower goes undetected. Besides, the discrimination between

muons and electrons or photons is possible: muons are less scattered in the atmosphere producing earlier,

quicker and higher amplitude signals when compared to those of electrons or photons [1]. Finally, to

select potentially physical events several triggers in time, space and quality are applied (see, for instance,

[11]) and, afterwards, interesting events are sent to the CDAS.

4.1.2 Fluorescence Detector

The Fluorescence Detector comprises four sites located on the top of small hills in the array's boundaries:

Los Leones, Los Morados, Loma Amarilla and Coihueco. Each site houses a six telescopes uorescence eye

overviewing the array 180 in azimuth and 28.6 in elevation [42]. The communication with the CDAS

and the SD is established through the antenna tower built on each uorescence site. Data taking began

at Los Leones and Coihueco eyes in January 2004 and at Los Morados in March 2005, while the Loma

Amarilla eye was only nished by February 2007 [42, 38]. Besides the optical telescopes, the atmosphere

is an integral part of the Fluorescence Detector as well. Since it was characterised in 3.2.2, this section

will only present a detailed description of this telescopes.

The optical devices used in the FD, protected from rain and wind by individual shutters, are Schmidt

telescopes with a eld of view of 30 in azimuth and 28.6 in elevation [24]. As exemplied in gure 4.3,

each telescope consists of several separate parts.

First of all, an 80 cm x 40 cm ultraviolet lter is installed after the shutter in order to transmit in the

280-430 nm band, thus blocking visible light. The lter reduces the night background light and ensures

an acceptable signal-to-noise ratio. Moreover, it acts as a window protecting the devices from dust and

rain.

Then, the ultraviolet radiation crosses a circular diaphragm with a radius of 0.85 m that is surrounded

by a corrector ring with outer radius 1.10 m. The corrector ring, composed of 24 ultraviolet transmitting

glass segments, almost doubles the eective area and is constructed in such a manner that it allows a

good quality of the optics [43, 44].

Between the diaphragm and the mirror there is the 440 PMT camera (described below) that allows

the imaging of the shower prole. The light is collected by a 3.5 m x 3.5 m spherical mirror with 3.4

m of radius [43, 42] presenting a mean reectivity around 90% in the 300-400 nm band [44]. The large

dimensions of the mirror require that it consists of several segments − hexagonal and square-shaped

segments were chosen in order to maximise light collection and sky coverage.

The PMT signals are sampled in 100 ns time bins and are then subjected to two basic levels of trigger −see, for instance, [44] − that rule out random coincidences and select potentially physical events. Besides,

23

(a) Schematic representation [43]. (b) Detailed view of the PMT cameraon the right and the mirror with square-shaped segments on the left [24].

Figure 4.3: The Schmidt telescope used in the FD.

the telescopes must undergo several calibration processes so that the energy reconstruction accuracy is

better than 15% [42] − this requirement is of extreme importance in the study of the far end of the

cosmic ray spectrum as discussed in section 3.3.

PMT camera

The PMT camera lies on the spherical focal surface of the telescope and consists of a 6 cm x 94 cm x 86 cm

aluminum body supported by two legs and housing 22 rows x 20 columns of hexagonal PMTs [43]. Each

PMT corresponds to a pixel so that the camera presents a total of 22x20=440 pixels. Light collection is

maximised surrounding each PMT with six mercedes stars: these are inclined reecting surfaces designed

to direct ∼ 90% of the incident light into the centre of the PMT, thus smoothing the transition between

pixels.

As the camera lies on a spherical surface, the pixels are not regular hexagons in spherical coordinates

but have variable size in order to best cover the focal surface [44]. So, to ease the data analysis it is

convenient to dene a coordinate system in which the camera is rectangular and the pixels regular. The

new coordinates (β, α), proposed in [45], are represented in gure 4.4 and their relation to spherical

coordinates (θ, φ) is given by the equations:

β = arcsin (sin (φt − φ) sinθ) (4.1)

α = αc − αm + arcsin

(cosθ

cosβ

)(4.2)

where φt and αm = 16 are respectively the azimuth and the elevation angles of the telescope axis and

αc =√

3

2 is the oset angle between the camera centre and the telescope axis [46].

In these coordinates, each pixel of the camera is a regular hexagon of radius rpix =√

3

2 , side length

lpix =√

3

2 and side-to-side dimension dpix = 1.5 as shown in gure 4.5. The mercedes structures form

a similar inner hexagon scaled by 0.8. For each telescope, the camera is a 22 rows x 20 columns grid of

24

Figure 4.4: The (β, α) coordinates and their relation to spherical coordinates (θ, φ). In this particularcase, φt = 90 (adapted from [45]).

pixels whose central points have coordinates

βij =

1.5 · (10− i) if j odd

1.5 · (10− i) + 0.75 if j even(4.3)

αij = 1.5√

3

2· (j − 11) (4.4)

being i ∈ [1, 20] the column number and j ∈ [1, 22] the row number. The whole camera is represented in

gure 4.6 and, as αc =√

3

2 = rpix, the telescope axis centre coincides with the centre of a mercedes star.

Moreover, the camera limits in (β, α) are given by:

βmin = 1.5 · (10− 20)− dpix2

= −15.75 βmax = 1.5 · (10− 1) + 0.75 +dpix

2= 15

αmin = 1.5√

3

2· (1− 11)− rpix ' −13.86 αmax = 1.5

√3

2· (22− 11) + rpix ' 15.16

The eld of view of the camera is then 30.75 in β and ∼ 29.01 in α.

For practical reasons, each pixel must be labeled by a number Npix ranging from 1 to 440. Usually

the relation Npix = 22 · (i− 1) + j is used to dene the pixel number given the respective column i and

row j. Inversely, there are also unambiguous relations that yield the column and row given the pixel

number Npix: i = int[Npix−1

22

]+ 1 and j = mod [Npix − 1, 22] + 1.

Having the detailed camera description presented in the above paragraphs, one may convert the (θ, φ)

direction of an incident photon into the number of the pixel it hit and, additionally, verify whether the

photon undergone a mercedes reection or not. This is an important tool when taking into account the

details of FD optics in order to analyse shower events, as done further ahead in this work.

Spot

A photon entering the diaphragm with a certain direction may be misplaced in the PMT camera. The

so-called spot is precisely the circle of least confusion that measures the degree of this misplacement. The25

Figure 4.5: The dimensions of an FD pixel and the mercedes structures in (β, α) coordinates.

main advantage in using Schmidt optics is that the spot is independent from the incident direction and

is not worsened (on the contrary, it is improved) by the introduction of the correction rings. On the focal

surface the spot radius is ∼7.5 mm or ∼ 0.25 which corresponds to ∼ 16 of the pixel size [43, 42]. This

eect is mainly due to mirror aberration and is expected to have radial behaviour or, in other words,

depend on the distance to the camera centre.

Further detail on the spot is achieved with simulation of FD optics. The Karlsruhe group (KG)

simulation [47] allows the study of the spot on dierent positions. Figure 4.7 shows the misplacement of

the incident photons in a particular region of the focal surface. The knowledge of this kind of information

is important in careful uorescence data analysis.

4.1.3 Laser facilities

There are two laser facilities at the Pierre Auger Observatory, both near the centre of the array as

represented in gure 4.8: the Central Laser Facility (CLF), nearly equidistant from Los Leones, Los

Morados and Coihueco eyes and working since July 2003, and the second Central Laser Facility (XLF),

nearly equidistant from Los Morados, Loma Amarilla and Coihueco sites and nished in January 2007

[48, 38]. This section will shortly describe the CLF which is thoroughly documented elsewhere [48, 49].

The Central Laser Facility, strongly based on the HiRes collaboration laser devices, is situated 26

km away from Los Leones, 30 km from Los Morados and Coihueco and 40 km from Loma Amarilla.

Figure 4.9 shows the facility and its adjacent SD tank, named Celeste. The CLF is a fully independent

unit, controlled wirelessly through a microwave internet link, and has its own weather station in order to

determine air temperature, pressure, humidity, wind speed and wind direction [49]. This station, together

with the ones installed at all FD eyes, allows a realistic weather monitoring over the whole site aperture.

A 355 nm linearly polarised laser is installed in the CLF. The laser wavelength is ideal for the

uorescence technique since it is approximately in the middle of the nitrogen uorescence spectrum;

therefore, two mirrors reecting only 355 nm radiation are used to eliminate other wavelengths. The

laser is red in 7 ns pulses up to a maximum energy of 7 mJ − at this energy the pulse mimics an EAS

originated by a 1020 eV cosmic ray [49]. In order not to privilege a certain direction, the initially linearly

polarised laser is depolarised before being red in vertical beams or steered in any direction with 0.2

26

Figure 4.6: An entire FD camera with its 440 pixels and mercedes stars. The telescope axis centre issignaled with a full circle and the origin with an open one.

precision [51]. As a beam crosses the atmosphere its light is scattered o by Rayleigh and Mie processes

and this scattered radiation is detected by the FD telescopes. Besides, a 40 m long optical ber may

drive the laser signal to the Celeste tank.

The potential of a laser facility within a large cosmic ray observatory is immense. The main motivation

to build the CLF was the atmosphere monitoring: a measurement of the aerosol optical depth as a function

of height is possible with the tracks of a vertical laser beam seen by the FD eyes. The behaviour of this

measurement with time and position in the array is monitored as well since it is crucial to the FD data

reconstruction. But atmospheric monitoring is not the only task assigned to the CLF. For instance, the

time synchronisation between the four FD eyes is performed with vertical beams. And almost horizontal

beams directed to each eye are used to determine the time oset between the FD eyes and the SD: 289±43ns for Los Leones, 363±43 ns for Los Morados and 307±49 ns for Coihueco [52] 2 , being the SD the

last to record the events. Moreover, as the position and direction of the laser beams are known with

higher accuracy than the expected FD resolution, the geometrical reconstruction and the alignment of

the telescopes may be tested and hybrid and FD-only reconstructions compared. Also the FD trigger

as a function of energy, shower direction and atmospheric conditions can be evaluated by computing

the fraction of laser events recorded by the eyes. Finally, a determination of the photometric resolution

becomes possible with the CLF if one compares the actual and reconstructed energy of the laser. This

comparison may also work as an atmospheric quality parameter.2The eye at Loma Amarilla was only nished in February 2007 and, thus, no estimate on its time oset to the

SD is available yet.

27

Figure 4.7: The spot on θ ∈ [0, 5] and φ = 2 as previewed by KG simulation [47]. The plot comparesthe incident direction of each photon (θin, φin) with the direction in the focal surface (θfs, φfs).

4.1.4 Atmospheric monitoring devices

As pointed out earlier in this work, the uorescence technique is extremely sensitive to all atmospheric

features. At the moment, the dependence of the atmospheric density, temperature and pressure on the

position is well monitored by radiosondes launched above the southern site and by the weather stations

at the uorescence sites and the CLF [42]. Besides, molecular processes such as Rayleigh scattering are

understood and accurately accounted for. However, the aerosol content of the atmosphere − low altitude

clouds, dust, smoke and other pollutants driven by the wind − must be constantly monitored since it

varies rapidly. It is known that aerosols scatter (Mie scattering) in a signicant manner both uorescence

and erenkov light. So, at the Pierre Auger Observatory there is a hourly monitoring of aerosol conditions

as a function of height, wavelength and position in the array − indeed, there are ve monitor stations (in

the CLF and in the four FD sites) [51]. Several other monitor devices are available and distributed around

the site according to gure 4.8: the CLF (as explained in the last section), backscatter LIDARs (LIght

Detection And Ranging), Horizontal Attenuation Monitors (HAMs), Aerosol Phase Function Monitors

(APFs), cloud cameras and star monitors.

A backscatter LIDAR is a steerable ensemble of a 355 nm laser and a telescope [53, 51]. After the

laser is red, the telescope records the elastic backscattered light providing a measurement of the aerosol

optical depth in the direction of ring. An interesting possibility of LIDAR systems is the 'shoot-the-

shower' mode: an FD eye detects an event and sends the geometry information to the LIDAR which

scans the event shower detector plane upon conrmation of the SD. In this way, detailed, updated aerosol

monitoring is achieved for a specic event. Presently, three LIDAR systems are operational at Los Leones,

Los Morados and Coihueco eyes and the fourth will be installed at Loma Amarilla. The Pierre Auger

Observatory is also equipped with Raman LIDARs, that record the light backscattered (and frequency

shifted) by Raman scattering. This system allows a more precise measurement of the aerosol transmission

and the identication of the dierent constituents of the atmosphere. But the required lasers are quite

intense and, therefore, pollute the eld of view of the telescopes − that is why Raman LIDARs are only

red when the FDs are not collecting data [51]. Further details on Raman LIDARS can be found in [54].

The Horizontal Attenuation Monitors were designed to quantify the attenuation length in horizontal

paths between FD sites. They consist of a UV light source and a corresponding detector. As of 2005,

28

Figure 4.8: The position of the laser facilities (CLF and XLF) in the southern site. Other atmosphericmonitoring devices are also signaled [50].

Figure 4.9: The Central Laser Facility (on the right) and the Celeste SD tank (on the left) [40].

one HAM system was implemented between Los Leones and Coihueco eyes [51].

As for APFs, their task is to study the Mie scattering cross section so that the aerosol scattered

erenkov light is correctly accounted for in the uorescence proles of FD events. This is achieved by

recording the FD signal originated by a horizontal, collimated beam shot in front of the eye. For now,

Los Morados and Coihueco sites are equipped with APFs.

Finally, cloud cameras and star monitors complement the atmospheric aerosol monitoring. The former

provide a map of cloud distribution over the array and on the FD eld of view, while the latter use the

attenuation of star light to determine the total optical depth until the top of the atmosphere.

The extensive programme of atmospheric monitoring at the southern site is expected to allow a good

control and determination of systematic errors− and this is one of the main advantages of the Pierre Auger

Observatory when compared to other experiments. Moreover, there are several redundant monitoring

instruments. For instance, CLF, LIDAR systems and star monitors are all capable of measuring the

vertical optical depth. The choice for redundancy is yet another way to perform an undoubted control

over systematics.

29

4.2 Event reconstruction

4.2.1 Surface Detector

An ultra high energy cosmic ray produces an EAS that triggers several tanks with dierent charge signals

(measured in VEM) and hit times. As stressed in 3.2.1, the SD event reconstruction is two-fold. The

rst step is that of computing the zenith and azimuth angles of the shower direction (θs, φs) given the

arrival times of the tanks. Assuming a plane front propagation, θs and φs are found with a 2 accuracy

[44]. Renements to this determination are achieved by using a spherical parameterisation of the shower

front that Monte Carlo simulations prove to lead to a better direction reconstruction [44, 55].

Fixed θs and φs, the next step is to t the signals measured in the tanks to a certain lateral distribution

function (LDF). The analysis of events from 2004 and 2005 shows that the following Nishimura-Kamata-

Greisen (NKG) distribution is the best t to the data [56, 57]:

S(rc) = S(1000) · 3.47β ·[rcrs

(1 +

rcrs

)]−β(4.5)

where rs = 700 m, β = 2.4 − 0.9 (secθs − 1), rc is the distance to the core and S(1000) is the signal in

VEM at 1000 m from the shower axis. The t to this function, which is valid up to zenith angles of 60,

yields an estimate on the core position and on S(1000).

In order to determine the energy of an SD event, one needs an energy estimator that must be almost

independent from primary nature, longitudinal shower development and geometry and simultaneously

present a clear behaviour with energy in the EeV range. Close to shower axis the features of an EAS have

somewhat important uctuations; besides, with an 1.5 km spacing, measurements less than 100 m from

the core are not frequent [44]. The measured signal at greater distances from the core is used in giant

ground arrays as energy estimator. Since Monte Carlo simulations show that S(1000) is the parameter

with smaller shower-by-shower uctuations, this is used at the Pierre Auger Observatory [44, 57, 63].

However, because S(1000) falls with the zenith angle θs due to the larger slant depths, the estimator

S38 = S(1000)/CIC (θs), where CIC (θs) is the so-called constant intensity curve (see [25] for further

details), is applied as energy estimator in SD analysis. One of the main aims of the Observatory is to

calibrate S38 using the energy reconstructed by the FD in order not to depend on shower simulations.

Finally, showers with zenith angles greater than 60 are deected by Earth's geomagnetic eld in a

signicant way. Therefore, the lateral distribution of such showers is asymmetric − in other words, it

is not just a function of the distance to the axis − and (4.5) becomes inadequate [44, 56]. Dierent

methodologies are needed to treat those cases.

4.2.2 Fluorescence Detector

The analysis procedure of an FD event comprises two separate steps: rstly, the geometry of the shower

is determined and then the reconstruction of the longitudinal prole and energy is performed. The whole

process will be referred to as standard reconstruction along this work.

Geometry reconstruction

The geometry reconstruction begins with the determination of the shower detector plane (SDP), repre-

sented in gure 4.10. Each triggered FD pixel views a direction ~ri and records a total signal qi. An

approximation on the SDP normal ~nSDP is found by minimising the quantity∑i qi (~nSDP · ~ri)2 where

30

the sum is over the triggered pixels [44]. Then, the pixels that are more than 2 away from the approxi-

mated SDP are excluded and the process is repeated iteratively [11]. This method has been tested with

laser beams generated by the CLF and it proved to determine the SDP with an accuracy less than ∼ 0.5

[44, 11, 42].

Figure 4.10: The FD geometry reconstruction setup (adapted from [63]).

Afterwards, it is necessary to constrain the shower axis within the SDP. In monocular events, when

only one eye is triggered, the timing information of the pixels is essential to accomplish this task. Each

pixel records signal during several 100 ns time bins, but only the centroid time ti,c is used in the standard

reconstruction procedure. Assuming that a shower evolves along a line at the speed of light c, one would

expect for the centroid times of each pixel (see gure 4.10):

ti = Temission + Tpropagation = T0 −Rpctg (χi − δ) +

d

c

= T0 +Rpc

(1− sin (χi − χ0 + π/2)cos (χi − χ0 + π/2)

)= T0 +

Rpc

(1− cos (χi − χ0)−sin (χi − χ0)

)= T0 +

Rpctg

(χ0 − χi

2

)(4.6)

where T0 is the time when the shower is at closest distance Rp from the eye, χi is the viewing direction of

the pixel projected onto the SDP and χ0 is the angle of the shower axis within the SDP. A t to equation

(4.6) letting the parameters T0, Rp and χ0 vary is performed by minimisation of∑i qi(ti − ti,c

)2[44]

− an example is shown in gure 4.11. In this way, the geometry of the shower is completely dened.

However, several technical problems may arise when certain events − usually distant ones − produce

short tracks in the eld of view of the eye [12]. In fact, these showers yield so reduced intervals of χ0−χi2

that the t to (4.6) may not be sensible enough to the curvature of tg(χ0−χi

2

)and a wide range of Rp

and χ0 values may be reconstructed. The problem becomes specially acute in the case of small χ0−χi2

values.

The just-mentioned ambiguity may be broken with stereoscopic observations. As explained in section

3.2.2, these events achieve improved geometrical precision by intersecting the SDPs from each eye, thus

avoiding the troublesome timing t to equation (4.6).

31

Figure 4.11: Timing t to equation (4.6) of the monocular event SD 2521005 (FD 2/1066/35) recordedon 2006/08/01 [58]. In this case, Rp ' 5.3 km, χ0 ' 65.5 and T0 ' 24800 ns.

Longitudinal prole and energy reconstruction

Once the geometry of the shower is xed, the longitudinal prole reconstruction may begin. It comprises

two phases: the computation of the light prole at the diaphragm as a function of time and the determi-

nation of the number of charged particles in the shower as a function of the slant depth, i.e. the proper

longitudinal prole.

In the rst phase, the ADC counts registered by the PMTs must be converted into equivalent 370 nm

photons at the diaphragm − this is done using the conversion factor CADC/370 ' 5 ADC−1 determined

during FD calibration and that includes both the detector eciency ε(λ) and the nitrogen uorescence

spectrum [59]. Then, one needs to calculate the contributions of the dierent pixels. Even though

the geometry reconstruction takes the shower propagation to be a line, the spot caused by imperfect

optics results in the distribution of signal over several pixels at the same instant. To decide which

PMTs contribute to the light prole, the signal-to-noise ratio is maximised. Firstly, each time tk yields a

direction in the sky χk by inversion of (4.6). Considering at each time slot all pixels pointing at directions

less than dSN degrees from χk as signal, the signal-to-noise ratio is computed and the value dSN,max that

maximises the ratio is discovered. Now the number of equivalent 370 nm photons − the light prole −at the diaphragm is determined, in a conservative manner, using at each time slot k the pixels within a

certain angle (related to dSN,max) from χk [59]:

Nγ (tk) =∑i

Nγ,ik =∑i

CADC/370 · (NADC,i (tk)−Nped,i) (4.7)

where NADC,i (tk) is the number of ADC counts of pixel i at time tk and Nped,i ' 100 ADC is the pedestal

or baseline value for the PMT i. An example of a light prole at the diaphragm is shown in gure 4.12.

The reconstruction of the proper longitudinal prole uses both the prole (4.7) and the shower ge-

ometry as inputs. The light prole at the telescope diaphragm, given by (4.7), is a mix of uorescence,

erenkov and scattered erenkov light (and eventually background noise that is taken to be small). To

isolate the uorescence component, an iterative method is used. The rst guess is that all light at the

diaphragm is of uorescence origin. The photons are propagated backwards until the emission points by

considering Rayleigh and Mie processes and then the uorescence yield is used to have a rst estimate

on the shower longitudinal prole, N (1)e,max (X). With this prole and the shower geometry, the expected

components of direct and (Rayleigh and Mie) scattered erenkov light at the diaphragm are determined32

Figure 4.12: The light prole at the diaphragm as a function of time for the event SD 2521005 (FD2/1066/35) recorded on 2006/08/01 [58]. The actual quantity represented is the light ux that crossedthe diaphragm. The several components of direct and scattered light are represented as well (see text).

and subtracted from the initial light prole (4.7). The method is iterated until convergence (up to four

times in total) [59]; it works adequately for showers that produce low erekov contamination at the

detector, but completely diverges otherwise [12]. Anyhow, the nal longitudinal prole N (l)e,max (X) is

tted to the Gaisser-Hillas formula:

Ne (X;Nmax, Xmax, X0, λGH) = Nmax

(X −X0

Xmax −X0

)Xmax−X0λGH

eXmax−XλGH (4.8)

where X0 and λGH are t parameters (weakly) correlated to the rst interaction point and the proton

interaction length in air, respectively [60]. In principle, the t is performed with four parameters − Nmax,Xmax, X0 and λGH − but the restrictions X0 = 0 gcm−2 and λGH = 70 gcm−2 are usually imposed for

the sake of convergence. Figure 4.13 presents the reconstructed longitudinal prole of a shower and the

corresponding Gaisser-Hillas t.

Figure 4.13: The reconstructed longitudinal prole and the Gaisser-Hillas t [44].

A non-iterative procedure to calculate the longitudinal prole given the light at the diaphragm was

recently proposed and is already in use [61, 58]. Since the uorescence production is proportional to the

energy deposited by the shower in the atmosphere dEdX , it is convenient to reconstruct the dE

dX (X) prole

rather than the Ne (X) mentioned in the above paragraphs. However, the erenkov light, both direct33

and scattered, depend on the Ne (X) prole 3 ; so, there is the need to relate dEdX to Ne. According to

[62, 46]:dE

dX(Xi) = Ne (Xi)αKGi (4.9)

where αKGi ≡ αKG (si) = 1.06724 ·(

43.2535(1.34508+si)

11.3005+2.44755+0.122845si

)MeVgcm−2 and si = 3

1+2Xmax/Xiis

the shower age parameter. At this point, one may relate in a matricial manner the dEdX prole of the

shower and the three components of detected light: uorescence, direct erenkov and scattered erenkov

light. Considering line propagation as in the geometry reconstruction, the rst two kinds of light arriving

at the detector at a certain instant come from a specic segment of the shower, while the scattered part

builds up during cascade development and must therefore be summed over all shower history [61]:

y = Cx (4.10)

being y and x vectors and C a matrix given by

Cij =

0 if i < j

cdi + csii if i = j

csij if i > j

where y is the light received at the diaphragm as a function of time, x is the prole dEdX (X), cdi is the

direct light contribution (uorescence and erenkov) at time slot i and csij is the scattered erenkov

fraction (due to Rayleigh and Mie scatterings) detected at time slot i but emitted in Xj slant depth

bin. The dEdX (X) prole is found by inverting (4.10) and, afterwards, one ts it to a (slightly) modied

Gaisser-Hillas formula:

dE

dX

(X;

dE

dX

∣∣∣∣max

, Xmax, X0, λGH

)=

dE

dX

∣∣∣∣max

(X −X0

Xmax −X0

)Xmax−X0λGH

eXmax−XλGH (4.11)

Further details on this procedure are given in [61].

Finally, the energy estimate is determined by integrating the best t to the reconstructed longitudinal

prole. In the case of the Ne (X) prole, the energy is given by

E '< dE

dX>

∫ ∞X0

Ne (X) dX (4.12)

where the mean energy deposit is < dEdX >' ε0

Xl' 2.2 MeV

gcm−2 [12, 9]. As for the dEdX (X) prole, the

estimated energy is simply the integral of the curve:

E =∫ ∞X0

dE

dX(X) dX (4.13)

As a nal remark, note that the energy calculated with (4.12) or (4.13) is about 10% less than the

actual energy of the cosmic ray − recall the discussion on the missing energy in section 3.2.2.

4.2.3 Hybrid

The hybrid technique is being used for the rst time ever at the Pierre Auger Observatory. It diers from

the FD-only reconstruction described in the previous section uniquely on the second step of the geometry

3While direct erenkov light is proportional to Ne (X), the scattered component grows with the integral ofNe (X).

34

(a) (b)

Figure 4.14: Comparison of monocular and hybrid reconstructions using laser shots [64]. On the leftthe dierence between the reconstructed Rp and the actual one is plotted, while on the right the samedierence for χ0 is presented.

reconstruction, i.e. on the denition of the shower axis within the SDP. If, besides the FD, one SD tank

is triggered − as in gure 4.10 − , the arrival time of the shower front to the ground, tgrd, constrains the

T0 parameter in the timing t to equation (4.6) [63]:

T0 = tgrd −~rgrd · ~sc

(4.14)

where ~rgrd is the position of the tank and ~s is the normalised shower direction. The constraint allows

a more accurate reconstruction of Rp and χ0 or, in other words, of the shower geometry. In the case of

more than one triggered SD stations, a better precision is achieved and eventual restrictions on the core

position are possible [44, 12]. Note that the measurements needed from the surface detector are arrival

times and not deposited charges which makes the hybrid technique rather clean and straightforward

even though a careful synchronisation between the FD eyes and the SD is crucial as reported in 4.1.3.

Moreover, this reconstruction works quite well even if the shower produces short tracks within the eld

of view of the FD. Indeed, the use of ground stations, that correspond to small χi, allows a better t to

equation (4.6).

Since the SD presents a duty cycle of 100%, FD events are almost all hybrid ones as well. Approx-

imately 10% of the SD data are hybrid and, when nished, the Pierre Auger Observatory is expected

to collect about 4000 of these events per month [64]. An important energy feature is that the hybrid

mode triggers at somewhat low energies when compared to the SD trigger, because only one active tank

is required. This means that events that would not have triggered the surface detector are recovered and

analysed with improved geometrical precision through the hybrid technique.

The hybrid detection was put to test using laser beams shot from the CLF, whose direction is almost

vertical (χ0 = 90±0.01) and the Rp (in this case, also the 'core' position) is known with a 5 m precision.

During CLF routine operation, several vertical laser beams are shot into the sky and one of them is also

driven into the Celeste tank, thus allowing the comparison of monocular and hybrid (with one tank only)

reconstructions [64]. The results of the resolutions in Rp and χ0 are plotted in gure 4.14 and are rather

impressive: the hybrid technique provides a geometry reconstruction with a ∼10 times better resolution

and no systematic shift.

The improved geometry precision makes hybrid data very clean to perform anisotropy studies and also35

allows a better accuracy in the determination of the shower longitudinal prole and primary energy. Thus,

most physical studies use preferentially hybrid data sets, even though they represent smaller statistics.

But maybe the most important feature of the hybrid technique is the possibility to calibrate the S38

parameter of SD analysis with the energy reconstructed by the FD, as demonstrated in gure 4.15 for

a specic set of hybrid data [25]. The calibration provides an energy conversion almost simulation-

independent and constitutes the rst step in getting an unbiased energy measurement and therefore a

reliable spectrum such as the one presented before in gure 3.7(b).

Figure 4.15: Calibration between S38 and the energy reconstructed by the uorescence technique [25].

4.3 Future steps

The next steps for the Pierre Auger Observatory are the installation of enhancements on the southern

site and the construction of the northern one. Brief details on these fronts are given below.

4.3.1 Southern site enhancements

The main physical motivation to enhance the southern site is the possibility of exploring the region of the

cosmic ray spectrum around the ankle, i.e. 1017 − 1019 eV. In this energy range cosmic ray sources are

believed to change from galactic to extragalactic ones and that transition is not yet totally understood

nor experimentally documented [65, 66, 67]. Presently, the FD and the SD achieve full eciency at 3·1018

eV [24, 68]. So enhancements were planned to aim at lower energies.

But lowering the threshold energy is not enough to study the ankle region − both a good energy

resolution and (statistical) capability to identify primaries are crucial. Primary identication may be

performed knowing the muon content of the air shower. Therefore, one of the southern site enhance-

ments is the Auger Muons and Inll for the Ground Array (AMIGA) which plans to install underground

muon counters and additional SD tanks, as explained below. Another reliable composition study is the

dependence of the shower maximum on energy. For lower energy cascades, the shower maximum occurs

higher in the atmosphere and, consequently, an extended FD eld of view is needed − the issue is being

delt with by the High Elevation Auger Telescopes (HEAT). Both HEAT and AMIGA were approved by

the Auger Collaboration and are now under construction.

The HEAT initiative will deploy three telescopes similar to the ones already in use behind of the

Coihueco site. The telescopes will be inclined so that the region between 28 and 58 in elevation is

36

covered [66]. The azimuth spacing between the telescopes is still under discussion, but they will work

independently from Coihueco eye in trigger matters. HEAT will allow the full detection of EAS with

primary energies below 1019 eV. In fact, those cascades emit fainter uorescence light and simultaneously

peak earlier in their development. So, extending the eld of view, i.e. looking at higher elevations closer

to the eyes, will make the FD sensible to lower energies. Specically, the threshold energy will be 7 · 1017

eV and may fall to 2 ·1017 eV with ground array inlls such as AMIGA and considering a hybrid threshold

[67]. Last but not the least, the composition study through the Xmax (E) dependence in the 1017 − 1019

eV range may help to unravel the mystery behind the transition that takes place around the ankle.

AMIGA, on the other hand, consists of an inll of both water erenkov tanks and underground muon

counters ∼ 6.0 km in front of Coihueco eye [69]; the additional detectors will be placed in 433 m and 750

m grids. The muon counters will contain several 400 cm x 4.1 cm x 1.0 cm strips buried ∼ 3 m under the

ground. AMIGA will achieve SD full eciency at lower energies than the current 3 ·1018 eV: 1016 eV with

433 m spacing and 3 · 1017 eV with 750 m [66]. Besides, AMIGA will improve the event reconstruction

and complement composition studies performed by HEAT.

Another southern site potential enhancement is the radio detection through antennas that aims at

recording . 100 MHz signals emitted during EAS development. This technique monitors the electro-

magnetic properties of a cascade as it crosses the atmosphere, being complementary to the SD and the

FD and presenting an 100% duty cycle [70]. Moreover, it exhibits a good pointing accuracy and, thus,

allows anisotropy and source studies. However, radio detection is not yet fully developed and is highly

inuenced by signicant background in the . 100 MHz frequency band due to storms and human-made

noise. Since November 2006 a prototype antenna has been operating near the CLF, but up to now it

proved to be too sensitive to storms ∼ 100 km away [38].

4.3.2 The northern site

In June 2005 the Auger Collaboration chose Lamar, Colorado, USA to be the site for the northern part of

the Observatory [71]. Located 1300 m above sea level, it is an approximately rectangular site of 134 km

x 77 km enclosing a total area of ∼ 10300 km2 and extension is possible up to 22000 km2. The northern

Fluorescence Detector will comprise two eyes with telescopes similar to those used in the southern site.

The Surface Detector will consist of a 4000 water erenkov tanks array in an 1.6 km spacing rectangular

grid. This second site will focus on the 1020− 1021 eV range, presenting a threshold energy of about 1019

eV. The construction is scheduled to start on 2009 and from then on signicant statistics will be added

to the PAO data, specially SD data since the northern ground array is more than three times bigger than

the one in Argentina. Besides, with full sky coverage, the Pierre Auger Observatory will become the rst

experiment able to perform comprehensive anisotropy and other astrophysical studies.

37

Chapter 5

The 3D FD reconstruction

The standard FD event reconstruction, described in 4.2.2, does not use all the information recorded by

the telescopes. In the geometry reconstruction, only the centroid times of the triggered pixels are used

and, so, the time structure of the signal inside each pixel is neglected. Besides, the line propagation

approach, that may be sucient for distant showers, is responsible for the mistreatment of the lateral

proles in close-by events. As for the prole reconstruction, at each time slot the contributions of nearby

pixels are summed, even though the same observation times in dierent pixels correspond to dierent

emission times.

It is therefore necessary to develop a new reconstruction procedure that deals properly with all avail-

able FD information. The 3D reconstruction [72] is a possible approach. Indeed, it uses all relevant time

bins inside each pixel and considers a disk (rather than line) shower propagation. This improved geometry

reconstruction allows the study of the 3D features in cascade development and the reconstruction of a

3D shower prole instead of separate longitudinal and lateral analyses.

5.1 Geometry reconstruction

5.1.1 The 3D method

While each pixel of the FD camera corresponds to a direction in the sky, the third dimension− the distance

to the eye − is determined by the timing information. The idea of the 3D geometry reconstruction is to

use both pixel direction and time to locate in space the volume from where the photons observed by each

pixel at each time slot were originally emitted. An important dierence from the standard procedure is

the shower propagation hypothesis: instead of considering line propagation, the cascade is assumed to be

a plane disk, moving at the speed of light in vacuum c.

The 3D geometry reconstruction is initiated by constraining ~nSDP , Rp and χ0 to the values given by

the standard method and retting the data to equation (4.6) with T0 as the only t parameter. Then,

as exemplied in gure 5.1(a), each triggered pixel i, of central direction (θi, φi), at each time slot k is

associated to an unambiguous point ~rik:

tk = Temission + Tpropagation = T0 −rikccosαi +

rikc⇔

rik =c (tk − T0)1− cosαi

(5.1)

where tk is the central time of slot k, αi is the angle between −~rik and the shower axis ~s and rik is the

distance from the eye in (θi, φi) direction. The point dened in this way is the central point of the volume

whose vertices are determined by (5.1) using tk ± 50 ns and the directions (θvi , φvi ) corresponding to each

38

r

r

A

AAAAAK

eye

~rik

~s

αi

T0

shower front

@@

@@

@@

@

@@

@@@

(a) The 3D geometry reconstruction setup. Eachpixel i at each time slot k corresponds to anunique volume with central point ~rik.

(b) 3D visualisation of event SD 2521893 (FD4/1731/2757) recorded on August 2006. The full cir-cles represent the central points of the volumes andthe color code is referred to observation times.

Figure 5.1: The 3D geometry setup and event visualisation.

vertex of the hexagonal pixel (v = 1, ..., 6). Therefore, for each pixel at each time slot there exists an

irregular volume in the space.

Fixed all 'detected' volumes for an event, a new estimate on the shower axis ~s and core position ~rsmay be computed through inertia calculations. Using the number of 370 nm photons at the diaphragm,

Nγ,ik = CADC/370 · (Ni,ADC (tk)−Ni,ped), as the 'mass', one may calculate the centre of mass of the

central points ~rik:

~rcm =

∑i,kNγ,ik~rik∑i,kNγ,ik

Moreover, the inertia matrix of the set of central points is given by

I =

Ixx Ixy Ixz

Ixy Iyy Iyz

Ixz Iyz Izz

(5.2)

being

Ixx =

∑i,kNγ,ik

(dy2ik + dz2

ik

)∑i,kNγ,ik

Ixy = −∑i,kNγ,ikdxikdyik∑

i,kNγ,ikIxz = −

∑i,kNγ,ikdxikdzik∑

i,kNγ,ik

Iyy =

∑i,kNγ,ik

(dx2

ik + dz2ik

)∑i,kNγ,ik

Iyz = −∑i,kNγ,ikdyikdzik∑

i,kNγ,ikIzz =

∑i,kNγ,ik

(dx2

ik + dy2ik

)∑i,kNγ,ik

with dxik = (~rik − ~rcm) · ~ex, dyik = (~rik − ~rcm) · ~ey and dzik = (~rik − ~rcm) · ~ez.The main inertia axis − the third eigenvector of I − is the new estimate on ~s and together with ~rcm

it yields the core position ~rs. From ~s, ~rs and the eye position, one easily nds the geometric parameters

~nSDP , Rp and χ0 that are once again used in the timing t with T0 as the only parameter. The method

is repeated iteratively until convergence of both ~s and ~rs.39

Figure 5.2: The distribution of dCP−eye and log10EKG in the data collected from January 2006 until

September 2006 with KG and 3D prole reconstructions and χ3D0 ≥ 45. The red and blue lines indicate

the border of the empirical cut (5.3) with d∗CP−eye = 5 km and d∗CP−eye = 10 km, respectively.

Note that the geometry generated by equation (5.1) is such that the constant observation times

(tk =constant) correspond to rikc (1− cosαi) =constant, i.e. the bisectrix between the shower axis and

the observation line.

The visualisation of the complex 3D output of the new geometry reconstruction is done using the

map3d package [73], as exemplied in gure 5.1(b). Besides all vertices of the volumes and the central

points, also the so-called near points may be represented. The near point of a volume is the point at

minimum distance from the shower axis; it represents a more physical position in the cascade structure

than the central point. Several features associated to each volume − observation time, Nγ,ik, eye − can

as well be represented using a color code.

The 3D method is supposed to be particularly accurate for events simultaneously close to the detector

and with high energies. The mean distance of the central points of the volumes to the respective eye

dCP−eye and the KG-reconstructed energy EKG are used to select the referred events:

log10EKG − 17.5 >

d2CP−eye

d∗2CP−eye∧ dCP−eye 6= 0 (5.3)

The distribution of dCP−eye and EKG as well as the empirical cut (5.3) with d∗CP−eye = 5 km and

d∗CP−eye = 10 km are presented in gure 5.2 for the data collected from January 2006 until September

2006 with KG and 3D prole reconstructions and χ3D0 ≥ 45. In order to check if the new geometry

reconstruction is working properly, a comparison between the values of Rp, χ0 and T0 as given by

the standard (KG) and 3D methods applied to the data set of gure 5.2 with d∗CP−eye = 10 km was

performed and is plotted in gure 5.3. Except for some (potentially interesting) events, the 3D approach

yields essentially the same geometry as the standard procedure, which is natural. There seems to be a

slight tendency of the new method to reconstruct smaller Rp, but further cross-checks must be done −see next section.

5.1.2 Some applications

The 3D shower structure allows a study on lateral proles. Indeed, minimum and medium lateral dimen-

sions rmin and rmed may be dened for each shower. The positions of the near points (np) and central40

(a) (b)

(c)

Figure 5.3: Comparison between the Rp, χ0 and T0 values as reconstructed by the standard and 3Dapproaches. Data collected from January 2006 until September 2006 with KG and 3D prole reconstruc-tions, χ3D

0 ≥ 45 and passing cut (5.3) with d∗CP−eye = 10 km was used to produce the plots.

41

points of the volumes yield these variables directly:

rmin =

∑i,k dPL (~rnp,ik, ~s, ~rs)

Nvolrmed =

∑i,k dPL (~rik, ~s, ~rs)

Nvol

where the sums are over all good volumes, Nvol is the number of those volumes and dPL (~r,~s, ~rs) is the

distance from point ~r to the shower axis given by ~s and ~rs.

Another lateral parameter is the so-called rmax which is of an essentially dierent kind from rmin and

rmed and represents roughly the radial distance that includes most detected volumes. Its denition is

as follows. A Monte Carlo method was designed to determine the volume of the irregular solid seen by

each pixel i at each time slot k. Each solid ik is surrounded by a cylinder of volume Vcyl,ik inside which

Ntot,ik points are randomly generated according to the volume element rdrdθdh = d(r2

2

)dθdh. Of the

total Ntot,ik, only Nin,ik points ~r(l)ik are in the detected volume ik and, therefore,

Vik '∑Nin,ikl=1 1Ntot,ik

Vcyl,ik =Nin,ikNtot,ik

Vcyl,ik (5.4)

σ (Vik) =∂Vik∂Nin,ik

√Nin,ik +

∂Vik∂Ntot,ik

√Ntot,ik = Vik

(1√Nin,ik

+1√

Ntot,ik

)(5.5)

where Ntot,ik was xed to 1000 independently from i and k. Then, a cylinder of radius r(m)cyl around the

shower axis is dened and the following estimator computed:

η(r

(m)cyl

)=

∑i,k Vact,ik∑i,k Vik

being Vact,ik the active volume ik, i.e. the volume of the solid ik inside the cylinder of radius r(m)cyl :

Vact,ik =∫Vik

factdV 'Nin,ik∑l=1

fact

(~r

(l)ik

) VikNin,ik

=Nin,ik∑l=1

fact

(~r

(l)ik

) Vcyl,ikNtot,ik

(5.6)

with fact(~r

(l)ik

)=

1 if dPL(~r

(l)ik , ~s, ~rs

)≤ r(m)

cyl

0 otherwise.

Physically, η(r

(m)cyl

)is simply the fraction of the detected volume within r

(m)cyl from shower axis.

Finally, rmax is found by looping on r(m)cyl with step ∆rcyl = 10 m:

rmax = r(m)cyl −

∆rcyl2

:η(r

(m)cyl

)− η

(r

(m−1)cyl

)∆rcyl

≤ 10−3 ∧ η(r

(m)cyl

)≥ 0.1 (5.7)

With this denition, all the signicant volumes ik of the event are inside the cylinder of radius rmaxaround the shower axis. It is obviously possible that there are volumes further away than rmax, but they

are assumed not to be physically signicant.

The comparison between rmin, rmed and rmax is presented in gure 5.4(a). As expected, rmin ∼ 0 since

the near points are supposed to be close to shower axis if this was correctly reconstructed, while rmed and

rmax have more sparse distributions. These three parameters and their dependence on geometric variables

constitute an useful tool in the identication of non-standard events. To begin with, the behaviour of

rmin, rmed and rmax with χ0 is plotted in gure 5.4(b). There is a decrease of both rmed and rmax until

χ0 = π2 and a stabilisation afterwards. The rmin parameter, however, is close to 0 along the whole range

in χ0. In other words, events with χ0 <π2 present large, spread volumes containing parts of the shower

axis − the 3D structure of such an event is shown in gure 5.5. The reason behind this situation is the42

(a) (b)

Figure 5.4: Distributions of rmin, rmed and rmax in (a) and their dependence on χ0 in (b). The data setused is the same as in gure 5.3.

signal pile up that occurs for showers approaching the FD eye, i.e. χ0 <π2 . In fact, in the extreme case

where a shower evolves towards the eye parallel to the viewing direction of a pixel i, both the standard

and the 3D methods are problematic. In the standard procedure, as χ0 = χi, (4.6) yields ti = T0: all the

photons produced along the shower axis arrive at pixel i at the same instant ti. Notice that the standard

geometry reconstruction does not fail because Rp, χ0 and T0 are known (Rp = 0, χ0 = χi and T0 = ti),

but the shower prole determination becomes impossible. The 3D approach, on the other hand, diverges

in the extreme case since αi = 0 and (5.1) reads rik →∞ − that is why smaller χ0 correspond to larger

volumes. Another important remark on events with χ0 <π2 is that the erenkov radiation emitted during

cascade development may arrive directly at the FD telescopes and, consequently, this contribution must

be taken into account together with uorescence light in the reconstruction of the shower prole and in

energy estimation.

Figure 5.5: 3D visualisation of event SD 2553671 (FD 2/1081/2704). In this case, Rp ' 9.7 km andχ0 ' 33.1. Note the dierence between the reconstructed volumes here and those of the event presentedin gure 5.1(b).

Figure 5.6 shows the relation between rmax and Rp with the quality cut χ0 ≥ π2 in order to eliminate

the events described in the previous paragraph. The plot seems to indicate that rmax ∝ Rp which is

natural since a pixel overviews greater volumes at greater distances from the eye. The deviation from

this behaviour is a possible criterion to classify cosmic ray events and identify strange ones.

43

Figure 5.6: Dependence of rmax on Rp for the same data set as in gure 5.3 but passing the quality cutχ0 ≥ π

2 .

Also the inertia coecients Ixx, Iyy and Izz, dened in the last section, play an important role in

the characterisation and classication of events. From the analysis of these parameters in data, the rst

conclusion is that Ixx/Iyy ∼ 1 as shown in gure 5.7. Therefore, the shower 3D structure is cylindrically

symmetric within a very good approximation. Moreover, the fact that Ixx ∼ Iyy means the use of the

observation time as a third dimension in FD analysis introduces no signicant bias in the 3D geometry

reconstruction.

Figure 5.7: The Ixx/Iyy and Izz/r2med distributions for the same data set as in gure 5.3 but requiring

rmed 6= 0 and at least one Iii 6= 0.

A study of the three dimensional shape of shower development may as well be performed using

inertia relations. If the shower structure were a massive, uniform cylinder of radius r and height h,

then the moments of inertia along the main inertia axes would be Icylxx (r, h) = Icylyy (r, h) = r2

4 + h2

12 and

Icylzz (r) = r2

2 . In the case of a cone of radius r and height h, Iconexx (r, h) = Iconeyy (r, h) = 320r

2 + h2

10 and

Iconezz (r) = 310r

2. The typical lateral dimension of a shower is given by rmed and so one may verify if the

3D structure is similar to that of a cylinder of radius rmed, a cone of radius rmed or a cone of radius

2rmed in which cases the following relations are expected: Izzr2med

= Icylzz (rmed)

r2med= 1

2 ,Izzr2med

= Iconezz (rmed)

r2med= 3

10

and Izzr2med

= Iconezz (2rmed)

r2med= 6

5 . Figure 5.7 shows the distribution of Izzr2med

in events from August 2006. The

data is consistent with cylinders of radius rmed and not with any of the cones considered above. However,

44

(a) (b)

Figure 5.8: The distributions of√Ixx + Iyy − Izz (a) and

√Izz (b) for the same data set as in gure 5.7.

the considerable dispersion around the mean value indicates a mix of dierent geometries − probably

not only cylinders or cones − present in the data set. Note that the inertia coecients, namely Izz,

are calculated with weight Nγ,ik that is not uniformly distributed as the mass is in a massive, uniform

cylinder or cone. Thus, the coecients are a convolution of the Nγ,ik distribution and the geometric

structure and should not be misunderstood as simple geometric parameters such as rmin, rmed or rmax.

Anyway, the trail of the Izzr2med

histogram, this is, events far away from cylinder-like behaviour, corresponds

to two dierent populations of showers: those badly reconstructed and those potentially interesting that

present remarkable moments of inertia.

Finally, the longitudinal and lateral dimensions of a shower may be accessed using√Ixx + Iyy − Izz

and√Izz respectively, since

√Ixx + Iyy − Izz ∝ h and

√Izz ∝ r in both a cylinder and a cone. The

distributions of such parameters are presented in gure 5.8 and they allow the identication of particularly

'fat' and/or extense events.

Many more geometric tests and studies will be done in the near future.

5.2 Prole reconstruction

The volumes Vik determined by the 3D geometry reconstruction correspond to the regions in space from

where the detected photons were originally emitted. Using these volumes and a 3D prole, one may

compute the expected signal at pixel i and time slot k taking into account uorescence light, direct

and scattered erenkov emission, light attenuation and the detailed FD optics. While in the standard

reconstruction the observed photons are propagated backwards to obtain the shower prole, the 3D

approach uses a given prole to calculate the expected signals at the PMTs and then a comparison

between observation and expectation determines the best 3D prole.

5.2.1 The 3D shower prole

As mentioned above, the 3D reconstruction procedure calls for a three dimensional prole that describes

the evolution of the shower at each point in space. Since the shower longitudinal and lateral develop-

ments are known to be best parameterised in depth units, each point ~r is associated to a pair (X,R) as

45

schematically represented in gure 5.9:

X =

∫∞h(~ps)

ρ(h)dh

cosθs

[gcm−2

](5.8)

R = ρ (~r) dPL (~r,~s, ~rs)[gcm−2

](5.9)

where ~ps is the closest point of the shower axis to ~r and θs is the angle between ~s and the vertical.

Expression (5.8) is valid for almost vertical showers. Furthermore, the denition of R does not contain

an integral between ~r and ~ps as (5.8) does. This is because the shower particles do not follow the path

from ~r to ~ps and so the local density ρ (~r) is taken into account instead of performing the integral. Note

the use of capital letters for depth-like parameters (measured in gcm−2) and small letters for distances

(in m) − an exception is Rp, dened in section 4.2.2.

r

rr@@R

A

AAAAAK

HHY

eye

~s

α

θs

R

X

At

~et

~r − ~reyeθ

ps

r

@@

Figure 5.9: The geometric setup used in the 3D prole reconstruction.

The 3D prole is simply the product of a longitudinal prole, that gives the number of electrons at

slices of dierent slant depths, and a lateral distribution function, that spreads the electrons around each

slice:

Ne (~r;Nmax, Xmax, X0, λGH) = Ne (X,R;Nmax, Xmax, X0, λGH)

= Ne (X;Nmax, Xmax, X0, λGH) · LDF (X,R;Xmax)[ em2

](5.10)

Whereas the longitudinal prole is parameterised from CORSIKA simulations by the Gaisser-Hillas

formula (4.8), there are several possible lateral distributions, that are required to be normalised, i.e.∫ 2π

0

∫∞0LDF (X, r) · rdrdφ = 1. The lateral spread is scaled by the so-called Molière radius, a natural

transverse length given by multiple scattering [13, 74]: RM = EsXlε0' 9.6 gcm−2 (recall the denition

of the electron radiation length in air Xl and the critical energy ε0 in chapter 3), or in distance terms

rM = RMρ(~r) , where Es =

√4παfs

mec2 ' 21 MeV is the scale energy for electrons and αfs = e2

4πε0~c '1

137 is

the ne structure constant. About 90% of the shower energy is contained in one Molière radius around

the axis [9].

A lateral distribution suitable for electromagnetic showers is described by the Nishimura-Kamata-

Greisen (NKG) formula [74]:

LDFNKG (X, r;Xmax) =1r2M

(r

rM

)s−2(1 +

r

rM

)s−4.5 Γ(4.5− s)2πΓ(s)Γ(4.5− 2s)

[1m2

](5.11)

46

being r the distance to the axis and s = 31+2Xmax/X

the shower age. The respective NKG cumulative func-

tion at shower maximum (s = 1) is FNKG (Xmax, r) =∫ 2π

0

∫ r0LDF (Xmax, r) · rdrdφ = 1−

(1 + r

rM

)−2.5

.

So, an universal lateral distribution based on this formula and on CORSIKA simulations of hadronic pri-

maries was proposed by D. Góra et al. [13]. Independently from primary energy, primary (hadronic)

particle and zenith angle, the shower lateral cumulative function was found to be adequately parame-

terised by:

FGora (X, r;Xmax) = 1−(

1 + a(s)r

rM

)−b(s)(5.12)

with

a(s) = 5.151s4 − 28.925s3 + 60.056s2 − 56.718s+ 22.331

b(s) = −1.039s2 + 2.251s+ 0.676

Since (5.12) presents cylindrical symmetry around the shower axis, its corresponding LDF is given

by:

LDFGora (X, r;Xmax) =1

2πrdFGoradr

=1

2πra(s)b(s)

rM

(1 + a(s) r

rM

)1+b(s)

[1m2

](5.13)

However, as pointed out in [13, 15, 74], the cascade particles at certain depth X are spread according

to the value rM computed at depth X − 2Xl. Thus, in depth-like variables X and R, (5.13) becomes

LDFGora (X,R;Xmax) =1

2πR/ρ (~r)a(s)b(s)

RM/ρ (~r′)(

1 + a(s) R/ρ(~r)RM/ρ(~r′)

)1+b(s)

=1

2πRa(s)b(s)ρ (~r) ρ (~r′)

RM

(1 + a(s) Rρ(~r′)

RMρ(~r)

)1+b(s)

[1m2

](5.14)

where ~r′ is the point corresponding to ~r but at slant depth X − 2Xl. Note that (5.14) is in units 1/m2,

as required for a distribution of particles in a plane.

5.2.2 Light at diaphragm

Given a hypothetical 3D prole, the number of photons at the diaphragm is determined integrating in

each volume Vik the density of emitted photons that arrive at the telescope. Similarly to rmax calculation,

a Monte Carlo method is used to perform the integration. Firstly, a cylinder of volume Vcyl around the

shower axis is dened with radius rcyl = max (rmax, 3rM,max), where rM,max is the maximum value of rMcalculated in all central points ~rik, and height hcyl enough to include all detected volumes. The cylinder

is such that most shower particles are inside it. Then, Nvol · N1 points ~r(q) are uniformly generated in

the cylinder according to the element volume rdrdθdh and, as in equation (5.6), the number of expected

photons at the diaphragm coming from Vik is

Nγ,ik (Nmax, Xmax, X0, λGH) =∫Vik

dNγdV

(~r;Nmax, Xmax, X0, λGH) dV

'Nin,ik∑l=1

dNγdV

(~r(l);Nmax, Xmax, X0, λGH

) VcylNvol ·N1

(5.15)

where ~r(l) are the Nin,ik generated points that are inside Vik. Ignoring the (small) contribution of multiple

scattering, the density dNγdV is a sum of uorescence

(dNfγdV

), direct

(dN

dγ

dV

)and Rayleigh

(dN

Rγ

dV

)and

47

Mie(dN

Mγ

dV

)scattered erenkov fractions:

dNγdV

=dNf

γ

dV+dNd

γ

dV+dNR

γ

dV+dNM

γ

dV(5.16)

Recall that, unlike erenkov emission, uorescence is isotropic and, consequently, its scattering is not

very important to the angular distribution of the light.

As the above described Monte Carlo integration is performed for each telescope triggered by the event,

the treatment of information coming from dierent telescopes (and eventually from dierent eyes as well)

is straightforward.

Fluorescence

Although all charged particles in a cascade are responsible for the production of uorescence light, elec-

trons present the most signicant contribution. The electrons that contribute to this kind of light have

energies lower than the critical energy ε0 ' 81 MeV and above the uorescence threshold Efthr ' 1.4

MeV; therefore, the density of uorescence photons at (X,R) in the wavelength band [λ1, λ2] is simply

given by

Ne (X;Nmax, Xmax, X0, λGH) · LDF (X,R;Xmax) ·∫ ln(ε0/MeV)

ln(Efthr/MeV)yfγ (E, h, λ1, λ2) fe(E, s)dlnE

[ γm3

](5.17)

where yfγ is the uorescence yield and fe = 1Ne

dNedlnE is the electron energy distribution in an EAS that is

parameterised from CORSIKA simulations in [17]:

fe(E, s) = a0(s)E

(E + a1(s)) (E + a2(s))s

with E in MeV units and

a0(s) =

[∫ ∞ln(Ecut/MeV)

E

(E + a1(s)) (E + a2(s))sdlnE

]−1

a1(s) = 6.42522− 1.53183s a2(s) = 168.168− 42.1368s

The yield dependence on E and h (through the density ρ(h) and the temperature T (h)) may be dis-

regarded at rst approximation and, integrating over the nitrogen uorescence spectrum, yfγ ' 4 γ/e/m.

So, (5.17) becomes simple to compute and∫ ln(ε0/MeV)

ln(Efthr/MeV) fe(E, s)dlnE represents the fraction of electrons

contributing to uorescence at shower age s. As a reference, at shower maximum (s = 1), this fraction is

about 0.7.

Afterwards, the light angular distribution must be taken into account: since uorescence is an isotropic

phenomenon, 1

Nfγ

dNfγdΩ = 1

4π . Thus, when computing the expected number of photons at the diaphragm,

one nds the term1

Nfγ

dNfγ

dΩ∆Ω (~r)

where ∆Ω is the solid angle seen by the telescope:

∆Ω (~r) =At

|~r − ~reye|2[stereorad]

being At = π · 1.12 m2 the total area of the diaphragm including the corrector ring. In principle, instead

of At, one should use the eective collection area Atcosθ (~r), where θ is the angle between the incident48

photon and the normalised telescope axis ~et − see gure 5.9. But the cosθ contribution is already included

in the calibration factor C370/ADC introduced in section 4.2.2.

Finally, the light attenuation from the emission point until the detector is accounted for using the

transmission coecients TR and TM , given respectively by equations (3.8) and (3.12). The densitydNfγdV

is then

dNfγ

dV(~r;Nmax, Xmax, X0, λGH) = Ne (X;Nmax, Xmax, X0, λGH) · LDF (X,R;Xmax) · yfγ ·

·∫ ln(ε0/MeV)

ln(Efthr/MeV)fe(E, s)dlnE ·

1

Nfγ

dNfγ

dΩ∆Ω (~r) · TR

(~r, ~reye, λ

feff

)· TM

(~r, ~reye, λ

feff

) [ γm3

](5.18)

where λfeff = 370 nm is the eective uorescence wavelength for which the detector eciency is unitary.

Notice thatdNfγdV has indeed units of photon density, so that (5.15) yields a number of photons.

Direct erenkov

Following a similar reasoning as in the uorescence case, the density of erenkov photons with λ ∈ [λ1, λ2]

emitted at (X,R) and that arrive directly at the detector is essentially

dNdγ

dV∼ Ne (X;Nmax, Xmax, X0, λGH) · LDF (X,R;Xmax) ·

∫ ∞ln(Ethr(h)/MeV)

yγ (E, h, λ1, λ2) fe(E, s)dlnE ·

· 1

Ndγ

dNdγ

dΩ∆Ω (~r) · TR

(~r, ~reye,

λ1 + λ2

2

)· TM

(~r, ~reye,

λ1 + λ2

2

) [ γm3

]being E

thr(h) = mec2√

2(n(h)−1)the electron threshold energy for erenkov emission and n the refraction

index. The yield yγ is described in [17] and approximately given by:

yγ (E, h, λ1, λ2) ' 2παfs

(2 (n(h)− 1)− m2

ec4

E2

)(1λ1− 1λ2

) [ γ

e.m

](5.19)

As for the angular dependence, the erenkov radiation is forward directioned in a cone centred along

shower axis. Since the light distribution presents azimuthal symmetry,

1

Ndγ

dNdγ

dΩ(α, h, s) =

1

Ndγ

dNdγ

2πsinαdα≡ A (α, h, s)

2πsinα

[1

stereorad

]where α is the angle between the shower axis and the viewing direction− see gure 5.9− andA (α, h, s) ≡

1

Ndγ

dNdγ

dα is parameterised in [17] for α ∈ [5, 60] using CORSIKA showers:

A (α, h, s) =as(s)αc(h)

e−α

αc(h) +bs(s)αcc(h)

e−α

αcc(h)

[1rad

](5.20)

being the second term specially important for large angles α (close to 60) and:

as(s) = 0.42489 + 0.58371s− 0.082373s2 bs(s) = 0.055108− 0.095587s+ 0.056952s2

αc(h) = 0.62694

(Ethr(h)MeV

)−0.60590

αcc(h) = (10.509− 4.9644s)αc(h)

The angles α, αc and αcc are all measured in rad for consistency of expression (5.20). Even though the

parameterisation (5.20) was obtained for the range [5, 60], it is used for α > 60 as well since that is the

procedure followed in the Oine analytical erenkov model [46]. Nevertheless, the erenkov emission is

in principle very small at large α angles when compared to the emission in the parameterisation range.49

The main dierence between the detection of uorescence and erenkov light is in the fact that the

FD telescopes were designed to record the former and that the calibration takes the nitrogen uorescence

spectrum into account. So, the erenkov spectrum must be convoluted with the detector eciency ε(λ)

in the lter wavelength band [280, 430] nm. In mean terms, the eciency in the detection of erenkov

radiation is somewhat low:

ε =(

1280 nm

− 1430 nm

)−1 430 nm∑λn=280 nm

(1

λn − ∆λ2

− 1λn + ∆λ

2

)ε(λn) ' 0.45

where ∆λ = 5 nm. Nevertheless, the erenkov light may be studied by itself, rather than being considered

simply as noise to uorescence collection. Besides, in order to write down the density of direct erenkov

photons at the diaphragm, one needs the eective erenkov wavelength given the detector eciency ε(λ):

λeff =1ε

(1

280 nm− 1

430 nm

)−1 430 nm∑λn=280 nm

(1

λn − ∆λ2

− 1λn + ∆λ

2

)ε(λn)λn ' 356 nm

Finally,

dNdγ

dV(~r;Nmax, Xmax, X0, λGH) = Ne (X;Nmax, Xmax, X0, λGH) · LDF (X,R;Xmax) ·

·∫ ∞ln(Ethr(h)/MeV)

430 nm∑λn=280 nm

yγ

(E, h, λn −

∆λ2, λn +

∆λ2

)ε(λn) · fe(E, s)dlnE ·

· 1

Ndγ

dNγd

dΩ(α, h, s) ∆Ω (~r) · TR

(~r, ~reye, λ

eff

)· TM

(~r, ~reye, λ

eff

) [ γm3

](5.21)

When a shower propagates towards the detector, this is when χ0 <π2 , the direct erenkov light is

important and quite intense because it is highly collimated. However, in the other cases the fraction of

this kind of radiation is very small since A (α, h, s) falls quickly with α.

Scattered erenkov

The erenkov light emitted during shower development is scattered o by Rayleigh and Mie processes

and, consequently, the scattered erenkov radiation becomes important to consider in all geometry setups.

The rst step in the calculation of this contribution at the telescopes is the construction of the beam of

erenkov photons produced along the cascade. The beam is assumed to have dependence uniquely on

the slant depth X and the propagation direction is taken to be parallel to the shower axis. Therefore,

for each event the beam is given by:

Nγ (X,λ1, λ2;Nmax, Xmax, X0, λGH) =

∫ X

X0

Ne (X ′;Nmax, Xmax, X0, λGH) ·

·∫ ∞ln(Ethr(h)/MeV)

yγ (E, h′, λ1, λ2) fe(E, s′)dlnE · TR(X ′, X,

λ1 + λ2

2

)· TM

(X ′, X,

λ1 + λ2

2

)dX ′

ρ(h)(5.22)

Now, as in the previous sections, the density of Rayleigh scattered erenkov photons is:

dNRγ

dV(~r;Nmax, Xmax, X0, λGH) =

430 nm∑λn=280 nm

Nγ

(X,λn −

∆λ2, λn +

∆λ2

;Nmax, Xmax, X0, λGH

)·

·LDF (X,R;Xmax) · 1

NRγ

d2NRγ

dldΩ(~r, λn) ∆Ω (~r) · TR (~r, ~reye, λn) · TM (~r, ~reye, λn) · ε (λn)

[ γm3

](5.23)

50

where the LDF is not the same as in equations (5.18) and (5.21) since in principle the scattered erenkov

light does not present the same lateral distribution as the electrons in an EAS. The angular term1

NRγ

d2NRγ

dldΩ follows from equations (3.6) and (3.7):

1

NRγ

d2NRγ

dldΩ(~r, λ) =

316π

(1 + cos2θs

) ρ(h)XR

(400 nm

λ

)4 [1

m.stereorad

]being θs = α because the beam photons are considered to propagate parallel to the shower axis. The

Rayleigh scattered light presents lobes in the forward and backward directions, but the angular distribu-

tion is not exceedingly dierent from an isotropic one. Indeed, < 316π

(1 + cos2θs

)>= 3

16π

(1 + 1

2

)= 9

32π

and this value is close to 14π .

As for Mie scattered erenkov light, an equation similar to (5.23) is obtained:

dNMγ

dV(~r;Nmax, Xmax, X0, λGH) =

430 nm∑λn=280 nm

Nγ

(X,λn −

∆λ2, λn +

∆λ2

;Nmax, Xmax, X0, λ

)·

·LDF (X,R;Xmax) · 1

NMγ

d2NMγ

dldΩ(~r, λn) ∆Ω (~r) · TR (~r, ~reye, λn) · TM (~r, ~reye, λn) · ε (λn)

[ γm3

](5.24)

where, according to (3.9) and (3.10),

1

NMγ

d2NMγ

dldΩ(~r, λ) = aM · e−

θsθM

e− hhM

lM (λ)

[1

m.stereorad

]The above formula was implemented with aM = 0.891, θM = 26.7, hM = 1.2 km and lM = 14 km

as in chapter 3, but these values correspond to a mean Mie contribution. Indeed, when integrated in the

Auger Oine machinery, the method will use the values updated in a daily basis.

5.2.3 Spot and mercedes

While travelling from the diaphragm until the PMT camera (after reection in the mirror), the photons

undergo two dierent eects due to FD optics: the spot and the mercedes reection. Another important

issue would be the camera shadow, but that is already included in the calibration factor C370/ADC .

A photon coming from a given volume ik may be misplaced in the PMT camera according to the spot

presented in section 4.1.2 − the photon may jump into a pixel i′ 6= i which means it will be associated

with another volume i′k of the same time slot. This eect must be described accurately in order to

calculate the expected photon distribution at the PMTs. In the Monte Carlo integration, each of the

Nvol ·N1 randomly generated points ~r(q) corresponds to a certain direction(θ(q), φ(q)

)in the camera. For

each direction, N2 new directions(θ(r), φ(r)

)=(θ(q), φ(q)

)+(δθ(r), δφ(r)

)are generated according to the

simulated spot and associated to the respective volumes.

As for the mercedes stars, since its reectivity is non-unitary and the calibration constant C370/ADC is

determined by illuminating the whole FD camera uniformly, there must be a correction factor fcorr (θ, φ)

to take into account the sensitivities of the dierent parts of the PMTs. Following [59],

fcorr (θ, φ) =

0.87 if (θ, φ) in mercedes

1.08 otherwise(5.25)

51

Finally, one may write the expected number of detected photons for volume Vik:

Nγ,ik (Nmax, Xmax, X0, λGH) = Nγ,ik (Nmax, Xmax, X0, λGH) · 1N2

∑(θ(r),φ(r))∈pixel i

fcorr

(θ(r), φ(r)

)+∑j 6=i

Nγ,jk (Nmax, Xmax, X0, λGH) · 1N2

∑(θ(r),φ(r))∈pixel i

fcorr

(θ(r), φ(r)

)(5.26)

where the second term represents the contribution from neighbour pixels.

5.2.4 Expected and observed signals

Following the previous sections, the expected signal for a given volume ik is a sum of the dierent light

contributions:

Nγ,ik = Nfγ,ik + Nd

γ,ik + NRγ,ik + NM

γ,ik (5.27)

where multiple scattering processes are overridden. The respective observed signal Nγ,ik is also a sum of

those fractions and eventually some noise and is measured in equivalent 370 nm photons at the diaphragm

as explained in section 4.1.2. To monitor the relative behaviour of the expected and observed quantities

(Nγ,ik and Nγ,ik respectively), several quality estimators were used.

Firstly, consider a certain parameter pik − for instance, the slant depth of the central points of the

volumes, X (~rik). Binning the pik distribution, the expected-to-observed ratio is dened as

e/o (p) =Nγ (p)Nγ (p)

(5.28)

Although useful, the ratio does not take into account the errors associated to expected and observed

values. Nor does it allow a volume-to-volume comparison, i.e. between Nγ,ik and Nγ,ik. Thus, the χikestimator is applied as well:

χik =Nγ,ik −Nγ,ik√

σ2(Nγ,ik

)+ σ2 (Nγ,ik)

(5.29)

being σ2(Nγ,ik

)= N2

γ,ik

(1√Nin,ik

+ 1√Nvol·N1

)2

(confer equations (5.5) and (5.15)) and

σ2 (Nγ,ik) = σ2(C370/ADC · (NADC,i (tk)−Nped,i)

)=√|Nγ,ik|

2

+ C2370/ADC · σ

2 (Nped,i)

' |Nγ,ik|+ C2370/ADC · σ

2 (Nped,i)

Nγ,ik is a sum of a signal Ns,ik and a background Nb,ik, where the pedestal Nped,i appearing on

equation (4.7) has already been subtracted to obtain Nb,ik. To nd the distribution of this background,

the negative-valued signals Nγ,ik in events recorded throughout July 2006 are tted to a gaussian as

shown in gure 5.10. Thus, the background Nb,ik is roughly a normalised gaussian distribution Pb with

mean µb = 0 and standard deviation σb = 25: Pb(x) = Gaus (x;µb, σb) ≡ 1√2πσb

e− (x−µb)

2

2σ2b . Besides, the

term C2370/ADC · σ

2 (Nped,i) is equal to the standard deviation σb and, hence, σ2 (Nγ,ik) ' |Nγ,ik|+ 252.

Using χik, a χ2 =∑i,k χ

2ik is dened for each event. Given Nvol and the number of t parameters

Npar, the number of degrees of freedom is Ndf = Nvol −Npar and χ2/Ndf represents an estimator of the

concordance between observation and expectation.

52

Figure 5.10: Gaussian t to the Nγ,ik distribution for Nγ,ik < 0. All data collected throughout July 2006was used to produce the plot.

However, when Nγ,ik and Nγ,ik are small, they do not follow gaussian distributions and χ2 is not valid.

So, one must proceed to a likelihood function, which is the product of the probabilities of observing Nγ,ikwhen Nγ,ik is the expected value:

L =∏i,k

P1

(Nγ,ik, Nγ,ik

)(5.30)

Since Nγ,ik = Ns,ik + Nb,ik, the probability P1

(Nγ,ik, Nγ,ik

)is the convolution of the probabilities

of observing Ns,ik as signal and Nb,ik = Nγ,ik −Ns,ik as background:

P1

(Nγ,ik, Nγ,ik

)=

Nγ,ik∑Ns,ik=0

Ps

(Ns,ik, Nγ,ik

)· Pb (Nγ,ik −Ns,ik) (5.31)

For Nγ,ik ≤ 10, Ps is a normalised Poisson distribution of mean νs = Nγ,ik: Ps (x, νs) = Poisson (x; νs) ≡νxs e−νs

x! . When Nγ,ik > 10, Ps is well described by a gaussian distribution with µs = σ2s = νs and, therefore,

(5.31) becomes a convolution of two gaussians:

P1

(Nγ,ik, Nγ,ik

)=

∫ Nγ,ik

0

Gaus

(Ns,ik; Nγ,ik,

√Nγ,ik

)·Gaus (Nγ,ik −Ns,ik; 0, 25) dNs,ik

= Gaus

(Nγ,ik; Nγ,ik,

√252 + Nγ,ik

)=

1√2π(

252 + Nγ,ik

)e− (Nγ,ik−Nγ,ik)2

2(252+Nγ,ik)(5.32)

After determining each P1

(Nγ,ik, Nγ,ik

), the likelihood function (5.30) is more easily analysed in

terms of its logarithm:

lnL =∑i,k

lnP1

(Nγ,ik, Nγ,ik

)(5.33)

5.3 Validation

The validation of both geometry and prole 3D reconstructions is performed using the low energy simu-

lation from the Lecce-L'Aquilla Auger group [60, 75]. The simulation consists of 1017, 1017.5, 1018 and

1018.5 eV protons showers generated with CORSIKA and then integrated in the PAO detector simu-

lation chain [46]. The shower cores were uniformly distributed on the ground in the eld of view of

telescope 4 from Los Leones eye and no Mie scattering was considered. Moreover, all light arriving at53

the telescopes was spread according to the Góra cumulative function (5.12) in order to mimic the shower

lateral prole. The longitudinal proles, on the other hand, follow a shifted Gaisser-Hillas formula

Ne (X −X1;Nmax, Xmax, X0, λGH), where X1 is the depth of the rst interaction point.

To illustrate the results from the 3D reconstruction some events are analysed xing Nmax, Xmax,

X0, X1 and λGH to the respective simulation values, considering no Mie attenuation factor TM nor Mie

scattered erenkov light and using the Góra LDF for uorescence, direct and scattered erenkov light.

The latter assumption is probably not true, but reproduces what was done in the detector simulation

chain when spreading all light according to the Góra cumulative function. Figures 5.11(a) and 5.11(b)

represent the observed (grey shaded area) and expected (red line) signals as functions of the depth-like

variables XNP ≡ X (~rnp,ik) and RCP ≡ R (~rik) for a 1018.5 eV simulated event. The green and blue lines

indicate, respectively, the direct and Rayleigh scattered erenkov contributions for the total expectation

in red. The concordance between observed and expected behaviours is evident. The event triggered

telescopes 3 and 4 from Los Leones eye and that is the reason of the peak at X ∼ 675 gcm−2: in this

region the signals detected by both telescopes add up. As for the depression at X ∼ 950 gcm−2, it is

due to the fact that two pixels did not pass the trigger requirements. Besides, notice that the observed

values may fall below 0 since the PMT pedestal uctuates; however, the expected quantities are always

non-negative because they are calculated as a physical number of photons.

In order to monitor the concordance between observed and expected signals for the referred event,

the ratio e/o is presented in gures 5.11(c) and 5.11(d), while gures 5.11(e) and 5.11(f) show the

χik distribution and the dependence of the mean lnP1

(Nγ,ik, Nγ,ik

)(confer equation (5.33)) on RCP .

Although with some uctuations, e/o is reasonably close to 1 in the whole range of X and R. As for χik,

one would expect a gaussian distribution centred in 0 and with unitary standard deviation, but there are

clearly some volumes that deviate from this behaviour. Nevertheless, most volumes correspond to almost

null χik values, which is a reection of the empirically good concordance shown in gures 5.11(a) and

5.11(b). Finally, the mean values of lnP1

(Nγ,ik, Nγ,ik

), plotted in gure 5.11(f), are upper limited by the

probability P1

(Nγ,ik, Nγ,ik

)that depends on Nγ,ik, but yields approximately −lnP1

(Nγ,ik, Nγ,ik

)'

4.1− 4.2. Furthermore, the highest RCP are associated to volumes where the expected signal is close to

0 and the observed one is in the order of the background, i.e. σb = 25 − the plateau at greater RCP has

then a value of ∼ lnP1 (25, 0) = −4.64.

Figure 5.12 illustrates a 1018.5 eV simulated event where the observed signal is dominated by Rayleigh

scattered erenkov light. In face of the narrow slant depth window overviewed by the telescope in

this case, it is encouraging that the 3D method predicts to a quite good approximation the observed

distribution, while the KG reconstruction measures an energy somewhat lower than that simulated (∼1017.89 eV).

54

(a) (b)

(c) (d)

(e) (f)

Figure 5.11: In (a) and (b), the dependence of the observed (grey shaded area) and expected (red line)signals on XNP and RCP are presented for the 1018.5 eV simulated event SD 78 (from job 0). The directand Rayleigh scattered erenkov expected fractions are signaled by the green and blue lines respectively.This event presents Rp ' 7.6 km and χ0 ' 90.8. The behaviour of the ratio e/o with XNP (c) and

RCP (d), the χik distribution (e) and the dependence of the mean lnP1

(Nγ,ik, Nγ,ik

)on RCP (f) are

also shown for the same event.

55

Figure 5.12: Observed (grey shaded area) and expected (red line) signals for the 1018.5 eV simulated eventSD 14 (from job 0). As in gure 5.11, the green and blue lines represent direct and scattered erenkovcontributions at the telescopes respectively. This event presents Rp ' 1.6 km and χ0 ' 139.3.

56

Chapter 6

Lateral prole measurements

In order to perform a measurement of the shower lateral prole, one must ensure in the rst place

that no signicant bias is being introduced by the 3D reconstruction. This is accomplished through a

systematic study of the method proposed in the last chapter using a sample of several simulated events.

But to proceed to an analysis of real data, where a priori there is no indication on the parameters

Nmax, Xmax, X0 or λGH as in the simulation, one needs to start from the parameters reconstructed

by the KG (standard) procedure. Using these parameters, the next step is to understand the lateral

sensitivity achieved by the 3D reconstruction and dene the set of quality cuts required to have reasonable

sensitivities.

6.1 Systematic study of the 3D reconstruction

The comparison between expectation and observation, already performed for isolated events in sec-

tion 5.3, is presented in gure 6.1 for 30 1018.5 eV events from the simulation of the Lecce-L'Aquilla

group. The expected signals computed with both the simulated and the KG-reconstructed parameters

(Nmax, Xmax, X0, λGH) are plotted. The sample of events was selected requiring 3D and KG prole re-

constructions, χ3D0 ≥ 45 and

∣∣log10Esim − log10E

KG∣∣ ≤ 0.5. The latter quality cut is needed to prevent

the events misreconstructed by the KG method from contaminating the 3D analysis. When processing

real data this cut is obviously inappropriate.

Firstly, the results with the simulated parameters seem to describe inaccurately the depths below

∼ 550 gcm−2 as evident in gures 6.1(a) and 6.1(c), but this feature is absent when using the KG

reconstruction. A misinterpretation or bad handling of the simulation depth parameters is probably the

reason for such dierence.

Another interesting feature is shown in gure 6.1(d): the ratio e/o corresponding to the use of simu-

lated parameters is slightly above 1 for RCP . 10 gcm−2 and below afterwards. Such a behaviour may

be explained by the s-dependence of the Góra LDF. Indeed, in the denition s = 31+2Xmax/X

one uses

the shower maximum Xmax from simulation and X from the reconstructed method, thus introducing a

bias. So, it is more adequate to use the KG-reconstructed Xmax − the result is presented in red in gure

6.1(d) and is more at and close to 1. Notice that above RCP ∼ 25 gcm−2, the wide uctuations on the

ratio e/o are due to the low number of detected volumes.

In what refers to gures 6.1(e) and 6.1(f), it is clear that the use of the KG parameters instead of the

simulation ones reduces the χ2/Ndf values. Besides, the χik distribution seems slightly better as well.

57

(a) (b)

(c) (d)

(e) (f)

Figure 6.1: In (a) and (b), the dependence of the observed (grey shaded area) and expected (red line)signals on XNP (a) and RCP (b) for 30 1018.5 eV simulated events. The direct and Rayleigh scatterederenkov expected fractions are signaled by the green and blue lines respectively. Dashed lines refer toexpected signals calculated with the KG parameters, while solid lines represent the use of the simulatedparameters. The value of RM was xed to 9.6 gcm−2 to produce these plots. The behaviour of the ratioe/o with XNP (c) and RCP (d), the χik distribution (e) and the χ2/Ndf values per event (f) are alsoshown for the same simulation set. Notice that in plot (d), for RCP & 25 gcm−2, the quantity of detectedvolumes is low and thus there are signicant uctuations.

58

(a) (b)

Figure 6.2: The behaviour of the estimators χ2 (RM ) /Ndf (in black) and lnL (RM ) (in red) with theeective parameter RM for the close simulated event SD 78 (from job 0) (a) and the distant one SD 10(from job 0) (b).

6.2 Lateral sensitivity

Analysing the lateral distribution functions presented in section 5.2.1 such as (5.13), one concludes that

a straightforward way to study the lateral sensitivity is to dene RM as an eective parameter. By

varying RM and computing the respective estimators χ2 (RM ) /Ndf and lnL (RM ), a preliminary t

is performed to nd the best parameter value R∗M for which χ2 (R∗M ) /Ndf and lnL (R∗M ) are mini-

mal. The uncertainties σ+ and σ− on R∗M are then xed by an unitary variation ∆(χ2 (RM ) /Ndf

)≡

χ2 (RM ) /Ndf − χ2 (R∗M ) /Ndf = 1, that corresponds to ∆lnL (RM ) ≡ lnL (RM )− lnL (R∗M ) = Ndf/2 −this correspondence is easily shown by replacing P1 in equation (5.33) by a gaussian distribution.

Figure 6.2 shows the study of the lateral sensitivity for two 1018.5 eV simulated events using the KG

reconstructed parameters. The events dier essentially on the mean distance of the central points of the

volumes to the respective eye, dCP−eye: event SD 78 (from job 0) presents dCP−eye ' 8.4 km, while

in event SD 10 (from job 0) dCP−eye ' 26 km. It is clear that the closer event enables a reasonable

determination of the eective parameter RM(RM = 9.6 + 4.1− 3.0 gcm−2

), whereas in the other one

the 3D reconstruction has no sensitivity(RM = 15.6+ > 15.0− 15.6 gcm−2

).

Notice that, in regard of the discussion in section 5.2.4 about the dierences between χ2/Ndf and lnL,the black and red curves in gure 6.2 do not present exactly the same behaviour as expected, but they

yield similar sensitivity results in those examples.

6.3 Data

The lateral sensitivity was studied in real data collected throughout July 2006 requiring KG and 3D

prole reconstructions and χ3D0 ≥ 45. Since in the data Mie scattering is important, the analysis takes

into account both the Mie attenuation factor (3.12) and the Mie scattered erenkov light fraction (5.24).

Moreover, in order to analyse events where the 3D method is potentially more sensitive to the lateral

distribution, the empirical cut (5.3) (with d∗CP−eye = 5 km and d∗CP−eye = 10 km) is applied to select

events simultaneously close to the detector and with high energies.

The simulation from the Lecce-L'Aquilla group does not contain many events passing the empirical

cut (5.3): only 3 showers for the cut with d∗CP−eye = 5 km and 15 with d∗CP−eye = 10 km. Consequently,

there is the need to generate a signicant number of events in the region where the 3D method is more

accurate. CORSIKA showers are being run and integrated in the Auger Oine detector simulation

59

at dierent distances away from the FD eyes. Nevertheless, the 15 showers passing the cut (5.3) with

d∗CP−eye = 10 km are presented in gure 6.3. Furthermore, gure 6.4 shows the expected and observed

signals of 50 showers passing the cut (5.3) with d∗CP−eye = 5 km, while gure 6.5 refers to 50 showers

passing the same cut with d∗CP−eye = 10 km. The most evident and interesting feature is that presented

by gures 6.3(d), 6.4(d) and 6.5(d): the ratio e/o decreases steeply with RCP . Indeed, both simulation

and data seem to present a more sparse lateral distribution than expected, with lower signals near the

axis(RCP . 5 gcm−2

)and higher ones away from it. The reason for this behaviour is still not clear.

Another remark should be made in regard of the χik distributions in plots 6.3(e), 6.4(e) and 6.5(e).

In the rst place, those distributions present only the volumes where Nin,ik 6= 0 (for consistency of the

χik value). Besides, the t presented in gure 6.3(e) is not totally rigorous since there seem to exist two

dierent populations of volumes ik. Anyhow, and taking into account 6.4(e) and 6.5(e) as well, the mean

value of χik is negative, which means that Nγ −Nγ is a fairly constant positive value. Such shift is not

yet fully understood but is probably linked to the way the observed number of photons is obtained.

As in the simulation analysis, the sensitivity to the RM eective parameter was studied in two real

showers and is shown in gure 6.6. In both cases 9.6 gcm−2 seems to be included in the measurement.

60

(a) (b)

(c) (d)

(e) (f)

Figure 6.3: In (a) and (b), the dependence of the observed (grey shaded area) and expected (red line)signals on XNP and RCP are presented for the 15 events from the Lecce-L'Aquilla simulation and passingthe cut (5.3) with d∗CP−eye = 10 km. The direct and Rayleigh scattered erenkov expected fractions aresignaled by the green and blue lines respectively. The behaviour of the ratio e/o with XNP (c) and RCP(d), the χik distribution (e) and the dependence of the mean lnP1

(Nγ,ik, Nγ,ik

)on RCP (f) are also

shown for the same simulation sample.

61

(a) (b)

(c) (d)

(e) (f)

Figure 6.4: In (a) and (b), the dependence of the observed (grey shaded area) and expected (red line)signals on XNP (a) and RCP (b) for 50 events collected in June/July 2006 and passing the cut (5.3) withd∗CP−eye = 5 km. The direct erenkov expected fraction is signaled by the green line, while Rayleighand Mie scattered erenkov components are represented in blue and magenta respectively. The value ofRM was xed to 9.6 gcm−2 to produce these plots. The behaviour of the ratio e/o with XNP (c) and

RCP (d), the χik distribution (e) and the dependence of the mean lnP1

(Nγ,ik, Nγ,ik

)on RCP (f) are

also shown for the same data set. Notice that in plot (c) the ratio e/o grows for XNP & 1150 gcm−2,possibly because the Mie light fraction (5.24) is over estimated in this region − recall that the componentshown in magenta is only a mean one. Besides, the statistics in that region is small and consequentlylarge uctuations are expected.

62

(a) (b)

(c) (d)

(e) (f)

Figure 6.5: In (a) and (b), the dependence of the observed (grey shaded area) and expected (red line)signals on XNP (a) and RCP (b) for 50 events collected in July 2006 and passing the cut (5.3) withd∗CP−eye = 10 km. The direct erenkov expected fraction is signaled by the green line, while Rayleighand Mie scattered erenkov components are represented in blue and magenta respectively. The value ofRM was xed to 9.6 gcm−2 to produce these plots. The behaviour of the ratio e/o with XNP (c) and

RCP (d), the χik distribution (e) and the dependence of the mean lnP1

(Nγ,ik, Nγ,ik

)on RCP (f) are

also shown for the same data set.

63

(a) (b)

Figure 6.6: The behaviour of the estimators χ2 (RM ) /Ndf (in black) and lnL (RM ) (in red) with theeective parameter RM for two real showers recorded in July 2006. Plot (a) corresponds to event SD2425381 with dCP−eye ' 4.7 km, while (b) corresponds to event SD 2425226 with dCP−eye ' 7.5 km.

64

Chapter 7

Conclusion and prospects

The original work presented in this thesis was entirely developed in the context of the Pierre Auger Ob-

servatory, that is today the leading experiment in the eld of ultra high energy cosmic rays. The technical

documents and tools produced by the PAO Collaboration were extremely valuable in understanding the

physics behind cosmic ray detection and in event analysis. Moreover, the data and simulation sets used

were also provided by the Collaboration.

A three dimensional method to reconstruct EAS from uorescence and hybrid data was conceived

for event analysis at the PAO. The innovative idea of this method consists in using the sampling time

at the telescopes as a third dimension in space. Hence, instead of assuming line propagation in shower

development, all the available information recorded by the FD telescopes is used to locate the shower

in space. An iterative method for the geometry reconstruction of extensive air showers was then built

based upon inertia considerations and a 3D event display was developed using existing software. Once

the geometry is xed, the reconstruction of the shower 3D prole follows. Such reconstruction was

accomplished through Monte Carlo calculations by considering the detailed optics of the telescopes and

by taking into account uorescence and (direct and scattered) erenkov light.

The 3D method constitutes a powerful tool in the reconstruction of EAS and related studies and it

was validated using the proton simulation from the Lecce-L'Aquilla group. Furthermore, the sensitivity of

the method to the eective parameter RM was performed in order to get a rst preliminary measurement

of the shower lateral prole. Both simulation and data samples were considered in that analysis.

The tool developed allows a systematic study of EAS features and a comparison between existing

simulation and data − recall that the Pierre Auger Observatory has already further exposure than

AGASA. Firstly, a more consistent analysis of larger simulation and data samples may open the possibility

of understanding the consequences of dierent hadronic models in shower development. Furthermore, the

lateral distribution of direct and scattered erenkov light must be understood and parameterised, but

such task requires a three dimensional simulation yet to be implemented. Another relevant channel is the

search for new physics by identifying particularly 'fat' or irregular events that may correspond to double

bangs, decays of new particles or other exotic phenomena.

65

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