Monocular measurement of the ultra-high energy cosmic ray spectrum

The Telescope Array Project was designed to observe cosmic rays with energies greater than 1018 eV. Its goals are to study the physics of cosmic rays by measuring their anisotropy, composition, and energy spectrum. This work makes a monocular measurement of the ultra high energy cosmic ray spectrum and analyzes the physics produced from that spectrum. The flux of cosmic rays observed on Earth follows a power law over 12 decades in energy and 32 decades in flux. At the highest energies, the spectrum has detailed structure. Studying these features can tell us about the astrophysics of the production and propagation of cosmic rays. First, it can tell us about the sources of cosmic rays such as they capable of producing a power law spectrum and the maximum energy of cosmic rays that they can produce. Second, the acceleration mechanisms that can boost cosmic rays to ultra high energies can be studied. Third, the spectral features themselves can tell us about their possible cause for formation. For example, the ankle feature in the ultra high energy regime can tell us if it is the galactic-extragalactic transition or if it is due to e+e− pair production. Fourth, the energy losses that cosmic rays incur can tell us about their physical interactions during propagation. Studying the physics of the cosmic ray spectrum in the ultra high energy regime with data from the Telescope Array Project is the goal of this analysis. The Telescope Array Project consists of three fluorescence detectors overlooking an array of 507 scintillation surface detectors. Due to their extremely low flux at these energies, cosmic rays can only be observed indirectly via an extensive air shower produced when they collide with the nucleus of an atom in the Earth's atmosphere. These charged secondary particles produce fluorescence light. The array of surface detectors observes the lateral footprint of the extensive air shower when it reaches the ground. The fluorescence detectors observe the longitudinal profile of this fluorescence light. This thesis analyzes the data from one of the fluorescence detectors, Middle Drum, using a different geometry reconstruction technique, the Time versus Angle geometry. The results of this analysis show an ultra high energy cosmic ray spectrum that is consistent with the results previously published by the High Resolution Fly's Eye (HiRes) experiment, the Telescope Array surface detectors, and other experiments in this energy region. Due to insufficient statistics at this date, the GZK cutoff cannot be confirmed in this analysis, but a fit shows the cutoff to be at log10 E (E/eV) = 19.56 ± 0.36, with a spectral index after the cutoff of -3.86 ± 2.0. This is within the range determined previously by other measurements. This analysis shows that the feature known as the ankle occurs at log10 E (E/eV) = 18.63 ± 0.09, with a spectral index of -3.27 ± 0.07 before the ankle and a spectral index of -2.81 ± 0.10 after the ankle. The normalized log likelihood per degree of freedom is 0.90. The ankle is observed at the 4−5! confidence level. The fit to the ankle is also in excellent agreement with previous measurements, and even more remarkable given that some other measurements use different techniques. While this study cannot tell us information about the sources or the acceleration mechanisms of cosmic rays, it does show us a feature and tell us about energy losses during propagation. The dip at the ankle is clearly visible in the spectrum. The results of this study are consistent with the energy loss model of extragalactic protons interacting with the cosmic microwave background radiation and supports the idea that the ankle is excavated due to e+e− pair production. The location of the ankle at a threshold greater than for e+e− pair production supports that the ankle is a composite feature where the redshift energy losses begin to dominate the e+e− pair production losses. The location of the ankle also implies that sources at larger distances than the GZK cutoff contribute to its formation.

MONOCULAR MEASUREMENT OF THE ULTRA-HIGH ENERGY COSMIC RAY SPECTRUM by Priti Dhanesh Shah A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Department of Physics and Astronomy The University of Utah December 2012 Copyright !c Priti Dhanesh Shah 2012 All Rights Reserved The Univers i ty of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Priti Dhanesh Shah has been approved by the following supervisory committee members: Gordon B. Thomson , Chair 7/31/2012 Date Approved Orest G. Symko , Member 7/28/2012 Date Approved Behrouz Farhang , Member 7/28/2012 Date Approved Mikhail E. Raikh , Member Date Approved Stephan L. LeBohec , Member Date Approved and by David B. Kieda , Chair of the Department of Physics and Astronomy and by Charles A. Wight, Dean of The Graduate School. ABSTRACT The Telescope Array Project was designed to observe cosmic rays with energies greater than 1018 eV. Its goals are to study the physics of cosmic rays by measuring their anisotropy, composition, and energy spectrum. This work makes a monocular measurement of the ultra high energy cosmic ray spectrum and analyzes the physics produced from that spectrum. The flux of cosmic rays observed on Earth follows a power law over 12 decades in energy and 32 decades in flux. At the highest energies, the spectrum has detailed structure. Studying these features can tell us about the astrophysics of the production and propagation of cosmic rays. First, it can tell us about the sources of cosmic rays such as they capable of producing a power law spectrum and the maximum energy of cosmic rays that they can produce. Second, the acceleration mechanisms that can boost cosmic rays to ultra high energies can be studied. Third, the spectral features themselves can tell us about their possible cause for formation. For example, the ankle feature in the ultra high energy regime can tell us if it is the galactic-extragalactic transition or if it is due to e+e− pair production. Fourth, the energy losses that cosmic rays incur can tell us about their physical interactions during propagation. Studying the physics of the cosmic ray spectrum in the ultra high energy regime with data from the Telescope Array Project is the goal of this analysis. The Telescope Array Project consists of three fluorescence detectors overlooking an array of 507 scintillation surface detectors. Due to their extremely low flux at these energies, cosmic rays can only be observed indirectly via an extensive air shower produced when they collide with the nucleus of an atom in the Earth's atmosphere. These charged secondary particles produce fluorescence light. The array of surface detectors observes the lateral footprint of the extensive air shower when it reaches the ground. The fluorescence detectors observe the longitudinal profile of this fluorescence light. This thesis analyzes the data from one of the fluorescence detectors, Middle Drum, using a different geometry reconstruction technique, the Time versus Angle geometry. The results of this analysis show an ultra high energy cosmic ray spectrum that is consistent with the results previously published by the High Resolution Fly's Eye (HiRes) experiment, the Telescope Array surface detectors, and other experiments in this energy region. Due to insufficient statistics at this date, the GZK cutoff cannot be confirmed in this analysis, but a fit shows the cutoff to be at log10 E (E/eV) = 19.56 ± 0.36, with a spectral index after the cutoff of -3.86 ± 2.0. This is within the range determined previously by other measurements. This analysis shows that the feature known as the ankle occurs at log10 E (E/eV) = 18.63 ± 0.09, with a spectral index of -3.27 ± 0.07 before the ankle and a spectral index of -2.81 ± 0.10 after the ankle. The normalized log likelihood per degree of freedom is 0.90. The ankle is observed at the 4−5! confidence level. The fit to the ankle is also in excellent agreement with previous measurements, and even more remarkable given that some other measurements use different techniques. While this study cannot tell us information about the sources or the acceleration mech-anisms of cosmic rays, it does show us a feature and tell us about energy losses during propagation. The dip at the ankle is clearly visible in the spectrum. The results of this study are consistent with the energy loss model of extragalactic protons interacting with the cosmic microwave background radiation and supports the idea that the ankle is excavated due to e+e− pair production. The location of the ankle at a threshold greater than for e+e− pair production supports that the ankle is a composite feature where the redshift energy losses begin to dominate the e+e− pair production losses. The location of the ankle also implies that sources at larger distances than the GZK cutoff contribute to its formation. iv For my family, Dhanesh, Smita, Janak, Pranav, Tamara, Peyton " Thank you for all of your love and support." CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix CHAPTERS 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Cosmic Ray Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1.1 The Extensive Air Shower Overview . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1.2 The Hadronic Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1.3 The Electromagnetic Cascade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.2 Galactic Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.3 Extragalactic Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Composition and Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2. PHYSICS OF THE UHECR SPECTRUM . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Acceleration Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.1 Statistical Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.2 Shock Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Spectral Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.1 Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.2 Ankle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.2.1 Possible Cause: Galactic-Extragalactic Transition . . . . . . . . . . . . . 30 2.3.2.2 Possible Cause: Pair Production . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4 Energy Loss Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4.2 Extragalactic Proton Propagation Model . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4.2.1 Correlation of Distance to Source . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.5 Mixed Composition Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.6 Other Relevant Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.8 Disappointing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.8.1 Fit to Spectrum Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.8.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.9 Energy Scale Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3. THE TELESCOPE ARRAY EXPERIMENT . . . . . . . . . . . . . . . . . . . . . . . 54 3.1 Measuring the UHECR Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2 Overview of The Telescope Array Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3 Surface Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3.1 Calibration and Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3.2 Lateral Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3.3 Energy Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.4 Fluorescence Detector Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.5 The Black Rock Mesa and Long Ridge Fluorescence Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.5.1 Black Rock Mesa and Long Ridge Calibration and Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.6 The Middle Drum Fluorescence Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.6.1 Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.6.2 Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.6.3 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.6.4 RXF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.6.5 Cloud Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.6.6 Running time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4. DATA ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.1 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.1.1 Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.1.2 Preprocessing: On-Time and Calibration . . . . . . . . . . . . . . . . . . . . . . . . 84 4.1.3 Pass 0: Time Matching and Event Building . . . . . . . . . . . . . . . . . . . . . . 84 4.1.4 Pass 1: Event Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.1.5 Pass 2: Rayleigh Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.1.6 Pass 3: Shower Detector Plane Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.2 Laser Calibration Removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4 Pass 4: Geometry Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4.1 Linearizing the Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.4.2 Additional Good Tube Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.5 Pass 5: Profile and Energy Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.5.1 Cerenkov Light Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.5.2 Rayleigh Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.5.3 Aerosol scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.5.4 Ozone Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.5.5 Profile Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.5.6 Energy Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.6 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5. THE MONTE CARLO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.1 Philosophy of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2 The EAS Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.3 Middle Drum Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.4 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.5 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 vii 5.6 Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6. DATA/MONTE CARLO COMPARISON . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.1 Quality Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7. RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.1 Calculating the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.2 The Energy Spectrum Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.3 Analysis of the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 7.3.1 Fit for the Ankle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 7.3.2 Comparison of the Fit to Other Experiments . . . . . . . . . . . . . . . . . . . . . 145 7.4 Comparison of Time versus Angle to Profile-Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.5 Contribution of Time versus Angle Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 146 8. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 viii LIST OF FIGURES 1.1 The energy spectrum of cosmic rays. Note that this is a log-log plot and covers a very wide range in both flux and energy. Also note that the plot follows a power law with a slope of about 3. Therefore, for each factor of 10 increase in energy, the flux falls by about 1000. Reprinted with permission from [43]. . . . 10 1.2 The spectrum of cosmic rays. Here the flux has been multiplied by E3 in order to take out the underlying slope and highlight the detailed structure of the spectrum. Reprinted with permission from [25]. . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 A sketch indicating some of the initial particle interactions in the development of an extensive air shower. Reprinted with permission from [36]. . . . . . . . . . . 12 1.4 The Heitler branching model of the electromagnetic cascade of the EAS. In this model, photons e+e− pair produce and electrons and positrons Bremsstrahlung until the energy falls below the critical energy to produce further particles. Reprinted with permission from [43]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Cosmic ray flux times E2.5 vs Energy as measured by KASCADE and inter-preted using the SIBYLL model. Note that the proton flux turns over around 3 × 1015 eV. Reprinted with permission from [44]. . . . . . . . . . . . . . . . . . . . . . 14 1.6 KASCADE-Grande's energy spectrum showing a possible Fe knee located at log10 E (E/GeV) = 7.9. The top graph is E2.5· J versus E. The bottom graph is the flux, scaled such that the features are more clearly visible, versus the log of the energy. A decline in the spectrum ends around 107.2 GeV and is not understood. At about 7.9 × 1016 eV, there is a cutoff. KASCADE-Grande claims that this is the Fe knee. Reprinted with permission from [44]. . . . . . . . 15 1.7 Akeno's analysis of their data showing no signs of an Fe knee. The Akeno array is represented by the open squares. Akeno reports what it calls a smooth connection from the knee region to about 1020 eV, albeit with some energy scale differences in the 1019 eV decade. Reprinted with permission from [73]. 16 1.8 Several experiments show evidence of a 2nd knee. The left graph shows that the Yakutsk [65], Akeno [73], Fly's Eye [27], and HiRes-MIA [19] experiments all show a flat portion in energy before a cutoff is observed. If the flat portions of all the spectrums are laid on top of each other by rescaling the energy, as shown in the right graph, all of the experiments demonstrate a cutoff, which is the 2nd knee. Reprinted with permission from [55]. . . . . . . . . . . . . . . . . . . . 17 1.9 Measurement of the chemical composition as measured by the HiRes-MIA and HiRes Stereo experiments. The mean depth of shower maximum (< Xmax >) as a function of energy is plotted. The bars show the predicted elongation rate (evolution development as a function of energy) for two models. The HiRes-MIA measurement indicates a changing composition from heavy to light between 1017 eV and 1018 eV. Above 1018 eV, the HiRes Stereo measurement indicates a constant light composition consistent with protons. Reprinted with permission from [19]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.10 A cartoon indicating the transition between the galactic and extragalactic contributions to the cosmic ray flux. The galactic contribution to the flux decreases, and while the extragalactic contribution to the flux increases, the overall flux is still decreasing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.11 The cosmic ray spectrum above 1017 eV as measured by the HiRes experiment and the Telescope Array Project. The spectrum has been multiplied by E3 to take out the predominant slope and show the detailed structure. The ankle is seen at log10 (E/eV) = 18.6, and the GZK cutoff is seen around 6 × 1019 eV. Reprinted with permission from [63]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.12 The cosmic ray spectrum above 1017 eV as measured by the HiRes and AGASA experiments. The HiRes experiment observes the ankle at log10 (E/eV) = 18.6 and the GZK cutoff around 6 × 1019 eV. The AGASA experiment does not observe the GZK cutoff. Reprinted with permission from [63]. . . . . . . . . . . . . 21 1.13 The energy loss mechanisms during propagation for cosmic ray protons. At the highest energies, the dominant energy loss mechanism is due to photo-pion production at 10's of Mpc. Pair production dominates at about 2000 Mpc at 1018.5 eV. Redshift becomes important at approximately 3000 Mpc at all energies. Reprinted with permission from F. Aharonian and [49]. . . . . . . . . . . 22 1.14 Energy loss due to spallation for iron nuclei. The axes are attenuation length versus log(Lorentz boost). At the highest Lorentz boosts, photo-erosion with the CMBR is the dominant energy loss mechanism. Reprinted with permission from D. Allard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1 Magnetic field strength versus size of possible UHECR sources. Objects below the diagonal lines do not have a sufficient combination of magnetic field strength and size to accelerate protons or iron nuclei to ultra high energies. The velocity of the shock wave, or efficiency of the acceleration mechanism, is represented by ". As seen fromthe plot, active galactic nuclei (AGN), neutron stars, radio galaxy (RG) lobes, and colliding galaxies are the best candidates for sources for UHECRs. Reprinted with permission from [43]. ........... 40 2.2 The total #p interaction cross section. The cross section starts with the $0 production threshold at $s =1.078 GeV. There is a rapid increase in cross section up to $s =1.232 GeV, which corresponds to the #+ resonance. The average cross section is ! = 0.12 mb. Reprinted with permission from [43]. . . 41 x 2.3 The cosmic microwave background radiation photon density. The total photon density is 412 photons/cm3, and the mean of the distribution is 6.4 ×10−4 eV. The threshold energy for photo-pion production for for the GZK cutoff to occur at 5 ×1019 eV is about 13.4 ×10−4 eV. Thus, most of the photo-pion production is due to the tail of the blackbody spectrum. Reprinted with permission from [43]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.4 The cosmic ray spectrum above 1018 eV as measured by the Pierre Auger Observatory compared to the spectra of HiRes-I and HiRes-II. The plot shows the fractional difference between the data and a spectrum with a spectral index of 2.6. The data from the HiRes experiments are shown by the open and closed blue triangles. The Auger Collaboration believes that there is an abrupt change in the spectral index around 4 × 1018 eV and a gradual suppression of the flux above 3 ×1019 eV. Reprinted with permission from F. Schussler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5 The cosmic ray spectrum above 1018 eV as measured by the HiRes and the Middle Drum Telescope Array Fluorescence Detectors. The ankle and the GZK cutoff feature are clearly seen. Reprinted with permission from [5]. . . . . 44 2.6 Modification factors for different nuclei. The left figure shows the modification factor for protons and iron nuclei, while the right figure shows the modification factor for protons and helium. Curves % = 1 corresponds to redshift losses in both plots. The proton modification factors are given by Curves 1 (redshift and pair production energy losses) and by Curves 2 (total energy losses). The nuclei modification factors are given by Curves 3 (redshift and pair production energy losses) and by Curves 4 (total energy losses including photodissocia-tion). The calculation of the modification factor for iron and helium nuclei show that even a small admixture of any nuclei is not in good agreement with the observed proton dip. The fraction of nuclei in the primary flux should be less than 10-20%. Reprinted with permission from V. Berezinsky and A. Gazizov. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.7 Predicted dip in comparison with the HiRes data (left) and TA (right). The straight line in both plots correspond to redshift losses, Curves %ee correspond to redshift and pair production energy losses, and Curves %tot correspond to all energy losses. Both HiRes and TA data fit well to the model. Data above %(E) % 1 shows that there is another component of cosmic rays, which is the galactic component. Reprinted with permission from [14] and V. Berezinsky. 46 2.8 Predicted dip in comparison with Auger data in 2007 and 2010. The straight line corresponds to redshift losses in both plots, Curve %ee corresponds to redshift and pair production energy losses, and Curve %tot corresponds to all energy losses. There is a mediocre fit in the data in the 1020 eV decade. Reprinted with permission from V. Berezinsky. . . . . . . . . . . . . . . . . . . . . . . . . 47 xi 2.9 The model prediction of cosmic ray fluxes from sources at different redshift shells and their contribution to the overall extragalactic proton energy spec-trum. It illustrates the fractionation of extragalactic events in energy by redshift. It can be seen how the GZK cutoff and ankle develop. A redshift of z = 0.3 contributes to the dip seen at 1018.5 eV around the region of the ankle. The low redshift shells shape the highest energy part of the spectrum. This result implies that there is a correlation between energy and distance. Reprinted with permission from D. Bergman. . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.10 A plot of flux · E3 obtained from the Mixed Composition Model and fit to data. In the case of a mixed composition with " = 2.2 − 2.3, there is no ankle feature. There is a good fit to the data for a pure proton model with " = 2.6 down to about 1018 eV. Reproduced with permission from !c ESO, [13]. . . . . 49 2.11 The relative abundance of iron in cosmic rays from &Xmax' and ! &Xmax' from the Auger experiment data as given two hadronic interaction models, QGSJETII and EPOSv1.99. The data shows a monotonic increase from proton towards iron for both &Xmax' and ! &Xmax'. Reprinted with permission from G. Wilk and Z. Wlodarczyk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.12 Auger data on Xmax as a function of energy. The mass composition indicates that UHECRs are getting heavier with increasing energy. Reprinted with permission from [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.13 A comparison of calculated proton spectra with the combined Auger spectrum for different Emax p . The two extreme cases, #g = 2.8 and #g = 2.0, are shown in the left and right figures, respectively. In the left figure, all curves with Emax % 10 EeV are below the data points at E > 5 EeV and hence compatible with the Auger energy spectrum. However, these curves are excluded by the prediction of the pure proton composition at E ( (4 -5) EeV due to the contradiction in mass composition in a narrow energy range. Reprinted with permission from [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.14 Pair production dip as an energy calibrator. The energies of several ex-periments are shifted in energy according to E ) &E assuming an energy-independent systematic error, and the resulting fluxes are compared. The recalibration factors are: & = 1.2 for Auger, & = 1.0 for HiRes, & = 0.75 for AGASA, and & = 0.625 for Yakutsk. The agreement of the fluxes after recalibration confirm the dip as a feature produced in the spectrum by the interactions of protons with the CMBR. Reprinted with permission from [55]. 53 3.1 A map of The Telescope Array Project. The three green boxes mark the locations of the fluorescence detectors, while the black boxes mark those of the surface detectors. The azimuthal field of view of the fluorescence detector stations is indicated by the arrows. Reprinted with permission from [75]. . . . 68 3.2 A photo of the Middle Drum Fluorescence Detector. There are seven bays with two telescopes in each bay. One telescope in each bay looks at a lower elevation (3" − 17" above the horizon) while the second telescope looks at the sky above the first (17" − 31" in elevation). The white box seen on the left side of the picture contains a xenon flasher used to monitor the optical calibration of the site. Reprinted with permission from J. N. Matthews. . . . . 69 xii 3.3 A schematic layout of the Middle Drum site. Reprinted with permission from S. B. Thomas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4 A photo of a Surface Detector deployed in the field. The "bed" has a steel cover over a stainless steel box containing two layers of half-inch scintillation plastic. Wavelength shifting optical fibers gather the light from the scintillators and deliver it to the PMTs, one per layer. Power is provided by a solar panel and deep cycle battery (behind the solar panel). The detector communicates with the rest of the array via a 2 GHz radio (antenna on pole). Reprinted with permission from J. N. Matthews. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.5 A picture of the Black Rock Mesa Fluorescence Detector. The station houses 12 telescopes and views about 108" in azimuth. There are three telescope bays housing four telescopes each. Two telescopes are mounted vertically on each stand, one above the other. The upper telescope views 3-17" in elevation, while the lower on views 17-30". The Long Ridge station is basically identical to this one. Reprinted with permission from J. N. Matthews. . . . . . . . . . . . . . 73 3.6 The Middle Drum telescopes. Left is a photo showing two adjacent telescopes in a MD FD bay. The telescope on the left observes 3" to 17" in elevation, while the one on the right observes 17" to 31" in elevation. The camera boxes in the front of the mirrors contain the arrays of PMTs. The electronics crates are behind the mirrors and are not visible. On the right is a picture showing the inside of a camera box. The UV band pass filter is open showing the array of 256 hexagonal PMTs which are camera pixels. Reprinted with permission from J. N. Matthews. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.7 An event display of a cosmic ray EAS. The elevation versus the azimuth is plotted for an event. The colors represent the timing information, and the size of the circle represents the signal size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.8 A measured mirror reflectivity curve. The reflectivity used in this analysis is 80%. Reprinted with permission from [66]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.9 Filter transmission as a function of wavelength for the UV filter in front of the PMT cluster at Middle Drum. The measured UV filter transmission is shown (dotted line) as a function of wavelength. It transmits well in the 300-400 nm region and lets through very little visible light: it has a hole in the far red/infra-red region. Under the filter transmission curve, the nitrogen fluorescence spectrum (solid/shaded) as calculated by Alan Bunner is shown. The filter is well matched to the fluorescence light emission. Reprinted with permission from [43]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.10 The Quantum Efficiency of a typical phototube at Middle Drum as measured by the manufacturer. This analysis uses a flat QE of 0.278, which is the nominal QE at 355 nm. The Bunner spectrum of N2 is also overlaid for comparison. Reprinted with permission from [43]. . . . . . . . . . . . . . . . . . . . . . . 78 3.11 The spatial response of a typical Philips PMT at Middle Drum. The response is relatively uniform across the face of the PMT. Reprinted with permission from [43]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 xiii 3.12 Picture of an electronics crate and a diagram showing its components. Reprinted with permission from [66]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.13 A diagram of the waveforms of the electronics, indicating the sequential timing of events. Reprinted with permission from [69]. . . . . . . . . . . . . . . . . . . . . . . . . 81 3.14 A plot showing the "on-time" for the first three years of operation of the Middle Drum fluorescence detector. This is the "on-time" used for this anal-ysis. The "Dark" solid line indicates the total possible hours with no sun and no moon. "All" indicates the actual data collection time. For example the detectors are not operated if the period of no sun and no moon is less than three hours for any night. The detector is also not operated if there is lightening, rain, snow, or high wind. "Good" indicates hours of collected data where the operator indicated good operating conditions (good visibility, low clouds). Reprinted with permission from [66]. . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.1 An illustration of the Shower Detector Plane where the EAS serves as a line and the detector serves as a point. The shower axis, Rp (impact parameter), and ' (angle within the shower detector plane) are indicated in the figure. Reprinted with permission from [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.2 A schematic of the shower core in the shower detector plane. The three geometric parameters, t0,Rp, and ' are shown. Reprinted with permission from [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.3 A Time vs Angle plot where the addition of a good tube extends the angle by 4". The red dots represent the tubes that were triggered at the end of Pass 3. The green dots represent the tubes that were determined to be "good" and thus added. The black dot represents a triggered PMT that is not a part of the event. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.4 A plot of residuals for the same event shown in Figure 4.3. The green circles represent the added tubes, and the red circles represent the original good tubes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.5 Plots of the temperature, pressure, and density profile of the Earth's atmo-sphere from the 1976 U.S. Standard Atmosphere. Reprinted with permission from [66]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.6 The Etterman model is used to determine the aerosol extinction length as a function of the scattered wavelength. Reprinted with permission from [66]. . . 103 4.7 The Longtin phase function used to determine the amount of light scattered as a function of angle. Reprinted with permission from [66]. . . . . . . . . . . . . . . 104 4.8 The ozone concentration as a function of altitude. Reprinted with permission from [66]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.9 The ozone attenuation coefficient as a function of wavelength. Reprinted with permission from [66]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.10 Light flux vs slant depth for an EAS. The fluorescence, Cerenkov, Rayleigh scattered, and Aerosol scattered light is plotted. The data, represented by the black dots with associated error bars, is fitted to the fluorescence light. Reprinted with permission from [62]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 xiv 4.11 The Bunner nitrogen fluorescence spectrum. Reprinted with permission from [43]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.1 Monte Carlo determination of the Gaisser-Hillas & = 60 g/cm2 parameter. Reprinted with permission from [66]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.2 Monte Carlo reconstructed energy bias. The ratio of Ethrown/Erecon versus the log10(Erecon) - 18 is plotted, and a linear function is then fit to the line. The reconstructed energy is about 15% too high from the thrown values, and this is corrected in the reconstruction of the data. . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.3 Resolution in Rp, the impact parameter for the EAS with respect to the telescope. The log of the reconstructed over the thrown Monte Carlo values for Rp gives a RMS value of about 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.4 Resolution in '. The difference between the reconstructed and the thrown Monte Carlo values for the ' angle, the angle the shower makes with the ground in the shower detector plane, gives a RMS value of about 8". . . . . . . . 122 5.5 Resolution in zenith angle. The resolution in zenith angle is better than 5". . 123 5.6 Resolution in energy. The energy resolution is about 18%. .. .. .. .. .. .. .. . 124 5.7 Cartoon of how the aperture is calculated. A homogeneous and isotropic flux, J(E), is assumed. The detector efficiency in reconstructing events is measured in all directions over a 2$ solid angle in area dA out to a distance rp that varies with energy. Reprinted with permission from [43]. . . . . . . . . . . . . . . . . 125 5.8 Aperture for the Time vs Angle geometry fit. The aperture decreases, as expected, as the energy decreases since low energy events do not have enough photons to generate a trigger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.1 Data/Monte Carlo comparison for the angle of the shower track in the shower detector plane, '. The three plots show three energy ranges: Top: 1018.0 eV * E < 1018.5 eV, Middle: 1018.5 eV * E < 1019.0 eV, and Bottom: E > 1019.0 eV. The black points with error bars show the data, while the Monte Carlo is shown by the red histogram. The Monte Carlo has been normalized to the same number of events as the data. For 1018.0 eV * E < 1019.0 eV, there are 30 bins between 0" and 180". For E > 1019.0 eV, there are 11 bins between 0" and 180". The data agree well with theMonte Carlo to 1!. . .. .. .. .. .. .. . 133 6.2 Data/Monte Carlo comparison for inverse angular speed. The three plots show three energy ranges: Top: 1018.0 eV * E < 1018.5 eV, Middle: 1018.5 eV * E < 1019.0 eV, and Bottom: E > 1019.0 eV. The black points with error bars show the data, while the Monte Carlo is shown by the red histogram. The Monte Carlo has been normalized to the same number of events as the data. For 1018.0 eV * E < 1019.0 eV, there are 30 bins between 0 μs/degree and 2.5 μs/degree. For E > 1019.0 eV, there are 11 bins between 0 μs and 100 μs/degree. The data agree well with the Monte Carlo to 1!. ............. 134 xv 6.3 Data/Monte Carlo comparison for timing (2 per degree of freedom, which is the (2 from the Time versus Angle fit. The three plots show three energy ranges: Top: 1018.0 eV * E < 1018.5 eV, Middle: 1018.5 eV * E < 1019.0 eV, and Bottom: E > 1019.0 eV. The black points with error bars show the data, while the Monte Carlo is shown by the red histogram. The Monte Carlo has been normalized to the same number of events as the data. For 1018.0 eV * E < 1019.0 eV, there are 30 bins between 0 and 60 for both (2's, and for E > 1019.0 eV, there are 5 bins between 0 and 60. The data agree well with the Monte Carlo to 1!. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.4 Data/Monte Carlo comparison for zenith angle. The three plots show three energy ranges: Top: 1018.0 eV * E < 1018.5 eV, Middle: 1018.5 eV * E < 1019.0 eV, and Bottom: E > 1019.0 eV. The black points with error bars show the data, while the Monte Carlo is shown by the red histogram. The Monte Carlo has been normalized to the same number of events as the data. For 1018.0 eV * E < 1019.0 eV, there are 30 bins between 0" and 90". For E > 1019.0 eV, there are 7 bins between 0" and 90". The data agree well with the Monte Carlo to 1!. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.5 Data/Monte Carlo comparison for Rp, the impact parameter which the shower track makes with the detector. The three plots show three energy ranges: Top: 1018.0 eV * E < 1018.5 eV, Middle: 1018.5 eV * E < 1019.0 eV, and Bottom: E > 1019.0 eV. The black points with error bars show the data, while the Monte Carlo is shown by the red histogram. The Monte Carlo has been normalized to the same number of events as the data. For 1018.0 eV * E < 1018.5 eV, there are 40 bins between 0" and 30". For 1018.5 eV * E < 1019.0 eV, there are 30 bins between 0" and 30". For E > 1019.0 eV, there are 12 bins between 0" and 90". The data agree well with theMonte Carlo to 1!. . . . . . . . . . . . . . 137 6.6 Data/Monte Carlo comparison plot for the profile (2 per degree of freedom, which is the (2 from the determination of the profile. The three plots show three energy ranges: Top: 1018.0 eV * E < 1018.5 eV, Middle: 1018.5 eV * E < 1019.0 eV, and Bottom: E > 1019.0 eV. The black points with error bars show the data, while the Monte Carlo is shown by the red histogram. The Monte Carlo has been normalized to the same number of events as the data. For 1018.0 eV * E , 1019.0 eV, there are 30 bins between 0 and 60 for both (2's, and for E > 1019.0 eV and above, there are 5 bins between 0 and 60. The data agree well with the Monte Carlo to 1!. . .. .. .. .. .. .. .. .. .. .. .. .. . 138 6.7 Data/Monte Carlo comparison for the time duration of an event. The three plots show three energy ranges: Top: 1018.0 eV * E < 1018.5 eV, Middle: 1018.5 eV * E < 1019.0 eV, and Bottom: E > 1019.0 eV. The black points with error bars show the data, while the Monte Carlo is shown by the red histogram. The Monte Carlo has been normalized to the same number of events as the data. For 1018.0 eV * E < 1019.0 eV, there are 30 bins between 0 μs and 100 μs. For energies 1019.0 eV and above, there are 11 bins between 0 μs and 100 μs. The data agree well with the Monte Carlo to 1!. ................... 139 xvi 6.8 Data/Monte Carlo comparison for tracklength. The three plots show three energy ranges: Top: 1018.0 eV * E < 1018.5 eV, Middle: 1018.5 eV * E < 1019.0 eV, and Bottom: E > 1019.0 eV. The black points with error bars show the data, while the Monte Carlo is shown by the red histogram. The Monte Carlo has been normalized to the same number of events as the data. For 1018.0 eV * E < 1019.0 eV, there are 30 bins between 15" and 50". For E > 1019.0 eV, there are 7 bins between 15" and 50". The data agree well with the Monte Carlo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.9 Data/Monte Carlo comparison for the energy. The three plots show three energy ranges: Top: 1018.0 eV * E < 1018.5 eV, Middle: 1018.5 eV * E < 1019.0 eV, and Bottom: E > 1019.0 eV. The black points with error bars show the data, while the Monte Carlo is shown by the red histogram. The Monte Carlo has been normalized to the same number of events as the data. For 1018.0 eV * E < 1019.0 eV, there are 15 bins. For E > 1019.0 eV, there are 9 bins. The data agree well with the Monte Carlo to 1!. . . . . . . . . . . . . . . . . . . 141 7.1 The exposure for Telescope Array Middle Drum Fluorescence Detector from December 16, 2007 to December 16, 2010. The on-time of the detector is 2406.15 hours in this period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.2 The number of observed events using the Time versus Angle geometry for the first three years of observation of the TA Middle Drum fluorescence detector is 2056. The log of the number of data events looks like a power law distribution. 149 7.3 The energy spectrum of UHECRs using the Time versus Angle method (red circles) as compared using the profile constraint fit. In the Time versus Angle plot, the 1019.3 eV and 1019.4 eV energy bins are combined, the 1019.5 eV and 1019.6 eV energy bins are combined, and the highest three highest energy bins were combined to produce this spectrum due to the low statistics in that region. The results of this work agree well with the results obtained by the profile constraint fit [5] (blue squares). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7.4 The energy spectrum of UHECRs using the Time versus Angle method (red circles) as compared to the spectrum resulting from analysis of the TA surface detector data (black triangles). The 1019.3 eV and 1019.4 eV energy bins are combined, the 1019.5 eV and 1019.6 eV energy bins are combined, and the highest three highest energy bins were combined to produce this spectrum due to the low statistics in that region. The results of the Time versus Angle method agree well with the results obtained with the scintillation surface detectors [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.5 The fit to the ankle obtained in the energy spectrum from the Time versus Angle geometry fit. The black line indicates the fit to the ankle. The dotted blue line is the lower error on the fit to the ankle. The dotted red line is the upper error on the fit to the ankle. There is not enough statistics to determine a GZK cutoff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.6 The difference in the reconstructed in-plane angle, ', between Time versus Angle geometry and PCF versus the thrown value. The Time versus angle reconstructs on average a lower ' angle than PCF for all values of '. ...... 155 xvii 7.7 The ratio of the energy obtained from Time versus Angle geometry divided by the energy obtained from PCF. In the top plot, there is no bias correction in the energy. PCF reconstructs higher in energy as the energy increases as compared to the Time versus Angle geometry. In the bottom plot, both reconstructions have had their appropriate energy bias corrections applied. There average ratio is about 1 for all energies. . . . . . . . . . . . . . . . . . . . . . . . . 156 7.8 The results of the Telescope Array Time versus Angle geometry energy spec-trum compared to other experiments [28, 67, 10, 1, 15, 73, 65, 17, 34, 53]. The red stars represent the results of this analysis. These results correlate well with other experiments indicating that a different method of determining the geometry on a different fluorescence detector is in agreement with different other measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 xviii LIST OF TABLES 1.1 A summary of cosmic ray composition measurements [19, 38, 74]. . . . . . . . . . 24 3.1 Table showing the good weather cuts. If the weather code for an event packet is listed below, the event is kept. Reprinted with permission from [66]. . . . . . 72 4.1 The numbers of events remaining after each stage of processing. The "d" after Passes 2 and 3 refer to downward-going events. Pass 3 is actually done in two stages: "a" and "b." For simplicity, the number of events after Pass 3bd is given. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.1 The number of events that remain after each cut. . . . . . . . . . . . . . . . . . . . . . . . 132 7.1 The flux at the ankle obtained by the Time versus Angle geometry for the Telescope Array Middle Drum Fluorescence Detector for the region around the ankle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.2 A comparison between the fits for the ankle for Fly's Eye [27], HiRes [78], TA Surface Detectors [7], and TA MD Time versus Angle monocular measurements.153 7.3 A comparison between the fits for the GZK cutoff for HiRes [78], TA Surface Detectors [7], and TA MD Time versus Angle monocular measurements. . . . . 153 7.4 A comparison between the number of events observed from the Time versus Angle (TvsA) fit and the Profile Constraint Fit (PCF) [66]. . . . . . . . . . . . . . . . 154 CHAPTER 1 INTRODUCTION Ultra High Energy Cosmic Rays (UHECRs) are charged particles of galactic and ex-tragalactic origin. The spectrum of energies with which UHECRs are observed on Earth provides information on the sources of cosmic rays and on mechanisms that exist in the universe which could accelerate these cosmic rays to energies one hundred million times that of the largest man-made accelerator in operation today, the CERN Large Hadron Collider. In 1966, two years after the discovery of the cosmic microwave background radiation [64], Kenneth Greisen [42], and independently from Greisen, Georgiy T. Zatesepin and Vadim A. Kuz'min [83], predicted the end to the cosmic ray energy spectrum, now known as the GZK cutoff. This cutoff was predicted based on the assumption of a protonic cosmic ray colliding with a cosmic microwave background radiation photon to produce a delta resonance which would then decay into a nucleon and a pion: p + #CMBR −) ##(1232 MeV ) −) ! p + $0 n + $+ (1.1) The cutoff occurs at about 6 ×1019 eV. Above this energy, the universe becomes opaque to UHECRs. The distance that an UHECR can travel is about 50Mpc (1 pc = 3.1 × 1013 km), which is about the size of our local supercluster of galaxies [30]. The Fly's Eye experiment, the High Resolution Fly's Eye (HiRes) experiment, the Akeno Giant Air Shower Array (AGASA), the Telescope Array Project (TA), and the Pierre Auger Observatory (PAO) were all designed to test this hypothesis. Since there are no known sources of cosmic rays within 50 Mpc of Earth, the GZK mechanism would prove to be correct if no cosmic rays are observed above 6 × 1019 eV. In 2008, the High Resolution Fly's Eye experiment reported the first observation of the GZK cutoff, at the 5! confidence level [1], at 5.6 × 1019 eV. This thesis presents the results of a monocular ultra high energy cosmic ray spectrum from one of the fluorescence detectors of the Telescope Array Project (TA), where the ultra 2 high energy range is defined to be E > 1018 eV. In particular, the analysis of a Time versus Angle geometry reconstructed is presented. 1.1 Phenomenology To better understand the scope of cosmic rays, a basic introduction to the high en-ergy cosmic ray spectrum is first presented. This is divided into two parts, galactic and extragalactic CRs, and illustrated with selected experimental results. 1.1.1 Cosmic Ray Physics Figure 1.1 [43] shows a plot of the energy spectrum for cosmic rays observed on Earth. The spectrum of cosmic rays spans 12 decades in energy (E) and 32 decades in flux! This is an enormous range for any natural phenomenon. For the most part, the plot follows a simple power-law spectrum. However, near 1015 eV, there is a change of slope from E−2.7 to E−3.0 at a structure called the "knee." Due to the wide range in energies and flux, other structures are difficult to discern on this particular plot. In order tomake the other features more visible, the overall slope is taken out by multiplying the flux by E3.0. This is shown in Figure 1.2 [25]. From Figure 1.2 it is apparent that many experiments have measured cosmic rays over these vast span of energies. However, since the energy spectrum is so large in range, each experiment is only able to measure a portion of it. Nonetheless, in particular energy regions, all of the experiments see the same features. There are four main features seen in this spectrum. The first feature is where the rising part turns over and the spectrum becomes flat around 1015.5 eV; it is known as the "knee." Above this energy the flat part slopes downward around 1017.5 eV, at the "2nd knee." The dip in the spectrum just about 1018.6 eV is called the "ankle." Lastly, there is a cutoff around 6 × 1019 eV, called the "GZK cutoff." Each of these four features will be discussed later in the chapter after a basic introduction to cosmic ray physics. At the lower energies, the flux of cosmic rays is relatively high. Therefore, balloons and satellites above the Earth can directly observe them. However, at the higher energies the flux falls dramatically. These CRs can be observed only indirectly via the Extensive Air Shower (EAS) which they induce when they enter the atmosphere. 3 1.1.1.1 The Extensive Air Shower Overview Figure 1.3 [36] sketches out the initial development of an EAS cascade. An EAS develops when a charged particle enters the Earth's atmosphere and collides with the nucleus of an atom. If the particle is a proton, it interacts to produce a roughly equal distribution of pions ($0,$+,$−). The $0 decays immediately into two gamma rays, each of which then pair produce to form e+e−. These electrons then Bremsstrahlung and pair produce again and again. This initiates the electromagnetic part of the shower. The charged pions initiate the hadronic part of the shower. The $+ and $− can either interact further or decay into its respective muons (μ) and neutrinos () or antineutrinos ¯)μ). If the initial charged particle happens to be a heavier element instead of a proton, then the total energy would be shared among each of the nucleons of that heavier element. The resulting shower would act like a superposition of lower energy protons. For example, for an iron (Fe) particle, the EAS would look like 56 proton showers each with 1 56 th of the primary energy. Lower energy protons produce showers higher in the atmosphere. Hence, more of the pions tend to decay rather than interact, thus producing more muons. Therefore, heavier elements are expected to produce showers higher in the atmosphere and with more muons. The EAS can be thought of as consisting of two components: the hadronic core and the electromagnetic (EM) cascade. We will now describe each. 1.1.1.2 The Hadronic Core The charged pions produced in the primary interaction initiate the hadronic cascade. Due to their relative heaviness, the charged pions have small changes in transverse momen-tum from the original particle [30]. This results in a hadronic core that is compact around the shower axis with a lateral extent of a few meters. Hadronic interactions continue until the energy of the charged pions falls below about 1 GeV. Below this energy, the hadronic core transfers energy to the electromagnetic component by producing $0, essentially fueling the EM subshowers. If the primary cosmic ray is a nucleus, the energy released in the first interaction with an atomic nucleus in the atmosphere is greater than the binding energy per nucleon in both nuclei. The nuclei break up, creating smaller hadronic showers, each feeding its own hadronic core and electromagnetic cascade [43]. Figure 1.1 [43] indicates that the highest energies for which accelerator data is available is about 1014 eV in the center of mass frame. Thus, the model for hadronic interactions 4 must be extrapolated to the ultra high energy regime from accelerator data and rely on models to provide particle cross sections at our energies of interest. 1.1.1.3 The Electromagnetic Cascade The $0's produced in the primary interaction decay promptly into two photons. At the highest energies, the cascade develops by e+e− pair production and Bremsstrahlung production of photons by electrons and positrons: e± −) # + e± (1.2) and is described well by the Heitler model. Figure 1.4 [43] shows the Heitler model of the electromagnetic cascade. In this figure, a photon first pair produces to form e+e−. The e+ and e− both Bremsstrahlung to produce a photon and an e+ or e−, respectively. In this model, the energy of the parent particle is equally divided into the daughter particles in each interaction length, &, until the daughter particle has less energy than the critical energy for further particle production. An estimated value for the critical energy is 81 MeV [72, 43]. From Figure 1.4, a rough estimate is that the the average particle energy is halved and the number of particles are doubled in each interaction length. 1.1.2 Galactic Cosmic Rays The KASCADE experiment measures the cosmic ray energy spectrum in the range from 1014 − 1017 eV, which is the region that contains the knee. KASCADE is an acronym for KArlsruhe Shower Core and Array DEtector, and the experiment is located at the Karlsruhe Institute of Technology in Germany. It consists of a 200 m × 200 m array of 252 scintillator detectors to measure the electrons. Underneath each scintillation detector, there is a muon detector, which is just a scintillation detector with an iron-lead absorber above it to filter out the electrons. KASCADE uses the muon-to-electron ratio to determine the composition of the primary particle. However, the muon-to-electron ratio does not unequivocally determine the composition of the primary particle. The data has to be matched to Monte Carlo simulations of EAS development. Different models result in different answers. KASCADE used several models; the analysis of their data with the SIBYLL model is shown in Figure 1.5 [44]. The plot in Figure 1.5 shows E2.5· J versus primary energy using the SIBYLL model. For the protons, a cutoff is seen at about 3 × 1015 eV. Helium, which has an atomic number 5 of 2, has a cutoff at 6 × 1015 eV, or twice that for protons. Carbon has an atomic number of 6, and a cutoff occurs around 2 × 1016 eV, or six times that of protons. Therefore, KASCADE sees a cutoff in energy that is proportional to charge. Of the many possible theories of a rigidity-dependent cutoff, two are discussed. The first is the ability of the galaxy to either contain or leak protons due to its magnetic field (B). The second is the maximum energy to which these CRs can be accelerated. The first possible cause of a rigidity-dependent cutoff is magnetic containment. The charged cosmic ray particles are bent by the galactic magnetic field. The galactic magnetic field is approximately 3 μG. It has a regular component and a random component, which are roughly equal in magnitude [39]. The coherence length, or the average size of a region with roughly uniform magnetic field, is about 100 pc. Setting the magnetic component of the Lorentz force equal to the centripetal force for relativistic particles, we can find the critical energy needed for a cosmic ray to escape our galaxy. The critical energy between which cosmic rays would be contained within our galaxy versus escaping from the galaxy is given by the formula: Ec = Z · (lc/kpc) · (B/μG) · (1018) eV. (1.3) The critical energy is proportional to the charge (Z), the coherence length (lc), and the magnetic field (B). Substituting a 3 μG magnetic field and a 100 pc coherence length, the critical energy is about 3 × 1017 eV for protons, and 26 times this value, or 8 × 1018 eV for iron. Recall that KASCADE sees a cutoff in energy that is about 3 × 1015 eV. Theoretical and experimental results should not be two orders of magnitude different! Their observed rigidity-dependent cutoff does not appear to be due to a failure of magnetic containment. A second possible cause of a rigidity-dependent cutoff is the ability of the accelerator to accelerate the CRs above certain energies. Galactic CRs are thought to be accelerated by SuperNova Remnants (SNR). A CR gains energy as it repeatedly crosses a SNR shock wave front. This naturally results in a power law spectrum, which would be in agreement with the plot of the observed energy dependent flux shown in Figure 1.1 [60]. However, given the finite lifetime of about 3000 years of SNRs, the maximum energy to which these CRs can be accelerated by this mechanism is calculated to be 1014 eV. However, the KASCADE result is an order of magnitude higher. Therefore, there are many theories that describe how CRs could be accelerated to energies greater than 1014 eV. Of the many possible theories, two that describe a collective effect will be discussed. The first theory is that CRs could amplify the magnetic field of 6 a SNR shock wave front as they cross it [29]. The second is a collective effect theory of superbubbles. Superbubbles form in OB associations [52, 32, 33, 47, 46]. These associations have stars that supernova close together spatially and temporally on a cosmological scale. When the shock wave front of one SNR merges with another, a bubble is formed. When at least five of these SNR shock wave fronts merge, they form a superbubble. Such a superbubble may be able to accelerate CRs into the 1015 eV decade. Recall that in Figure 1.5 the KASCADE result shows a different cutoff for different elements. Thus, for each successive element, there could be a knee, up to the iron (Fe) knee. To observe the Fe knee, KASCADE expanded their experiment to the KASCADE-Grande experiment by adding 37 new scintillator detectors to create roughly a 700 m × 700 m array, extending the upper energy limit from 1017 to 1018 eV. Unfortunately, no new muon detectors were added. The results of the KASCADE-Grande experiment are shown in Figure 1.6 [44, 50, 17]. The top graph is flux times E2.5 versus log E. The bottom graph is the flux, scaled such that the features are more clearly visible, versus the log of the energy. A decline in the spectrum ends around 107.2 GeV and is not understood. At about 7.9 × 1016 eV, there is a cutoff. KASCADE-Grande claims that this is the Fe knee. However, other independent experiments are needed to confirm the result. The Akeno experiment also measured energy in this region. The Akeno array is a precursor to and a subset of the AGASA array. The Akeno results are shown in Figure 1.7 [73]. Akeno reports what it calls a smooth connection from the knee region to about 1020 eV, albeit with some energy scale differences in the 1019 eV decade. To conclude the discussion of the knee feature, KASCADE-Grande observes an Fe knee, however Akeno does not. One remaining question in the 1017 eV decade is that of the 2nd knee which is above the KASCADE-Grande Fe knee. The mass composition, which provides information on the identity of the source, is not known in this region, and thus the cause is unknown. In Figure 1.8 [55], the left graph, a E3·J vs E plot, shows that the Yakutsk [65], Akeno [73], Fly's Eye [27], and HiRes-MIA [19] experiments all show a flat portion in energy before a cutoff is observed. If the flat portions of all the spectrums are laid on top of each other by rescaling the energy, as shown in the right graph, all of the experiments demonstrate a cutoff, which is the 2nd knee. Recall that KASCADE-Grande expanded its experiment to reach the 1018 eV decade, but it does not have enough statistics in the high energy region. Furthermore, the systematics of current experiments do not provide an explanation of the 7 2nd knee; however, one speculation is that the 2nd knee could be the critical energy of the galactic magnetic field. A better measurement is needed in this energy region, which is the aim of the Telescope Array Low Energy Extension (TALE). 1.1.3 Extragalactic Cosmic Rays In the 1017 eV decade, the HiRes-MIA experiment observes a transition in primary par-ticle composition changing from heavy or iron to a light or protonic as shown in Figure 1.9 [19]. At the beginning of the 1018 eV decade, the HiRes Stereo experiment measures a light composition consistent with protons. Thus it appears that the transition is complete by 1018 eV. This is thought to be the galactic to extragalactic transition because as depicted in Figure 1.10, the galactic part decreases the flux, and the extragalactic part, although increasing, still decreases the flux. The sources of extragalactic CRs are still unknown, although some details of their propagation are known. These CRs escaped from the galaxy in which they originated, traversed the intergalactic medium, and entered our galaxy. In their journey, they lost energy. They also lost energy due to the expansion of the universe. In addition, these CRs lost energy by interacting with the Cosmic Microwave Background Radiation (CMBR) photons. If the CR is a proton, with 6 × 1019 eV, then the mean free path for this interaction is 50 Mpc, after which the proton is likely to interact with a CMBR photon to produce pions. This mechanism produces the GZK cutoff. This same type of interaction can also produce e+e− pairs, however with a lower threshold. This interaction is expected to excavate the spectrum around 4 × 1018 eV and result in a feature which is commonly called the ankle. If the charged particle is a nuclei instead, then at distances greater than 50 Mpc, there would not be an expectation to see any Fe due to spallation at thresholds of about 4 × 1019 eV. Before considering the details of CR propagation, it is useful to observe these features in the spectrum. In Figure 1.11 [3], both the HiRes and TA experiments see a dip in the spectrum around 1018.6 eV. This is the feature known as the ankle. Recall that the HiRes-MIA experiment indicated that the transition was complete near the beginning of the 1018 eV decade, and the HiRes Stereo experiment saw a composition consistent with protons. Therefore if extragalactic cosmic rays are protons, the cause of the ankle is most likely due to e+e− pair production, although an older interpretation is that the ankle is the galactic-extragalactic transition. Figure 1.12 [1] shows a spectrum measurement by HiRes and AGASA. It is an E3· J versus log E plot, and AGASA, represented by the upside down blue triangles, sees a 8 spectrum that continues upward. The HiRes data shows a significant fall off in the flux above 6 × 1019 eV. This is the first observation of the predicted GZK cutoff. This observation was subsequently confirmed by the PAO and the TA Project. Returning to the details of the energy loss mechanisms in propagation, Figure 1.13 [76, 49] shows a plot of the mean free path versus log of the energy for protons. At the highest energies, the dominant energy loss mechanism is due to photo-pion production at 10's of Mpc. At lower energies, e+e− pair production dominates at about 2000 Mpc at 1018.5 eV. At the lowest energies, redshift becomes important and is approximately 3000 Mpc at all energies. Figure 1.14 [11] shows a plot of energy loss due to spallation for iron nuclei. The axes are attenuation length versus log(Lorentz boost). At the highest boosts, photo-erosion, or the loss of nucleons due to photonuclear interactions, with the CMBR is the dominant energy loss mechanism. Therefore any iron nuclei from sources greater than 50 Mpc away are not expected to be observed. 1.2 Composition and Anisotropy There are three things that the Telescope Array Project can tell us about cosmic rays: composition, anisotropy, and energy spectrum. This thesis focuses on the energy spectrum, which will be considered in detail. However, it is necessary to know a little about composition and anisotropy to understand the whole picture and interpret the energy spectrum. In particular, since this thesis interprets the cause of the ankle feature in the spectrum around 4 ×1018 eV, it is useful to understand the composition and anisotropy in that energy region. There are several experiments that have measured chemical composition as a function of energy. Figure 1.9 [19] showed the HiRes-MIA experiment indicating a changing com-position from heavy to light between 1017 eV and 1018 eV. Above 1018 eV, the HiRes Stereo measurement indicated that the change in composition is complete and consistent with protons [19]. The Fly's Eye [27, 71] experiment saw a composition that is changing from heavy to light beginning around 1017.5 eV and a correlated change in energy spectrum and composition between 1018 eV and 1019 eV. The Auger experiment [38] observes a composition that is consistent with protons between 1018.0 eV and 1018.5 eV. However around the middle of the 1018 eV decade, Auger observes a composition that is changing from protonic to heavy as the energy increases [14]. 9 Table 1.1 summarizes the composition measurements. HiRes-MIA [19] sees a compo-sition that is getting lighter from 1017.5 eV to 1018 eV. Furthermore, HiRes Stereo [19], Auger [38], and TA [74] all see a composition that is consistent with protons from 1018.0 eV to 1018.5 eV. Above 1018.5 eV, the HiRes Stereo experiment observes a composition that is consistent with protons while Auger sees a composition that is getting heavier. The change in composition has implications on where the galactic-extragalactic transition occurs. The composition of cosmic rays at the energy of the ankle determines what physical processes are causing its formation. All of the galactic protons and most of the heavier nuclei should have escaped the galaxy at these energies. It becomes a question of whether all of the heavy nuclei has escaped or not. We will discuss this more in detail later. From all of the data available from cosmic ray experiments, galactic anisotropy is not visible in the galactic plane [80]. With the known galactic magnetic field strength, higher energy particles would hardly be deflected and should point back to their source. There are no known sources which produce ultra high energy cosmic rays within our galaxy. 1.3 Summary We have discussed the phenomenology of cosmic rays from the knee region and above. The processes discussed to explain the spectrum of cosmic rays are: the maximum energy of the accelerator, the ability of the galactic magnetic field to contain versus leak cosmic rays, lower energy cosmic rays originating within the galaxy and higher energy cosmic rays originating from outside the galaxy, and energy losses in propagation. There are three things that the Telescope Array Project can tell us about cosmic rays: composition, anisotropy, and energy spectrum. We will now turn our focus to the physics of the ultra high energy region of the cosmic ray energy spectrum. 10 Energy (eV) 9 10 10 10 11 10 12 10 13 10 14 10 15 10 16 10 17 10 18 10 19 10 20 10 Flux (m2 sr GeV sec)-1 -28 10 -25 10 10-22 -19 10 -16 10 -13 10 -10 10 10-7 10-4 10-1 102 104 -sec) 2 (1 particle/m Knee -year) 2 (1 particle/m Ankle -year) 2 (1 particle/km -century) 2 (1 particle/km FNAL Tevatron (2 TeV) CERN LHC (14 TeV) LEAP - satellite Proton - satellite Yakustk - ground array Haverah Park - ground array Akeno - ground array AGASA - ground array Fly's Eye - air fluorescence HiRes1 mono - air fluorescence HiRes2 mono - air fluorescence HiRes Stereo - air fluorescence Auger - hybrid Cosmic Ray Spectra of Various Experiments Figure 1.1. The energy spectrum of cosmic rays. Note that this is a log-log plot and covers a very wide range in both flux and energy. Also note that the plot follows a power law with a slope of about 3. Therefore, for each factor of 10 increase in energy, the flux falls by about 1000. Reprinted with permission from [43]. 11 Figure 1.2. The spectrum of cosmic rays. Here the flux has been multiplied by E3 in order to take out the underlying slope and highlight the detailed structure of the spectrum. Reprinted with permission from [25]. 12 Figure 1.3. A sketch indicating some of the initial particle interactions in the development of an extensive air shower. Reprinted with permission from [36]. 13 Figure 1.4. The Heitler branching model of the electromagnetic cascade of the EAS. In this model, photons e+e− pair produce and electrons and positrons Bremsstrahlung until the energy falls below the critical energy to produce further particles. Reprinted with permission from [43]. 14 Figure 1.5. Cosmic ray flux times E2.5 vs Energy as measured by KASCADE and inter-preted using the SIBYLL model. Note that the proton flux turns over around 3 × 1015 eV. Reprinted with permission from [44]. 15 Figure 1.6. KASCADE-Grande's energy spectrum showing a possible Fe knee located at log10 E (E/GeV) = 7.9. The top graph is E2.5· J versus E. The bottom graph is the flux, scaled such that the features are more clearly visible, versus the log of the energy. A decline in the spectrum ends around 107.2 GeV and is not understood. At about 7.9 × 1016 eV, there is a cutoff. KASCADE-Grande claims that this is the Fe knee. Reprinted with permission from [44]. 16 Figure 1.7. Akeno's analysis of their data showing no signs of an Fe knee. The Akeno array is represented by the open squares. Akeno reports what it calls a smooth connection from the knee region to about 1020 eV, albeit with some energy scale differences in the 1019 eV decade. Reprinted with permission from [73]. 17 Figure 1.8. Several experiments show evidence of a 2nd knee. The left graph shows that the Yakutsk [65], Akeno [73], Fly's Eye [27], and HiRes-MIA [19] experiments all show a flat portion in energy before a cutoff is observed. If the flat portions of all the spectrums are laid on top of each other by rescaling the energy, as shown in the right graph, all of the experiments demonstrate a cutoff, which is the 2nd knee. Reprinted with permission from [55]. 18 Figure 1.9. Measurement of the chemical composition as measured by the HiRes-MIA and HiRes Stereo experiments. The mean depth of shower maximum (< Xmax >) as a function of energy is plotted. The bars show the predicted elongation rate (evolution development as a function of energy) for two models. The HiRes-MIA measurement indicates a changing composition from heavy to light between 1017 eV and 1018 eV. Above 1018 eV, the HiRes Stereo measurement indicates a constant light composition consistent with protons. Reprinted with permission from [19]. 19 Galactic Cosmic Rays Extragalactic Cosmic Rays Fe Knee Protons leaking in Energy Flux Figure 1.10. A cartoon indicating the transition between the galactic and extragalactic contributions to the cosmic ray flux. The galactic contribution to the flux decreases, and while the extragalactic contribution to the flux increases, the overall flux is still decreasing. 20 Figure 1.11. The cosmic ray spectrum above 1017 eV as measured by the HiRes experiment and the Telescope Array Project. The spectrum has been multiplied by E3 to take out the predominant slope and show the detailed structure. The ankle is seen at log10 (E/eV) = 18.6, and the GZK cutoff is seen around 6 × 1019 eV. Reprinted with permission from [63]. 21 Figure 1.12. The cosmic ray spectrum above 1017 eV as measured by the HiRes and AGASA experiments. The HiRes experiment observes the ankle at log10 (E/eV) = 18.6 and the GZK cutoff around 6 × 1019 eV. The AGASA experiment does not observe the GZK cutoff. Reprinted with permission from [63]. 22 10 100 1000 10000 100000 18 18.5 19 19.5 20 20.5 21 21.5 (Ep -1 dEp/dx)-1 [Mpc] log10 Ep [eV] redshift e+e- creation π production Figure 1.13. The energy lossmechanisms during propagation for cosmic ray protons. At the highest energies, the dominant energy loss mechanism is due to photo-pion production at 10's of Mpc. Pair production dominates at about 2000Mpc at 1018.5 eV. Redshift becomes important at approximately 3000 Mpc at all energies. Reprinted with permission from F. Aharonian and [49]. 23 Figure 1.14. Energy loss due to spallation for iron nuclei. The axes are attenuation length versus log(Lorentz boost). At the highest Lorentz boosts, photo-erosion with the CMBR is the dominant energy loss mechanism. Reprinted with permission from D. Allard. 24 Table 1.1. A summary of cosmic ray composition measurements [19, 38, 74]. log10 E (E/eV) 17 - 18.0 18.0 - 18.5 18.5 - 20 # Lighter HiRes/MIA - - HiRes Proton - Auger HiRes TA # Heavier - - Auger CHAPTER 2 PHYSICS OF THE UHECR SPECTRUM In this thesis, a different technique to reconstruct the geometry is used to produce the UHECR spectrum. We begin by exploring what we can learn from the UHECR energy spectrum. The energy spectrum of UHECRs gives us information about their sources, the acceler-ation mechanisms, and their energy loss mechanisms. First, we learn details of the source by finding the highest energy events and examining the spectrum. Second, the features of the spectrum, such as the cutoff and the ankle, tell us about the possible cause for their formation, such as photo-pion production and e+e− pair-production. Third, by modeling the evolution of the universe, we can learn how cosmic rays lose energy during propagation. We begin by examining the information we can learn. 2.1 Sources By finding the highest energy events in the spectrum, we can learn details about the sources of cosmic rays. First, there exists an accelerator that is capable of accelerating cosmic rays to ultra high energies. At the highest energies, the flux of cosmic rays at the Earth is of order one particle/km2/century. By integrating the flux with respect to the energy observed, we find that the energy density in UHECRs is very large. If we assume that 1020 eV cosmic rays fill the local supercluster of galaxies isotropically and have a lifetime of 108 years, the source must produce about 5 × 1041 eV per second to keep the flux constant [70]. Thus, cosmic ray sources do not follow a blackbody radiation spectrum, and there must be nonthermal mechanisms for acceleration. Second, the spectrum of the source must generate a power law for cosmic rays observed on Earth as shown in Figure 1.1 [43]. Possible extragalactic candidate sources within our supercluster of galaxies (30 - 50Mpc, where 1 pc = 3.1 × 1013 km) are radio galaxies, blasars, and active galactic nuclei. However, because cosmic rays lose energy during propagation, the maximum energy of the source must have been greater than the observed highest energy 26 events. We know that the Emax of the source had to be greater than the GZK cutoff since this is observed. The Monte Carlo thrown for this analysis is generated with an Emax of 1020.5 eV. 2.2 Acceleration Models Figure 2.1 [43] shows the size and magnetic field strength of astrophysical objects relative to what is required to boost cosmic rays to 1020 eV [48, 43, 26]. The plot shows the magnetic field versus the size of candidate sources on a log-log scale. The velocity of the shock wave, or efficiency of the acceleration mechanism, is represented by ". Objects belowthe diagonal lines do not have a sufficient combination of magnetic field strength and size to accelerate protons or iron nuclei to ultra high energies. Hence astrophysical objects, with the realistic assumption of a shock velocity of " = 1/300, would need to lie above the top solid blue line to accelerate protons to 1020 eV. As seen from the plot, there are no known candidate sources above this line. The dashed blue line represents the extreme assumption for shocks traveling at the speed of light, " = 1. Under this unrealistic condition, the few candidate sources are Active Galactic Nuclei (AGN), neutron stars, Radio Galaxy (RG) lobes, and colliding galaxies. The lowest green solid line corresponds to 1020 eV iron nuclei at " = 1. It is thought that cosmic rays are created with relatively lower energies at their source and then accelerated. To fit observations, acceleration models need to produce a power law spectrum. There are two major acceleration mechanisms: statistical (slow) and shock (fast) acceleration. A brief summary from the book by Longair [60] is given. 2.2.1 Statistical Acceleration In statistical acceleration, the final energy is gained slowly over many decades of energy [70, 30]. A model was first proposed by Fermi where charged particles scattering off magnetic clouds in the interstellar medium would gain or lose energy depending on the angle at which they enter the cloud. In a one-dimensional case, a particle of mass, m, with velocity, v, collides with amagnetic cloud of infinite mass moving with velocity, V [60, 70, 30]. In this model, the center of mass velocity is V, and the energy of the cloud should be greater than that of the particle. For a head-on collision, the particle's energy is conserved, but its momentum is reversed. The change in the particle's energy is given by: E$$ = #2V E " 1 + 2V vcos* c2 + # V c $2 % . (2.1) 27 Expanding to second order in terms of V/c and solving for E$$ - E: E$$ − E = #E + 2V vcos* c2 + 2 # V c $2 , (2.2) we obtain the change in energy for the particle before and after the collision in the laboratory frame. Fermi showed that the probability of a head-on collision and the fractional change in energy is greater than the "following collision," where the fractional energy gain per collision is given by & #E E ' = 8 3 # V c $2 . (2.3) The gain in energy is second order in V/c and is hence called second order Fermi acceleration. However, random velocities of interstellar clouds are much smaller than the speed of light, and the mean free path of cosmic rays in the interstellar medium is on the order of 1 pc [60]. This results in a very slow energy gain. Furthermore, energy losses have not been considered, and particles must either be injected into the accelerating region with energies greater than the maximum energy loss rate or else the initial acceleration process must be rapid enough to overcome the energy losses. Lastly, there is nothing in the theory which would explain why the resulting energy spectrum should have our observed spectral index. 2.2.2 Shock Acceleration There is a modified version of first-order Fermi acceleration which produces a power-law spectrum naturally. This happens in the case of a strong shock, caused by a supernova explosion for example [60]. If the particles have high energies such that the velocity of the shock is much less than the velocity of the particles, the particles hardly notice the shock since the thickness of the shock is much less than the gyroradius of the particle. Due to turbulent magnetic fields on either side of a shock wave front, a cosmic ray bounces back and forth across it. Unlike second-order Fermi acceleration, the particle gains energy each time it crosses the shock wave front, both from diffusing from behind the shock to the upstream region and then returning to the downstream region of the shock. Thus, there is never a crossing where it loses energy. If the average energy after one collision is given by E = "E0, which is first order in velocity, and the probability that the particle stays within the accelerating region is P, then the fractional number of particles left in the cloud is N N0 = # E E0 $lnP ln! . (2.4) 28 Taking the derivative of this equation with respect to E results in a power law for the spectrum. For a round trip across the shock, the fractional energy increase is & #E E ' = 4 3 V c , (2.5) which is first-order in V. This model of shock acceleration produces a spectral index of 2, which does not quite fit observations. However, first-order Fermi acceleration is attractive because it produces a power law spectrum naturally. 2.3 Spectral Features Studying the energy spectrum of UHECRs reveals two interesting features, the ankle and the GZK cutoff. We begin at the highest energies. 2.3.1 Cutoff The GZK cutoff describes a sharp steepening of the spectrum [24] which is model-dependent in that the cutoff of a single source depends on its distance to that source. For a wide range of generation spectral indices, 2.1 * #g * 2.7, the cutoff energy is E + 5.3 × 1019 eV [24]. It is important to consider the cases of whether a cosmic ray is a proton or an iron nucleus. As stated earlier, for a protonic cosmic ray, Kenneth Greisen [42], and independently from Greisen, Georgiy T. Zatesepin and Vadim A. Kuz'min [83], predicted the end to the cosmic ray energy spectrum, now known as the GZK cutoff. This cutoff was predicted based on the assumption of a protonic cosmic ray colliding with the cosmic microwave background radiation photon to produce a delta resonance which would then decay into a nucleon and a pion: p + #CMBR −) ##(1232 MeV) −) ! p + $0 n + $+. (1.1) Using isospin considerations with Clebsh-Gordan coefficients, the ratio of $0 to $+ produced is 2:1. Figure 2.2 [43] shows the total cross section versus the center of mass energy, $s, for this process. The cross section starts with the $0 production threshold at $s = 1.078 GeV. There is a rapid increase in cross section up to $s = 1.232GeV,which corresponds to the #+ resonance. The #+ resonance is 120 MeV wide with a lifetime of 10−23 seconds. The average cross section is ! = 0.12 mb. The #+ resonance is an energetically favorable state where the J = 1/2 nucleon transi-tions to a J = 3/2 state. The other peaks at higher energies correspond to similar nuclear 29 resonances. From the cross section, we can estimate the mean interaction length of an ultra high energy proton traveling through intergalactic space. The mean free path is given by & = (n!)−1, where n is the the CMBR photon density and ! is the cross section. The CMBR has a blackbody spectrum at about 2.73 K as shown in Figure 2.3 [43]. The photon density is calculated by integrating the distribution and yields a density of 412 photons/cm3. Therefore the mean free path is & = 2× 1024 cm. However at the highest energies, pions impart more kinetic energy to secondary particles, resulting in a & = 9.2×1024 cm [43]. This distance corresponds to a time of + = 3.1 × 1014 seconds whereas the age of the universe is 4.3 × 1017 seconds. Thus, cosmic rays with trans-GZK energies are not likely to be related to the birth of the universe. In the laboratory frame, a 2.73 K photon in the CMBR has a mean energy distribution of roughly 0.0006 eV. However, in the rest frame of the proton, the photon appears to be a gamma ray of about 145 MeV. The estimated threshold for pion production is about 6 × 1019 eV. For the GZK cutoff to occur at its observation, at 5 ×1019 eV, the threshold energy in the center of mass frame for the CMBR photon is: E! = m2 "0 + 2mpm"0 4Ep + 0.00134 eV, (2.6) where mp and m" are the proton and pion masses, respectively, and Ep is the energy of the proton. As seen in Figure 2.3, this photon energy lies at the upper energy tail of the blackbody spectrum. Thus, most of the pion production is due to the tail of blackbody spectrum, with some pion production occurring at lower proton energies. The mean free path for this process is estimated to be about 50 Mpc. Thus cosmic rays traveling from a distance greater than 50 Mpc and a threshold of 6 × 1019 eV should be observed from "nearby" sources. Heavier ions with a charge, Z, are accelerated to Z times the maximum energy of the source. Due to spallation at such distances, most of the heavy nuclei are expected to have disappeared before arriving here at Earth. If cosmic rays are iron nuclei, the GZK cutoff would be higher in energy for larger Z by ZE2. This is difficult to observe experimentally [20]. The Fly's Eye experiment, the High Resolution Fly's Eye (HiRes) experiment, the Akeno Giant Air Shower Array (AGASA), the Telescope Array Project (TA), and the Pierre Auger Observatory (PAO) were all designed to test this hypothesis. In 2008, the High Resolution Fly's Eye experiment reported the first observation of the GZK cutoff, at the 5! confidence 30 level [1], at 5.6 ± 0.5 (statistical) ± 0.9 (systematic) × 1019 eV. The Telescope Array Surface Detectors observe a cutoff at 4.8 ± 0.1(statistical) × 1019 eV. 2.3.2 Ankle The cause of the ankle is not as easily predicted as the GZK cutoff, and it is therefore an important feature in helping us understand the physical mechanisms that are occurring. At lower energies, galactic cosmic rays dominate the flux. As they accelerate to higher and higher energies, they escape the galaxy. The extragalactic component now begins to dominate the flux. At the transition, it is expected that galactic iron is escaping from the galaxy and extragalactic protons are entering the galaxy. However the point of this transition is not known. The oldest measurements, such as those of Fly's Eye, led to the interpretation of the ankle being formed by the galactic to extragalactic transition. However, when the composition data of the HiRes-MIA experiment was added to the picture, this led to other interpretations such as pair production from cosmic ray protons excavating the ankle. 2.3.2.1 Possible Cause: Galactic-Extragalactic Transition It was first thought that the ankle was caused by the galactic-extragalactic transition. The dip in the Fly's Eye energy spectrum [40] was explained as the sum of two components: a heavier component of galactic origin dominating below 1018 eV, and a light extragalactic component taking over at energies greater than 3 ×1018 eV based on correlated results of composition, energy spectrum, and anisotropy. By plotting the mean Xmax as a function of energy, the elongation rate, or the change in the mean Xmax per energy decade, was greater than expected for any fixed composition above 1017.5 eV. Since different models give different predictions for the mean Xmax for proton and iron, Fly's Eye concluded that the inference of a changing composition from heavy to light was more robust than a determination of the actual composition itself. The Fly's Eye experiment also found that the higher energy particles were lighter than the lower energy population. Since the lightest particles above the transition energy would not be deflected much by the galactic magnetic field, these particles should point back to their source. However, the Fly's Eye data did not detect any anisotropy, and it was concluded that the higher energy component did not originate in the galactic disk. One Fly's Eye event was detected at 3 ×1020 eV; it was concluded that particle originated in the contemporary era of the Universe and was not left over from the Big Bang. The 31 Fly's Eye experiment thus concluded that the ankle represented a transition to a population of cosmic rays of different extragalactic origin. 2.3.2.2 Possible Cause: Pair Production When a cosmic ray traverses space, it loses energy from three mechanisms: pion produc-tion, pair production, and redshift. As seen in Figure 1.13 [76, 49, 13, 59], a proton with energy E > 1020 eV needs to travel about 50 Mpc before it has a good probability of losing energy due to pion production. This is the strongest energy loss mechanism for protons with energies greater than the GZK cutoff. However, pair production and redshift energy losses are still contributing. The protons that remain have energies less than the threshold necessary to produce the cutoff and are seen to the left of the cutoff in the spectrum plot. The result is a cutoff of cosmic rays above this energy and a "pile-up" in the flux of protons with energies just below this. The protons that did not have enough energy for pion production propagate for 1000's of Mpc losing energy mainly via pair production and some via redshift. The threshold energy for e+e− pair production: p + # −) p + e+ + e− (2.7) is given by Ep = (mp + 2me)2 − m2 p 4E! + 7.980 × 1017 eV, (2.8) where mp is the mass of the proton, me is the mass of the electron/positron, and using E! = 0.6 meV. This is less than the currently accepted location of the ankle, around 1018.5 eV to 1018.6 eV. Thus, the ankle is likely a composite feature. It is usually modeled as a sharp intersection of two lines, but it may have a small curvature at the minima. The ankle is excavated mainly due to a buildup of protons with energies less than necessary for pair production. However, the constant redshift energy loss begins to dominate over the e+e− pair production loss and contributes to the shape and location of the ankle. The protons that remain have energies less than 1018.5 eV. These protons lose energy primarily due to redshift, or expansion of the universe, and some due to pair production from 10,000's to 100,000's of Mpcs as seen in Figure 1.13 [76, 49, 13, 59]. 2.4 Energy Loss Model Energy loss models are used to explain the observed features in the UHECR spectrum. By changing the parameters in models, the features change, and it is compared to data. We 32 begin by looking at what the data tells us. 2.4.1 Data Recall that Figure 1.12 [1] showed a spectrum measurement by HiRes and AGASA that shows the ankle and the GZK cutoff. This observation was subsequently confirmed by the Pierre Auger Observatory as shown in Figure 2.4 [79] and the Telescope Array Project shown in Figure 2.5 [5]. Figure 2.4 shows the cosmic ray spectrum above 1018 eV measured by the Auger experiment. It is a plot of the fractional difference between their data and a spectrum with a spectral index of 2.6. The HiRes data are also plotted for comparison. On this plot, there is an abrupt change in the spectral index around 4 ×1018 eV and a gradual suppression of the flux above 3 ×1019 eV. Figure 2.5 shows the cosmic ray spectrum above 1018 eV measured by the HiRes and Telescope Array Middle Drum Fluorescence Detector. Two features can be seen in all of the experiments in the ultra high energy regime, the GZK cutoff and the ankle. All of the experiments see the GZK cutoff, but a debate remains whether it is due to a proton or iron primary cosmic ray. A debate also remains as to the cause of the ankle, and details of models are used to predict its shape. The details of the spectral shape can give us information about the Emax of the sources, the spectral index, and the evolution of the sources [37]. With simple assumptions such as a proton flux, a constant density and luminosity of the source, a power law at the source, an evolution parameter, and an overall intensity constant, it is possible to model what the observed cosmic ray spectrum should look like. The two predictions are that UHECRs can be either extragalactic protons or heavier nuclei. Let us examine both of these possibilities. 2.4.2 Extragalactic Proton Propagation Model Recall that the Fly's Eye experiment [40] saw a dip in the energy spectrum. Based upon composition measurements at the time, Fly's Eye interpreted the ankle as a transition from galactic to extragalactic cosmic rays. Since then, the HiRes-MIA experiment saw a composi-tion that was changing from heavy to light from about 1017 eV to 1018 eV. Combining these results with the HiRes Stereo experiment, which saw a composition that was consistent with protons from the beginning of the 1018 eV decade, led to the interpretation that the ankle is excavated due to e+e− pair production. V. Berezinsky et al. [23] argue that the dip is a more reliable signature of proton interactions with the CMBR than the GZK cutoff [23]. This is because the shape of the GZK is strongly-model dependent: it is more flat in the case of overdensity of sources and 33 more steep if there is a local deficit of sources. There is also a dependence on the discreteness of the source distribution, fluctuations on the distance to the sources, and fluctuations on luminosities of the sources. In contrast, the dip is a reliable signature of the interaction of protons with the CMBR since its shape is fixed and is difficult to imitate with other mechanisms unless they have many free parameters. The protons in the dip come from distances of about 1000 Mpc, and this assumption of a uniform distribution of sources within this volume is justified. Berezinsky analyzes the dip in terms of a modification factor [23] given by: %(E) = Jp(E) Junm p (E) , (2.9) where Junm p (E) = KE−!g includes only adiabatic energy losses (redshift). #g is the spectral generation index of the source. This equation is the ratio of the spectrum, with all energy loses taken into account, to the unmodified spectrum where only redshift energy losses are included. This makes the dip less model-dependent than Jp(E). It depends very weakly on #g and Emax, a rectilinear or diffusivemode of propagation, large-scale source inhomogeneity, source separation within 50 Mpc, and local source density. It is modified by the presence of nuclei and the cosmological evolution of sources. Figure 2.6 [23] shows the modification factors for nuclei as a function of energy. The calculation of the modification factor for iron and helium nuclei show that even a small admixture of any nuclei is not in good agreement with the observed proton dip. The fraction of nuclei in the primary flux should be less than 10-20% to observe the dip. Figure 2.7 [21] shows the results of a pair production dip fit to the HiRes and TA data. Based on comparison with the HiRes and TA data, there is good agreement on the predicted shape of the dip and the predicted modification factor. Data points above the %(E) = 1 line indicate that the galactic component plays a dominant role and contributes significantly to the flux of cosmic rays. With two free parameters [24], #g and a flux normalization constant, the dip describes about 20 energy bins with a good (2/d.o.f. + 1 froma fit to the data. The values of the generation index parameter are #g = 2.7 for HiRes and #g = 2.6 for TA with uncertainties of 2.55 - 2.75 [21], consistent with observations. The excellent agreement of the data with the dip supports the model of protons interacting with the CMBR. Figure 2.8 [22] shows a fit to the PAO data in 2007 and in 2010. There is a mediocre fit near the cutoff, but it does not contradict the dip. 34 2.4.2.1 Correlation of Distance to Source It is possible to construct a simple model of the evolution of the universe putting in details of the interactions between the CMBR and extragalactic protons and the Hubble expansion of the universe. The assumptions are that there is a heavy component which is galactic in origin and a light component which is extragalactic. All extragalactic sources follow the same power law spectral index, have a maximum energy of Emax = 1021 eV, and have an isotropic distribution that is modified by a factor (1 + z)m, where z is the redshift and m is an evolution parameter takes into account the recent evolution of the sources. Fitting to the spectrum data, the results indicated that the region of the ankle is sensitive to the spectral index, and the region just below the ankle is sensitive to the evolution parameter [37]. Thus these parameters can be measured independently and becomes a powerful tool in modeling. This model was then applied to sources in different shells in redshift. Figure 2.9 [37] shows the decomposition of the extragalactic spectrum from the energy loss model for sources grouped in shells of redshift, z. The model was then compared to the HiRes results. The sum of the components are shown in black. It can be seen how the GZK cutoff and ankle develop. This figure shows the fractionation of extragalactic events in energy by redshift [37]. There is a correlation between cosmic ray energies and the average redshift of their origin. Protons originating at the largest redshifts lose a significant amount of their initial energy and affect the low energy part of the spectrum. A redshift of z = 0.3 contributes most to the dip seen at 1018.5 eV around the region of the ankle. The low redshift shells shape the highest energy part of the spectrum. Furthermore, if the pair production mechanism is turned off in the model, the resulting model prediction shows the ankle region to flatten. In this scenario, cosmic rays with a certain energy and distance shift to lower energies. This result implies that there is a correlation between energy and distance. Recall that Figure 1.13 also showed three energy loss mechanisms for cosmic rays based upon distance. The energy loss mechanisms, photo-pion production and e+e− pair production, are the same for both proton are iron nuclei. With iron nuclei, however, there is also spallation where the iron nucleus releases a nucleon with energy E/A, but the main difference is that the features are seen at a different energy. Let us consider what would happen if the ankle is caused by heavier nuclei. 35 2.5 Mixed Composition Model The Mixed Composition Model [13, 12] investigates how the interpretation of the ankle changes with heavier nuclei. This model used data below the ankle. In this model, it is assumed that: 1. Composition: Extragalactic cosmic rays have the same relative source abundances as low-energy galactic cosmic rays. 2. Generation Spectrum: The energy per nucleon (E/A) is ,i = xiA#−1 i , where - is the spectral index with a source spectrum of Ni(E) , ,iE−#. 3. Spectral Index at High Energy ("): The source has a different spectral index at high energy which is given by xiA#−1 i E−$ to account for lower energy protons not reaching our galaxy. 4. Maximum Energy: Energy losses and photo-fragmentation inside source are neglected so that all nuclei with the same gyroradius will behave the same way (rigidity-dependent cutoff): Emax(AZ X) = Z × Emax(11 H). The fit to the spectrum shown in Figure 2.10 [13, 12] indicates a good fit for a pure proton model with " = 2.6 down to about 1018 eV. This implies that the transition from galactic to extragalactic sources should occur before the ankle. However, shock acceleration processes produce a spectral index of " + 2.2 - 2.3. Given these physical processes, Allard et al. [13, 12] argue that the transition should occur at the ankle. 2.6 Other Relevant Models We will consider three independent analysis to explain the viability of heavier nuclei that includes data above the ankle. Wilk and Wlodarczyk analyzed Auger and HiRes data to help solve the question of composition. They studied the mean Xmax, &Xmax', which is the penetration depth in the atmosphere at which the shower reaches its maximum number of secondary particles, and !(Xmax), which is the root mean square fluctuation of Xmax from event to event. Figure 2.11 [82] shows the energy dependence of the relative abundance of iron in cosmic rays from &Xmax' and ! &Xmax' from the Auger experiment data with two hadronic interaction models, QGSJETII and EPOSv1.99. The data show a monotonic increase from proton towards iron composition for both &Xmax' and ! &Xmax'. With an energy increase, the 36 Xmax dependence can be interpreted by a two component cosmic ray composition with the relative abundance of iron nuclei, -, and a proton contribution of (1 - -) given by: &Xmax' = (1− -) &Xmax'p + - &Xmax'Fe , (2.10) where &Xmax'p and &Xmax'Fe are the shower maxima for pure proton and iron nuclei, respectively. This equation has a monotonic dependence on -. For !(Xmax), there is a non-monotonic dependence on -: !2 = (1− -) !2p + - !2 Fe + - (1 − -) ( &Xmax'p − &Xmax'Fe )2 . (2.11) This leads to an inconsistency. There is a different chemical composition with energy from proton dominated &Xmax' to an iron dominated ! &Xmax'. A similar study was done by Shaham and Piran [68] where they find a similar result to Wilk and Wlodarczyk [82]. In addition, they show that the observation would require a iron:proton ratio of 1:50 at the source and a very hard spectrum to fit the observations. Fur-thermore, they find that replacing iron with helium does not work either. They say that the lack of natural sources with such metallicity, a hard spectrum, and overall incompatibility of the full data set are a problem. Another study by Taylor [77] found that sources with intermediate-to-heavy nuclei consisting of silicon and iron to be consistent with the observed spectra and composition above the ankle. For this consistency, there must be sources within 60 Mpc consisting only of silicon and 80 Mpc consisting only of iron. The (2 for this model is not good. 2.7 Remarks There is no satisfactory model that can explain a heavier composition with increasing energy. Furthermore, the dip structure can be reproduced only in models with protons; even helium does not produce the dip. If the fraction of heavier nuclei is more than about 15%, then the dip is not produced. It is expected that only the heavier galactic components remain in our galaxy near the transition, and if the dip cannot be produced with heavier nuclei, then it is unlikely that the ankle is caused by the galactic-extragalactic transition. Moreover, all experiments see a composition that is consistent with protons in the energy region between 1018.0 eV to 1018.5 eV, which is the region that contains the ankle. Thus the ankle is most likely excavated due to e+e− pair-production. In addition, spallation of heavier nuclei occurs in the extragalactic medium. There is a correlation between the distance that cosmic ray nuclei travel and the amount of energy 37 they lose in propagation. Hence iron nuclei are not expected to be observed after traveling about 50 Mpc, and there are no known sources within 50 Mpc. This implies that the GZK cutoff is due to protons interacting with the CMBR. However, this does not explain the Pierre Auger Observatory data. 2.8 Disappointing Model Amodel was developed to explain the Auger data [14] which was termed the Disappoint-ing Model. The assumptions are based upon the Auger composition and energy spectrum results that the mass composition becomes heavier with increasing energy from 3 × 1018 eV to 35 × 1018 eV. The basic assumptions of the Disappointing Model [14] are: 1. Composition: There is a protonic composition in the energy range 1 × 1018 eV to 3 × 1018 eV, which is consistent with both Auger and HiRes observations, but that gets progressively heavier at the highest energies. 2. Generation Spectrum: The generation spectrum is Qg(E) , E−!g, with Emax = E0. 3. Acceleration: There is rigidity-dependent acceleration in sources. The maximum energy is given by Eacc max = ZE0, where E0 is determined from the data and Z is the nuclear charge number. 2.8.1 Fit to Spectrum Data The approach used to fit the model to data [14] was to calculate the extragalactic diffuse proton flux with a power law generation spectrum, Qg(E) , E−!g with Emax = E0, and normalize this flux by the Auger flux between 1 × 1018 eV to 3 × 1018 eV. This determines the maximum acceleration energy for protons, Emax = E0. Then by varying #g in the range 2.0 - 2.8, the maximum value of E0 allowed by the Auger mass composition data and energy spectrum was searched. Increasing E0 beyond this limit, there was a contradiction either with mass composition or with the energy spectrum. Figure 2.12 [14] shows the Auger Xmax distribution as a function of energy. The mass composition becomes heavier at higher energies and narrower in width, which was difficult to falsify [14]. Figure 2.13 [14] shows the comparison of calculated proton spectra with the combined Auger spectrum for different Emax p . The two extreme cases, #g = 2.8 and #g = 2.0, are shown in the left and right figures, respectively. In the left figure, all curves 38 with Emax % 10 EeV are below the data points at E > 5 EeV and hence compatible with the Auger energy spectrum [14]. However, these curves are excluded by the prediction of the pure proton composition at E ( (4 -5) EeV due to the contradiction in mass composition in a narrow energy range. Thus the Auger composition and spectrum are not self-consistent in this model. 2.8.2 Consequences The Disappointing Model has other consequences [14]. Since the average energy per nucleon for all nuclei are less than (2 − 4) × 1018 eV, there is not enough energy for photo-pion production on the CMBR. 1. This means that a cutoff in the spectrum is not due to photo-pion production as predicted by the GZK mechanism. The cutoff observed in the spectrum would be provided by nuclei photo-disintegration and strengthened by the acceleration cutoff. 2. The GZK mechanism predicts an accompanying neutrino flux. There are several processes that contribute to cosmogenic neutrino production. The delta resonance mechanism (Equation 1.1) has a 1/3 probability of creating a positively charged pion. This charged pion decays into a neutrino and a charged muon via: $+ −) μ+ + )μ (2.12) The charged muon then decays into a neutrino, antineutrino, and a position via: μ+ −) e+ + )e + ¯)μ (2.13) The decay of a secondary neutron produces a proton, an electron, and an antineutrino via: n −) p + e− + ¯)e (2.14) If the GZK cutoff is absent, then the associated cosmogenic neutrinos and photons are also absent. The JEM-EUSO experiment may be able to detect this flux in the future. 3. Correlation with nearby sources is absent even at the highest energies due to nuclei deflection in galactic magnetic fields. Hence another model would be needed to explain the Auger data, and none is present at this time. 39 2.9 Energy Scale Calibration Since the position and shape of the dip is fixed by proton interactions with the CMBR, it can be used to calibrate the energies of various detectors [21]. Assuming an energy-independent systematic error, the energies of several experiments are shifted by E ) &E to obtain the minimum (2 and compared with the calculated dip. This results in & = 1.0 for HiRes, & = 0.625 for Yakutsk, & = 1.2 for Auger, and & = 0.75 for AGASA. Figure 2.14 [22] shows the resulting flux. The equality of the fluxes after this energy calibration in all of the experiments confirms the dip as a feature produced in the spectrum by the interactions of protons with the CMBR [22]. 2.10 Summary In this chapter we have discussed what we can learn from the UHECR energy spectrum. We can learn some of the details of the sources, the acceleration mechanisms, the features at the ultra high energies, the ankle and the GZK cutoff, and energy losses during propagation. One of the goals of the Telescope Array Project is to learn about physics from the ultra high energy cosmic ray spectrum. This thesis presents the results of a monocular measurement of the ultra high energy cosmic ray spectrum from one of the fluorescence detectors of the Telescope Array Project (TA), where the ultra high energy range is defined to be from 1018 eV to above 1020 eV. In particular, the analysis of a Time versus Angle geometry reconstruction is presented. We now move forward to describing the Telescope Array experiment. 40 Figure 2.1. Magnetic field strength versus size of possibleUHECRsources. Objects below the diagonal lines do not have a sufficient combination of magnetic field strength and size to accelerate protons or iron nuclei to ultra high energies. The velocity of the shock wave, or efficiency of the acceleration mechanism, is represented by ". As seen from the plot, active galactic nuclei (AGN), neutron stars, radio galaxy (RG) lobes, and colliding galaxies are the best candidates for sources for UHECRs. Reprinted with permission from [43]. 41 Figure 2.2. The total #p interaction cross section. The cross section starts with the $0 production threshold at $s =1.078 GeV. There is a rapid increase in cross section up to $s =1.232 GeV, which corresponds to the #+ resonance. The average cross section is ! = 0.12 mb. Reprinted with permission from [43]. 42 Figure 2.3. The cosmicmicrowave background radiation photon density. The total photon density is 412 photons/cm3, and themean of the distribution is 6.4 ×10−4 eV. The threshold energy for photo-pion production for for the GZK cutoff to occur at 5 ×1019 eV is about 13.4 ×10−4 eV. Thus, most of the photo-pion production is due to the tail of the blackbody spectrum. Reprinted with permission from [43]. 43 Figure 2.4. The cosmic ray spectrum above 1018 eV as measured by the Pierre Auger Observatory compared to the spectra of HiRes-I and HiRes-II. The plot shows the fractional difference between the data and a spectrum with a spectral index of 2.6. The data from the HiRes experiments are shown by the open and closed blue triangles. The Auger Collaboration believes that there is an abrupt change in the spectral index around 4 × 1018 eV and a gradual suppression of the flux above 3 ×1019 eV. Reprinted with permission from F. Schussler. 44 Figure 2.5. The cosmic ray spectrum above 1018 eV as measured by the HiRes and the Middle Drum Telescope Array Fluorescence Detectors. The ankle and the GZK cutoff feature are clearly seen. Reprinted with permission from [5]. 45 Figure 2.6. Modification factors for different nuclei. The left figure shows the modification factor for protons and iron nuclei, while the right figure shows the modification factor for protons and helium. Curves % = 1 corresponds to redshift losses in both plots. The proton modification factors are given by Curves 1 (redshift and pair production energy losses) and by Curves 2 (total energy losses). The nuclei modification factors are given by Curves 3 (redshift and pair production energy losses) and by Curves 4 (total energy losses including photodissociation). The calculation of the modification factor for iron and helium nuclei show that even a small admixture of any nuclei is not in good agreement with the observed proton dip. The fraction of nuclei in the primary flux should be less than 10-20%. Reprinted with permission from V. Berezinsky and A. Gazizov. 46 Figure 2.7. Predicted dip in comparison with the HiRes data (left) and TA (right). The straight line in both plots correspond to redshift losses, Curves %ee correspond to redshift and pair production energy losses, and Curves %tot correspond to all energy losses. Both HiRes and TA data fit well to the model. Data above %(E) % 1 shows that there is another component of cosmic rays, which is the galactic component. Reprinted with permission from [14] and V. Berezinsky. 47 Figure 2.8. Predicted dip in comparison with Auger data in 2007 and 2010. The straight line corresponds to redshift losses in both plots, Curve %ee corresponds to redshift and pair production energy losses, and Curve %tot corresponds to all energy losses. There is a mediocre fit in the data in the 1020 eV decade. Reprinted with permission from V. Berezinsky. 48 Figure 2.9. The model prediction of cosmic ray fluxes from sources at different redshift shells and their contribution to the overall extragalactic proton energy spectrum. It illustrates the fractionation of extragalactic events in energy by redshift. It can be seen how the GZK cutoff and ankle develop. A redshift of z = 0.3 contributes to the dip seen at 1018.5 eV around the region of the ankle. The low redshift shells shape the highest energy part of the spectrum. This result implies that there is a correlation between energy and distance. Reprinted with permission from D. Bergman. 49 0,1 1 10 18 18,5 19 19,5 20 20,5 Hires 1 (monocular) Hires 2 (monocular) Stereo Fly's eye AGASA (E-20%) !=2.6 (protons only) !="=2.2 (mixed) !="=2.3 (mixed) #(E) $ E3 (1024 eV2m-2s-1sr-1) log 10 E (eV) protons and nuclei (uniform distribution) E max (p) = 3 1020 eV Figure 2.10. A plot of flux · E3 obtained from the Mixed Composition Model and fit to data. In the case of a mixed composition with " = 2.2 − 2.3, there is no ankle feature. There is a good fit to the data for a pure proton model with " = 2.6 down to about 1018 eV. Reproduced with permission from !c ESO, [13]. 50 1018 1019 1020 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ! E [eV] EPOSv1.99 QGSJETII from <Xmax> from "(Xmax) ! Figure 2.11. The relative abundance of iron in cosmic rays from &Xmax' and ! &Xmax' from the Auger experiment data as given two hadronic interaction models, QGSJETII and EPOSv1.99. The data shows a monotonic increase from proton towards iron for both &Xmax' and ! &Xmax'. Reprinted with permission from G.Wilk and Z.Wlodarczyk. 51 Figure 2.12. Auger data onXmax as a function of energy. The mass composition indicates that UHECRs are getting heavier with increasing energy. Reprinted with permission from [14]. 52 Figure 2.13. A comparison of calculated proton spectra with the combined Auger spectrum for different Emax p . The two extreme cases, #g = 2.8 and #g = 2.0, are shown in the left and right figures, respectively. In the left figure, all curves with Emax % 10 EeV are below the data points at E > 5 EeV and hence compatible with the Auger energy spectrum. However, these curves are excluded by the prediction of the pure proton composition at E ( (4 -5) EeV due to the contradiction in mass composition in a narrow energy range. Reprinted with permission from [14]. 53 Figure 2.14. Pair production dip as an energy calibrator. The energies of several experiments are shifted in energy according to E ) &E assuming an energy-independent systematic error, and the resulting fluxes are compared. The recalibration factors are: & = 1.2 for Auger, & = 1.0 for HiRes, & = 0.75 for AGASA, and & = 0.625 for Yakutsk. The agreement of the fluxes after recalibration confirm the dip as a feature produced in the spectrum by the interactions of protons with the CMBR. Reprinted with permission from [55]. CHAPTER 3 THE TELESCOPE ARRAY EXPERIMENT Since UHECRs can be only observed indirectly, a detector must observe physical proper-ties of the EAS, which then provides information about the primary cosmic ray. Two main considerations for such a detector are how the UHECR spectrum can be measured and the subsequent specifications of the detector itself. The Telescope Array Project is used as an example for the rest of this thesis to describe the observation and spectrum of UHECRs. This chapter describes the Telescope Array experiment, the largest operational UHECR detector in the northern hemisphere. First, different techniques to measure the cosmic ray energy spectrum is discussed. Then, an overview of the TA Project is given. Lastly, specifics of the Surface Detectors and Fluorescence Detectors are presented. 3.1 Measuring the UHECR Spectrum Up to the present time, two independent techniques have been used to study extensive air showers and then subsequently determine the UHECR spectrum. One technique is to use Surface Detectors (SDs) to measure the lateral profile of an EAS. Another technique is to use Fluorescence Detectors (FDs) to measure the longitudinal profile of an EAS. When one of these techniques is used independently and for one detector site, it is a monocular measurement. This thesis uses only the information from one FD site, the Middle Drum site, and is thus a monocular measurement of the UHECR spectrum. A combination of these two techniques can also be used to measure the spectrum, with each technique having its own advantages and disadvantages. One such technique uses information from two FD sites to measure the spectrum; this is known as a stereoscopic measurement. A stereoscopic measurement results in a more precise geometry, albeit with a fewer number of events. A hybrid measurement uses both the SD and FD data. If a Time vs Angle geometry is used to analyze the FD information, then the SDs can be thought to increase the tracklength of the shower. Since the core location is known more accurately with SDs, 55 this information further constrains the possible geometry parameters, resulting in a more precise measurement of the spectrum. However, this method limits the statistics since an event must be seen by both the SD and the FD. A more quantitative description of the geometry is given later in the thesis. Now a brief overview of TA is presented. 3.2 Overview of The Telescope Array Project The TA experiment is located in the western Utah desert in Millard County. The origin of the array is located at the Central Laser Facility which is at 39" 17$ 48$$ in latitude and −112" 54$ 31$$ in longitude. The Middle Drum Fluorescence Detector is located at the GPS coordinates 39" 28$ 22$$ in latitude and −112" 59$ 39$$ in longitude, at the relatively close distance of about 180 miles from the host institution, The University of Utah. This location is at the moderately high altitude of approximately 4700 feet above mean sea level and has the relatively low humidity of approximately 50% (yearly average). This location is well-situated for many reasons. There is about 825 g/cm2 of atmosphere above the detector, which results in a Xmax for a vertical event at around 1020 eV. Thus the detector is located at about the right altitude to observe events around 1018 −1019 eV since Xmax is observed. With the low humidity, fluorescence light generated by the EAS is not absorbed by the moisture in the air, making it possible to make a measurement. The site is remote enough to practically eliminate the glow of urban lights, and an additional advantage of the location was that an infrastructure of roads existed. Figure 3.1 [75] shows a map of Telescope Array Experiment. The three green boxes are the three fluorescence detectors, and the Middle Drum Site is represented by the green box at the top center of the map. The FDs overlook about a 700 km2 area containing an array of surface detectors that are represented by the black squares. The TA experiment consists of three fluorescence detectors: Black Rock Mesa (BR), Long Ridge (LR), and Middle Drum (MD), overlooking approximately a 700 km2 array of 507 scintillation surface detectors that are located on a 1.2 km square grid. The BR and LR sites are each instrumented with 12 telescopes while the MD site is composed of 14 telescopes. The MD telescopes view 115" in azimuth, while BR and LR each view about 108" in azimuth. This thesis will concentrate on the data from the MD telescope site. Figure 3.2 shows a picture of the MDFD site. Figure 3.3 shows the basic layout of the Middle Drum site indicating the mirror pointing directions. 56 3.3 Surface Detectors The surface detectors measure the lateral footprint of the EAS on the ground. The TA SDs are scintillation detectors. The 507 SDs are located about 1.2 km apart and cover about 700 km2. Figure 3.4 shows a picture of a SD deployed in the field. There are five major components to each SD: two layers of 3 m2 and 1.2 cm thick scintillation material, a battery that holds enough charge to power the detector overnight and during cloudy periods, a 1 m2 solar panel that can generate up to 125 W of power to charge the battery, an antenna that transmits data to one of the three communication towers via a microwave link, and a 50 MHz FADC readout system. 3.3.1 Calibration and Trigger Each SD has two layers of scintillation material that interacts with secondary EAS particles. The scintillation material is a sheet of polyvinyltoluene plastic doped with a fluor. Embedded in the sheet are fiber optic cables that connect to a PMT. When a secondary particle from the EAS excites a molecule in the scintillation plastic, fluorescence photons are produced. These photons are captured by a fiber optic cable and guided to a PMT. A signal is produced by the PMT and digitized using a Flash Analog to Digital Conversion (FADC) in 20 ns bins. There are a maximum of 4096 FADC counts per bin [6, 18]. Single muons are used to calibrate the SDs. As discussed earlier, low energy cosmic rays produce showers higher in the atmosphere and produce more muons; the flux is also greater. The SDs are constantly hit by these atmospheric muons at an average rate of 700 per second. The amount of energy deposited corresponds to a minimum ionizing energy for muons. The one Minimum Ionizing Particle (MIP) signals are used to calibrate the SDs. When 15 FADC counts are recorded within 160 ns (8 bin window) in coincidence between the upper and lower layers, the SD is triggered at the zeroth level trigger, and a 2560 ns waveform is recorded. In 10 minute intervals, the SD level zero signals are summed over a 12 by 20 ns window and scored, and the peak of the resulting histogram represents the detector response from one muon. This peak allows for the conversion of an FADC count to the energy deposited in units of a minimum ionizing particle. This calibration is used by the SDs to count the number of particles produced by the EAS that reach the ground. Additionally, the "pedestal," or background noise, is subtracted from the signal. This value is determined by the peak of the pedestal histogram of every SD, made by scoring the signals in 8 by 20 ns windows which do not achieve a 0 level trigger. 57 If 45 FADC counts are recorded within 160 ns, the SD stores a waveform and informs its communication tower that a level one trigger occurred. These are local level triggers by the SDs. If a communication tower receives a 3 MIP or level one trigger from three or more adjacent SDs within an 8 μs window, then the tower saves the waveforms from all SDs with a minimum of 0.3 MIP (level zero or level one trigger) in a ± 32 μs time window [6, 18]. This is an event level two trigger that will be analyzed further. 3.3.2 Lateral Distribution The EAS is intrinsically three-dimensional, and the particles in the shower have a lateral width when they reach the Earth's surface. This distribution is sampled by the SD array and reconstructed. This lateral distribution is modeled by the Nishimura-Kamata-Greisen (NKG) function: .(r) = N r2 f # s, r rM $ , (3.1) where N is the total number of electrons, s is the shower age, rM is the Moliere radius for multiple scattering, and f is the Nishimura-Kamata function: f # s, r rM $ = # r rM $s−2 # 1 + r rM $s−4.5 $(4.5 − s) 2$$(s)$(4.5 − 2s) . (3.2) s = 3 1 + 2Xmax x , (3.3) s is the shower age where a shower age of 1 corresponds to x = Xmax [57]. This model is included in Monte Carlo shower simulations that will be discussed later. 3.3.3 Energy Scale The energy scale determined solely from the SDs has a relatively high degree of un-certainty. The SD energy reconstruction depends on the density of particles at a certain distance from the shower core. This density is calculated from CORSIKA Monte Carlo simulations. It is known that the data reveals a much higher density of particles than predicted by models. Since Earth-based accelerators cannot measure hadronic interactions at the ultra-high energies, the SD energy reconstruction is model-based. As seen with KASCADE, there must be some caution used. The FD energy reconstruction, on the other hand, provides a more reliable method of determining the energy. As will be discussed later, the charged particles from the shower produ