Analysis of simulation andreconstruction methods used by thePierre Auger ObservatorySteven J. SaffiSchool of Physical SciencesUniversity of AdelaideThis dissertation is submitted for the degree ofDoctor of PhilosophyMay 2020DeclarationI certify that this work contains no material which has been accepted for the award of anyother degree or diploma in my name, in any university or other tertiary institution and, to thebest of my knowledge and belief, contains no material previously published or written byanother person, except where due reference has been made in the text. In addition, I certifythat no part of this work will, in the future, be used in a submission in my name, for any otherdegree or diploma in any university or other tertiary institution without the prior approval ofthe University of Adelaide and where applicable, any partner institution responsible for thejoint-award of this degree.I give permission for the digital version of my thesis to be made available on the web, viathe University’s digital research repository, the Library Search and also through web searchengines, unless permission has been granted by the University to restrict access for a periodof time.I acknowledge the support I have received for my research through the provision of anAustralian Government Research Training Program Scholarship.March 2020AbstractThough a great deal of knowledge has been gained on the physics of ultra-high energy cosmicradiation, many questions remain unanswered. The largest experiment to date, the PierreAuger Observatory, has been built by an international collaboration to detect and measurethe properties of cosmic rays at the highest energies with unprecedented statistics. ThePierre Auger Observatory is a hybrid detector, containing both a surface detector array, and afluorescence detector. The surface detector array covers an area of 3000 km2, consisting of∼1600 water Čerenkov detectors, operating with a nearly 100% duty cycle. The Fluorescencedetector overlooks the surface detector array with 27 telescopes, each containing 440 PMT’s,to measure the development of extensive air showers calorimetrically, operating on clearmoonless nights. The two detectors each benefit each other, with the surface detectorimproving the geometric reconstruction of showers measured by the fluorescence detector,and the fluorescence detector providing an energy scale calibration for the surface detector.After over 15 years collecting data, the Pierre Auger Observatory is undergoing a number ofupgrades, including upgrades to the surface detector to distinguish different particle types tobetter resolve the primary particles composition.This thesis contains a number of studies related to the detection of cosmic rays. First is anexamination of the pointing directions of the fluorescence telescopes, where misalignmentsin both the pointing directions and in camera positions are found. Second, a new method ofdethinning is developed, reducing biases introduced into detector simulations when thinningis employed in extensive air shower simulations. Third, with the ability to distinguish themuon component of extensive air showers being introduced in the upgrades, the shape of themuon shower front has been examined in simulated showers, to determine the feasibility ofusing the shape of the muon front for composition determination.AcknowledgementsMy time as a PhD student has been long, and although I have had may great experiences,and gained so much knowledge and experience, it has also been filled with many adversities.It has only been with the support of so many of the people around me that I have been ableto overcome these, and complete my research. It is here that I would like to express mygratitude to these people.My supervisors, Bruce Dawson and José Bellido, have been an unwavering source ofguidance and support throughout the years. They have provided a great amount wisdom andenthusiasm, without which completion of my PhD would not have been possible.I would also like to extend my gratitude to the Adelaide High Energy Astrophysics Group.In addition to their advice and and input, they have also been great friends, particularly MaxMalacari.I am grateful for the opportunity provided by the Pierre Auger Observatory collaboration,allowing me to be a part of such an unprecedented experiment. I thank the researchers andstaff not only for providing and maintaining the observatory, but also for their input andresources, particularly on our visit to the observatory.The influence of my family has been crucial in allowing me to get to this point. Thesupport provided by my family, particularly my parents, Paul and Tiz, have allowed meto pursue my passion for science from a very young age. I would also like to extend mygratitude to my sister Stephanie and brother-in-law Matthew, as well as my extended family,some of whom are no longer with us to see this. My family’s constant belief in me, throughboth good and bad times, has given me the confidence and determination needed to overcomethe hardships I have faced, that may have otherwise prevented me from achieving my goals.I will never be able to repay them for the opportunities they provided to me.Lastly, I would like to thank my girlfriend, Emily. The love she has provided me over thelast two years has given me the motivation required to complete my PhD. Our time togetherhas given me a new outlook on life, and made me a better person. I would say that she helpedkeep me sane as I was writing my thesis, but I never was to start with!Table of contents1 Introduction 11.1 Cosmic Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Discovery of cosmic radiation . . . . . . . . . . . . . . . . . . . . 11.1.2 Detection of extensive air showers . . . . . . . . . . . . . . . . . . 11.1.3 Sources of cosmic rays . . . . . . . . . . . . . . . . . . . . . . . . 21.1.4 Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.5 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.6 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.7 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2 Extensive air showers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2.2 Electromagnetic component . . . . . . . . . . . . . . . . . . . . . 171.2.3 Hadronic component . . . . . . . . . . . . . . . . . . . . . . . . . 201.2.4 Muonic component . . . . . . . . . . . . . . . . . . . . . . . . . . 251.3 Detection of ultra-high energy cosmic ray showers . . . . . . . . . . . . . 261.3.1 Surface detector arrays . . . . . . . . . . . . . . . . . . . . . . . . 261.3.2 Fluorescence detection . . . . . . . . . . . . . . . . . . . . . . . . 301.4 History of UHECR detectors . . . . . . . . . . . . . . . . . . . . . . . . . 331.4.1 Volcano Ranch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.4.2 Haverah Park . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.4.3 SUGAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.4.4 Yakutsk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.4.5 AGASA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.4.6 Cornell experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 391.4.7 Fly’s Eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.4.8 High Resolution Fly’s Eye (HiRes) . . . . . . . . . . . . . . . . . 421.4.9 Telescope Array (TA) . . . . . . . . . . . . . . . . . . . . . . . . . 45x Table of contents1.4.10 FAST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471.4.11 JEM-EUSO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 Pierre Auger Observatory 512.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.2 Surface detector array (SD) . . . . . . . . . . . . . . . . . . . . . . . . . . 522.3 Fluorescence detector (FD) . . . . . . . . . . . . . . . . . . . . . . . . . . 562.4 Shower reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.4.1 SD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.4.2 FD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.4.3 Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762.5 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782.5.1 FD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782.5.2 SD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.6 Upgrades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.6.1 High Elevation Auger Telescopes (HEAT) . . . . . . . . . . . . . . 872.6.2 AugerPrime upgrade . . . . . . . . . . . . . . . . . . . . . . . . . 892.7 Current results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.7.1 Energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.7.2 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972.7.3 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003 FD telescope alignment study 1113.1 Previous methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113.1.1 Using star trails . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113.1.2 Using CLF laser shots . . . . . . . . . . . . . . . . . . . . . . . . 1133.2 Determining relative offsets with cosmic ray showers . . . . . . . . . . . . 1143.2.1 Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1143.2.2 SDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.3 Determining absolute offsets with hybrid showers . . . . . . . . . . . . . . 1223.3.1 SDP method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1233.3.2 Timing method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1243.3.3 Core bias effect on SDP method . . . . . . . . . . . . . . . . . . . 1263.3.4 SDP resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283.4.1 Relative offsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283.4.2 SDP method for absolute offsets . . . . . . . . . . . . . . . . . . . 133Table of contents xi3.4.3 Timing method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393.4.4 Comparison of timing and SDP methods . . . . . . . . . . . . . . 1423.4.5 Comparison between relative and absolute offsets . . . . . . . . . . 1493.5 Comparison with other alignment studies . . . . . . . . . . . . . . . . . . 1513.6 Additional errors in telescope geometry . . . . . . . . . . . . . . . . . . . 1533.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1554 Simulation of extensive air showers 1574.1 CORSIKA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1574.2 Thinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594.3 Dethinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1604.4 Simulation of detector response for the Pierre Auger Observatory . . . . . . 1655 An improved dethinning method to reduce biases in detector simulations 1695.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1695.1.1 Dethinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1695.2 Biases caused by resampling . . . . . . . . . . . . . . . . . . . . . . . . . 1715.3 The new resampling method . . . . . . . . . . . . . . . . . . . . . . . . . 1715.4 Performance comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . 1725.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1776 Muon shower front shape and correlation with first interaction depth, Xmax andX µmax 1816.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1816.2 Muon timing study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1826.3 Muon time fitting study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1866.3.1 Resolution of shower parameters using shower front shape . . . . . 1926.3.2 Detector resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 1976.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1987 Conclusions 199References 203Chapter 1Introduction1.1 Cosmic Radiation1.1.1 Discovery of cosmic radiationPrior to the discovery of cosmic radiation, it was thought that the only source of backgroundradiation was from the decay of radioisotopes present in the environment. It was noticed thatbackground radiation levels were higher at higher altitudes above sea level, but it was unclearwhether or not this was due to higher concentrations of radioisotopes at locations with ahigher altitude. In 1912, Victor Hess [55] carried out measurements of ionising radiationlevels on a series of balloon flights to observe how radiation levels varied with increasingaltitude (while also increasing distance from the ground) (see figure 1.1). It was expectedthat radiation levels would drop with increasing distance from the ground as the atmospherewould block radiation from radioisotopes in the ground. This drop was observed with a smallincrease in altitude up to about 1000 m, but at even higher altitudes, radiation levels increased.At 5000 m, radiation levels were approximately twice as high as at ground level. From this,it was concluded that there was a source of ionising radiation coming from space. Severalascents were made, at great risk to himself, during both day and night, as well as during asolar eclipse, finding no significant difference in the radiation increase at high altitude. Thisexcluded the sun as the likely source of this radiation. The only suitable explanation was acosmic origin.1.1.2 Detection of extensive air showersIn 1939 Pierre Auger was experimenting with coincident detections in particle counters [12].He was able to detect when two or three particle counters were triggered within less than 10−62 IntroductionFig. 1.1. Victor Hess undertaking a balloon flight.s of each other, for detector spacings up to 300 m. With this set-up, it is very unlikely thattwo unrelated particles would simultaneously trigger two detectors. Given the backgroundrate of single detections, approximately 1 accidental trigger per hour was expected for twodetectors. Three detectors being triggered simultaneously would be exceptionally rare sowhen this occurred, the only possible explanation for this is a particle shower originating froma single high energy particle. Experiments shielding one of the detectors with an increasingthickness of lead led to the conclusion that most of the ionising particles in these showerswere electrons. Simultaneous triggers were detected at distances of up to 300 m. Longerdistances were not tested as 300 m is the approximate distance light travels in 1 microsecond,so the geometry of the shower may not cause simultaneous triggers with a 1 microsecondresolution. It was concluded from this that showers of particles were being initiated by aprimary particle with an energy of at least 1015 eV.Over the decades following this discovery, a series of air shower detection experimentshave led to the discovery and study of cosmic ray particles, primarily protons and atomicnuclei, with an energy range that extends past 1020 eV. This means that cosmic rays havenow been studied over an energy range spanning an enormous 11 orders of magnitude.1.1.3 Sources of cosmic raysThe high energy of cosmic ray particles is not attainable by any known mechanism other thanacceleration in electro-magnetic fields. Particles cannot gain energy in a static magnetic field,but in the interstellar medium, magnetised plasma is constantly in motion. When chargedparticles move between regions of magnetised plasma with differing velocities, energy canbe exchanged between the magnetic field region and cosmic rays. This process is stochastic,with it being possible for a particle to either gain or lose energy in an interaction, but the1.1 Cosmic Radiation 3overall effect over many interactions is an increase in energy. This is especially prevalentwhere there are strong shocks, such as supernova blast waves. This mode of acceleration isknown as Fermi acceleration.Consider a region of space with a strong shock. It is assumed that each interaction with anaccelerating region leads to an increase in particle energy of ∆E = ξ E. After n interactions,the energy will be:En = E0(1+ξ )n (1.1)With each interaction, there is also a possibility that a given particle will leave the accelerationregion, Pesc. This means that the probability of a particle remaining in the accelerating regionafter n interactions is (1−Pesc)n. From this, with N0 particles injected with energy E0, weget the number of particles with energy above E:N(> E) = N0(EE0)γ(1.2)whereγ =ln(1−Pesc)ln(1+ξ )≈ Pescξ(1.3)In Fermi’s original theory, the source of acceleration is from gas clouds in the interstellarmedium. These gas clouds have velocities of ∼15 kms−1 relative to the average motionorbiting the galaxy. Cosmic ray particles approaching the cloud are assumed to have anisotropic distribution in the galaxy frame. Upon entering the cloud, the particles are scatteredand exit the cloud with an isotropic distribution in the cloud frame (see figure 1.2). In thecloud frame, a particle does not gain or lose energy, but in the galaxy frame, a particle canlose or gain energy depending on the angle that the particle has with respect to the cloudmotion upon entering and exiting. A head on collision will result in an energy gain, whereasan overtaking collision will result in an energy loss. Due to relativistic transformations, headon collisions are more likely. The average energy gain in an interaction with the cloud is:⟨∆E⟩E≃ 43β2 (1.4)where β = vclc , for a cloud velocity vcl . This is known as second order Fermi acceleration,since the energy gained is second order with respect to cloud velocity. With typical cloudvelocities of 15 kms−1, β ≪ 1, so the average energy gain in each interaction is negligible.The spectral index obtained with reasonable estimates of the escape time from the galaxy andthe rate of interactions is far steeper than the observed index. Thus second order acceleration4 Introductionfrom clouds in the interstellar medium is unlikely to be source that accelerates cosmic raysto ultra-high energies.Fig. 1.2. Cosmic ray interaction with a moving cloud, showing the mechanism behindsecond order Fermi acceleration. From [84].Another mechanism where cosmic rays can experience acceleration is at a strong shock,such as those produced by supernovae. In this case, the shock is approximated as a planefront, with fast moving supernova ejecta causing the interstellar medium to pile up at theshock front. Supernova ejecta have velocities in the order of 104 kms−1. Calculation of theenergy gain of the cosmic ray is similar to Fermi’s original theory of magnetised clouds, byconsidering the supernova ejecta as one of the magnetised clouds. The difference is that toa particle on either side of the shock front, the plasma on the other side of the shock frontappears to be approaching the shock front, so every crossing acts as a head on collision.This can be seen in figure 1.3. Because of this, the energy gain averaged over all possibleinteraction angles is:⟨∆E⟩E≃ 43β (1.5)where β = vpc , for an ejecta velocity vp. Cosmic rays may undergo repeated interactions, orescape into the interstellar medium by diffusion. Calculations of the rate of crossing andescape of particles, and the energy gain of a shock crossing, give an energy spectrum with aspectral index γ =−2. This spectral index is flatter than the index observed at Earth, thoughthis can be explained. As Earth is (fortunately) not in the immediate vicinity of a supernovashock, cosmic rays produced by a supernova need to diffuse through the interstellar mediumfor a long time before arrival at Earth. Higher energy cosmic rays are more likely to escapethe magnetic field of the galaxy before then, resulting in a steeper spectral index at Earth.1.1 Cosmic Radiation 5Fig. 1.3. Cosmic ray interaction with a strong shock, showing the mechanism behind firstorder Fermi acceleration. From [84].Sources of cosmic ray acceleration have limits on the energy of particles they can produce.Though there are a number of factors that may limit the energy of a cosmic ray, the mostrelevant are the strength and extent of the magnetic fields. A requirement for cosmic rayacceleration is that, to an approximation, the accelerator must be larger than the magneticgyroradius of the cosmic ray. The gyroradius of a particle of charge Z is rL = E/ZeB, givinga maximum energy E < ZeBL for an accelerator of size L. As a result of this, supernovaremnants are not capable of accelerating cosmic rays to the highest energies. The velocityof shocks within the accelerating region also plays a significant role, with an energy limitE < ZeBLβ , for scattering centres with velocity βc [57]. A plot of the strength and extent ofmagnetic fields for a number of possible accelerators is given in figure 1.4.1.1.4 PropagationCosmic rays are subject to a number of effects when propagating from a source to Earth.Magnetic fields have an effect on charged cosmic rays of all energies, though the extent doesvary. In a given magnetic field, the gyro-radius of a particle of charge Z is proportional torigidity R = pc/Ze. For an ultra-relativistic particle of energy E, rigidity is proportional toE/Z, so for a given energy, heavier nuclei experience a greater deflection when compared toprotons. At low energies, cosmic rays have a gyro-radius much shorter than their propagationdistance to Earth, and as a result propagate diffusively through galactic magnetic fields. Asrigidity increases, particles diffuse more rapidly and escape galactic magnetic fields in ashorter time. It is only at the very highest energies that magnetic deflections are expected tobe small enough for cosmic ray arrival directions at Earth to be traceable to a source. Fora rigidity of 1020 V, deflections of a few degrees are expected for cosmic rays propagatingacross intergalactic distances.6 IntroductionFig. 1.4. The Hillas plot, showing the strength vs extent of magnetic fields for astrophysicalobjects. The lines show the requirements to accelerate cosmic rays to 1020 eV, with limits forprotons at β = 300, and protons and iron at β = 1. Plausible regions need to be above thelines. From [57].At the highest energies, cosmic rays propagating across intergalactic distances sufferfrom energy losses. Soon after the discovery of the cosmic microwave background (CMB),Greisen [47], and Zatsepin and Kuzmin [118] predicted that there would be a cutoff in theobserved cosmic ray spectrum due to interactions with CMB photons. At the highest energies,the low energy CMB photons are blue-shifted to sufficient energy for particle production.Cosmic ray protons can undergo photo-pion production by the following interactions:p+ γCMB → n+π+ (1.6)p+ γCMB → p+π0 (1.7)The threshold energy for this interaction is ∼ 5× 1019 eV [47, 118]. Protons above thisenergy have an interaction length of around 6 Mpc [14]. Protons will also undergo pairproduction at lower energies:p+ γCMB → p+ e++ e− (1.8)The proton threshold energy for pair production is ∼ 1018 eV [72]. Though both interactionswill cause an energy loss, there is a crucial difference between the two. The energy loss1.1 Cosmic Radiation 7for a pair production interaction is ∼ 0.1% of the proton energy, so pair production can beseen as a continuous, but slow energy loss, still allowing propagation over great distances.On average, photo-pion production interactions will cause a proton to lose ∼ 20% of itsenergy. As a result of this, a proton with an initial energy above the photo-pion productionthreshold will rapidly lose energy to photo-pion production, until the energy is below thephoto-pion production threshold. This occurs within a distance of ∼ 100 Mpc, regardless ofinitial energy (see figure 1.5).Fig. 1.5. The mean energy of protons propagating through the CMB, with starting energiesof 1020 eV, 1021 eV, and 1022 eV. From [32].Heavier nuclei will also interact with the CMB, with the dominant process being pairproduction and photo-disintegration. The energy threshold for pair production increases inproportion to the mass, A. The photo-disintegration energy threshold also increases withincreasing A, with energies being similar to the proton photo-pion threshold. The energy losslength due to CMB interactions for protons and heavier nuclei are shown in figure 1.6.The energy loss of cosmic ray particles at energies above ∼ 1020 eV, regardless of com-position, means that any cosmic rays observed above this energy must have been producedwithin ∼ 100 Mpc. Since particles above this energy have been observed, this implies that anaccelerator is nearby.1.1.5 SpectrumThe range of energy of cosmic rays observed is very broad, from the very numerous particleswith ∼GeV energy up to exceptionally rare particles with an energy above 1020 eV. The8 IntroductionFig. 1.6. Energy loss length χ of nuclei vs energy. Also shown is the effective energy losslength from the expansion of the universe. From [4].energy spectrum of cosmic rays approximately follows a power law:dNdE∝ E−γ (1.9)where N is the number flux of particles arriving at the top of the Earth’s atmosphere. Atthe lowest energies, the flux is suppressed, due to the outward moving solar winds. Theflux at the lowest energies also varies depending on solar activity, with high solar activitycorresponding to lower cosmic ray flux, as stronger solar winds reduce the inward diffusionof low energy cosmic rays. The value of γ is approximately 2.7, but varies with energy, withnotable changes in the hardness of the spectrum. This can be seen in Figure 1.7, and a closerview of the highest energy part of the spectrum in figure 1.8. Notable features in the energyspectrum include a steepening of the energy spectrum at ∼ 1015.5 eV, known as the knee, afurther steepening at ∼ 1017 eV, known as the second knee, a flattening of the spectrum at∼ 1018.5 eV, known as the ankle, and a cutoff in the spectrum beyond ∼ 1019.5 eV. Thereis also a recently discovered steepening of the energy spectrum between the ankle and thecutoff [110].At the energy where the knees in the spectrum are observed, cosmic rays are likely dueto galactic sources, such as supernova remnants. The first knee has been found to be asteepening of the spectrum for light nuclei [6], while the second knee is due to a steepening1.1 Cosmic Radiation 9Fig. 1.7. The cosmic ray energy spectrum, with measurements from a number of observato-ries combined. From [88].PoS(ICRC2019)234Auger and TA UHECR energy spectrum Olivier DelignyE [eV]16101710181019102010]2eV-1sr-1yr-2J(E) [km3E37103810TA, ICRC 2019Auger, ICRC 2019Figure 1: ICRC 2019 energy spectra of the Pierre Auger Observatory and the Telescope Array scaled byE3. In each experiment, data of different detection techniques are combined to obtain the spectrum over awide energy range.1. IntroductionCosmic rays compose less than one particle out of ten million in the interstellar gas. Still, theiraverage energy density is similar to that of the gas. A small proportion of particles has thereforeappropriated a substantial part of the available energy. The study of the energy spectrum of cosmicrays, providing the differential intensity (flux per steradian) of cosmic protons and nuclei as afunction of energy, is thus one of the cornerstones of astroparticle physics.Because of the very small value of the cosmic-ray intensity at high energies – less than oneparticle per km2 yr sr above 10 EeV – the construction of giant observatories has been necessaryto collect an increased influx of events. The Pierre Auger Observatory, located in the provinceof Mendoza (Argentina) and covering 3000 km2, has been allowing since 2004 a scrutiny of theUHECR intensity – except in the northernmost quarter. Another scrutiny, mainly of the Northernsky, has been provided by the Telescope Array (TA), located in Utah (USA) and covering 700 km2,operating since 2008. These latest-generation experiments have allowed an unprecedented sensi-tivity in measuring the UHECR energy spectrum.In this joint contribution, we review the different energy spectrum measurements made at theseobservatories in the last decade in the quest to decipher the UHECR origin. Both observatories arehybrid cosmic-ray detectors that consist of fluorescence telescopes overviewing an array of surfacedetectors (SD). The fluorescence detectors (FD) provide an accurate determination of the cosmic-ray energies by measuring the longitudinal developments of the extensive air showers in a nearlycalorimetric manner. Their duty cycle is however limited to about 15%. By contrast, the SD dutycycle is quasi-permanent, allowing for a large and uniform exposure. It is thus advantageous forboth Auger and TA to use their SD arrays to measure the energy spectrum at the highest energies,2Fig. 1.8. The cosmic ray energy spectrum at the highest energies, measured with the twolargest experiments, the Pierre Auger Observatory, and the Telescope Array. From [35].in the spectrum of heavier nuclei [7]. The reason for the knees is not entirely clear. Theprominent theory is that the knees represent an upper limit to the energy of the sources of10 Introductioncosmic rays within the galaxy. For ultra relativistic particles, the deflection due to magneticfields is proportional to the charge. As a result of this, for acceleration due to magnetic fields,the maximum energy attainable will be proportional to Z. The first knee would thereforebe an acceleration limit of the light nuclei, H and He. As H and He are the most abundantcosmic ray nuclei below the knee, this is seen in the overall spectrum. As iron nuclei are theheaviest with a significant contribution to the cosmic ray flux, a second knee is expected at ahigher energy. The energy ratio of the two knees is consistent with an acceleration limit for Hand He for the first, and iron for the second. The steepening of the energy spectrum betweenthe knees may be expected due to elements of intermediate mass, but nuclei of intermediatemass are likely to be less abundant than iron. Another theory is that the knees are due tocosmic rays escaping the galaxy. Cosmic rays of lower energy are likely to remain trappedwithin galactic magnetic fields. At higher energies, cosmic rays have a larger gyro-radius andthus diffuse out of the galaxy more rapidly. This would similarly occur at higher energies forheavier compositions, explaining the knee and second knee.The next prominent feature in the energy spectrum is the ankle, occurring at an energyaround 1018.6 eV, where the spectrum flattens out to a spectral index of ∼2.2. There area number of theories explaining the ankle in the spectrum. Anisotropy studies suggestthat around this energy, cosmic rays are transitioning from galactic to extra-galactic inorigin. Composition studies suggest a light composition near the energy of the ankle, withcomposition becoming heavier with increasing energy above the ankle energy. There are anumerous theories to explain the ankle. These include:• The ankle represents a transition from galactic to extra-galactic cosmic rays. In thiscase, the galactic spectrum is steeper than the extra-galactic spectrum, and dominatesat low energies, with the extra-galactic component dominant above the ankle [5]. Ifthis were true, a strong anisotropy would be expected at ankle energies in the directionof the galactic centre due to galactic cosmic rays. Only a very small anisotropy isobserved near the ankle energy. A transition from galactic to extra-galactic cosmic raysat the ankle also has difficulties explaining the light composition observed around theankle, as galactic cosmic rays at this energy are expected to be heavy due to acceleratorlimits. Additional galactic sources with higher rigidity limits would be required toexplain the light composition at the ankle energy.• Since the composition is light near the ankle, it has been suggested that the flux at ankleenergies is suppressed by pair production interactions with the cosmic microwave back-ground. In this model [5], the extragalactic component is mostly protons, becomingthe dominant component of cosmic rays above the second knee. This explains the light1.1 Cosmic Radiation 11composition around the ankle, as well as the lack of a strong anisotropy towards thegalactic centre below the ankle energy.• A recent model has shown that the ankle can be explained by photo-disintegrationnear the accelerator [109]. In this model, the accelerators exist in a region of highphoton density and strong magnetic fields. The accelerators have a rigidity limit, soheavier nuclei can accelerate to higher energy. Heavier nuclei are subject to photo-disintegration at both lower and higher energies. Cosmic rays are trapped in thisregion of strong magnetic fields, and escape this region by diffusion. Because themean escape time is energy dependent, nuclei at lower energies are likely to sufferphoto-disintegration, while higher energy nuclei are more likely to escape, creating anabundance of protons around the ankle energy, while also predicting a transition to aheavier composition at higher energy due to the rigidity limit of the accelerators andthe escape of high energy nuclei.The final feature in the energy spectrum is the cutoff, where the spectrum drops off witha very high spectral index (∼ 5), above 5×1019 eV. This has previously been consideredas a gradual steepening in the spectrum above the ankle energy, but recent results [110]suggest two steepenings in the spectrum: one from γ = 2.2 to γ = 3.2 at 1.2×1019 eV, andanother to γ = 5.4 at 5× 1019 eV. There are two possible explanations for the observedcutoff in the spectrum. One is that it is a suppression of higher energy cosmic rays due toGZK interactions. In this case, most of the cosmic rays at energies below the cutoff comefrom distant sources (> 100 Mpc), with cosmic rays of higher energy attenuated due tointeractions with CMB photons. This could be photo-pion production for light nuclei, orphoto-disintegration for heavier nuclei. The other possibility is that the sources of the highestenergy cosmic rays have a rigidity limit, so the cutoff would represent the maximum rigidityof the accelerator. For a given rigidity, heavier nuclei have a higher energy, so compositionwould be expected to transition from light to heavy approaching the cutoff, consistent withthe observed mass composition results.1.1.6 CompositionThe composition of cosmic rays is crucial to gain an understanding of possible sources ofcosmic rays and of their propagation to Earth, as well as explaining features in the cosmicray spectrum. Cosmic rays at low energies have a sufficiently high arrival rate that directdetection experiments are possible, allowing a determination of the composition of eachindividual cosmic ray particle. A plot of the relative abundance of cosmic ray nuclei isshown in figure 1.9, compared to the abundance of elements in the solar system. Overall12 Introductionthe relative abundance of cosmic ray nuclei are similar to those in the solar system, with theexception of elements with a very low solar system abundance, such as lithium, beryllium,and boron. The abundance of these nuclei can be attributed to spallation of heavier nucleias they collide with matter in the interstellar medium. These observed abundances of theseotherwise rare elements, and abundances of unstable radionuclides in cosmic rays, allow forthe determination of the time taken to propagate to Earth, and the amount of matter traversedfrom the source. This has shown that cosmic rays diffuse not only through the galactic disk,but also in the less dense galactic halo [43].Fig. 1.9. Relative abundance of cosmic ray nuclei, compared to the abundance of elementsin the solar system. From [43].At higher energies, the low arrival rate makes direct detection unfeasible, so observa-tions of extensive air showers must be used to deduce mass composition. Due to inherentfluctuations in shower properties, it is not possible to deduce the exact composition of eachindividual cosmic ray, though average composition can be deduced over a large number ofevents. At the highest energies, an additional complication arises due to uncertainties in the1.1 Cosmic Radiation 13physics of hadronic interactions at the highest energies. The most robust measure comesfrom the depth of shower maximum, Xmax, the details of which will be discussed later. Xmaxresults from the Pierre Auger Observatory and Telescope Array are shown in figures 1.10and 1.11.Fig. 1.10. (left) 〈Xmax〉 vs energy measured by the Auger FD. (right) σ(Xmax) vs energymeasured by the Auger FD. The lines shown are the predictions for proton and iron primaries.From [117].Fig. 1.11. (left) 〈Xmax〉 results from Telescope Array, with Monte Carlo predictions usingQGSJetII-04. (right) σ(Xmax) vs energy measured with Telescope Array, with Pierre Augermeasurements and Monte Carlo predictions using QGSJetII-04. From [48].The Pierre Auger FD (fluorescence detector) results indicate a light composition at 1018.32eV, with composition becoming heavier at both higher and lower energies. The results fromTelescope Array seem to show a lighter composition at the highest energies, though it should14 Introductionbe noted that values of ⟨Xmax⟩ obtained by Telescope Array are not directly comparable tothe Pierre Auger results due to the analysis methods used. In the Pierre Auger analysis,events with a limited field of view are removed from the analysis as they may cause a bias inthe Xmax distribution, so the values of ⟨Xmax⟩ and σ(Xmax) are estimates of the true ⟨Xmax⟩and σ(Xmax). The Telescope Array analysis does not exclude these events, as the smallerexposure does not allow for data to be rejected. Instead, the data is compared to detectorsimulations of pure primary compositions, which are expected to include an identical bias.To obtain a comparison between the two datasets, the Xmax distributions obtained by thePierre Auger are injected into the detector simulations of Telescope Array [48]. A graph ofthis comparison is shown in figure 1.12, showing that the observed Telescope Array ⟨Xmax⟩is reasonably consistent with the Pierre Auger Xmax distribution when simulated throughTelescope Array.Fig. 1.12. Comparison of Pierre Auger and Telescope Array ⟨Xmax⟩ measurements. Thered points are ⟨Xmax⟩ obtained from Telescope Array, while the blue points are the ⟨Xmax⟩obtained from simulating the Pierre Auger Xmax distributions in Telescope Array. From [48]Composition and spectrum data from a number of cosmic ray observatories has beencombined to give a spectrum separated into different mass components, covering a wideenergy range [36]. This is shown in figure 1.13.1.1.7 AnisotropyDue to deflections by magnetic fields, anisotropies at small angular scales are only expectedat the highest energies. The two largest observatories, the Pierre Auger Observatory and the1.2 Extensive air showers 15Fig. 1.13. Combined cosmic ray spectrum with spectra separated for different mass compo-nents. From [88], updated from [36].Telescope Array have both detected significant anisotropies at high energies. The TelescopeArray has detected a hotspot [60] with a local significance of 5.1σ , within a 25◦ circle forenergies above 57 EeV, which equates to a 2.9σ significance when accounting for scansin position, angular scale, and energy. The TA hotspot does not coincide with any knowncandidates for production of ultra-high energy cosmic rays. The Pierre Auger Observatoryhas observed a hotspot [28] for energies above 38 EeV within a 27◦ radius, with a localsignificance of 5.6σ . This hotspot is very close to Centaurus A, the nearest AGN (activegalactic nucleus) to Earth. The significance for an excess at Centaurus A, accounting forscans in energy and angular scale, is 3.9σ . A combined anisotropy map, using data fromboth observatories, is shown in figure 1.14. Anisotropy results will be discussed further insection 2.7.2.1.2 Extensive air showersFor energies up to about 1014 eV, the cosmic ray flux is high enough that a detector with acollecting area in the order of ∼1 m2 will detect a sufficient number of particles to be studied.This allows for direct detection experiments which are done at high altitude, in balloon orsatellite based detectors. Direct detection experiments are capable of accurately determiningparticle energy and composition and so are preferred when possible. Above this energy, amuch larger collecting area is required. The only feasible way this has been done is to utilisethe atmosphere as a detector by observing the cascade of particles produced in an extensive16 IntroductionFig. 1.14. Map of the local flux of cosmic rays within a 20◦ radius (top), and 15◦ radius(bottom), measured by the Pierre Auger Observatory and the Telescope Array. The energycutoff for these events is 40 EeV for the Pierre Auger events, and 53.2 EeV for telescopeArray, to account for different energy scales between the two observatories. Flux is shownon the left, with the local significance shown on the right. From [37].air shower. With a huge number of shower particles spread over a large area, detectors needonly cover a small fraction of the total collecting area, with the fraction required gettingsmaller with increasing cosmic ray particle energy. Detection of cosmic rays via extensiveair showers does come at a cost. Indirect detection does not allow for cosmic ray energy andcomposition to be determined simply. Cosmic ray particle properties must be deduced fromthe properties of the extensive air shower.1.2.1 TerminologyIn this section, some terminology will be introduced that is commonly used when discussingcosmic ray shower physics.Particles in a cosmic ray shower typically travel in approximately the same direction asthe primary particle. This is especially true at higher energies. This means that the majorityof particles are concentrated near a line that the primary particle would have taken had it not1.2 Extensive air showers 17interacted. This line is referred to as the shower axis. Shower development along this axis isreferred to as longitudinal development. Since the development of a cosmic ray shower ispredominantly caused by interactions with air nuclei, the level of development of a shower isprimarily dependent on the depth of matter traversed. The total column density (or "depth")of matter encountered along a path is:X =∫ρdl (1.10)for a matter density ρ Considering that particles in a cosmic ray shower travel in nearlystraight lines, and are generally close to the shower axis, we can define a depth at a givenheight for an atmospheric shower of zenith angle θ at height h:X(h) =1cosθ∫ infhρ(z)dz (1.11)The unit for X typically used in cosmic ray physics is g cm−2.Particles within a shower do not necessarily lie exactly on the shower axis. This ismostly relevant for surface detectors, as it allows showers to be detected without having adetector exactly at the shower core (the location where the shower axis intersects with theground). The density of particles arriving at the ground is primarily dependent on distancefrom the shower axis. Since showers are unlikely to be exactly vertical, the distance from theaxis is not the same as the distance to the core. Because of this, it is useful to introduce ashower plane coordinate system, where a position on the ground is projected onto a planeperpendicular to the shower axis. Shower plane coordinates are typically expressed as a polarcoordinate system, with a distance from the core in the shower plane, r, and an azimuth, ζ ,where ζ = 0 is usually chosen to be in the upstream direction (underneath the shower axis).The distribution of particles in the shower plane is often referred to as the lateral distribution.1.2.2 Electromagnetic componentElectromagnetic showers are cascades initiated by either electrons, positrons, or photons.Such cascades almost exclusively consist of electrons, positrons, and photons. With thecomposition of cosmic rays at the highest energy being hadronic with only very rare or noexceptions, pure electromagnetic showers are not applicable in most studies of ultra-highenergy cosmic rays. However, photons, electrons, and positrons are produced in hadroninitiated cascades and so electromagnetic cascades are an important component of hadronicshowers. The electromagnetic component of an ultra-high energy cosmic ray shower willtypically contain the majority of particles and is responsible for the majority of fluorescence18 Introductionand Čerenkov light. Electromagnetic cascades can be understood with the Heitler model [53].In the Heitler model of an electromagnetic cascade, the two dominant particle interactionsare considered, pair production and bremsstrahlung radiation. A high-energy photon in thevicinity of a massive charged particle (in this case predominantly air nuclei) can undergo pairproduction to produce an electron-positron pair:N + γ → N + e−+ e+ (1.12)Electrons and positrons at high energy predominantly interact in the vicinity of air nuclei toproduce photons by bremsstrahlung radiation:N + e → N + e+ γ (1.13)simulations. Nevertheless, Heitler!s EM model pre-dicted accurately the most important features ofelectromagnetic showers.Heitler!s model (Fig. 1a) has e+, e!, and pho-tons undergoing repeated two-body splittings,either one-photon bremsstrahlung or e+e! pairproduction. Every particle undergoes a splittingafter it travels a fixed distance related to the radi-ation length. After n splittings there are 2n totalparticles in the shower. Multiplication abruptlyceases when the individual e± energies drop belowthe critical energy nec, where average collisional en-ergy losses begin to exceed radiative losses.This simplified picture does not capture accu-rately all details of EM showers. But two veryimportant features are well accounted for: the finaltotal number of electrons, positrons, and photonsNmax is simply proportional to E" and the depth ofmaximum shower development is logarithmicallyproportional to E".We approximate hadronic interactions similarly[4]. For example, Fig. 1b shows a proton strikingan air molecule, and a number of pions emergingfrom the collision. Neutral pions decay to photonsalmost immediately, producing electr ma n ticsubshowers. The p± travel some fixed distanceand interact, producing a new generation of pions.The multiplication continues until individualpion energies drop below a critical energy npc ,where it begins to become more likely that a p±will decay rather than interact. All p± are then as-sumed to decay to muons which are observed atthe ground.This first approximation assumes that interac-tions are perfectly inelastic, with all the energygoing into production of new pions. We will studythe more realistic case which includes a leadingparticle carrying away a significant portion of theenergy later (Section 4).The important di!erence between a hadroniccascade and a pure EM shower is that a third ofthe energy is ‘‘lost’’ from new particle productionat each stage from p" decay. Thus the total energyof the initiating particle is divided into two chan-nels, hadronic and electromagnetic. The primaryenergy is linearly proportional to a combinationof the numbers of EM particles and muons.We examine the model in detail below. In par-ticular, we will look at its predictions for measur-able properties of extensive air showers,attempting to assess which predictions are reliableand which may not be. First, we review the specif-ics of Heitler!s electromagnetic shower model andthen develop the hadronic analogue. In all that fol-lows, the term ‘‘electron’’ does not distinguish be-tw en e+ and e!.2. Electromagnetic showersAs seen in Fig. 1a, an electron radiates a singlephoton after traveling one splitting length(a) (b)!e+ e_n=1n=2n=3n=4p" +_ " on=1n=2n=3Fig. 1. Schematic views of (a) an electromagnetic cascade and (b) a hadronic shower. In the hadron shower, dashed lines indicateneutral pions which do not re-interact, but quickly decay, yielding electromagnetic subshowers (not shown). Not all pion lines areshown after the n = 2 level. Neither diagram is to scale.388 J. Matthews / Astroparticle Physics 22 (2005) 387–397Fig. 1.15. The Heitler model for electromagnetic air showers. Scattering angles have beenexaggerated.At high energy, electrons will give on average about half of their energy to the photonproduced by bremsstrahlung. The electron and positron produced by pair production willalso each have, on average, half of the energy of the parent photon. Particles will interactwith a mean interaction depth λ .1.2 Extensive air showers 19At high energies, the interaction length for pair production and bremsstrahlung areapproximately equal. Additionally, the interaction length of these two interactions areroughly independent of energy for high energy particles.Given these assumptions, an electromagnetic shower can be thought of as having a totalnumber of particles doubling with each interaction length, starting from a single parentparticle. With each generation of interactions, the energy per particle is halved. Thus, after ninteraction lengths, the total number of particles is:N = 2n (1.14)and the average energy of each particle is:E =E02n(1.15)where E0 is the energy of the primary particle. Figure 1.15 shows an electromagnetic showerwith a photon as the primary particle. This process cannot continue indefinitely though.Electrons will lose energy continuously through ionisation loss. This loss is very small overone interaction length for high energy particles but below 81 MeV electrons are more likelyto lose their energy through ionisation loss before a bremsstrahlung interaction. This isknown as the critical energy, ξ ec . An electromagnetic shower will have its maximum numberof particles when the average energy is equal to the critical energy with the maximum numberof particles being:Nmax =E0ξ ec(1.16)The maximum number of particles occurs at the following atmospheric depth:Xmax = λrln(E0ξ ec)ln(2)(1.17)With the value of λr of about 37 g cm−2 in air.Due to bremsstrahlung interactions converting an electron into an electron and a photon,and pair production converting a photon into two electrons, we expect that the ratio of photonsto electrons to be:Ne = 2Nγ (1.18)More detailed simulations, as well as observations of electromagnetic showers, do showsome discrepancies with this simplified model. During bremsstrahlung, multiple photons may20 Introductionbe produced. Additionally, it is only electrons and positrons that lose energy to ionisation loss,meaning that for particles with energy near or below the critical energy, a deficit of electronsand positrons occurs. At depths near the maximum in shower development, the majority ofparticles are of low energy, so the number of electrons present is greatly overestimated in theHeitler model. In reality, photons outnumber electrons by an approximate factor of 6 at themaximum [71]. This is quite relevant as many detectors, including scintillation detectors, aremuch more sensitive to electrons. Despite this inaccuracy of the Heitler model, it does stillcorrectly predict that the maximum shower size is proportional to E0, and that the depth ofshower maximum increases logarithmically with energy.Particles in an electromagnetic shower can be deflected away from the shower axis bycollisions. The scattering angle is smaller for high energy particles so most of the deflectionoccurs with particles near the critical energy. The scale of the particle distribution is givenby the Moliére radius. The Moliére radius is the product of one radiation length of anelectromagnetic particle and the scattering angle of an electron at the critical energy. Nearsea level, this is approximately 100 m. 90% of all electromagnetic particles are containedwithin one Moliére radius of the shower axis.1.2.3 Hadronic componentHadronic showers, typically initiated by a proton or heavier nucleus, will develop intoa larger variety of particles, with the 3 most significant components being the hadronic,electromagnetic, and muonic component. The shower begins as purely hadrons, with decaysfeeding the electromagnetic and muonic components. When a high energy hadron interactswith an air nucleus, a number of pions are produced, with a smaller number of kaons andmore rarely other exotic hadrons.p+N → p+π0 +π++π−+ ... (1.19)The number of pions produced is typically about 10-20 for protons but does show anenergy dependence of approximately E1/5 [71]. The interaction length for such a collisionalso depends on energy, and is approximately 40 g cm−2 at the highest energies. For heavierprimary particles, the interaction length is much shorter. For iron nuclei, the interactionlength is approximately 15 g cm−2. Additionally, the primary interaction from a heaviernucleus produces many more particles when compared to a proton primary. Of the pionsproduced, approximately 1/3 of the pions are neutral pions and the other 2/3 charged pions.1.2 Extensive air showers 21π0 particles have an extremely short half-life, so decay will almost always occur beforeinteraction. π0 particles decay via the electromagnetic interaction into a pair of photons.π0 → γ + γ (1.20)The photons produced by this decay will then go on to create electromagnetic sub-showers.With each successive generation of hadronic interactions, about 1/3 of the energy remainingin the hadronic component is transferred to the electromagnetic component. For ultra-highenergy showers, there will be several generations of hadronic interactions. Because of this,the majority of the shower energy will be transferred to the electromagnetic component.At high energy, charged pions will most likely interact with another air nucleus. Theinteraction length for pions in air is approximately 120 g cm−2, though like protons, this alsodecreases with increasing energy. The number and type of particles produced in a pion-airnucleus interaction are very similar to that of a proton-air nucleus interaction:π±+N → π0 +π++π−+ ... (1.21)The less commonly produced kaons can also decay into either muons or pions:K± → µ±+ν∓µ (1.22)K± → π±+π0 (1.23)K0S → 2π0 (1.24)K0L → 3π0 (1.25)Like with the electromagnetic shower, the hadron component also has a critical energy. Inthis case, the critical energy is the energy that decay is more likely than a nuclear interaction.Charged pions undergo the following decay:π± → µ±+ν∓µ (1.26)In the case of an electromagnetic shower, the interaction rate and the rate of ionisation lossare both proportional to air density, so the critical energy was not dependent on air density.With the hadronic shower, the interaction rate is proportional to density but the decay rate isnot density dependent. As a result of this, the critical energy will be dependent on air density.Assuming an ultra-relativistic pion (a reasonable assumption for Earth’s atmosphere), thecritical energy is inversely proportional to air density. Critical energy will increase with22 Introductiondecreasing shower energy, and also increase with increasing zenith angle, due to the higheraltitude of the shower maximum. For ultra-high energy cosmic ray showers the pion criticalenergy is typically close to 10 GeV.High-energy cosmic rays 1149+-K , K0+-K , K0+-πe- e+ e+ e-π0+-µµ+- µ -νµνµµ+ +e-e+e+-π+-, K,p, n,e+ e--eπ , π+ -hadronicγ γγ γnuclear fragmentsγ γhadronicneutrinosmuonic component,component componentCherenkov +fluorescencecascadePrimary Particlewith air moleculenuclear interactionradiationelectromagneticFigure 2. EAS progeny.Most of the particles (figure 2) produced in hadronic collisions with air nuclei are pionsand kaons, which can decay into muons and neutrinos before interacting, thus producingthe most penetrating component of atmospheric showers: the muon component. The mostintense component—electrons and photons—originates mainly from the fast decay of neutralpions into photons, which initiate electromagnetic showers, thus distributing the originallyhigh energy of one primary particle over millions of charged (and neutral) secondary particles.The longitudinal development of the electromagnetic component shows a growth, amaximum and a decay as the energy of the shower is dissipated. In contrast the muon cascade(called the penetrating component) grows and maximizes, but the decay is only slow as aconsequence of the relative stability of the muon and small energy losses by ionization andpair production. The backbone of an air shower is the hadronic component of nucleons, pionsand other particles, which feeds the electromagnetic and muonic components. It is often statedthat the hadronic component is well concentrated around the shower axis. Nevertheless due tomultiple scattering, neutrons in particular, are also distributed far off the centre.The longitudinal EAS profile, i.e. the development of the number of charged particles(shower size) with the cumulated atmospheric depth X (the atmospheric thickness alreadycrossed) can be adequately parameterized by the Gaisser–Hillas function [16], for the electronsize, e.g.Ne(X) = Nmaxe ·(X − X0Xmax − X0)(Xmax−X0)/λ· exp(Xmax − Xλ)(1)with X the depth at observation, X0 the depth of the first interaction, and Xmax the depth of theshower maximum. The attenuation parameter λ is about 70 g cm−2. The difference (Xmax−X0)depends on the energy E0 and the nature of the primary and the difference (X − Xmax) is anindicator of the stage of development and increases approximately logarithmically with theenergy. According to the superposition model which considers a heavy primary A as a swarmFig. 1.16. Diagram of the processes involved in a hadron initiated air shower (not to scale).From [50].The Heitler model can be applied to the hadronic particles in a shower [71]. It will beassumed that the number of charged pions produced in each interaction Nch = 10. Thus, therewill also be Nch/2 neutral pions. After n interactions, we have a total number of chargedpions:Nπ = Nchn (1.27)Since the neutral pions will take away 1/3 of the total en rgy in the hadronic component witheach interactio , the average energy of each pion will be:Eπ =E0(32Nch)n (1.28)1.2 Extensive air showers 23For a given critical energy, ξ πc , the number of interactions required for the average pionenergy to reach the critical energy is:n =ln [E0/ξ πc ]ln[32Nch] (1.29)= 0.85log10[E0/ξπc ] (1.30)From this, we obtain the maximum number of charged pions Nπ±:Nπ± = Nchn (1.31)=(E0ξ πc)β(1.32)whereβ =Nchln[32Nch] ≈ 0.85 (1.33)These charged pions decay into muons and thus the number of muons produced, Nµ =Nπ± .Next, the electromagnetic component can be considered. By conservation of energy,all of the shower energy that does not produce critical energy pions will be in the electro-magnetic component. From this, we get the fraction of shower energy contributing to theelectromagnetic component:EEME0=E0 −Nπ±ξ πcE0(1.34)= 1−(E0ξ πc)β−1(1.35)At ultra-high energy, this will be the vast majority of the shower energy. Because of this,and also the lower critical energy, electrons will make up the vast majority of chargedparticles. This means that fluorescence detection (see section 1.3.2) will predominantlymeasure electrons. It is therefore relevant to find the depth of maximum development, Xmax,of the electromagnetic shower. This can be approximated by considering the electromagneticsub-showers initiated by the first interaction as these will have the largest share of the energyin the electromagnetic component. We assume Nch2 neutral pions are produced in the firstinteraction and will decay into Nch photons each with energyE03Nch. Using the equation for the24 Introductiondepth of shower maximum for these electromagnetic sub-showers, we get:X pmax = X0 +λr ln[E03Nchξ ec](1.36)Where X0 is the depth of the first interaction, for a primary proton p. This simple modeldoes show a discrepancy with more detailed simulations. One reason is that only photonsfrom the first generation of hadronic interactions is considered. Another important factorto consider is that energy is not uniformly distributed between the hadrons produced in ahadron-air collision. Though it is not well known at the highest energies, the highest energyparticle produced in a collision may take half of the energy of the parent particle. This willcause showers to develop more slowly giving a larger value for XmaxThe Heitler model can also be extended to handle nuclear primary particles using thesuperposition model. In the superposition model, a nucleus of mass number A and energy E0can be treated as A individual nucleons, each with energy E0/A. The resulting air shower isthen treated as the sum of A separate proton showers. Applying this to the equations for Xmaxand the number of muons, and comparing to a proton shower of the same energy, we get:NAµ = NpµA0.15 (1.37)XAmax = Xpmax −λr lnA (1.38)In addition to the shallower value of Xmax , there is also a difference in the shape of theXmax distribution at a given energy. Due to the stochastic nature of nucleon-air collisions,two showers with identical primaries will not necessarily have the same value of Xmax .Most of this variation between showers is due to the depth of the first interaction, with themultiplicity of particles from the first interaction and secondary interaction lengths havingsmaller contributions to this variability. Since the interaction length for nuclear primariesis shorter than the proton interaction length, it is expected that the variance in Xmax is alsosmaller. For a width of the proton Xmax distribution σ(Xmax)P, using the superposition model,we get:σ(Xmax)A =σ(Xmax)P√A(1.39)Though the width of the Xmax distribution does reduce with increasing A, the superpositionmodel overestimates this effect. More detailed simulations typically put σ(Xmax) for iron atabout 3 times smaller than for protons, whereas the superposition model predicts the width tobe approximately 7.5 times smaller. This is due to the assumption in superposition model that1.2 Extensive air showers 25the individual nucleon showers occur independently to each other. In reality, the nucleons arebundled in a nucleus where a collision between a nuclear primary particle and an air nucleuswill likely affect many of the nucleons. This results in the shower properties of each nucleonbeing correlated to each other. Due to this correlation, the variance in shower properties for aheavy nucleus shower will be higher than the superposition model prediction.1.2.4 Muonic componentThe muonic component of an extensive air shower is primarily produced by the decay ofhadrons that have an energy near or below the critical energy. Most common are chargedpion and kaon decays:K± → µ±+ν∓µ (1.40)π± → µ±+ν∓µ (1.41)It is also possible for muons to be created via pair production when a photon interacts withan air nucleus:N + γ → N +µ++µ− (1.42)The small cross-section for muon pair production means that this process is negligible inhadronic air showers but can be significant when considering the small number of muonsproduced in purely electromagnetic showers.Muons are unstable with a mean lifetime of 2.2 µs at rest. Due to relativistic time dilation,muons produced in hadronic showers are likely to reach the ground. Though muons dolose energy through ionisation loss in air, muons produced by pion decay have an energy inthe order of ∼10 GeV. A muon produced in a vertical shower near the maximum in muonproduction may lose ∼1 GeV through ionisation before reaching the ground, so ionisationloss will not prevent the majority of muons from reaching the ground. In inclined showers,there will be higher ionisation losses as muons pass through a greater depth before the ground,but the critical energy is higher owing to the lower air density around the maximum depth formuon production.Since muons are unlikely to transfer a large fraction of their momentum in a collision inair, they generally travel in nearly straight lines. Additionally, the parent hadrons typicallydo not deviate from the shower axis. When the parent hadron decays into a muon, a neutrinois also produced. This allows muons to have a component of momentum perpendicular to theshower axis. So although muons travel in straight lines, they arrive at the ground spread overa much larger area than hadrons. Due to muons travelling in straight lines it is also possible to26 Introductionobserve a muon at the ground and deduce the production point of the muon, if the geometryof the shower is also known. This can make muons detected at ground level potentially veryuseful for probing the hadronic component of the shower. Due to their instability, not allmuons will make it to the ground. Muons may decay into electrons in the following process:µ± → e±+ν±µ +ν∓e (1.43)The electron produced in this decay can produce a small electromagnetic shower. This issometimes referred to as the muon halo. Close to the shower core, the electromagneticcomponent caused by π0 decay dominates, with the muon halo being negligible. Due tothe narrower lateral distribution of the electromagnetic component caused by π0 decay, atlarger core distances the muon halo can make up a significant fraction of the electromagneticcomponent.1.3 Detection of ultra-high energy cosmic ray showersConsidering the increasing rarity of cosmic rays with increasing energy, ultra-high energycosmic rays require an exceptionally large collecting area, far larger than any apparatus thatcould be practically built. Cosmic ray showers produced by ultra-high energy cosmic rayscan contain many particles at ground level, with the highest energy cosmic ray showerscontaining over 100 billion particles. This makes it possible do detect and determineproperties of showers with detectors that are sparsely distributed over a vast area, with atotal collecting area far larger than the detection equipment. For example, the Pierre AugerObservatory covers an area nearly 200,000 times larger than the sum of all of the detectionequipment. There are a number of possible ways to detect cosmic ray showers. The two thatwill be detailed in this thesis are the surface detector array and fluorescence detection.1.3.1 Surface detector arraysSurface detector arrays have been around for longer than any other method for detectingcosmic ray showers, dating back to when Pierre Auger first discovered extensive air showers.In 1938, Pierre Auger found that two or three particle counters would on occasion besimultaneously triggered. This would occur at rates much higher than could be explainedby random coincidences of unconnected sources triggering each detector. The rate ofthese simultaneous triggers was highest for separations less than 10 metres and reducedas separation distance increased, with simultaneous triggers still observed with detectorsseparated by hundreds of metres. This observation of extensive air showers is how the1.3 Detection of ultra-high energy cosmic ray showers 27existence of cosmic rays at ultra-high energy were discovered. Cosmic ray primary particleswith energies above approximately 1014 eV will produce a sufficiently large number ofparticles at the ground to allow for a reconstruction of the original properties of the primaryparticle. At the ground level, muons, electrons, and photons are the 3 types of particlepredominantly detected due to their broad lateral distribution. There may still be a significantnumber of hadrons at ground level, but hadrons usually land within 10 metres of the showercore, making their detection in a sparse detector array difficult. Figure 1.17 shows the lateraldistribution of particles for a 1019 eV shower. If at least 3 detector stations are able tomeasure particle density, it is possible to reconstruct both the position of the shower core,and the number of particles in the shower, which can then be used to determine the energy ofthe shower.Figure The left plot shows the simulated lateral distributions of the three main shower components at ground level for a eV proton shower In the right frame the particles have beenconvolved with the response of a m deep water Cerenkov detector such as that proposed forthe Auger Observatory Time Structure of the Shower FrontParticles scatter from the region of the shower axis throughout its development The showercore eectively acts as a moving point source of both uorescence photons and particleswhich make their way to detectors far from the core The plane tangent to the shower frontat the axis is the shower plane The shower front itself is slightly curved resembling a coneParticles far from the core will arrive behind the shower plane due to simple geometryGenerally speaking those particles arriving at the ground rst at point some givendistance from the core originated higher and thus earlier in the cascade this can be simplyseen by considering the geometry of pathlength dierences Electrons and photons diuseaway from the shower axis throughout the shower development Thus far from the coreparticles are spread in time with the time spread roughly proportional to the distance fromthe axis This time spread helps to distinguish distant large showers from nearby smallshowers and is thus useful in triggering the surface array The time spread becomes greateras the depth of shower maximum increasesMuons tend to arrive earlier than electrons and photons because they suer much lessscattering and so have more direct paths to the ground Iron showers which are both muonrich and have developed higher in the atmosphere relative to proton showers thus have asignal which arrives over a shorter time than that from a proton shower with the same totalenergy Risetime measures based on this eect are among the most robust diagnostics ofcomposition for the surface array Fig. 1.17. The lateral distribution of particles at ground level due to a 1019 eV cosmic ray(left) and the signal in a 1.2 m deep water Čerenkov tank (right). Muons have the greatestwater Čerenkov signal per particle because the average energy of a muon is higher than anelectron or photon, and muons leave a track through the entire detector. Electrons have ashort interaction length and only travel a short distance. Photons contribute by pair producingelectrons in the water. From [95].Since most particles in a sh wer ravel at speeds close to t e speed of light, with relativelysmall angular deflections from the shower core, the particles in a shower will form a frontin the shape of a pancake centred about the shower axis. At small core distances, this canbe approximated as being flat, but for larger core distances the shower front does have acurvature which must be taken into account. If the arrival time of particles in at least 3detectors is known, it is possible to reconstruct the direction of propagation of the showerfront and hence the direction of the primary particle.28 IntroductionThe spacing of the detectors in a sparse array is predominantly based on the energy ofshowers being observed. The type and size of detectors, and the quality of the reconstructionrequired, also affects the required detector spacing but to a smaller degree. In order toreconstruct the energy and direction of a cosmic ray shower, at least 3 stations need to havereasonable estimates of both arrival time and particle density. Lower energy cosmic rayshowers have too few particles to be reliably detected at large distances from the shower core.Because of this, detector stations must be arranged with a small spacing between detectors toget a reasonable signal in 3 or more detectors. Lower energy showers do have a fairly higharrival rate, so a smaller array area is sufficient to detect a large number of showers. At thehighest energies, detectors at a larger core distance can collect a sufficient number of particlesto determine timing and density, so a larger detector spacing can still allow the propertiesof the cosmic ray shower to be reconstructed. A smaller detector spacing would give amore accurate reconstruction of an ultra-high energy cosmic ray shower, but considering theextremely low arrival rate of ultra-high energy cosmic rays, it is usually desirable to have thelargest possible detector spacing to maximise the collecting area of the array.There are different types of detectors that may be used in a surface detector array. The twopredominant types will be discussed in this section. They are scintillation detectors, and waterČerenkov detectors. Scintillation detectors rely on a scintillation medium that is luminescentwhen exposed to ionising radiation. Light is then detected using a photomultiplier tube(PMT). Chemically-doped plastic is typically chosen as the scintillation material due to itsrelatively low cost and ability to be used for a large detector area. The scintillation medium isusually a few centimetres thick, and is surrounded by reflective materials designed to ensurelight produced anywhere in the scintillation medium is detected by a PMT. Scintillationdetectors can detect muons and electrons directly, as they are ionising particles. Photonsmay also be detected if they interact and pair produce in the detector. Sometimes a layer oflead may be used to cause pair production so that photons are more likely to be detected,since photons are the most numerous particles at the ground. Scintillation detectors areequally sensitive to both muons and electrons, but given the higher particle density of theelectromagnetic component of a shower, scintillation detector signals are usually dominatedby the electromagnetic component.Another type of detector, currently being used in the Pierre Auger Observatory, is thewater Čerenkov detector. In a dielectric medium, light at optical wavelengths travels at aspeed slower than the speed of light in a vacuum. For a refractive index n, this is given as:vn =cn(1.44)1.3 Detection of ultra-high energy cosmic ray showers 29It is therefore possible for a massive particle to have a speed greater than the speed of light ina refracting medium (at optical wavelengths). If the particle is charged, the result of this isthat light is emitted in a cone shaped front analogous to a sonic boom. The emission angle ofthe light cone relative to the particle direction, for a particle velocity v is given as:cos(θ) =cnv(1.45)Čerenkov light forms a continuous spectrum, with a spectral intensity approximately propor-tional to frequency. Considering that this only occurs at wavelengths where the medium isrefracting, Čerenkov emission will predominantly be at ultraviolet wavelengths. In water,the refractive index is 1.33, so the minimum particle velocity to produce Čerenkov lightis 0.75c. The majority of muons and electrons in a cosmic ray shower have a velocitygreater than 0.75c, so water Čerenkov detectors will detect most of these. Photons can alsobe detected when they interact in the water by pair producing electrons, or by Comptonscattering. Considering that water depth in a water Čerenkov detector is typically about 1m, corresponding to 100 g cm−2 of matter, these interactions are quite likely. Because ofthis, water Čerenkov detectors are highly efficient at detecting photons provided the photonenergy is sufficient to pair produce electrons with a speed above the Čerenkov thresholdvelocity. Muons at the ground level have an average energy in the order of a few GeV. Withan ionisation loss over the depth of a water Čerenkov detector being a few hundred MeV,most muons will produce Čerenkov light over the entire depth of the detector. Electrons fromthe electromagnetic component often have a much lower energy, and given a relatively shortinteraction length for bremsstrahlung, many electrons will not make through the depth ofthe detector due to energy loss from ionisation and bremsstrahlung. Because of this, waterČerenkov detectors have a higher sensitivity to muons than they do to EM particles.It is also possible to estimate the composition of the primary particle with a surfacedetector array. The simplest method of doing this is by determining both the density of EMparticles and the density of muons. Considering that the number of muons in a shower isproportional to A0.15, for a nucleus of mass number A, it is possible to determine A if boththe EM and muon densities are known. This does require an array designed to be able soseparate the EM and muon components, which may require additional detectors. Detectorscan be made sensitive to only the muon component by shielding them to block out the EMcomponent. Burying detectors below about 1 m of earth is sufficient to block out nearly allof the EM component. It is more difficult to isolate the EM component of a shower, but if adetector above ground is used in conjunction with a buried detector, the muon componentdetermined by the buried detector can be subtracted from the above ground detector. It is30 Introductionalso feasible, though not as straight-forward, to deduce both the EM and muon componentwith two above ground detectors if the two detectors have a different ratio of EM:muonsensitivity. Examples of this may include stacked scintillation detectors with some shieldingin between them to reduce EM sensitivity of the lower detector, segmenting a water Čerenkovdetector into two levels so that the bottom level is less EM sensitive than the top, and placinga scintillation detector on top of a water Čerenkov detector.There are some other ways to deduce composition without detectors that can discriminatemuons from EM particles. The lateral distribution of muons is wider than the lateral distribu-tion of EM particles. Additionally, the lateral distribution of each of the particle types mayalso depend on the specifics of the longitudinal development of a shower. Thus, the lateraldistribution of particles can be indicative of the overall muon content, and also may give cluesabout the longitudinal development of a shower. Another method to deduce compositioninvolves studying the arrival time distribution of particles at the detectors. Muons travel innearly straight lines at close to the speed of light. Given that the hadrons that decay intomuons lie very close to the shower axis, the point of production of the muon can be deducedfrom the arrival time of the muon at the ground. The timing of the muon will be delayedrelative to the plane front of the shower by τ , where:τ =√d2 + r2 −dc≈ r22dc(1.46)for a muon produced at a distance d from the shower core along the axis, at a core distance r.1.3.2 Fluorescence detectionAs charged particles pass through the atmosphere, they excite air molecules, which thenisotropically emit fluorescence light. The amount of fluorescence light emitted is proportionalto the energy deposited, so if this fluorescence light can be measured, it is possible to get acalorimetric measurement of the energy of an air shower. Most of the fluorescence light isemitted from electronic transitions in N2 molecules and N+2 ions. The majority of fluorescencelight emission occurs in the near ultraviolet at wavelengths between 300 nm and 400 nm(seefigure 1.18)[10]. Though this emission is small for a single particle, the large number ofparticles present in the core of an ultra-high energy shower is sufficient for observation atnight with sensitive equipment. This method of detection is possible for showers with anenergy above approximately 1017 eV.Though the quantity of fluorescence light produced is proportional to energy deposit,there are other factors that affect the observation of fluorescence light. Excited nitrogenmolecules are capable of de-excitation through collisions. This collisional de-excitation1.3 Detection of ultra-high energy cosmic ray showers 31by the rotational structure.4 The set of bands connecting a givenpair of electronic states is named a band system.In our spectral range, nitrogen fluorescence comes basically fromthe Second Positive system C3Pu ! B3Pg of N2 and the FirstNegative system B2Sþu ! X2Sþg of Nþ2 (see Fig. 1) which in the air-fluorescence community are usually denoted as 2P and 1N systems,respectively. Notice that while the 2P system is generated after the N2X1Sþg ! C3Pu excitation, 1N fluorescence takes place as a conse-quence of the X1Sþg ! ðNþ2 ÞB2Sþu molecular ionization, leaving thenitrogen ion in a specific excited state. The wavelengths of the mole-cular bands of nitrogen are well known (see for instance Ref. [30]).Apart from the 1N and 2P systems the weak bands of the N2Gaydon–Herman (GH) system have been observed in the air-fluorescence spectrum [29,31].A spectrum typically observed at high pressure between 280and 430 nm for air is depicted in Fig. 2 [32]. The labels mark 21major transitions. All important transitions and the correspondingwavelengths between 290 and 430 nm are compiled in Table 1.The cross-section for excitation of the upper electronic levels ofboth systems as a function of electron energy is displayed in Fig. 3.The curve for the 2P system shows a sharp maximum at about 15 eVfollowed by a fast E2 decrease, as expected from the opticallyforbidden nature of this transition. On the contrary, the excitationcross-section for the 1N system shows a much softer maximum atabout 100 eV followed by a much slower ðlog EÞ=E decrease whichbecomes a soft growing behavior at relativistic energies [33,34].For a given electronic state the cross-section for the excitationto a vibrational level v is proportional to the Franck–Condon factorqX!v, defined as the overlapping integrals between the vibrationalwave functions of the lower and upper levels of the excitationprocess. The Einstein coefficients Avv0 give the probability per unittime of radiative de-excitation v–v0. Therefore, the probability ofemission of a fluorescence v–v0 photon by electron impact isproportional to the optical cross-section defined assvv0 ¼ svAvv0Pv0Avv0¼ svBvv0(6)and therefore, in the absence of other effects, the relative intensityof a molecular band with respect to a reference transition (e.g.0–0) of the same system is given byI0vv0I000¼svv0s00¼qX!vqX!0Bvv0B00. (7)ARTICLE IN PRESSFig. 2. Air-fluorescence spectrum excited by 3 MeV electrons at 800 hPa as measured by the AIRFLY Collaboration [32].Table 1Transitions and corresponding wavelengths of the air-fluorescence spectrum [32]Transition l (nm)2P(3,1) 296.22P(2,0) 297.7GH(6,2) 302.0GH(5,2) 308.02P(3,2) 311.72P(2,1) 313.62P(1,0) 315.9GH(6,3) 317.62P(4,4) 326.82P(3,3) 328.52P(2,2) 330.92P(1,1) 333.92P(0,0) 337.1GH(0,4) 346.32P(2,3) 350.02P(1,2) 353.72P(0,1) 357.7GH(0,5) 366.12P(3,5) 367.22P(2,4) 371.12P(1,3) 375.62P(0,2) 380.52P(4,7) 385.8GH(0,6) 387.71N(1,1) 388.51N(0,0) 391.42P(2,5) 394.32P(1,4) 399.82P(0,3) 405.02P(3,7) 414.12P(2,6) 420.01N(1,2) 423.62P(1,5) 427.01N(0,1) 427.84 See e.g. Ref. [29] for some illustrative examples.F. Arqueros et al. / Nuclear Instruments and Methods in Physics Research A 597 (2008) 1–22 5Fig. 1.18. The air fluorescence spectrum excited by 3 MeV electrons at 800 hPa. The labelsindicate the transitions associated with the prominent spectral lines. From [10].occurs predominantly with molecules that are not nitrogen, such as oxygen and water vapour.Thus, at higher atmospheric pressure, the yield of fluorescence light per unit of energy depositis lower due to the higher rate of collisions between molecules. Humidity will also havean effect on fluorescence light emission, with a 20% difference between dry air and 100%humidity at room temper ture and atmospheric pressure [10]. Temperature also plays a role,as at higher temperature, for a constant density, the rate of collisions between moleculeswill increas . T e velocity of molecules colliding will affect the probability of a collisionalde-excitation, so temperature will have a further effect on fluorescence yield because of this.Detecting a extensive air shower can be likened to being able to see a 100 W light bulb,at a distance of 30 km, travelling at the speed of light. This requires a detector that is verysensitive to ultraviolet light as well as having a high temporal resolution. This is usuallyachieved by using an optical telescope to focus light onto photomultiplier tubes. Filters thatonly pass UV light are also incorporated into the optical system to reduce the amount ofbackground lig t. The optical telescopes need to have a large collecting area, preferably atleast a few square metres, in order to gather enough light. A fairly wide field of view is alsodesi able, to obse ve a long length of sh wer track, and to observe a large volume of theatmosphere. A good angular resolution is useful for reconstructing the shower geometry, andfor reducing the amount of background light collected by each pixel. It is somewhat difficult32 Introductionto cover a very wide field of view with a large area telescope, so a large number of telescopesare usually used to cover a large solid angle of the sky. Taking into account differences inthe observed track direction due to different positions (typically a very small correction), theimages from each telescope can be combined to be equivalent to one "eye" with a large fieldof view. Operation of a fluorescence detector requires a clear, moonless night, so they willtypically have a duty cycle of about 10%.Since the majority of particles lie close to the axis of a shower, the fluorescence signalcan be treated as a point source of light, travelling at the speed of light in a straight line. Thelateral distribution of particles is only visible in exceptionally close high energy events, andwould not ordinarily require consideration. Reconstructing the geometry of the shower isdone in multiple steps. The first step is to examine the directions of pixels that observed thefluorescence light. A plane can be defined, centred on the detector, that contains the directionsof the pixels observing the shower. This plane is referred to as the SDP (shower-detectorplane), within which the shower axis must lie. Reconstructing the position and direction ofthe shower axis within the SDP requires knowledge of the timing of the pulses observed ineach pixel. Given the direction of each pixel and the time that the spot passed through thatpixel, a fit can be done to the pixel timing to determine the direction of the shower axis withinthe SDP, the distance from the shower axis to the detector, and the timing of the showeras it crosses the point of closest distance to the detector. Details of this procedure will bediscussed in section 2.4.2.The timing fit to determine the position and direction of the shower axis can have arelatively low accuracy when compared to the SDP fit. This is especially true if the track isviewed over a short length. Short track lengths are likely when viewing the highest energyshowers, where a long distance from the shower to the detector is more likely. A shorttrack length leads to a degeneracy between the direction of the shower axis and the distancebetween the detector and the shower axis, leading to an inaccurate determination of energyand Xmax , as well as the primary particle direction. There are however, ways to compensatefor this. The first method is stereo detection. If a shower is observed by two fluorescencedetectors that are separated from each other, a timing fit may not be required. The showeraxis can be found simply by finding the intersection between the two SDP’s. The othermethod is hybrid detection. This method is possible if the fluorescence detector is situated sothat it overlooks a surface detector array. The arrival time of the shower front at a surfacedetector station is able to constrain the geometry of the shower axis, removing the degeneracyfrom the timing fit. This also has an added benefit for the surface detector array, as the energydetermined by the fluorescence detector can be used to calibrate the surface detector.1.4 History of UHECR detectors 33Once the shower geometry is known, the fluorescence emission from the shower can bededuced along the shower axis. Attenuation in the atmosphere between the shower axis anddetector must be taken into account. Particles in a shower emit Čerenkov light as well asfluorescence. Since Čerenkov light is heavily forward directed in air, most of the Čerenkovlight is not observed, but a small number of particles moving towards the detector can produceobservable direct Čerenkov light, and forward directed light can be scattered toward thedetector. The Čerenkov contribution is much smaller than the fluorescence contribution forshowers viewed from side on, but the Čerenkov contribution must still be calculated for anaccurate reconstruction. Given the fluorescence light emission profile, and the fluorescenceyield, the energy deposit profile can be determined. From this, Xmax can be found, and thetotal energy deposit can be deduced by integrating the energy deposit profile. A small fractionof the primary particle energy will not show up in this profile, from particles that do notrelease their energy in the atmosphere, such as hadrons, neutrinos, and high energy muons.This ’invisible energy’ must be added to the total energy deposit to determine the energyof the primary cosmic ray. Though the invisible energy correction is not easy to quantify,relying on either simulations or data from muon detection, the invisible energy as a fractionof total shower energy is small, so inaccuracies in simulations will only contribute a smallerror to the primary particle energy.Fluorescence detectors, when compared to surface detectors, are able to obtain a muchmore direct measurement of primary energy, due to their calorimetric nature. Surface detectorarrays rely much more heavily on simulating ultra-high energy particle interactions, whichare not accessible by particle accelerator experiments. Though corrections are necessaryfor fluorescence detectors, they are based on well understood physical processes, with theonly exception being small invisible energy correction. The low duty cycle of fluorescencedetectors (∼10%) does mean that fluorescence detectors are able to detect far fewer eventswhen compared to a similarly sized surface detector array.1.4 History of UHECR detectors1.4.1 Volcano RanchThe Volcano Ranch array [69] consisted of 19 plastic scintillation detectors, each with acollecting area of 3.3 m2 (see figure 1.19), operating from 1959 to 1978. The scintillationsignals were picked up with a 5 inch photomultiplier. It was initially operated with a spacingof 442 m giving a collecting area of 2 km2. It was also operated with a spacing of 884 m togive a larger collecting area of 8.1 km2. In addition to this, there was also an additional 3.334 Introductionm2 detector shielded with 10cm of lead in order to record the muon density for some of theevents. The Volcano Ranch array produced an energy spectrum for cosmic ray events above1018 eV, showing a flattening of the spectrum referred to as the ankle [68]. An exceptionallyhigh energy event was also observed, with an energy of 1.4× 1020 eV [67]. It was notrealised at the time that the energy of this event exceeded the predicted spectral cutoff as thecosmic microwave background had not been observed.VOLUME 10, NUMBER 4 PHYSICAL RK VIEW LKTTKRS 15 I'EBRUARY 196)cleon-nucleon scattering see, for example, M. L. Gold-berger, Q. T. Grisaqu, S. %'. MacDow'ell, and D. Y.Kong, Phys. Rev. 120, 2250 (1960). Other methods ofcalculating phase shifts in terms of scalar and vectorparticle exchanges have been considered by a numberof authors. See, for example, R. Bryan, C. Dismukes,and W. Ramsay (to be published).3R. Blankenbecler and M. L. Goldberger, Phys.Rev. 126, 766 (1962); G. F. Che~ and S. C. Frautschi,Phys. Rev. Letters 7, 394 (1961);S. Frautschi,M. Gell-Mann, and F. Zachariasen, Phys. Rev. 126,2204 (1962); D. %'ong, Phys. Rev. 126, 1220 (1962).4H. Stapp (private communication).SM. Hull, K. Lassila, H. Ruppel, F. McDonald, andG. Breit, Phys. Rev. 122, 1606 (1961).6C. de Vries, R. Hofstadter, and R. Herman, Phys.Rev. Letters 8, 381 (1962).7J. Ball and D. %'ong (to be published).EVIDENCE FOR A PRIMARY COSMIC-HAY PARTICLE WITH ENERGY 10 eV~John LinsleyLaboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge, Massachusetts(Received 10 January 1963)Analysis of a cosmic-ray air shower recordedat the NIT Volcano Ranch station in February1962 indicates that the total number of particlesin the shower (Serial No. 2-4834) was 5x10'0.The total energy f the primary particle whichproduced the shower was 1.0x10~ eV. The show-er was about twice the size of the largest we hadreported previously (No. 1-15832, recorded inMarch 1961).'The existence of cosmic-ray particles havingsuch a great energy is of importance to astrophys-ics because such particles (believed to be atomicnuclei) have very great magnetic rigidity. It isbelieved that the region in which such a particleoriginates must be large enough and possess astrong enough magnetic field so that REI» (1/300)x(E/Z), where R is the radius of the region (cm)and H is the intensity of the magnetic field (gauss).E is the total energy of the particle (eV) and Z isits charge. Recent evidence favors the choiceZ = 1 (proton primaries) for the region of highestcosmic -ray energies. ' For the pr esent event oneobtains the condition RB» 3 x 10' . This conditionis not satisfied by our galaxy (for which RH ~ 5x10", halo included) or known objects within it,such as supernovae.The technique we use has been described else-where. ' An array of scintillation detectors isused to find the direction (from pulse times) andsize (from pulse amplitudes) of shower eventswhich satisfy a triggering requirement. In thepresent case, the direction of the shower wasnearly vertical (zenith angle 10+ 5'). The valuesof shower density registered at the various pointsof the array are shown in Fig. 1. It can be ver-ified by close inspection of the figure that thecore of the shower must have struck near thepoint marked "A," assuming only (1) that showerparticles are distributed symmetrically about anaxis (the "core"), and (2) that the density of par-ticl.es decreases monotonically with increasingdistance from the axis. The observe densities0.6KlLOMETERSFIG. 1. Plan of the Volcano Ranch array in February1962. The circles represent 3.3-m2 scintillation de-tectors. The numbers near the circles are the showerdensities (particles/m ) registered in this event, No.2-4834. Point A is the estimated location of theshower core. The circular contours about that pointaid in verifying the core location by inspection.146Fig. 1.19. Layout of the Volcano Ranch array in February 1962. The numbers above each ofthe points are the particle densities (in particles m−2) measured for the highest energy eventrecorded in the detector. The letter A represents the estimated core position of the event andthe circles are contour lines of particle density. From [67].1.4.2 Haverah ParkThe Haverah Park array, unlike its predecessors, relied primarily on water Čerenkov detectors,operating from 1962 to 1987. The array initially started with 4 detectors, each with anarea of 34 m2 [94]. Each of the detectors was made up of fifteen smaller units, made of1.85×1.24×1.29 m galvanized steel tanks, filled with 1.2 m of water, with the water sourcedfrom a nearby bore drilled into limestone. The tanks each had a 5-inch photomultiplier, withthe bottom of the photomultiplier submerged in the water to improve light collection. Thedetectors were configured with one in the centre, and three surrounding it in a triangle with a500m spacing from the central detector to the outlying detectors. For each of these detectors,the signals from the fifteen photomultipliers were combined in a summing circuit to give onesignal for each of the 4 stations. In 1968, [66], additional detectors were placed at a distance1.4 History of UHECR detectors 35of 2 km from the centre. Due to land use restrictions, the detectors were arranged into sixsub-arrays of four 13.5 m2 detectors, with the sub-arrays having 50 m & 150 m spacing.Some additional detectors were placed to fill in the largest gaps in the detector, and also aninfill array of thirty 1 m2 detectors in a lattice with a 150 m spacing. The detectors in theinfill array were constructed out of expanded plastic foam, with an depth of 1.2 m, and filledwith deionised water. A diagram of the array layout is given in figure 1.20.Energy spectrum above 4 x lOI7eV 735 t ” 30 . P G $0 U 0 1 km A B m e 1. The Haverah Park array. The triggering detectors A1 . . , A4 are of area 34m2. The sub-arrays B . . . G comprise 4 x 13.5 m2 detecton. At H there is 13.5 mz, at J, K, L 2.25 mz and at the three detectors 150 m from AI there is 9 m2. Within the shaded area is a lattice (-l5Om spacing) of 30X 1 mz detecton, the ‘infilled‘ array. There are muon detectors at the positions inside the shaded area marked with open circles. The azimuthal angle, m, is measured clockwise with respect to North. near the centre of the array (Al). The 30 x 1 mz modules and those at J , K and L were linked to A1 by twisted-pair. A 7.791 GHz microwave link was used to send a command signal from A1 to the recording systems of the sub-arrays Data from all of the detectors inside a circle defined by the points J, K and L were recorded in the central laboratory building, A l . Data from the other detectors (B . . . H) were recorded locally either by photographing oscilloscope traces or using a purpose-built digital system. Records from the remote sites were collected during weekly visits for monitoring and maintenance. The condition of each Cerenkov module was monitored automatically (several times per day but with a frequency which varied from detector to detector and was altered from time to time). No evidence for any change in the performance of a module was found which was attributable to degradation of the water/Darvic combination. Indeed, apart from one module to which biological contamination was inadvertently introduced, the water in each module remained unchanged throughout Table 1. Details of the periods of operation and recording modes of the various detector units of the Haverah Park array. In addition 10 m2 of 500 MeV muon detector was operated beside detectors A3 (1963-65) and A1 (1965-87). Detector system Operating period A I . . . A4 (34 m’) December 1962- 31 July 1987 A150 (3 x 9mZ) January 197- 31 July 1987 31 July 1987 H(13.5 mZ) September 1970- August 1984 I , K, L (2.25 m’) July 1976- 31 July 1987 30x1m2 July 1976- lune 1981 B . . . G July 1968- (each 4 x 13.5 m2) Recording mode and densitv saturation level Oscilloscope photography, 60 m-2 or 10’ m-a Photagraphy/digital, 120 m-’ Photography/digital, 60 m-‘ Photography. 60 m-’ Digital, 104m? Digital, 10‘ m-’ Fig. 1.20. Map of the Haverah Park array, as it was from 1976 onwards. Detectors A1-A4are the triggering detectors with an area of 34 m2. Detectors B-G are each made up from four13.5 m2 detectors. At H there is a single 13.5 m2 detector. J-L are 2.25 m2 detectors. Thethree points 150 m from A1 are 9 m2. The shaded area contains an infill array of thirty 1 m2detectors. The open circles inside the infill array are ddit onal mu n detectors. From [66].The method of energy reconstruction did evolve over the life of the experiment. Theinitial method was to estimate the total energy loss of the shower into a 1.2 m layer of waterat core distances between 100 m and 1000 m. It was later found that a better energy estimateis to find the detector signal at fixed core distance. In the case of Haverah Park, this waschosen as the signal at a core distance of 600 m. [66]. Through their 20+ years of operation,the detectors demonstrated that the water quality and lining of water Čerenk v detectorscould be maintained over an extended period, with only one of the tanks showing issuescaused by contamination from that specific ank being o ened repeate ly throughout theexperiment. The most notable proof of the quality of the water came when the collabo atorsdrank water from the 20 year old tanks upon decommissioning.36 Introduction1.4.3 SUGARThe Sydney University Giant Air Shower Recorder was located near Narrabri, New SouthWales, and was the only large cosmic ray detector in the southern hemisphere before thePierre Auger Observatory [112]. It was in operation from 1968 until 1979. This array differedfrom others by only having detectors that are buried below ground. There was a total of 54stations deployed over an area of 60 km2. The stations were made up of two 6 m2 liquidscintillation detectors buried 1.7 m below the ground and 50 m apart. Most of the stationswere arranged in a square array with a 1600 m spacing, as well as a smaller number with800 m and 400 m spacing. Unlike previous arrays, which required each detector stationto be connected by cables to a central recorder, SUGAR stations operated autonomously.This allowed for a large array without long cable runs. A station would store the timingand intensity of pulses when triggered. The data is then collected and checked against datafrom other stations for coincidences consistent with a cosmic ray shower. In order to reducethe number of pulses recorded from low energy cosmic rays, a detector station would betriggered when signals were detected in both scintillators within 350 ns of each other. Thetwo scintillators were separated by 50 m for this reason. Accurate timing was maintained bycontinuously transmitting a radio timing signal to all of the stations.M M Winn et a1 0 0 0 0 O t" 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /l~;i€ 0 30"32'S 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Figure 1. Map of the SUGAR array showing the places where stations were operated at some time during the life of the experiment. I 1 I I 1 l o o o m the signal above the threshold was timed using a 10 MHz clock. Ideally, the width is proportional to the logarithm of the charge deposited by the PM tube. This arrangement, known as a logarithmic height to time converter, is taken from Suga et a1 (1961). Its advantages are that it covers a wide dynamic range of signals and the number of clock pulses provides a convenient record of pulse size. The principal deficiency of this converter is that it makes the detector system prone to errors if the photomultiplier afterpulses. This was investigated in detail by Bell (1976a, b). We describe in 0 3 how this problem was handled. The log height to time converter gave signals greater than the threshold at a rate of 30 s-' ; this was constantly monitored, as was the rate of pulses corresponding to more than eight simultaneous vertical muons passing through the tank. Identical signal processing was used on both detector channels. If both discriminators at a station fired within 350ns, a master trigger was generated and the following information was then recorded by a local tape recorder: (i) the time of the event as determined from the transmitted timing signal; and (ii) the widths of the pulses at the output of the discriminators (recorded with 100 ns resolution). Fig. 1.21. Map of SUGAR showing station positions. From [112].1.4 History of UHECR detectors 37Since the particle density near the shower core is orders of magnitude larger than theminimum density detectable with detectors designed for a sparse array, it is desirable to havedetectors capable of measuring both very high and very low particle densities, thus requiringa high dynamic range. Electronics to detect the magnitude of a pulse from photomultipliersused in cosmic ray shower experiments were not capable of such a high dynamic range.Detectors at SUGAR avoided this problem by using a logarithmic time to height converter.Charge from the photomultiplier anode would be fed into a capacitor in parallel with aresistance to give a decay constant of 3.0 µs. The signal would then be fed through anamplifier to a discriminator which would trigger when voltage was above a certain threshold.Assuming the photomultiplier pulse duration is much shorter than the decay constant, theamount of time that the voltage is above threshold will be proportional to the logarithm ofthe charge in the pulse. Thus, the stations would simply record the length of time for whichthe discriminator was triggered, which in turn could be used to deduce the size of the pulse.This meant that the detectors had a very high dynamic range, though with an unfortunate sideeffect; photomultipliers of this type were capable of afterpulsing. Afterpulsing is an effectwhereby the photomultiplier produces a spurious secondary pulse several microseconds afterthe original signal. On occasion this afterpulse could significantly extend the time that thediscriminator was triggered, substantially increasing the estimated charge, leading to a vastlyoverestimated particle density.The very large detector spacing meant that for the majority of events, only three stationswere triggered. The small number of stations triggered, even for the highest energy events, inaddition to the problems caused by photomultiplier afterpulsing meant that the reconstructedenergy of events were of poor resolution. SUGAR did contribute to the field with arrivaldirections of events in the southern hemisphere and pioneered the use of autonomous detectorstations.1.4.4 YakutskThe Yakutsk array, located in Russia, began taking data in 1970, and is still in operation tothis date [58], [72]. The array is made up of a number of different types of detector. Thereare 58 above-ground scintillation detector stations each with two scintillation detectors withan area of 2 m2, and thickness 5 cm. There are also 6 underground scintillation detectorsto measure muons. The underground stations are also plastic scintillators with a collectingarea of 20 m2, with one of them having a collecting area of 192 m2. In addition to thearray of particle detectors, there are also a series of air Čerenkov detectors, able to measurethe Čerenkov emission from shower particles and give information about the longitudinaldevelopment of an air shower. This allows for a better calibration of energy from the surface38 Introductiondetector array, with reduced reliance on shower simulations. There are currently 48 of thesedetectors spread across the array. The total area of the array has varied over the lifetime of theexperiment, with a maximum area of 17 km2 in 1990. In 1995, the array was contracted to10 km2, in order to gather more lateral detail in showers. The array currently has a collectingarea of 8.2 km2. The Yakutsk energy spectrum does show results consistent with a cutoff due–2000–1000010002000–2000 –1000 0 1000 2000X (m)Y (m)Fig. 1.22. Arrangement of detectors in the Yakutsk array. From [59].to the GZK effect, though the size of the array means that there is a low statistical significanceto this result.1.4.5 AGASAThe Akeno Giant Air Shower Array ran from 1990 until 2004, and at the time, was the largestcosmic ray observatory in the world [30]. It consisted of 111 2.2 m2 plastic scintillationdetectors, with a spacing between detectors of approximately 1 km. The array had a totalcollecting area of 100 km2. The plastic scintillation medium was 5 cm thick and viewedby a 125 mm photomultiplier. In addition to the scintillation detectors, there were also 27muon detectors. The muon detectors are made up of proportional counters 10 cm wide and10 cm high, with a length of either 5 m or 2 m. A set of these proportional counters wasplaced underneath some of the scintillators, as well as a few of larger area placed adjacent todetectors. The proportional counters were shielded by either 1 m of concrete, 30 cm of iron,or 5 cm of lead above 20 cm of iron. There were fifteen with an area of 2.8 m2, three of 3.61.4 History of UHECR detectors 39m2, one of 7.2 m2, six of 10 m2, one of 15 m2, and one of 20 m2. A prototype of a waterČerenkov detector for the Pierre Auger Observatory was also tested at this site.The array was divided into four smaller branches, each having a branch centre to handlethe acquisition of data. The detectors were linked to their branch centre by fibre optic cables.Cables had to be hung on utility poles of electric and telecommunication companies, whichdid lead to long cable runs, and complicated data acquisition. Initially, each of the 4 branchesoperated independently. This limited the total collecting area of the observatory as eventsthat landed near the boundary between branches may not trigger one or more of the necessarybranches, and lack of precise timing between branches made reconstruction difficult. Theresult was an effective area 1.7 times smaller than the area of the whole array. In 1995, thefour branches were unified to allow rapid communication of triggering signals and accuratetiming between each branch, bringing the effective collecting area up to 100 km2.The energy spectrum of cosmic rays measured by AGASA did not show an expectedcutoff in the spectrum due to the GZK effect, with a spectrum continuing well beyond 1020eV, up to the highest energy event recorded at 2.46×1020 eV [91]. This is in contradictionwith results from other large observatories, that show a substantial suppression at energiesabove the GZK cutoff.1.4.6 Cornell experimentThe Cornell experiment was an early attempt at measuring air fluorescence from cosmic rayshowers [27]. The experiment was constructed in 1967. The detector was comprised of 10fluorescence detector modules. Each of the modules had a 0.1 m2 Fresnel lens covered by afilter to pass UV light. 50 PMT’s were placed at the focal plane of each module, with eachPMT covering a 6 degree wide section of the sky (see figure 1.24). An event was triggeredby two or more adjacent pixels having signals with a delayed coincidence. Pulses were thendisplayed on a large bank of 3 inch cathode ray tubes and photographed with 70 mm film.Ultimately, the small collecting area of the detector modules, and the levels of aerosolsand water vapour in the atmosphere meant that the experiment was unable to reliably detectcosmic ray showers.1.4.7 Fly’s EyeThe Fly’s Eye detector was the first to reliably detect extensive air showers using theatmospheric fluorescence technique [13, 72]. Work began on the detector in the 1970’s, withthe detector becoming operational in 1981. The detector was situated in the desert at theDugway Proving Grounds, Utah. The original detector, called Fly’s Eye I, was comprised of40 IntroductionFig. 1.23. Map of AGASA. The circle points represent the surface scintillation detectors.The squares represent muon detectors. The lines connecting the detectors show the routes ofthe optical fibers. The dotted lines are the boundaries between the four branches. From [76].67 telescopes, each with a 1.575 m diameter spherical mirror. Each mirror focused light on toan array of either 12 or 14 photomultiplier tubes with a diameter of 90 mm, giving each pixelan angular diameter of 5◦. Hexagonally faced reflective funnels were used to channel lightinto the photomultipliers that would otherwise go in between them. Each of these moduleswere housed in a steel drum with a length of 2.13 m and a diameter of 2.44 m. The drumswould point downwards to protect the mirror and electronics from light and weather, andturned upwards at night to a pre-determined position. Fly’s Eye I had a total of 880 pixels,covering the entire night sky. Fly’s Eye had a collecting area of about 1000 km2, significantlylarger than any surface detectors of the time. The detector was only able to operate on clear,moonless nights, so the duty cycle of the detector was only about 10%.1.4 History of UHECR detectors 41Fig. 1.24. Diagram of one of the fluorescence detector modules in the Cornell experiment.From [27].Fig. 1.25. Picture of Fly’s Eye I.Signals from each PMT are connected by cable to a central building, containing theelectronics and computers to control the observatory and acquire data. Due to varyinglevels of background light, triggering levels for each pixel were constantly adjusted to give42 Introductiona constant trigger rate for each pixel. When the triggering conditions were met for theobservatory, the pulse integrals and timing for all triggered pixels were read and stored. Tomonitor the sensitivity of the detectors, optical pulsers were installed in each telescope. Tomonitor atmospheric conditions, 28 vertical flashers were installed. Nitrogen lasers were alsofired periodically to study the scattering of light in the atmosphere.A second detector, Fly’s Eye II (FE II), began operating in 1986. It was situated at adistance of 3.4 km from Fly’s Eye I(FE I), with 36 mirrors and a total of 464 photomultipliertubes. The telescopes covered half of the sky in the direction facing FE I. This made stereodetection of showers possible, with a substantial improvement in geometric resolution. Whenan event was triggered in FE I, an infra-red triggering signal would be sent to FE II to recordany signal triggered simultaneously.The Fly’s Eye detector provided an energy spectrum which showed a precise measurementof the ankle (discussed in section 1.1.5) in the cosmic ray energy spectrum at ∼ 3× 1018eV [21]. With the ability to measure Xmax , composition was also studied. A change incomposition with energy was found, with heavy primaries at 0.1 EeV, changing to a lightercomposition at 15 EeV [21]. Fly’s eye also has the honour of observing the highest energycosmic ray ever detected to this day, with an energy of 3.2±0.9×1020 eV, on October 15,1991 [20] (see figures 1.26, 1.27). Operation was ceased in 1992 to begin work on a higherresolution fluorescence detector.Figure 1: The pointing directions of the 22 phototubes which triggered in connection with this event areshown projected into the xz-plane. The x-axis points east, the y-axis north, and the z-axis upward. Thetriggered phototubes have positive y-components.14Fig. 1.26. Pointing directions of the pixels triggered from the highest energy cosmic rayever detected, projected into the xz-plane. From [20].1.4.8 High Resolution Fly’s Eye (HiRes)HiRes was a significant upgrade over the Fly’s Eye experiment, being situated in the samelocation [56]. There were detectors on two sites, separated by 12.6 km. The telescope mirrors1.4 History of UHECR detectors 43Figure 3: The 3-parameter best-t shower prole is shown along with points obtained from the data in5-degree intervals. The size at maximum is greater than 200 billion particles.16Fig. 1.27. Fit of the shower profile for the highest energy cosmic ray ever detected. From[20].had an area of 5.2 m2, translating to an effective area of 3.72 m2 when considering theobscuration by the central PMT cluster. Each telescope had an array of 256 photomultipliersat the focal plane giving a pixel size of 1 degree. This meant that events could be observed ata much greater distance when compared with the original Fly’s Eye. HiRes telescopes werepointed at low elevation in order to observe a large volume of atmosphere, since the highestenergy events would be visible at very large distances, in excess of 30 km. Upward pointingtelescopes would only observe a comparatively small volume of atmosphere, so would notsubstantially increase the collecting area of the observatory.HiRes-1 began operations in 1997, and was situated at the same site as Fly’s Eye I. Itconsisted of 22 telescopes, observing elevations from 3 to 16.5 degrees, and a full 360 degreesin azimuth.HiRes-2 was completed in 1999, 12.6 km away from HiRes-1. Like HiRes-1, HiRes-2 had 360 degree coverage in azimuth, but utilized a total of 42 telescopes to cover alarger range of elevations, from 3 to 30 degrees. This allowed for more accurate geometricreconstruction of monocular events. Having the second detector operational also allowedfor stereoscopic detection of showers, with much more accurate geometry. Though thetelescopes and photomultipliers were the same as those in HiRes-1, HiRes-2 had significantupgrades to the electronics [24]. HiRes-1, like Fly’s Eye, used ADC’s (analogue to digitalconverters) that would only allow the integrated pulse signal to be read for each pixel, with44 Introductionthe timing of the pulse also being stored as a single value for each pixel. HiRes-2 utilisedan FADC (flash analogue to digital converter) that allowed for an instantaneous reading ofthe photomultiplier with very short time intervals. Signals from each PMT were digitisedat a rate of 10 MHz, and passed through a 820 µs buffer. This allowed for better triggeringcriteria based on information from distributed sources, and signals could then be read outfrom all pixels in a detector. The full shape of the pulse was also recorded, which allowedfor a more accurate reconstruction.Calibration of PMT sensitivity was done with a portable xenon flash lamp directlyilluminating the PMT cluster. Changes in PMT response were also monitored using afrequency tripled YAG laser. The laser could illuminate the PMT cluster directly, as wellas by reflecting off of the mirror to monitor mirror reflectivity using optical fibres. Laserswith steerable beams were used at each of the two sites to measure atmospheric extinction,and could also give an indication of any cloud cover. Infrared cloud monitors also gaveinformation on cloud cover. Xenon flashers were also placed between the sites to giveinformation about the atmosphere between sites [56].3Sibyll [24] find a correction within 2% [13] of that foundvia QGSJet.The measurement of the cosmic-ray flux requires a re-liable determination of the detector aperture. The aper-ture of the HiRes detectors has been calculated using afull Monte Carlo (MC) simulation. The MC includes sim-ulation of shower development (using CORSIKA), fluo-rescence and Čerenkov light production, transmission oflight through the atmosphere to the detector, collectionof light by the mirrors, and the response of the PMTs,electronics and trigger systems. Simulated events arerecorded in the same format as real data and processed inan identical fashion. To minimize biases from resolutioneffects, MC event sets are generated using the publishedmeasurements of the energy spectrum [25] and composi-tion [26, 27, 28].To ensure the reliability of the aperture calculation,the MC simulation is validated by comparing key distri-butions from the analysis of MC events to those from theactual data. Several of these comparisons were shown inreference [29]. Two comparisons are especially notewor-thy. The data-MC comparison of the distances to showersshows that the simulation accurately models the cover-age of the detector. The comparison of event brightnessshows that the simulations of the optical characteristicsof the detector, and of the trigger and atmospheric con-ditions, accurately reproduce the data collection environ-ment. The excellent agreement between the observed andsimulated distributions shown in these cases is typicalof MC-data comparisons of other kinematic and physicalquantities, and this agreement demonstrates that we havea reliable MC simulation program and aperture calcula-tion. Figure 2 shows the result of the aperture calculationfor both HiRes-I and HiRes-II in monocular mode.log10(E (eV))Aperture (km2 sr)HiRes-2 MonocularHiRes-1 Monocular11010 210 310 417 17.5 18 18.5 19 19.5 20 20.5 21FIG. 2: The apertures (defined as the product of collectionarea and solid angle) of the HiRes-I and HiRes-II detectorsoperating in monocular mode.Figure 3 shows the monocular energy spectra from thetwo HiRes detectors [30]. The data included in the figurewere collected by HiRes-I from May, 1997 to June, 2005,and by HiRes-II from December, 1999 to August, 2004.Figure 3 shows the flux multiplied by E3, which does notchange the statistical interpretation of the results buth ghlights featur s more clearly. Two prominent featuresseen in the figure are a softening of the spectrum at theexpected energy of the GZK threshold of 1019.8 eV, andthe dip at 1018.6 eV, commonly known as the “ankle”.Theoretical fits to the spectrum [31] show that the ankleis likely caused by e+e− pair production in the same in-teractions between CMB photons and cosmic-ray protonswhere pion pro uction produces the GZK cutoff. The ob-servation of both features is consistent with the p blishedHiRes results of a predominantly light composition above1018 eV [28].log10(E (eV))Flux*E3 /1024 (eV2 m-2 s-1 sr-1 )AGASAHiRes-1 MonocularHiRes-2 Monocular11017 17.5 18 18.5 19 19.5 20 20.5 21FIG. 3: The cosmic-ray energy spectrum measured by theHiRes detectors operating in monocular mode. The spectrumof the HiRes-I and HiRes-II detectors are shown. The highesttwo energy bins for each detector are empty, with the 68%confidence level bounds shown. The spectrum of the AGASAexperiment is also shown [7, 8].At lower energies, the cosmic-ray spectrum is well fit bya piece-wise power law model. A similar fit also gives anexcellent representation of the spectrum in Figure 3. Thethree straight line segments shown represent the resultof a fit of the measured flux to a triple-power law. Thefit contains six free parameters: one normalization, theenergies of two floating break points, and three powerlaw indices.We performed a binned maximum likelihood fit [32] tothe data from the two detectors. The fits include twoempty bins for each monocular dataset. We found thetwo breaks at logE (E in eV) of 19.75±0.04 and 18.65±0.05, corresponding to the GZK cutoff and the ankle,respectively. When the datasets were made statisticallyindependent by removing events seen by both detectorsfrom the HiRes-I dataset, we obtained a χ2 of 35.1 in thisfit for 35 degrees of freedom (DOF). In contrast, a fit to amodel with only one break point, while able to locate theankle (at the same energy), yielded a χ2/DOF=63.0/37[33].A measure of the significance of the break in the spec-tral index at 1019.8 eV can be made by comparing theactual number of events observed above the break to theexpected number for an unbroken spectrum. For the lat-Fig. 1.28. Energy spectrum obtained by HiRes showing the GZK cutoff. From [1].With the large collecting area of the observatory, in combination with the calorimetricenergy measurement from fluorescence detection, HiRes was the first experiment to observea suppression in flux at energies above 6×1019 eV, consistent with the predicted GZK cutoff(see figure 1.28) [1]. Results from the analysis of Xmax distribution found that a proton-richomposition is favoured at th highes energies [89]. HiRes ceased op rations in April 2006.1.4 History of UHECR detectors 451.4.9 Telescope Array (TA)The Telescope Array Experiment is the most recently constructed observatory designed todetect ultra-high energy cosmic rays, with data taking beginning in 2007 [92]. It is also thelargest ultra-high energy cosmic ray experiment in the northern hemisphere. TA is a hybriddetector, consisting of both a surface detector array and fluorescence detectors.Fig. 1.29. Map of Telescope Array. The arrows indicate the field of view of the fluorescencetelescopes.The surface detector array consists of a total of 507 stations covering an area of 762km2, arranged in a square grid with a 1.2 km spacing [2]. The detector stations consist oftwo layers of plastic scintillators, with an area of 3 m2 and thickness 1.2 cm for each layer.Light produced by the scintillation medium is directed to a 98 inch photomultiplier via 96wavelength shifting fibres. The PMT signal is then low-pass filtered with a cutoff frequencyof 9.7 MHz and read out at a rate of 50 MHz by a 12 bit FADC. The array is divided into 3sub-arrays, each controlled wirelessly from 3 communication towers that handle triggering.Individual stations will transmit a signal to the tower if triggered, and if 3 adjacent stationsare triggered, the tower will store the event, collecting traces from any station that detected asignal. Synchronisation of stations is handled via GPS. Stations are powered by a 125 W46 Introductionsolar panel and a lead-acid battery. Detector calibration is monitored by observing pulsesfrom minimally ionising particles in background radiation.The Fluorescence detector consists of 3 sites overlooking the surface detector array,separated by about 35 km [106]. The north-west site, called Middle Drum (MD), contains 14fluorescence telescopes that were relocated from HiRes-1. These telescopes are described inthe previous section. The two southern sites, Black Rock Mesa (BMR), and Long Ridge (LR)each have 12 telescopes newly constructed for TA. The new telescopes are each made from18 hexagonal segments, with a total diameter of 3.3 m, and a focal length of 3 m. The PMTcameras are composed of 256 PMT’s, with a pixel size of approximately 1 degree. EachPMT camera covers 15 degrees in elevation and 18 degrees in azimuth. PMT signals aredigitised by a 12 bit FADC at a rate of 40 MHz. Calibration of PMT’s is achieved with a N2laser, and monitored with a YAP pulser on three PMTs in each camera. The other PMTs arecalibrated relative to the three calibrated PMTs by uniformly illuminating the camera with axenon flasher. Atmospheric monitoring is done using a central laser facility in the center ofthe array, as well as a LIDAR at each of the three sites. A more complete calibration of thefluorescence detector is also done with a linear particle accelerator fired into the atmospherein front of the FD sites [93].An extension called TALE (Telescope Array Low energy Extension) has been addedto TA to observe showers at lower energy [107]. TALE consists of additional fluorescencetelescopes and surface detectors designed to detect showers down to energies below 1017eV, an order of magnitude lower than TA. The fluorescence detector components consistsof 11 telescopes at MD. These telescopes were refurbished telescopes from HiRes-2. Thetelescopes are pointed at elevations between 31 and 59 degrees, and can operate in conjunctionwith the low elevation telescopes at MD. Low energy showers are only observable at closedistances, where the shower maximum will only be observable at high elevation angles. Thesurface detector component uses a similar station design to the TA SD. An additional 37stations were added to extend the surface detector array towards MD. 76 stations were placedat close distance in front of MD, with the closest 45 counters having a 400 m spacing, andthe remaining stations having a 600 m spacing.An expansion of the array is currently in progress called TAx4, which involves increasingthe area of the detector to 3000 km2 [62]. 500 surface detector stations, of similar designto the previous TA SD, will be utilised in a square grid with a 2.08 km spacing. Additionalfluorescence telescopes will be installed at Middle Drum and Black Rock overlooking theexpanded surface detector array.1.4 History of UHECR detectors 471.4.10 FASTFAST (Fluorescence detector Array of Single pixel Telescopes) is a new design for a flu-orescence detector with the ability to achieve a very large collecting area at a lower cost[42, 41]. It is currently being designed, with prototypes undergoing testing. This designutilises smaller telescopes, with a collecting area of ∼1 m2, and much larger pixels, withangular sizes of 15◦. This design can be built at a significantly lower cost compared totraditional fluorescence telescopes, making it feasible to build a vast array of these detectors.An array spacing of 20 km is being considered, allowing for a density of stations ∼100 timeslower than for a surface detector array.Fig. 1.30. An array of 60 FAST stations with Telescope Array and Pierre Auger Observatoryshown for scale. Also included are the simulated signals for a 57 EeV event. From [41].This design does present some drawbacks, mostly being the lower sensitivity and thedifficulty reconstructing geometry. A smaller telescope collecting area means that fewerphotons are collected. In addition to this, the larger pixels collect much more of the nightsky background light, further reducing sensitivity. This will mean a much higher energythreshold when compared to existing fluorescence detector designs, but the large collectingarea of FAST is intended for shower energies above ∼ 1019.5 eV, which are bright enough tobe observed with FAST telescopes.48 IntroductionA more difficult issue to overcome is accurately reconstructing the geometry of theshower. Since a shower is observed by a few pixels (potentially only one), a shower-detectorplane cannot be determined. There are possible remedies to this. One is to use a surfacedetector array to reconstruct shower geometry. The fluorescence light observations couldbe used to determine an energy deposit profile if the shower geometry is known. A surfacearray would come at a significant cost though, as a large number (∼100) of surface detectorstations would be required for each FAST station. A method of geometry reconstructionwithout a surface detector array is also being considered. If multiple FAST stations observe ashower, the relative intensities and pulse shapes of fluorescence light observed can be used todetermine geometry.Designs are currently being tested at the Telescope Array site, and at the Pierre AugerObservatory. The design being tested uses reflecting telescopes, each covering a 30 degree by30 degree field of view with four pixels. These are situated next to the current fluorescencetelescopes, and rely on them for triggering and for testing geometry reconstruction.1.4.11 JEM-EUSOJEM-EUSO is a space based fluorescence detector currently being planned, to observe cosmicrays with energies above the GZK cutoff with an unprecedented collecting area [40, 3]. Theobservatory will be located on the Japanese experiment module on the ISS. Due to the ∼400km altitude of the ISS orbit, JEM-EUSO will be able to observe a very large area, ordersof magnitude above current cosmic ray detectors. A single telescope will be used, utilisingFresnel lenses, with an area of ∼4.5 m2, and a 60 degree field of view. A substantially smallerpixel size is required when compared to ground based fluorescence detectors. Multi-anodePMTs each containing 64 pixels may be used to give a total of 2304 pixels, with the pixelsize corresponding to ∼500 m on the ground. A simulated shower image from this camerais shown in figure 1.31. Silicon-based photo-sensors are also being evaluated as a possiblealternative to PMT’s, with a higher quantum efficiency. With the telescope pointing directlydownwards, the instantaneous collecting area will be ∼ 2×105 km2. The long distance fromcosmic ray showers being observed means that the detector will have a very low efficiencyfor energies below ∼ 5× 1019 eV, but this observatory’s purpose is to study cosmic raysabove that energy. With an orbital inclination of 51.6 degrees, JEM-EUSO will have a nearlyuniform exposure across the northern and southern hemispheres. Tilting the detector cancover a much larger area, as much as 7×105 km2, though the detector would require evenhigher energies to become fully efficient. Prototype versions are currently being tested, witha ground based version at TA, and additional prototypes being tested on balloons.1.4 History of UHECR detectors 490 100 mm:>30:24! 30:16! 24:8! 16:4! 8:2! 4:<2[counts]PDMb)Fig. 1.31. A simulated image of a 1020 eV shower seen by JEM-EUSO. From [3].Chapter 2Pierre Auger Observatory2.1 OverviewThe origin of the highest energy cosmic rays has been a mystery for many decades, since theearliest experiments with surface detectors discovered cosmic ray showers at exceptionallyhigh energies. Due to the rarity of such high energy events, cosmic ray observatorieswith increasingly large area are desirable. Many scientists saw the need for a cosmic rayobservatory with a collecting area far greater than any previous experiments. With such agreat task at hand, a collaboration spanning 18 countries was formed to pursue this goal. Itwas decided that the observatory should be named after the man who discovered extensiveair showers, so from then it was named the Pierre Auger Observatory. Data taking began in2004, and construction of the observatory was completed in 2008. The site that was selectedis in Mendoza Province, Argentina, near the town of Malargüe.The Pierre Auger Observatory is a hybrid detector, consisting of a 3000 km2 surfacedetector, in combination with a series of fluorescence telescopes overlooking the surfacedetector. The surface detector, with a nearly 100% duty cycle, provides the largest exposureof any cosmic ray observatory in the world. Although the fluorescence detector has a similarcollecting area to the surface detector, it can only operate on clear, moonless nights, givinga ∼15% duty cycle. The fluorescence detector, even with its far lower exposure, providesa number of important functions for the observatory. The fluorescence detector providesa calorimetric measurement of energy, with low systematic and statistical errors, that canbe used to calibrate the surface detector energy. The ability to measure Xmax also meansthat the fluorescence detector is far more sensitive to the primary particle composition. Thefluorescence detector also benefits from the surface detector, since surface detector stationsallow for a hybrid geometry reconstruction.52 Pierre Auger ObservatoryFig. 2.1. Map of the Pierre Auger Observatory2.2 Surface detector array (SD)The surface detector array is comprised of 1660 stations spread over an area of 3000 km2[101, 77]. Detector stations are configured in a uniform triangular grid, with a spacing of1500 m (see figure 2.1). Most of the array is situated on a plane, with an average altitude of∼1400 m, though the altitude for individual stations varies between 1340 m and 1610 m. Inaddition to this array, there is also an infilled portion of the array, in front of the Coihuecofluorescence detector site. The infill array has 61 detectors with a spacing of 750 m, and isdesigned to detect lower energy showers, in conjunction with the high elevation fluorescencetelescopes at Coihueco.The detector stations are water Čerenkov detectors, with the water reservoir having a 3.6m diameter and a height of 1.2 m [77]. The tank is composed of polyethylene, produced byrotomoulding, coloured on the outside to blend in with the environment, with a carbon blackpigment on the inside to ensure light cannot penetrate. The total height of a tank is ∼1.6m. Three 9 inch photomultipliers are used to pick up Čerenkov light, and are distributedsymmetrically over the water at a distance of 1.2 m from the centre of the tank. The water is2.2 Surface detector array (SD) 53contained within a liner composed of Tyvek and polyethylene films. The liner is also lighttight, with a reflective interior. The liners are filled with ultra-pure distilled water, producedlocally at a water plant owned by the Auger collaboration. The liner has three clear windowsthat the PMTs sit in. The PMTs are covered with a light tight plastic casing. Above the threePMTs, there are hatches that allow access to the inside of the tank. The hatches have lighttight and water resistant coverings, with one being larger to accommodate the electronics.The stations are powered by two 55 watt solar panels feeding two 105 Ah, 12 V lead-acidbatteries wired in series. The solar panels are mounted on top of the tank using aluminiumbrackets. The brackets also support the communication and GPS antenna mast.Fig. 2.2. A Pierre Auger surface detector. (Image by the author).The photomultiplier tubes have a 9 inch photocathode and eight dynode stages, designedto optimise linearity. The anode is operated at a positive voltage, with the photocathodegrounded due to the proximity to water. Two signals are recorded from each PMT. One is anAC-coupled anode signal. The other is the last dynode signal inverted and amplified to give asignal with a gain 32 times higher than the anode signal. Each of the 6 channels are fed to acircuit where signals are filtered and read into 10-bit semi-flash ADCs operating at 40 MHz.The use of two signals with a gain difference of 32 gives a dynamic range of 15 bits. AnLED flasher is mounted in a test port in the liner, containing two LEDs. The LEDs can bepulsed simultaneously or independently, to allow the linearity of the PMTs to be monitored.Time synchronisation is achieved with a GPS receiver at each station, with an antennamounted on top of the communications mast. The GPS receiver outputs a signal once persecond. This, together with a custom built timing circuit, allows the synchronisation of datafrom the ADCs to within a 10 ns RMS. The timing signals from the GPS unit may be offset54 Pierre Auger Observatoryfrom the true GPS time by up to 50 ns, which is periodically corrected for. A programmablelogic device analyses the ADC outputs for trigger patterns, stores data in a buffer, and informsthe station controller when a trigger occurs.There are two levels of triggering that occur locally at one station [78]. The first level,T1, is set off by two possible signals. The first, a threshold trigger (TH), is if all 3 PMTssimultaneously record a signal greater than 1.6 times the VEM peak intensity (VEM beingthe signal equivalent to one vertical muon going through the detector). The second, calledtime over threshold (ToT), is when at least two PMTs record an intensity greater than 0.2times the VEM peak for 13 time bins within 3 µs. ToT triggers occur at a lower rate thanTH triggers. TH triggers are useful for inclined showers, where muon signals dominate, andmay produce a pulse too short for a ToT trigger. ToT only triggers are more likely for lowenergy showers near the core, or high energy showers of small zenith angle at larger coredistance. If a second level trigger (T2) occurs, the timing of the T2 trigger is transmitted tothe central data acquisition system (CDAS). While all T1-ToT triggers automatically becomeT2, T1-TH triggers are only promoted to T2 level if all 3 PMTs record an intensity greaterthan 3.2 times the VEM peak. Since June 2013, additional triggering conditions have beenadded, that build upon the ToT condition, to lower the energy threshold of the SD [101]. Thefirst method, time over threshold deconvolved (ToTd), deconvolves the signal, removing thelong exponential tails from pulses before applying the ToT condition. This has the effectof reducing the influence of a single muon by compressing its signal to 1-2 time bins. Thesecond new trigger, multiplicity of positive steps (MoPS), counts the number of time binswhere at least two PMTs record an increase in signal greater than a small threshold of about5 times the RMS noise, but smaller than about 0.5 times the size of a vertical muon step(to reduce the influence of muons). Both ToTd and MoPS require a 0.5 VEM integratedsignal. There are also additional triggers for signals that are used for recording the scale ofbackground radiation, and for calibration purposes that are only used by the station locally.Stations must be calibrated so that signals for each station can be found in units ofVEM (vertical equivalent muons) [18]. Due to the limited bandwidth available, stationsmust perform this calibration locally. This is done by examining background pulses fromsingle muons. A histogram of the height of each PMT pulse is made, and the peak of thisdistribution is found. The gains of each PMT are adjusted so that the peak is at 50 ADCchannels. A histogram of the total charge in each pulse is also found. The peak of thisdistribution (charge-peak) is used to convert the integrated PMT signal into units of VEM,with the relation between the charge-peak and the VEM unit found using a muon hodoscope[101] (see figure 2.3).2.2 Surface detector array (SD) 55Figure 5: Charge spectrum obtained when a surface detector is triggered by a threefold coincidence amongits photomultipliers (open histogram). The hatched histogram shows the spectrum when triggered on centralvertically aligned plastic scintillators. The bin containing the peak of the scintillator triggered spectrum isdefined as a vertical equivalent muon. The leftmost peak in the open histogram is due to low energy andcorner-clipping muons convolved with the threefold low threshold coincidence.ory. The PLD implements firmware that monitors the ADC outputs for interestingtrigger patterns, stores the data in a buffer memory, and informs the station microcon-troller when a trigger occurs. There are two local trigger levels (T1 and T2) and aglobal third level trigger, T3. Details of the local triggers are described in Section 3.5.The front end is interfaced to a unified board which implements the station con-troller, event timing, and slow control functions, together with a serial interface to thecommunications system. The slow control system consists of DACs and ADCs usedto measure temperatures, voltages, and currents relevant to assessment of the operationof the station.The station controller consists of an IBM PowerPC 403 GCX-80MHz, with a32 MB DRAM bank to store data and executable code, and a 2 MB Flash EPROMfor the bootstrap and storing of the OS9 operating system. The data acquisition systemimplemented on the station controller transmits the time stamps of the ∼20 T2 eventscollected each second to CDAS (Central Data Acquisition System; see Section 6).CDAS returns T3 requests to the station within ∼8 seconds of the event (includingcommunications delays due to retransmission). The station controller then selects theT1 and T2 data corresponding to the T3 requests and builds it into an event for trans-mission to CDAS. Calibration data are included in each transmitted event.3.4. Calibration of the surface detectorThe Cherenkov light recorded by a surface detector is measured in units of thesignal produced by a muon traversing the tank on a vertical trajectory (Figure 5). Thisunit is termed the Vertical Equivalent Muon (VEM). The goal of the surface detectorcalibration is to measure the value of 1 VEM in hardware units (i.e., in integrated FADCchannels). During shower reconstruction, the signal recorded by the tanks is convertedinto units of VEM, and the total shower energy and arrival direction are fitted using alateral distribution function and energy conversion based upon hybrid analysis usingthe FD. The conversion to units of VEM is done both to provide a common referencelevel between tanks and to calibrate against the detector simulations.19Fig. 2.3. The distribution of charges obtained from a threefold coincidence of photomulti-plier triggers. The filled histogram shows the spectrum obtained when triggered by verticallyaligned plastic scintillators above and below the centre of the tank, corresponding to a verticalmuon, used to relate the peak of the threefold triggered spectrum to a VEM. From [101].The surface detector stations communicate via a WLAN system, operating in a band from902 to 928 MHz, to one of four communication towers located at the fluorescence detectorstations. These towers communicate via a microwave link to the central data acquisitionsystem (CDAS) [101]. CDAS gathers triggering data from the SD stations and decides onglobal triggering of the array, and collecting and storing pulses and calibration data fromstations. When stations have a T2 trigger, the timing of the trigger is sent to CDAS, wheretriggering data from stations is combined to decide on a global trigger for the SD array, calleda T3 trigger. The main T3 trigger is based on the timing and positions of station T2 triggers.For each T2 triggered station, a series of concentric hexagons of stations are defined centredon that station. If three stations are coincidentally T2-ToT, it is required that for one of thetriggered stations, one of the other stations is in the first hexagon, and the third station is nofarther than the second hexagon. If four stations display a coincidence of any kind of T2,one other station must be in the first hexagon, the third within the second hexagon, and thefourth may be within the fourth hexagon. A diagram of the hexago s is shown in figure 2.4.Timing is also checked, where timing of trigger must be within (6+5n) µs of each otherfor a station in the n’th hexagon. If the T3 condition is met, a message is sent to all stationsrequesting trace data for any T1 or T2 triggers within 30 µs of a T2 trigger that meets the T3criteria.T3 triggers may also be from other criteria. There are a small number of doublet stations(two stations about 10 m apart) which cause a T3 trigger if they trigger within 1 µs of eachother. A random single tank T2 is also selected about every 30 minutes and elevated to T3 for56 Pierre Auger ObservatoryFig. 2.4. A Diagram showing the hexagons used for SD T3 triggers. The red circles showan example of a 3-fold coincidence, and the blue squares show an example of a 4-foldcoincidence. From [101].monitoring purposes. CDAS will also receive triggering information from the fluorescencedetector and will also produce a T3 trigger in order to obtain relevant information for a hybridevent.2.3 Fluorescence detector (FD)The main fluorescence detector of the Pierre Auger Observatory consists of 24 telescopesat 4 sites, each site having 6 telescopes covering 180 degrees in azimuth, and 30 degrees inelevation, with a minimum elevation of 1.5 degrees. The four sites sit along the boundariesof the surface detector array and overlook the stations (see figure 2.1). The northern site isnamed Loma Amarilla (LA), the eastern site is named Los Morados (LM), the southern siteis named Los Leones (LL), and the western site is named Coihueco (CO) [97].The telescopes are located inside climate controlled buildings. The side walls of thebuildings contain the apertures through which light can enter, with a radius of 1.1 m. Theaperture is covered with a filter which passes UV light. The apertures sit behind shutters,which only open when conditions are suitable for observation. Inside the aperture is acorrecting lens ring to reduce spherical aberration near the edges of the aperture, with aninner radius of 850 mm. Light is focussed by a 13 m2 mirror, of spherical shape with a radiusof curvature of 3400 mm. The mirrors are constructed of smaller segments to reduce thecost and weight. The mirrors at Los Morados and Los Leones are made from 36 rectangularsegments, and the mirrors at Coihueco and Loma Amarilla are made from 60 hexagonal2.3 Fluorescence detector (FD) 57Fig. 2.5. A Schematic of a fluorescence detector station. From [97].Fig. 2.6. A fluorescence detector station (Los Morados) in operation. Note that there are 6telescopes at this site, but only 4 are visible due to the proximity of the camera. (Image bythe author).58 Pierre Auger Observatorysegments. The focal surface has a radius of curvature of 1700 mm. The camera body ismachined from an aluminium block, with a curvature to match the focal surface. Eachcamera contains 440 hexagonal PMTs arranged with 22 rows and 20 columns. Each pixel isa hexagon with a size of about 45.6 mm. PMTs are separated by Winston cones that directlight into the PMTs that would otherwise land in between them. The camera is placed suchthat the focal surface is at the outer edges of the Winston cones. This gives a pixel angularsize of approximately 1.5 degrees.Fig. 2.7. One of the Fluorescence telescopes at Los Leones. The UV filter and correctorring are visible in the upper right corner of the image. (Image by the author).The PMTs are hexagonal with a 40 mm diameter with eight dynode stages. The powersupplies for the PMTs each supply a group of 44 PMTs. Since the high voltage is adjusted tomaintain the correct gain, PMTs were tested for their gain properties, and grouped in lotsof 44 with similar gain properties to be used on the same high voltage supply. PMT signalsare fed into analogue boards, with each analogue board managing signals from 22 PMTs.The analogue boards can fine tune the gains of each individual PMT by a factor of up to1.9. There is also an anti-aliasing filter that is designed for the 10 MHz sampling rate of theADC. The anti-aliasing filter is a fourth order Bessel filter with a 3.4 MHz cutoff. The finalstage in the analogue circuit is designed to extend the dynamic range to 15 bits, using 12bit ADCs. Virtual channels are made by summing the signals from 11 non-adjacent pixels,with a gain that is a factor of 20 lower than for the individual pixel channels. When a close2.3 Fluorescence detector (FD) 59enough shower with a sufficiently high energy occurs, pixels may saturate, but at any giventime bin, only one pixel in each virtual channel is likely to be saturated, so the saturatedpixel’s signal can be recovered from the virtual channel. The signals are then passed to thefirst level trigger (FLT) module.Each FLT module reads in signals from one 22 channel column. Signals are digitisedusing a 12-bit pipelined ADC at a rate of 10 MHz. The FLT board stores the digitised signalsin a circular buffer with the ability to hold 100 µs of data. This data can be read out if thereis a second level trigger. A field programmable gate array is used to handle first level triggers.A moving sum of the last n ADC samples from a pixel is found, where n can be adjustedfrom 5 to 16, and the sum is compared to an adjustable trigger threshold level. The thresholdis adjusted dynamically to give a trigger rate of 100 Hz. A moving sum going above thetrigger level causes a first level trigger (FLT) in that pixel. After a pixel moving sum dropsbelow the threshold, a retriggerable mono-flop extends the pixel trigger for a period of 5µs to 30 µs common to all pixels, to increase the chance of coincident triggers. To analysebackground light levels, the variance of each PMT is also found every 6.5 ms using 65536consecutive ADC samples. The number of pixels above the threshold is tracked and passedon to the second level trigger board (SLT).Each second level trigger board handles the FLTs from the 20 FLT boards from onetelescope. The SLT board examines FLTs for coincident triggers of pixels. A second leveltrigger (SLT) requires four out of five pixels in a straight line segment to be triggered. Thedifferent allowable pixel configurations of straight line segments are shown in figure figure2.8. A full scan of all FLTs occurs every 1 µs, with FLT information being stored in a ringbuffer. The SLT board also creates the time stamp for each event, using synchronisationsignals sent from the GPS module.Fig. 2.8. Configurations of five triggered pixels that can cause a SLT. From [97]Information from the FLT and SLT memory is read out by a MirrorPC. Each MirrorPC isconnected to the FLT and SLT boards with a firewire interface and communicate with theEyePC via LAN. Second level triggers from air showers are processed by a third level trigger(TLT). Third level triggered showers, as well as second level triggers triggered externally, aresent to the EyePC, where an event is built and a T3 trigger is generated for the SD array. The60 Pierre Auger Observatorythird level trigger examines the time evolution of the FLT multiplicity to reject backgroundevents, primarily caused by distant lightning, but also random triggers and muon impacts.This is done in less than 50 µs. A simple reconstruction of events are done to determine anapproximate core location and timing, which is sent to CDAS for the T3 trigger to checksurface detectors for triggers at an appropriate time.An absolute calibration of the response of all PMTs in the FD is achieved by doing adrum calibration on each telescope [25]. A drum is placed over the aperture of each telescopeto uniformly illuminate the camera (see figure 2.9). The drum contains a UV LED thatcan be pulsed to illuminate the drum. The drum is designed to internally scatter the lightfrom the LED, and transmit it diffusely at a uniform intensity. The intensity of the drum iscalibrated with a PMT, which itself is calibrated against a photodiode calibrated by NIST.Uncertainties resulting from all steps in this calibration result in a 9% uncertainty in theabsolute calibration of the FD. This has also been checked against a calibration using verticallaser shots of known intensity.There is also a system to monitor time variations in telescope calibration. 3 differentlight sources are used on each telescope for this. One is an LED mounted in the centre ofthe mirror which directly illuminates the PMTs. Another is a xenon flasher that is sent viaoptical fibres to the sides of the PMT camera, where they reflect light off the mirror andinto the camera. The third source is a xenon flasher whose light is reflected off the insideof the telescope doors, and thus must also travel through the UV filter as well as reflect offthe mirror to reach the PMT. A monochromator drum source was also used to calculate therelative spectral efficiency from 270 nm to 430 nm.There are a number of systems in place for atmospheric monitoring. They include:• In the centre of the SD array, there is a central laser facility (CLF) [9]. The CLFcontains a frequency tripled YAG laser with a wavelength of 355 nm. This laser ispulsed upwards with a light output equivalent to a 100 EeV shower. The observedbrightness at the detector sites allows for the atmospheric aerosol profile to be deducedso that atmospheric extinction of light from showers can be compensated for. Anothersimilar facility, called the XLF, has also been installed further north to provide a sitecloser to Loma Amarilla. Some of the light from the laser is also directed into anadjacent surface detector station to assist in monitoring any time offsets between theFD and the SD. CLF shots for atmospheric monitoring are done every 15 minutesduring FD operation. The beam can also be inclined to assist in monitoring of telescopepointing (this is discussed in chapter 3).• A horizontal attenuation monitor (HAM) [96] also provides an atmospheric extinctionmeasurement, with a high intensity discharge light source at Coihueco, and a filtered2.3 Fluorescence detector (FD) 61Fig. 2.9. Schematic of a fluorescence telescope showing the calibration drum mounted infront of the Aperture. From [97]CCD camera at Los Leones, measuring extinction at five wavelengths between 350 nmand 550 nm.• At each FD site there is a LIDAR to monitor cloud height, cloud cover, and aerosolsin the atmosphere around the site (see figure 2.10) [17]. LIDAR stations can alsoprovide ground level aerosol attenuation measurements in the direction of the CLF,and may also be used to scan the atmosphere along the SDP (shower-detector plane)of an interesting cosmic ray shower candidate immediately after it is detected.• A Photometric Robotic Atmospheric Monitor (FRAM) [98] is installed at Los Leones,consisting of a 0.3 m diameter optical telescope to measure starlight to determinethe wavelength dependence of Rayleigh and Mie scattering. FRAM also has a widerangle camera with a 4 degree field of view, that uses stellar photometry to determineatmospheric extinction along a recently detected SDP.• Infrared cameras are installed at each FD site to monitor cloud cover, that could maskor scatter light from cosmic ray showers (see figure 2.10). The camera pans and tilts tocover the field of view of the FD every 5 minutes, and the full sky every 15 minutes[99].62 Pierre Auger Observatory• A balloon launching station (BLS) [61] is located towards the western side of the arraywhere radiosonde launches have been performed to measure temperature, pressure,and humidity profiles, in order to validate the data from the global data assimilationsystem (GDAS).Fig. 2.10. (top) The Central Laser Facility (CLF). (left) An infrared cloud camera inoperation, situated on the roof of Los Leones (Image by the author). (right) A LIDAR inoperation, located adjacent to Los Leones (Image by the author).2.4 Shower reconstruction2.4.1 SDReconstruction of a shower measured by the surface detector relies on both the timing andmagnitude of signals recorded by the surface detector stations. Firstly, T3 events recordedby the surface detector undergo a selection process to filter out random triggers. This isprimarily based on coincidences between adjacent stations having timing consistent with a2.4 Shower reconstruction 63propagating shower front. Events that pass this are triggered off-line to the T4 level. Stationswith incompatible timing, primarily due to random muons, are rejected as random signals. Afiducial cut is also applied to events to ensure a quality reconstruction. Typically, a cut knownas a 6T5 trigger is applied, where the station with the highest signal must be surrounded by 6operating neighbours, to ensure the event is well contained within functioning stations, andto allow for a simple calculation of detector exposure. Studies without such requirementsmay use a less strict cut such as 4T5 or 5T5.To accurately reconstruct the direction of the shower, the position of the shower coremust be known, since shower fronts observed with a relatively large array spacing cannotbe accurately approximated as planar. Finding the position of the shower core accuratelyrequires the direction of the shower to be known so that station distances from a showeraxis can be found for a lateral distribution fit. Since the axis direction and core position fitsboth rely on each other, a more approximate arrival direction is first found, using a planarapproximation. If there are sufficient stations, a simple spherical approximation may be used,where the shower front is approximated as a sphere expanding at the speed of light, giving:c(ti − t0) = |⃗xsh − x⃗i| , (2.1)where ti and x⃗i are station times and positions, and t0 and x⃗sh is the start time and position ofvirtual origin for the shower.With this approximate direction, the core position can be found by performing a fit tothe lateral distribution function (LDF) of SD station signals. The LDF fit is done using thelog likelihood method, which can take into account stations that did not trigger, as well assaturated stations. The function used to fit the lateral distribution is:S(r) = S(ropt)(rropt)β ( r+ r1ropt + r1)β+γ(2.2)where ropt is an optimum distance based on the array spacing, and r1 = 700m. S(ropt) isthe signal at a distance of ropt [73]. For a 1500 m array spacing, ropt is optimally 1000 m,making the shower size S(1000), which is used to determine energy. β and γ can be freeparameters if there are sufficient stations, but they may be fixed to a value based on zenithangle and shower size if there are too few stations. Once the position of the shower core isknown, the direction of the shower axis can be determined. Given a shower core position x⃗gr,the direction of the shower axis, â, can be found:â =x⃗sh − x⃗gr∣∣⃗xsh − x⃗gr∣∣ (2.3)64 Pierre Auger ObservatoryThe angular resolution achieved is typically better than 1.6 degrees for showers with onlythree stations, down to 0.9 degrees for showers with six or more stations [22].x [km]12 14 16 18 20 22 24 26y [km]024681012141618r [m]500 1000 1500 2000 2500 3000 3500 4000Signal [VEM]110210310Stage: 4.5/Ndf: 17.9/ 182χcandidatesnot-triggered r[m]500 1000 1500 2000 2500 3000Time [ns]02004006008001000120014001600 /NdF: 79.8/ 172χtime residual Fig. 2.11. A reconstruction of an SD event. The shower has a reconstructed energy of(9.52±0.20±0.06)×1019 eV, and a zenith angle of (44.5±0.1) degrees. The top figureshows a map of triggered stations coloured according to their time, and sized by signal. Thedirection of the axis is shown by a black line extending out from the core. The left figure isthe LDF fit. The right figure is the timing fit showing the timing residual of each station froma plane fit.After direction and S(1000) has been determined, the primary energy can be determined.For a given primary energy, S(1000) is dependent on zenith angle, with S(1000) being lowerat larger zenith angles due to particle attenuation and geometric effects. To reconstructenergy, an attenuation curve must first be found to compensate for this attenuation. Theattenuation curve, fCIC(θ), is determined by the Constant Intensity Cut (CIC) method [54].In the CIC method, it is assumed that the arrival rate of cosmic rays above a given energy isnot dependent on zenith angle. Given this, at a given zenith angle θ , a value for S(1000)mincan be found such that the arrival rate of a larger shower size than S(1000)min is equal to aconstant. Applying this constant to all zenith angles allows fCIC(θ) to be found:2.4 Shower reconstruction 65S(1000)min(θ) = fCIC(θ)S(1000)min(θ̄) (2.4)where θ̄ is chosen as the median zenith angle, 38 degrees, and fCIC(θ) is normalisedto give fCIC(θ̄) = 1. The fitting function used is fCIC(θ) = 1 + ax + bx2 + cx3, wherex = cos2θ − cos2θ̄ . Originally the parameters a, b, and c had fixed values but have recentlybeen updated to account for an energy dependence. a, b, and c are now expressed as asecond degree polynomial in the variable k = log10(S38/40VEM) [110]. From this S(1000)is converted to a zenith angle independent version known as S38:S38 =S(1000)fCIC(θ)(2.5)Fig. 2.12. Plot of S(1000) vs sec(θ) for a constant intensity, for three intensity thresholdscorresponding to approximate energies of 3 EeV (I1), 8 EeV (I2), and 20 EeV (I3). From[110].The final step is to convert S38 into primary particle energy. In previous surface detectorexperiments, the relationship between observed particle densities and primary energy wouldhave relied on methods highly prone to systematic errors, such as shower simulations. Witha hybrid detector, a subset of showers detected by the surface detector array are also detectedby the fluorescence detector, and thus have an independent calorimetric measurement ofenergy. To obtain the relationship between energy and S38, high quality hybrid events areselected, with energy above the threshold for full SD triggering efficiency. A maximum66 Pierre Auger Observatorylikelihood fit is then performed using a single power law function:EFD = A(S38VEM)B(2.6)for FD energy EFD. The values in use for the parameters are A = (1.90±0.05)×1017 eV,and B = 1.025±0.007. This gives the energy determined by the SD as:ESD = A(S(1000)fCIC(θ)×VEM)B(2.7)The SD energy systematic error due to this method of calibration is less than 2% over thewhole energy range, so the systematic in SD energy is dominated by the systematics in theFD energy scale. The energy resolution of the SD energy can also be determined using hybridshowers. By examining the distribution of ESDEFD , if the resolution of the FD is known, the SDresolution can be deduced. The resolution of the SD obtained by this method is found to be(16±1)% at lower energy, and (12±1)% at the highest energies.Fig. 2.13. S38 vs EFD fit used to determine the SD energy scale. From [110].The standard SD energy reconstruction is used for zenith angles up to 60 degrees. Itis desirable to extend event reconstruction to larger zenith angles, as this can increase theexposure of the observatory by up to 30 %. This has been achieved with a separate method ofreconstruction [80]. Showers with zenith angles larger than 60 degrees are called horizontalair showers. Due to the large amount of atmospheric attenuation, horizontal air shower2.4 Shower reconstruction 67signals in surface detectors are dominated by muons, and much of the electromagnetic signalis muon halo (electromagnetic particles produced by muon decay and interactions). Thoughhorizontal air shower signals are typically smaller at a given core distance when compared tovertical showers of the same energy, the density of stations in the shower plane is higher, so alarge number of stations can be triggered. The lateral distribution of particles in a horizontalair shower can be highly asymmetric, due to there being significant longitudinal developmentbetween the early and late stations, as well as muon deflection due to the geomagnetic field.Geomagnetic deflection also means that the lateral distribution of particles is dependent onthe azimuth of the shower axis. As a result of this, a more complex 2-dimensional lateraldistribution function is used, depending on the azimuth and zenith angle of the shower. Fromthe lateral distribution fit, the shower size parameter obtained is known as N19. The relationbetween energy and N19 is found using hybrid data, with the same method as for verticalevents.x [km]15− 10− 5− 0 5y [km]05101520Fig. 2.14. A map of triggered stations for a horizontal air shower. The shower has a zenithangle of (75.7±0.1) degrees, and an energy of (5.2±0.26)×1019 eV.2.4.2 FDReconstruction of a shower measured by the fluorescence detector is done in steps, startingwith geometry and then continuing to profile and energy reconstruction. The geometry68 Pierre Auger Observatoryreconstruction utilises both the magnitude and timing of the signals in each pixel, alongwith the known pointing direction of each pixel. For energy and profile reconstruction, thecalibration of the fluorescence detector, and the properties of the atmosphere must be used.First, triggered pixels have their traces examined to find a shower signal. The trace isscanned to find a start and stop time for the pulse that maximises the signal to noise ratio(SNR). A pixel is only selected for geometry reconstruction if the SNR is ≥ 5. With this startand stop time, the integrated signal for the i’th pixel is given as:qi = ∑ksik (2.8)where sik is the charge of the k’th ADC bin of pixel i, and the sum is run from the previouslydetermined start and stop time. The timing of the centroid of the pulse is:ti =∑k τiksikqi(2.9)where τ ik is the time of the k’th ADC bin. The error in ti and qi can be determined byconsidering the background variance and Poisson fluctuations in the signal.Next, the SDP (shower-detector plane) is determined using the integrated signal in eachpixel, and the pointing directions of each pixel, p⃗i. This is done by minimising the offsetof pixel pointing directions from the SDP, weighted by integrated signal. The function tominimise is:S =1∑i qi∑iqi(π2 −arccos(p⃗i · n⃗SDP⊥ )σSDP)2(2.10)with the vector n⃗SDP⊥ defined as the vector normal to the SDP. The parameter σSDP is set toallow S to be interpreted as a χ2 function for error analysis. The value of σSDP was determinedto be 0.35 degrees by studying CLF shots of known geometry. The free parameters for theSDP fit are θSDP and φSDP, the direction of n⃗SDP⊥ in spherical polar coordinates.The next step in reconstruction is to determine the geometry of the shower axis withinthe SDP. For this, an angle known as χ is defined for a direction within the SDP as beingthe angle with respect to the horizontal within the SDP. As there are two opposite horizontaldirections within the SDP, the one chosen is in the direction of n⃗SDP⊥ × ẑ, and the directionof n⃗SDP⊥ is chosen (again, from 2 possible opposing directions) such that the cross productis in front of the detector (within the detector field of view). Each triggered pixel has anassociated angle, χi. Since pixel pointing directions do not necessarily lie exactly on theSDP, pixel directions are projected onto the SDP before χi is determined. The geometry ofthe shower axis within the SDP is defined by three parameters. They are χ0, Rp, and t0 (see2.4 Shower reconstruction 69figure 2.15). χ0 is the angle of the direction of origin of the shower. Rp is the distance of theshower axis at its closest point to the FD. t0 is the time at which the shower passes its closestpoint to the FD. The expected observation time of the shower ti at a pixel angle χi is derivedunder the assumption that the shower propagates at the speed of light, while also consideringthe propagation time of light from a point along the shower axis to the FD. Consider a pointSi along the shower axis, corresponding to an angle χi. The distance along the shower axisrelative to the shower at time t0 is:dSi =Rptan(χ0 −χi)(2.11)giving an arrival time of the shower at Si as:tSi = t0 −Rpc · tan(χ0 −χi)(2.12)Similarly, the propagation time to the eye from Si is:ti − tSi =Rpc · sin(χ0 −χi)(2.13)This gives an arrival time at the eye, ti, as:t(χi) = t0 +Rpctan(χ0 −χi2)(2.14)The parameters χ0, Rp, and t0 are determined by minimising the following function:χ2 = ∑i(ti − t(χi))2σ(ti)2(2.15)Events observed by the Pierre Auger Observatory FD often have only short tracksobserved. For a fit based solely on FD pixel timing and directions, there can be a largedegeneracy between Rp and χ0. This is especially true if dχ/dt does not vary greatly alongthe observed track. This error in geometry can be quite large, and lead to an inaccurateenergy as well. Previous experiments have relied on stereoscopic observations of showers toconstrain geometry. Relying on stereoscopic observations does result in a reduced exposurethough, with the reduction also depending on the configuration of the FD stations. Forthe Pierre Auger Observatory, the large detector spacing means that showers must be ofexceptionally high energy, and land near the centre of the observatory, making it unsuitableas a reliable method of geometry reconstruction.70 Pierre Auger ObservatoryFig. 2.15. Diagram of the coordinates used in the reconstruction of shower geometry. From[65].The Pierre Auger Observatory solves the geometry reconstruction in a different way, thatdoes not rely on stereo observation. Since the FD stations are situated along the edges ofthe SD array, and face inwards with a 180 degree field of view, most events observed by thefluorescence detector will also be picked up by the surface detector. If just one SD station istriggered, the timing of that station can constrain the geometry of the shower, significantlyimproving the accuracy of the timing fit. The requirement of only one SD station meansthat a hybrid station can be found at energies far lower than the full trigger efficiency of theSD, as the FD can provide a T3 trigger for the SD. For a given trial shower geometry, the χangle of the shower as the shower front passes the SD station is found, χSD, and the expectedtime that the shower front would pass the SD station is found, t(χSD). The determination oft(χSD) must take into account a time delay due to shower front curvature. The function tominimise then becomes:χ2 = ∑i(ti − t(χi))2σ(ti)2+(tSD − t(χSD))2σ(tSD)2(2.16)With the geometry of the shower determined, the longitudinal profile, and hence theenergy of the shower can be found. This does require the sensitivity of the detector, theatmospheric attenuation, and contribution of other light sources to be known [108]. The lightflux at the detector can be found for each ADC time bin, by adding the signals in pixelsdetecting light from the shower. Pixels are chosen for this if they are within an angle ζ of theexpected position of the shower at the corresponding time. With a profile of light observed bythe fluorescence detector, the next step is to determine the energy deposit profile. The light2.4 Shower reconstruction 71Fig. 2.16. A shower observed by all 4 FD stations. The shower energy is ∼ 8×1019 eV. Thelines emanating from the FD stations represent the directions of triggered pixels, coloured bythe pixel timing. The red lines are the shower axes determined by the FD stations. The blueline is the axis determined by the SD. The red spots along the axes are the locations of Xmaxreconstructed by the FD stations.observed at the detector can come from a number of different mechanisms. Fluorescencelight is produced in an amount proportional to the energy deposited in the atmosphere, and isis emitted isotropically. At a given slant depth, Xi, the number of photons produced in a slantdepth interval ∆Xi can be written as:N fγ (Xi) = Yfi wi∆Xi (2.17)72 Pierre Auger ObservatoryChapter 4Shower Detector PlaneGround Plane(a) Illustration of the SDP. The geometry is re-duced to a planar problem. Different showers stillfit to one SDP.]ged[ htumiza09 001 011 021 031 041 051pixel elevation [deg]050151025203(b) Light track of event 3308259 as seen by two ad-jacent fluorescence cameras (Los Morados). Dif-ferent colors indicate the arrival time at the tele-scope. The filled black square at the bottom ofthe telescope denotes the position of the stationused within the reconstruction (cf. Sec. 4.6).Figure 4.7: Determination of the SDP.and Čerenkov light contributions can be subtracted from the apparent brightness of theshower and the energy deposit of the electromagnetic cascade as a function of showerdepth is determined.4.5.1 Shower detector planeThe shower detector plane is defined as the plane, containing the shower axis and thecenter of the eye. The reconstruction procedure mainly uses the trace of triggered pixelswhere high signal PMTs are expected to be more reliable than noisy ones. First a twostep pre-selection of pixels is done [73]:1. It is required that the pixel is not isolated in space and time by requiring that validpixels should not be more than four camera rows or columns away from any other.The barycenter of reconstructed pulses should not be more than 6 μs away fromother pixels.2. It is required that pixel times are correlated with shower candidatesThe orientation of the SDP is specified by a unit normal vector n referred to as the“SDP vector”. Since every plane has two normal vectors, one opposite to each other, aconvention is used to remove this ambiguity. The common definition is that the crossproduct of the SDP vector with the local vertical of the detector points in the directionof the core [74]. For this convention only, the core is defined as the intersection of theshower axis and the detector’s horizontal plane. The direction of the shower is not taken32Pierre Auger Observatory(a) The shower energies reconstructed by threeeyes and the surface array of event 3351072. Theyellow band represents a weighted average of thesub-detectors. Its width combines the statisticaluncertainty σE.Los MoradosCoihuecoLos Leones(b) Two dimensional plot of the surface array ofevent 3351072. Stations used for SD reconstruc-tion are shown in red colors. Two lines from eachFD with a successful FD reconstruction show the±1σ projections of the SDPs on the array plane. [deg]0χ20 25 30 35 40 45 50 [km]pR111213MonoHybrid(c) Solution for the axis for mono and hybridreconstruction (1σ accuracy). The large uncer-tainty of the monocular reconstruction is brokenusing the timing information from the surface de-tector. The stars indicate the solution that mini-mize the χ2 for the axis reconstruction (cf. [81]).Figure 4.13: Advantages of the hybrid technique.39Fig. 2.17. The degeneracy in the geometry reconstructi n is shown in the upper figur . Theleft figure shows a comparison of a mono time fit to a hybrid time fit of the same shower. theright figure shows the allowable values of Rp and χ0 for a mono vs hybrid reconstruction.where wi is the energy deposited per unit depth at a depth Xi, and Yfi is the fluorescence yield,which is dependent on the pressure, temperature, and humidity at slant depth Xi. Taking intoaccount atmospheric attenuation, a fraction Ti will make it to the detector. With a detectoraperture area A, at a distance ri from the detector, and a light detection efficiency ε , themeasured flux of fluorescence light is given as:y fi = diYfi wi∆Xi (2.18)di = εTiA4πr2i(2.19)This signal, if isolated is the best measure of energy deposit, but this signal is contami-nated by Čerenkov light. Due to the refractive index of air, Čerenkov light is directed in a2.4 Shower reconstruction 73very narrow cone (less than 1.5◦) in the direction of the particle producing it. Since the vastmajority of particles in a shower lie very close to the shower axis, and are directed nearlyparallel to the shower axis, the majority of Čerenkov light exists in a narrow beam close tothe shower axis. With fluorescence detected showers being viewed side-on, this beam cannotbe detected directly, but does contribute to the observed signal when Čerenkov photons inthe beam are scattered in the direction of the detector. Both Rayleigh scattering due to airmolecules, and Mie scattering due to aerosols, can contribute to this observed signal. A smallfraction of particles in a shower can obtain a large transverse momentum and be directedtowards the FD aperture producing a direct Čerenkov signal (see figure 2.18).Fig. 2.18. Diagram showing the different light contributions to the observed signal. The leftdiagram shows the direct fluorescence and direct Čerenkov contributions. The right diagramshows the scattered Čerenkov contribution. From [108].The direct Čerenkov contribution observed at the detector can be expressed as:yCdi = di fC(βi)YCi ∆XiNei (2.20)where fC(βi) is the fraction of Čerenkov photons emitted per unit solid angle at angle βi withrespect to the shower axis, and Nei is the number of electrons in the shower at depth Xi abovethe Čerenkov threshold energy. The number of Čerenkov photons in the beam depends onthe number of electrons at shallower depths as the beam is built up as the shower progresses.The number of Čerenkov photons in the beam is:Nbeamγ =i∑j=0τ jiYCj ∆X jNej (2.21)74 Pierre Auger Observatorygiving an observed scattered Čerenkov signal:yCsi = di fs(βi)Nbeamγ (2.22)where τ ji is the fraction of light transmitted from the j’th position to the i’th position, andfs(βi) is the fraction of photons scattered at angle βi with respect to the shower axis per unitsolid angle. From this, the total signal observed at the FD can be expressed as:yi = yfi + yCdi + yCsi (2.23)Since there are two unknowns that need to be deduced, energy deposit and number ofelectrons, and only one observable, photons at the detector, an additional assumption must bemade. Since the vast majority of energy deposited in the atmosphere is by electrons, it canbe assumed that the number of electrons and the energy deposit are related. This relation isfound to be dependent on shower age, and is universal. With this, the solution can be writtenin matrix form. Let y be a vector of light observed at the detector yi, and let w be a vectorof the corresponding energy deposits wi. The energy deposit profile can be found using theexpression:y = Cw (2.24)where C is the Čerenkov - fluorescence matrix [108]. The coefficients of this matrix areobtained from the above equations for fluorescence and Čerenkov light. Inverting C allowsthe energy deposit profile to be found (see figure 2.19). There is also wavelength dependenceto consider. The atmospheric transmission and scattering coefficients are both dependent onwavelength, as is the detector response. To handle this wavelength dependence, this methodis done in a number of wavelength bins.The energy of the shower could be found by integrating the energy deposit profile, butthis requires a very large range of depths to be observed. Since only a limited range of depthis typically observed, the energy deposit profile is fit to the Gaisser-Hillas function:fGH = wmax(X −X0Xmax −X0)(Xmax−X0)/λe(Xmax−X)/λ (2.25)The free parameters are wmax, the maximum rate of energy deposit, Xmax , the depth ofmaximum energy deposit rate, and two parameters that describe the shape of the energydeposit profile, X0 and λ . Since some showers only observe a small range of depths, the shapeparameters can have a large uncertainty. This was previously handled by placing constraintson the fit to fix the values of X0 and λ to their average values, determined from showers with2.4 Shower reconstruction 75time slots [100 ns]250 260 270 280 290 300 310 320 330 340/100ns]2detected light [photons/m050100150200250datatotal lightMie Cherenkovdirect CherenkovRayleigh Cherenkovmultiple scatteredFig. 2.19. Profile reconstruction for a shower with an energy of 1.10±0.04×1019 eV. Theleft plot is of the light profile at the FD, showing the estimated Čerenkov contributions. Theright plot is the reconstructed energy deposit profile.a high quality profile. Recently, the Gaisser-Hillas function has been re-parameterised to thefollowing function:fGH = (dE/dX)max(1+RL(X −Xmax))1/R2(exp)(−X −XmaxRL)(2.26)with L and R representing a shower profile width and asymmetry, respectively. With this newparameterisation, all four parameters (including the two shape parameters, R and L) are bothfitted with loose constraints on R (0.257± 0.055) and L ((227.3+ 7.44log(Ecal/EeV))±11.5 g cm−2), with the central values determined from showers with a high quality profile,and the ranges determined from simulations to account for shower-shower fluctuations anddifferent mass compositions [33].Determination of the primary particle energy must also take into account invisible energy.Some of the energy in the shower goes into particles that do not deposit energy in thedepths associated with atmospheric showers, such as high energy muons, and neutrinos.The invisible energy fraction is dependent on primary energy, as well as composition, andinteraction model. The uncertainty due to composition and interaction model can be reducedif the number of muons in a shower is known, since the number of muons is related tothe invisible energy universally. The invisible energy fraction used for the Pierre AugerObservatory reconstruction has been parameterised as a function of calorimetric energy froma combination of inclined shower reconstructions (where muon signals dominate), and alsovertical golden hybrid showers, comparing SD and FD signals [104]. The invisible energyfraction used is shown in figure 2.20.With suitable quality cuts applied, the energy resolution is 8%, and Xmax resolution is 20g cm−2, for energies of 1-10 EeV [34].76 Pierre Auger ObservatoryFig. 2.20. Invisible energy parameterisation used by the Pierre Auger Observatory, deter-mined by data-driven methods, compared to model predictions and the TA parameterisation.From [104].The FD is capable of detecting events with a far lower energy if they are directed towardthe telescope due to the high Čerenkov light emission. These events are more difficult toreconstruct, as these events are not hybrid. With the shower being dominated by Čerenkovlight, even small errors in geometry can propagate into large errors in energy deposit due tothe angular dependence of Čerenkov emission. To reconstruct the shower geometry, a profileconstrained geometry fit (PCGF) can be used, where a fit is done simultaneously to both thegeometry and energy deposit profile, with a constraint that the energy deposit profile must becompatible with a Gaisser-Hillas function [75].2.4.3 SystematicsWith many steps involved in calibration and reconstruction of showers, there are systematicerrors incurred [33], which must be accounted for in the FD reconstruction, the possiblesources are:• The fluorescence yield. This includes uncertainties in the scale of the fluorescenceyield, and the relative spectrum. There are also uncertainties in atmospheric conditions,such as humidity, that contribute to this uncertainty in fluorescence yield.• Atmospheric attenuation and scattering. Uncertainties in aerosol optical depth, andatmospheric density profile, contribute to uncertainties in atmospheric attenuation and2.4 Shower reconstruction 77Systematic uncertainties in energy scaleFluorescence yield 3.6 %Atmospheric uncertainty 3.4 - 6.2 %FD calibration 9.9 %FD profile reconstruction 6.5 - 5.6 %Invisible energy 3 - 1.5 %SD Energy scale stability 5 %Total 14 %Table 2.1 Summary of the systematics in the energy scale. Where errors are shown as asingle value, the errors are not strongly dependent on energy. Where there is a strong energydependence, the uncertainties are given at energies of 3×1018 eV and 1020 eV. From [33].particulate scattering. There are also uncertainties in the phase function and wavelengthdependence of aerosol scattering.• Errors in the calibration of the FDs. The steps in performing the absolute drum calibra-tion, and also in the relative calibrations performed nightly. The optical efficiency ofthe telescope also contributes to this error.• Errors in the energy deposit profile reconstruction. Some of the light from the showeris spread onto pixels that are not selected for the reconstruction by both the telescopeoptics, and spread of light in the shower. Uncertainties in this lead to systematicuncertainty in the energy scale.• Invisible energy. Though this error has been reduced with muon detection studies,there is still a small uncertainty in the invisible energy fraction. Any uncertainty in theinvisible energy fraction will systematically propagate onto the energy scale.Since the SD energy scale is calibrated using the FD, the SD inherits the systematics in theFD energy scale. This also means that the SD calibration and reconstruction steps donebefore the calibration fit do not affect the SD energy scale systematic. There is an additionalsource of systematic error in the SD energy associated with the stability of the SD energyscale. Results from measuring the stability of the energy scale can be used to give an estimateof energy scale uncertainty due to stability. The total uncertainties are summarised in table2.1.78 Pierre Auger Observatory2.5 Composition2.5.1 FDThe fluorescence detector measures the energy deposit profile of a shower, and can deter-mine the depth of maximum energy deposit, Xmax. Xmax is the best available indicator fordetermining composition, and is used for determining the overall changes in compositionat a given energy. Due to fluctuations in the first few generations of interactions, the Xmaxvalue of an individual event cannot be used to determine the composition of that event, butthe distribution of Xmax can be used to infer the composition of a large set of showers.Many events observed by the FD only have a limited portion of the shower track in thefield of view of the FD telescopes. Some events may also be of low quality, due to a lowsignal to noise ratio, high levels of aerosols, and clouds. These can result in biases in theXmax distribution. The Pierre Auger Observatory prevents this by applying cuts to the eventsused to find the Xmax distribution, to remove events that may cause a bias [100]. The cutsthat are applied are:• Hardware status. Events are only selected if they occur during good data takingconditions. There must be a good calibration, the timing relative to SD timing must bewithin acceptable limits to ensure a good geometry reconstruction, and the telescopeoptics must be properly aligned.• Aerosols. There must be a measurement of aerosols in the atmosphere within one hourof the observed shower. The VAOD (vertical aerosol optical depth) from the ground toan altitude of 3 km must be less than 0.1, to reduce possible biases due to poor viewingconditions.• Hybrid geometry. For an accurate reconstruction, it is required that there is a hybridgeometry reconstruction. This also cuts events caused by spurious signals, such asdistant sheet lightning.• Clouds. Cloud layers can cause inaccurate profile reconstruction by reflecting orshadowing light. Information from lidars, the two laser facilities, the cloud cameras,and cloud data from GOES (Geostationary Operational Environmental Satellites) areused in determining if a profile is likely affected. Events are selected if no cloudis detected in the shower direction as seen by the FD station (using cloud cameradata), as well as having no cloud visible in the ground level projection of the shower(seen by GOES). Sufficiently high clouds will not have a significant effect on profilereconstruction if the majority of shower light production is below the cloud layer.2.5 Composition 79Events during cloudy conditions are therefore accepted if both the lidars and the laserfacilities measure a cloud height above the geometrical field of view or 400 g cm−2above the fiducial field of view (fiducial field of view will be discussed shortly).• Energy. For analysis with the standard FD telescopes (not including HEAT), an energythreshold is applied so only events with E > 1017.8 eV are accepted. For the analysisusing HeCo, the energy threshold is lowered to 1017.2 eV.• Hybrid probability. At low energies, the SD may not trigger a detector for hybridanalysis with full efficiency. If not fully efficient, there may be composition biases,since trigger probability may be composition dependent.• Xmax observed. It is required that Xmax is within the field of view of the telescope, sinceprofile reconstructions using only the tails of the energy deposit profile may lead toinaccurate Xmax reconstructions.• Quality cuts. Events must have an expected Xmax resolution of better than 40 g cm−2,determined by propagating statistical uncertainties from photo-electrons in expectedPMT signals, geometry, and atmospheric errors to an Xmax uncertainty. Geometrieswhere the shower light is observed at small angles with respect to the shower axis arealso rejected, due to the large contribution of Čerenkov light, that is highly dependenton geometry, and therefore sensitive to small geometry errors, at small observationangles.• Fiducial field of view. The fiducial field of view is defined by the minimum andmaximum values of Xmax for which a shower is expected to pass the quality and Xmaxobserved cuts. The minimum and maximum observable Xmax are referred to as Xlowand Xup. The values of Xlow and Xup are required to encompass the majority of theXmax distribution, to reduce biases caused by a differing acceptance of events withXmax. The fiducial cuts will be further discussed shortly.For the fiducial cuts, the values of Xlow and Xup are initially not known as the true Xmaxdistribution is not yet known. Xlow and Xup are found for each energy bin by analysing thedifferential behaviour of ⟨Xmax ⟩ as Xlow and Xup are adjusted. The values of Xlow and Xupare chosen to give a value of ⟨Xmax ⟩ that is within 5 g cm−2 from the asymptotic value(see figure 2.21). This method produces a flat Xmax acceptance for values of Xmax betweenXlow and Xup. This leaves the Xmax distribution with minimal detector bias, so that the Xmaxdistribution can be directly compared to the distributions predicted by various compositionmixes and interaction models [100].80 Pierre Auger ObservatoryFig. 2.21. ⟨Xmax ⟩ for showers in the energy interval from 1018.1 to 1018.2 eV, with differ-ing values of Xlow and Xup. The dashed lines indicate the chosen values of Xlow and Xupcorresponding to a 5 g cm−2 bias in ⟨Xmax ⟩ from the asymptotic value. From [100].For values of Xmax outside of the interval from Xlow to Xup, the acceptance is not constant,reducing for values of Xmax values farther outside the flat acceptance range. To compensatefor this, the detector acceptance is determined by a detailed simulation, and events areweighted by the inverse of their acceptance (see figure 2.22).Fig. 2.22. The simulated detector acceptance as a function of Xmax in the energy intervalfrom 1019.0 eV to 1019.1 eV. The open points show the simulated detector acceptance withoutfiducial cuts, and the filled points show the simulated acceptance with fiducial cuts applied.Note that both have been normalized to have a peak acceptance of 1. From [100].The simplest method for extracting overall composition information from the Xmaxdistribution is to look at the mean and width of the distribution, ⟨Xmax⟩ and σ(Xmax). Aheavier composition will correspond to a shallower ⟨Xmax⟩, with ⟨Xmax⟩ being linearly relatedto ⟨lnA⟩, where A is the nuclear mass, though this is also dependent on the interaction modelused. The rate of change of ⟨Xmax⟩ with increasing energy, known as the elongation rate,is also indicative of composition changes. For a constant composition, the elongation rate2.5 Composition 81is expected to be ∼ 60 g cm−2 per decade in energy, and is roughly independent of thecomposition and interaction model.For a pure composition, σ(Xmax) is larger for lighter composition, due primarily to thelonger first interaction length, and the larger variance in the multiplicity of particles producedin the first interaction. Mixed compositions are a more complicated matter. Unlike with⟨Xmax⟩, σ(Xmax) does not just depend on the average composition ⟨lnA⟩, but also the varianceof the composition σ2(lnA). For a given ⟨lnA⟩, σ(Xmax) will be the smallest for a purecomposition (σ2(lnA) = 0), and become larger with a more mixed composition (increasingσ2(lnA)). For the Xmax distributions obtained by the Pierre Auger Observatory, the detectorresolution must be considered, as the observed σ(Xmax) will be made larger than the truevalue. To deal with this, the Xmax resolution has been carefully studied, and the detectorresolution is subtracted in quadrature from the observed σ(Xmax) to give a true σ(Xmax).The ⟨Xmax⟩ for a given composition is dependent on the hadronic interaction model used,and it is uncertain which, if any, correctly predict ⟨Xmax⟩. Because of this, it is difficult withthe Xmax distribution alone to differentiate between a more pure but lighter mass composition,and a heavier but more mixed mass composition. By including data from the SD, it is possibleto differentiate these cases with a lesser dependence on the interaction model. For a purecomposition, a shower of a given energy with a deeper Xmax will have a larger S(1000), dueto the shower developing closer to the ground. When considering a mixed mass composition,heavier primary particles have a higher muon content, meaning that a larger S(1000) isexpected. Heavier primaries will, on average, also have a shallower Xmax. This meansthat in a mixed mass composition, deeper showers can have a smaller S(1000). Thus, byexamining the correlation between Xmax and S(1000), it is possible to deduce the purity ofthe composition mix [81, 117]. To compensate for the dependence of S(1000) on energy andzenith, and Xmax on energy, the values of Xmax and S(1000) are scaled to a reference energyof 10 EeV, and S(1000) is scaled to a zenith angle of 38◦. These scaled values are known asS⋆38 and X⋆max.2.5.2 SDThe surface detector array offers a vastly increased data set when compared to the FD, beinga factor of 30 larger at energies above 3 EeV. The surface detector does not provide a compo-sition measure that can be directly compared with other detectors or shower simulations, likethe measured Xmax from the FD. The SD observations could be compared against detectorsimulations to infer composition, but this relies heavily on hadronic interaction models,but the models currently available do not describe showers well at the highest energies,especially when considering the muon density at the ground. Despite this, there are useful82 Pierre Auger Observatorymeasurements of composition information extending to higher energies than the FD canprovide.RisetimeParticles produced at different depths will take a different path to get to an off-axis detector.As a result of this, not all particles in a shower will arrive at a detector at the same time. It istherefore possible to extract information about the longitudinal development of a shower byexamining the time distribution of the observed signal in a surface detector station.The signal observed by the Pierre Auger Observatory surface detector stations consistsof both a muon and electromagnetic component. Muons are typically energetic enoughthat they can be approximated as traveling in straight lines at the speed of light. As muonsare produced very close to the core, and are not likely to be scattered traveling from theirproduction point to the ground, the arrival time of a muon at the ground is dependent only ongeometrical effects (see figure 2.23). Away from the core, the electromagnetic componentconsists of particles at lower energies, that are likely to undergo multiple scatterings beforereaching a detector, so the electromagnetic component arrives at the detector later than themuon component.Fig. 2.23. Diagram showing the arrival time difference between muons produced at differentdepths. From [103].A measure that has been found to be useful for composition analysis is known as risetime(t1/2), the time taken for the integrated signal to increase from 10% to 50% of the finalvalue [103, 105]. Risetime will depend on zenith angle, as the longer distance from muonproduction to the ground will produce smaller risetimes for more inclined showers. There is2.5 Composition 83also an asymmetry that must be accounted for, where stations with the same core distancein the shower plane will have a risetime dependent on azimuth, as stations underneath theshower axis sample an earlier part of the shower development. Most importantly, showers thatdevelop earlier, with Xmax occurring at a higher altitude, will have a smaller risetime whencompared to a later developing shower, as the time spread of muons due to geometric effectsis smaller. The last effect allows for the determination of the depth of shower development,giving a measure of composition.Currently, risetimes measured by the SD are analysed using the Delta method. In thismethod, a risetime benchmark, tbench1/2 is found in a narrow energy range, as a function of zenithangle and core distance, as well as containing corrections for asymmetry. A parameterisationof the uncertainty, σ1/2, is also found, using measurements from twin stations (two stationsthat are separated by 11 m), and pairs of stations with a similar core distance. Thus for agiven station, we define ∆i as:∆i =t1/2 − tbench1/2σ1/2(2.27)This is also illustrated in figure 2.24. For an event where there are N stations with a risetimemeasurement, a shower Delta, ∆s can be obtained as an average of each of the stationestimates:∆s =1N ∑stations∆i (2.28)Fig. 2.24. Diagram of how ∆ is obtained in the Delta method. From [103]For the Delta method analysis, a number of selection criteria are applied to events. Theyare:84 Pierre Auger Observatory• 6T5. These events have the detector with the highest signal surrounded by 6 workingdetectors to ensure an accurate reconstruction.• Energy cuts. There is a lower bound on energy to ensure a 100% triggering efficiencyto avoid a composition related triggering bias. The lower energy bound is 1018.5 eVfor the 1500 m array and 1017.5 eV for the 750 m array.• Zenith angle. At large zenith angles, t1/2 becomes too small for the 25 ns resolutionof the FADCs. The maximum value of secθ is 1.30 for the 750 m array. Initially themaximum secθ was 1.45 for the 1500 m array, but this has recently been extended to2.0.• Bad periods. Events from data taking periods where the array performance was notoptimal are rejected.• 3 stations. Events were required to have 3 stations that fulfilled the selection criteriafor a risetime analysis. Stations were required to have signals of a least 3 VEM forthe 750 m array, and at least 5 VEM for the 1500 m array. Stations must also not besaturated since the risetime cannot be determined for a saturated station.After cuts are applied, the benchmark functions are determined. Benchmark functions aredone for the low gain and high gain electronics channels separately. After testing a numberof functions, the following functions are used:t low−gain1/2 = 40ns+√A(θ)2 +B(θ)r2 −A(θ) (2.29)thigh−gain1/2 = 40ns+N(θ)(√A(θ)2 +B(θ)r2 −A(θ))(2.30)(2.31)where A and B are free parameters, determined with a fit to the low gain channels. A fitto the high gain channels is then used to determine N. The benchmark fits are done in theenergy range from 1017.7 eV to 1017.8 eV for the 750 m array and 1019.1 eV to 1019.2 eV forthe 1500 m array. This energy range is chosen as it contains data over a good range of coredistances for both low and high gain channels. An example of this fit is shown in figure 2.25.Finally, using these fits applied over numerous zenith angles, A, B, and N are parameterised2.5 Composition 85as a function of θ :A(θ) = a0 +a1(secθ)−4 (2.32)B(θ) = b0 +b1(secθ)−4 (2.33)N(θ) = n0 +n1(secθ)2 +n2esecθ (2.34)Fig. 2.25. (left) An example of a benchmark fit to risetimes. The solid line shows the fit forthe low-gain trace, and the dashed line shows the fit to the high gain trace. (right) residualsto the example benchmark fit. From [103]Muon production depthSince muons travel in approximately straight lines at near the speed of light, the arrival timeof a muon can be traced back to the production point, if the shower geometry is well known.For vertical showers, the electromagnetic component makes it difficult to do this with asurface water Čerenkov detector. For heavily inclined events, the electromagnetic componentis strongly attenuated, while the muons are not. This makes it possible to determine a profileof muon production depth without the requirement for underground muon detectors[31, 70].Muons will have a time delay relative to the plane front due to a number of possiblefactors, shown in figure 2.26. At larger core distances, the geometric contribution is dominant,which is desirable for determining production height. For a muon produced at a position zalong the shower axis, travelling a distance l to reach the ground, they reach a point (r,ζ ),where r and ζ are shower plane distance and azimuth (see figure 2.26). The muon production86 Pierre Auger ObservatoryFig. 2.26. (left) Diagram showing the geometry causing the time delay. (right) Contributionsto the muon delay from the plane front, as a function of distance. From [31]position z can be found with the following equation.z 12(r2ctg− ctg)+∆−〈zπ〉 (2.35)where tg is the geometric delay, with:tg t −〈tε〉 (2.36)where tε is the kinematic delay due to the muon travelling slower than the speed of light. ∆ isthe distance from the muon ground impact point to the the shower plane, given by:∆ = r tanθ cosζ (2.37)〈zπ〉 is to take into account the decay length of the parent pion. The kinematic delay isdependent on the muon energy, which is not directly measurable with the surface detectors. Aparameterisation is used to estimate the kinematic delay. With the muon production positiondetermined, a muon production depth can be found:X µ(z) =∫ ∞zρ(z′)dz′ (2.38)The distribution of muon production depth (MPD) contains information about the de-velopment of the hadronic component of the cascade. The maximum of this distribution isknown as X µmax, and can be used similarly to Xmax, measured with the FD.Due to the time resolution of the detector, and the large contribution of kinematic delay,core distances smaller than 1200 m are not used. At small zenith angles, this could introducea bias, since the muons produced deeper are produced close to the ground and will not spread2.6 Upgrades 87out to large core distances. For more inclined events, almost all muons are produced far fromthe ground, and cover a large area on the ground.The Function used to fit the MPD distribution is:dNdX=(1+RL(X −X µmax))R−2exp−X−XµmaxLR (2.39)for a total number of muons produced N, at depth X . X µmax, R, and L are the fit parameters.Due to the small number of muons detected (∼50 at 1019.5 eV), the parameter R is fixed,and L is given an initial value. The R and L values used are dependent on zenith angle, so aparameterisation is used for the fits. To take into account a dependence of X µmax on zenithangle, the value of X µmax for each event is normalised to ⟨θ⟩= 55◦.A number of quality cuts are applied to ensure reliable MPD reconstructions. They are:• 6T5. The detector with the highest signal must have all six closest neighbours operating,to ensure a good reconstruction.• Number of stations. At least 5 stations with a signal greater than 3 VEM mustcontribute to the reconstruction, to avoid trigger fluctuations, and to minimise theimpact of accidental signals.• Fit uncertainty. The MPD fit must converge and have an uncertainty δX µmax/Xµmaxsmaller than a certain threshold εmax. This value varies from 28 % at low energy, to 18% at the highest energy.• L. The value of the parameter L must be between 130 g cm−2 and 415 g cm−2. Valuesoutside of this range are associated with the first part or the tail of the MPD being verypoorly reconstructed.2.6 Upgrades2.6.1 High Elevation Auger Telescopes (HEAT)The standard FD configuration, observing elevations of up to 30◦, is well suited to observingthe highest energy showers. At energies below ∼ 1018 eV, showers can only be observedat a close distance. At a close distance, the standard FD will only observe a limited rangeof depths, with Xmax being above the field of view. HEAT was added to the observatoryto extend the field of view of the FD to a 60◦ elevation, extending the range of observableenergy down to ∼ 1017 eV [101, 63].88 Pierre Auger ObservatoryFig. 2.27. Diagram of HEAT in horizontal mode and in tilted mode. From [101]Fig. 2.28. HEAT in operation. (Image by the author).HEAT consists of three telescopes similar in design to the other FD telescopes, with onecontainer for DAQ, slow control, and calibration hardware. One notable difference is thatthe electronics are capable of sampling at up to 40 MHz, compared to 10 MHz for the otherFD telescopes, to better reconstruct showers with high angular velocities. Each telescopeis housed in a separate building, located 180 m away from Coihueco (see figure 2.28). Thebuildings can be tilted using a hydraulic system (see figure 2.27). The telescopes are keptin the horizontal position to allow maintenance, and to allow for a cross calibration withCoihueco. HEAT is designed to operate normally in the tilted mode, observing elevations2.6 Upgrades 89up to 60◦. HEAT is designed to be used in conjunction with the infilled array of the surfacedetector, allowing fully efficient hybrid triggering at low energies. HEAT began operationsin 2009.Fig. 2.29. A 3.72±0.18×1017 eV shower seen by Heat and Coihueco.2.6.2 AugerPrime upgradeA series of upgrades are being applied to the Pierre Auger Observatory [102]. The mainmotivation for this is to determine the composition mix at the very highest energies. Amuch larger dataset is required than the current set available with the FD. The best way tosubstantially increase the number of events with composition measurements is to make thesurface detector sensitive to composition. The upgrades include scintillation detectors on topof every SD station for µ-EM discrimination, improved electronics for the SD array, buriedmuon detectors in the infilled array for direct measurements of muon content, radio detectionfor inclined showers, and a new operation mode for the FD to increase duty the cycle.Auger Muon and Infilled Ground Array (AMIGA)AMIGA is a detector designed to directly measure the muon content of extensive air showers[101, 23]. It consists of a denser array of water Čerenkov surface detectors (known as theinfilled array), and a series of buried scintillator detectors. The infilled array is nested withinthe 1500 m array, with a spacing of 750 m, and is situated 6 km in front of Coihueco, toallow hybrid reconstruction at lower energies for HEAT. The infilled array is fully efficientdown to 3×1017 eV for zenith angles below 55◦. The area of the infilled array is 23.5 km2.Though this area is small, the flux of cosmic rays increases rapidly with decreasing energy,so the event rate at lower energies is more than adequate. An additional array has also been90 Pierre Auger Observatoryconstructed with a 433 m spacing covering an area of 1.9 km2, lowering the energy thresholddown to 1016.5 eV.Fig. 2.30. Map of the AMIGA engineering array. From [23].Adjacent to each SD station in the infilled array, 30 m2 of plastic scintillators will beburied underground, to be sensitive only to muons. These are currently undergoing testing,with units installed at 7 stations operating since March 2015. The plastic scintillators areburied under 2.3 m of soil corresponding to ≈ 540 g cm−2 of vertical mass overburden. Thismeans that there is a cutoff of 1 GeV for vertical muons. The 30 m2 of scintillators at eachstation is made up of 10 m2 and 5 m2 units. Each unit is made up of 64 polystyrene strips4.1 cm wide and 1.0 cm thick. Each strip has a wavelength shifting optical fibre to directthe light to a photo-detector at the center of the module. For detection of the scintillationlight, 64 channel multi-anode PMTs, and arrays of 64 silicon photomultipliers (SiPMs) weretested, with the SiPMs being chosen for production. Two of the stations have two sets of 30m2 stations, to analyse the experimental accuracy of the detectors. There is also an additional20 m2 detector buried under 1.2 m of soil to better analyse the shielding properties of soil.Each photomultiplier and SiPM channel is digitised at 320 MHz with just 1 bit perchannel, being either 1 or 0 depending on whether the signal is above a set threshold. Withthe detector being highly segmented, and with a fast readout the detectors act as muoncounters. Counting muons means that there is no need to carefully calibrate the gain of eachdetector. Signals from the muon detectors are transmitted via a dedicated system based on a2.6 Upgrades 91WiFi 802.11g standard. Data from the muon detectors is sent to CDAS only when there is aT3 trigger.Radio research programIndividual particles in an air shower will often travel at a speed very close to the vacuumspeed of light, producing Čerenkov light. Čerenkov light emission is proportional to λ−2, somost Čerenkov emission is expected near the shortest wavelengths for which the mediumrefracts light (typically UV wavelengths). Čerenkov emission at radio wavelengths dueto individual particles would be expected to be virtually impossible to detect. However,in a cosmic ray shower, there can be a charge excess induced at the shower front, alsotravelling at very near the vacuum speed of light. At wavelength scales longer than thesize of the charge excess, the charge excess acts like one particle with a very large charge.With Čerenkov light emission also being proportional to q2, where q is the charge, Čerenkovemission is greatly boosted at longer wavelengths. This charge excess can be caused byphotons in the shower front scattering electrons from air molecules. This is known as theAskaryan effect [11]. Magnetic deflection of shower particles can also form charge excessesperpendicular to the direction of shower propagation. The formation of these charge excessesalso mean that electric currents will be produced in a shower. The electric currents alsoproduce electromagnetic radiation at radio wavelengths. Recent studies have found thatthe geomagnetic deflection of shower particles is the dominant source of radio emission inextensive air showers, though the Askaryan effect must still be accounted for [79]. Radiodetection of extensive air showers is possible during both day and night, and is not dependenton weather. Radio detection is primarily sensitive to the electromagnetic component of ashower, and can be used to determine a longitudinal profile.A test array, known as the Auger Engineering Radio Array (AERA) has been constructed,and has been measuring the radio properties of extensive air showers [102]. AERA consistsof 153 stations, spread across an area of 17 km2 (see figure 2.31). Each station has a dualpolarisation antenna, able to observe electric fields in the north/south and east/west direction.Nine stations also have a vertically aligned antenna. AERA stations are sensitive between30 and 80 MHz, chosen due to this range being fairly radio quiet. AERA is capable of aself-trigger, as well as an external trigger.Since the radio emission is highly forward directed, vertical air showers have only a verysmall radio footprint on the ground, requiring a very dense array to detect air showers. Forhighly inclined showers, the long distance from Xmax to the ground means that the radiofootprint is very large, making radio detection possible with a large array spacing [46]. Aspart of the Auger Prime upgrade, radio detectors will be placed on all of the surface detector92 Pierre Auger ObservatoryFig. 2.31. (left) Map of AERA. (right) Picture of an AERA station. From [102].stations [83]. This will allow for measurements of both the electromagnetic (radio) andmuonic (SD) component for inclined air showers, while the surface scintillation detectorprovides this for vertical events. A 1.2 m diameter short aperiodic loop antenna (SALLA)will be installed on top of each SD station, with electronics integrated with the other detectorcomponents (see figure 2.32). The energy threshold for full efficiency is dependent on zenithangle, ranging from 1018.3 eV for the highest zenith angle, and increasing with decreasingzenith angle.SD station upgradesA number of upgrades to the surface detector array are being undertaken, to allow for bettermass composition sensitivity for the SD array. As well as the previously discussed radiodetector, a small PMT will be added to the WCD, and a scintillation detector will be installedabove the detector.It is desirable to extend the dynamic range of the water Čerenkov detector (WCD), toallow measurements to be made closer to the shower core [74]. Near the center of the tank,there is a 30 mm window in the liner, originally intended for a spare LED. A small PMT(SPMT), with a much smaller cathode can be placed over this window. The SPMT will havea greatly reduced sensitivity compared to the 3 large PMTs (LPMTs). The SPMT is notsensitive enough to record a single muon, so a calibration based on a VEM is not possible.Instead, the SPMT will be cross-calibrated to the LPMTs using pulses that are below thesaturation threshold for the LMPTs low gain channel.The electronics are also being upgraded for the surface detector stations, with a factorof 10 increase in processing power and memory, allowing for the handling of the additional2.6 Upgrades 93Fig. 2.32. An upgraded SD station. From [83].hardware, as well as more advanced local processing. The ADCs for the surface detector arealso being upgraded, operating at 120 MHz as opposed to the original 40 MHz. The LPMThigh gain channel was previously derived from the last dynode stage, but this signal is toonoisy for the upgraded dynamic range, so both low and high gain signals will be derivedfrom the anode with two parallel amplifiers. The dynamic range of the upgraded stations willbe extended to 20,000 VEM.The most significant upgrade to the SD station is the scintillator surface detector (SSD)[82, 102]. To provide better composition sensitivity, it is desirable to be able to separate theelectromagnetic and muon component from each other. After a number of proposals wereevaluated, it was decided that the best way to achieve this is by placing a scintillator on topof the existing WCD. The WCD is more sensitive to muons than it is to the electromagneticcomponent, whereas the scintillator is equally sensitive to both muons and electrons. Withthe electron density being higher than the muon density for vertical showers, the scintillatoris more sensitive to the electromagnetic component of a shower. A combination of WCDand SSD signals allows for the two components of the air shower to be disentangled. TheSSD is composed of 48 plastic bars of dimensions 160 cm long, 5 cm wide, and 1 cm thick.The scintillators are surrounded by a layer of reflective TiO2. Wavelength shifting fibres areused to direct light into a single PMT. The detector area of the SSD is approximately 3.84 m294 Pierre Auger Observatory(see figure 2.33). The PMT signal is split to a high and low gain channel, with a gain ratioof 128. Each channel is digitised with a 12 bit 120 MHz ADC. Calibration is achieved in asimilar fashion to the WCD, by measuring signals from single particles passing through thedetector. The units in this case are MIPs (Minimum Ionising Particles). The dynamic rangeof the SSD extends to 20,000 MIP. The SSD is mounted on top of the detector with a rigidaluminium frame (see figure 2.32). Deployment of the SSDs should be completed in 2020.Fig. 2.33. Drawing of an AugerPrime SSD. From [82].Extended FD operationThe FD is capable of the most direct measurement of the energy deposit profile, but suffersfrom a big problem. The duty cycle of the FD is approximately 15 %, as operations currentlyonly occur on dark, moonless nights. Excessive levels of background light, with high anodecurrents, can lead to the deterioration of the PMTs.If FD operation can be extended into periods with higher night sky background, whenthe moon is up or during twilight, the duty cycle may be increased by up to 50 % [102]. Tocompensate for the high night sky background, the high voltage supply can produce a lowervoltage, reducing the gain of the PMTs. This does reduce the sensitivity of the detector,meaning only the highest energy events are detected, but the upgrade is primarily to increasethe number of events at the highest energy, so this operation mode is still very useful. Tests ofthis show good quality data can be obtained at energies above 1019.5 eV, and that the PMTswill not be aged to an excessive degree. With the higher allowed night sky background, theFD duty cycle will be increased to a theoretical maximum duty cycle of 29 % up from 19 %(these do not take into account weather and malfunctions).2.7 Current results 952.7 Current results2.7.1 Energy SpectrumThere are a number of different spectrum measurements, covering differing energy ranges.The largest exposure belongs to the 1500 m SD array for vertical events, with an exposuregreater than 60,000 km2sr yr, covering energies above 1018.4 eV. The spectrum obtained withthe 1500 m SD array has also been unfolded to compensate for energy resolution effects.Results are shown in figure 2.34. The spectrum shows the transition at the ankle and thecutoff at high energy. The spectrum is best described with an additional transition betweenthe ankle and the cutoff, with the J1234 function, as opposed to the J12∆ function describing asmooth transition in the spectral index from the ankle to the cutoff.Fig. 2.34. (left) The energy spectrum measured by the SD. (right) The SD energy spectrumunfolded to compensate for energy resolution effects. From [110].The data from the other observation modes of the observatory can also be combined withthe 1500 m SD vertical results to extend the energy range and exposure [110]. The additionalmodes are:• SD horizontal air showers, with θ > 60deg. This has the second largest exposure, at17,447 km2sr yr, and covering energies above 1018.6 eV.• FD hybrid, with an exposure of 2248 km2sr yr at 1019 eV (increasing with increasingenergy), and a minimum energy threshold of 1018 eV.• SD 750 m array, with an exposure of 105.4 km2sr yr, and a minimum energy thresholdof 1017 eV.96 Pierre Auger Observatory• FD Čerenkov events, with PCGF used for geometry and energy determination. Theexposure for this mode is just 2.86 km2sr yr, but with a minimum energy threshold of1016.5 eV.The results of these observation are shown in figure 2.35.Fig. 2.35. (left) The energy spectrum of the different observation modes. (right) Thecombined energy spectrum, with a J01234 fit. From [110].With the extension into lower energies covering the second knee, the spectrum is welldescribed by a J01234 fit, with changes in spectral index at the second knee, ankle, and cutoff,as well as the additional change between the knee and cutoff. The equation for this fit is:J01234 ∝ E−γ01+(E/E01)γ01+(E/E01)γ11+(E/E12)γ11+(E/E12)γ21+(E/E23)γ21+(E/E23)γ31+(E/E34)γ31+(E/E34)γ4(2.40)The values of the parameters are: E01 = 0.15±0.02 EeV, E12 = 6.2±0.9 EeV, E23 = 12±2EeV, E34 = 50±7 EeV, γ0 = 2.92±0.05, γ1 = 3.27±0.05, γ2 = 2.2±0.2, γ3 = 3.2±0.1,γ4 = 5.4±0.6. These results are the first to show the point of inflection at ∼12 EeV, betweenthe ankle and the cutoff.Comparing the Pierre Auger energy spectrum results to those obtained by TelescopeArray, there is a notable discrepancy between the two, as seen in figure 1.8. With oneof the experiments being in the southern hemisphere and one in the northern hemisphere,differences due to anisotropy have been considered, by comparing spectra in a commondeclination band, observed by both experiments, but the differences still persist in this band[35]. Most of the discrepancy can be attributed to differences in FD energy scale betweenthe two observatories, due to the use of different fluorescence yield and invisible energycorrections. Even with this rescaling, the spectra require an additional energy dependent2.7 Current results 97rescaling to bring the spectra into agreement. The placement of Auger detectors at TA, andthe use of scintillation detectors at Auger will allow for a cross checking of the two detectorsin the future.[35]2.7.2 AnisotropyAnisotropy has been measured over a range of energies. At the lowest energies, magneticdeflection means that the direction of an incoming cosmic ray cannot be traced back to thepoint of production. However, anisotropy at lower energies may still be expected. For cosmicrays of galactic origin, a dipole anisotropy would be expected as cosmic rays diffuse outfrom the galactic centre. For cosmic rays of extra galactic origin, the galactic magnetic fieldsmay reduce the flux of cosmic rays coming from the direction of the galactic centre. Thefull dipole moment has been examined for energies above 4×1018 eV, where the array isfully efficient [86]. The exposure, including both vertical and horizontal air showers, is92,500 km2sr yr. For energies below this, the array is not fully efficient, and it is difficultto determine the declination dependence to a sufficient accuracy. As the array has a near100 % duty cycle, the exposure is highly uniform in right ascension, so at lower energies,the component of the dipole moment perpendicular to Earth’s axis of rotation can be found.Using the denser infill array, it is possible to extend the right ascension anisotropy studydown to 3×1016 eV.Fig. 2.36. Map of cosmic rays above 8×1018 eV, averaged with a 45◦ radius. From [86].Events with energy above 4×1018 eV show a dipole moment with an excess pointingaway from the galactic centre. The dipole moment shows an increasing amplitude withincreasing energy with a significance of 5.1 σ over a constant amplitude. This suggests an98 Pierre Auger Observatoryextra galactic origin for cosmic rays at this energy. The results at low energies, looking atanisotropies in right ascension, smaller dipole moments are observed at low energies. Thephase of the dipole moments at low energies consistently indicate a small excess in thedirection of the galactic centre, switching to an excess pointing away from the galactic centreat ∼ 1018 eV (see figure 2.37). This suggests a transition from galactic to extra-galacticorigin at ∼ 1018 eV.Fig. 2.37. Amplitude and phase of the equatorial dipole moment measured by Auger, alongwith other EAS experiments. From [86].At higher energies, the magnetic deflections may be small enough that cosmic ray arrivaldirections can be used to determine a source. Additionally, at energies above ∼ 1019.5 eV,any cosmic rays observed must be from a nearby source (within ∼100 Mpc), meaningthat sources may be visible (This has been discussed in section 1.1.4). In the latest studyby the Pierre Auger collaboration [28], both vertical and horizontal SD events are used atenergies above 32 EeV, with a total exposure of 101,400 km2sr yr. 2157 events meet theselection criteria. Since the magnetic deflection of cosmic rays is not known, the angularspread of cosmic ray events due to a source is also not known. Due to this, a blind search isperformed over the whole sky, searching for an excess in a circular region in 1◦ incrementsfor both position and size of the region, and also for varying minimum energies. The greatestsignificance was found for a cutoff energy of 38 EeV, and a radius of 27◦. A map of thesignificance is shown in figure 2.38. The pre-trial significance is 5.6σ , with a p-value of0.025 accounting for the scan. The excess is within 2◦ of Centaurus A.Considering that Centaurus A is an excellent candidate for an ultra high energy cosmic raysource, the significance for an excess at the position of Centaurus A can be found. Scanningthrough the size of the region and minimum energy gives a maximum pre-trial significanceof 5.1σ , with a post trial significance of 3.9σ , for a minimum energy of 37 EeV and radiusof 28◦ (see figure 2.39).2.7 Current results 99Fig. 2.38. Map of the local significance for an excess with a 27◦ radius, and a minimumenergy of 38 EeV. From [28].Fig. 2.39. (left) significance for Cen A against radius and minimum energy. (right) numberof events vs search radius for Cen A for a minimum energy of 37 EeV with the expectationand 1, 2, and 3 σ significance. From [28].The directions of observed cosmic rays can also be tested against object catalogues forpossible sources of UHECRs. When comparing the observed cosmic ray distribution withthe catalogues, the brightness of each object, and attenuation of cosmic rays due to CMBinteractions are taken into account to give an expected number of cosmic rays observed atEarth. The brightness and distance of catalogue objects are used to make a probability mapfor cosmic ray arrival directions, that also has angular distribution width and an isotropicfraction as free parameters. A likelihood fit is performed and compared to an isotropicdistribution. The 4 catalogues used for the likelihood analysis are:• The 2MRS infra-red catalogue, tracing the nearby matter, with sources closer than 1Mpc removed. The flux in the K-band is used as a proxy for UHECR flux.100 Pierre Auger Observatory• Swift-BAT AGNs, both radio loud and radio quiet, within 250 Mpc. The X-ray fluxfrom 14 keV to 195 keV is used as a proxy for UHECR flux.• γ-AGNs, measured by Fermi in the 3FHL catalogue. The flux from 50 GeV to 1 TeVis used as a proxy for UHECR flux.• Starburst galaxies, with the continuum emission at 1.4 GHz used as a proxy for UHECRflux.The results are shown in figure 2.40, with the best fit being obtained with the starburstgalaxies for a cutoff energy of 38 EeV, and angular width of 15◦. The 15◦ angular width isequivalent to a ∼24◦ radius using the top-hat function in the blind search. The p-value of thismaximum is 4×10−7, corresponding to a post-trial significance of 4.5 σ . For the other threecatalogues, the maximum post-trial significance is: 3.1 σ for γ-AGNs, 3.7 σ for Swift-BAT,and 3.7 σ for 2MRS. For all four catalogues, the object with the largest contribution to theexpected UHECR flux is very near the largest excess from the blind search. The largestpredicted contribution for the starburst galaxies is from NGC 4945, 6◦ away from the centreof the maximum excess. For the other three catalogues, the largest contribution is predictedto be Cen A, 2◦ from the centre of the maximum. The second largest contribution in thestarburst model is NGC 253, near the southern galactic pole, where there is also a smallerexcess in UHECR flux (see figure 2.38).Fig. 2.40. Maximum likelihood vs energy threshold for the four catalogues. The dashedlines show the maximum likelihood if attenuation is not considered. From [28].2.7.3 CompositionDepth of shower maxumum, XmaxThe ⟨Xmax⟩ result (see figure 2.41) suggests a composition initially becoming lighter withenergy, but becoming heavier with increasing energy for higher energies [117]. The data is2.7 Current results 101well described by a broken linear fit, having an elongation rate of 80±1 g cm−2 per decadefor E < E0, where E0 = 1018.32±0.02 eV, and 26±2 g cm−2 per decade for higher energies,with the higher elongation rate indicating composition becoming lighter with increasingenergy, and the lower elongation rate indicating composition becoming increasingly heavywith increasing energy. The measurements of σ(Xmax) made by the Pierre Auger Observatoryshow a large value for energies up to E0, with σ(Xmax) gradually decreasing with increasingenergy above E0 (see figure 2.41).Fig. 2.41. (left) 〈Xmax〉 vs energy measured by the Auger FD. (right) σ(Xmax) vs energymeasured by the Auger FD. The lines shown are the predictions from three interaction modelsfor proton and iron primaries. From [117].The Pierre Auger Observatory results for 〈Xmax〉 and σ(Xmax) appear highly inconsistentwith the results from Telescope Array, with Telescope Array showing results consistent witha lighter composition at the highest energies. A possible explanation for the inconsistencylies in the differing analysis methods used. The Pierre Auger data has fiducial field ofview cuts applied to produce unbiased Xmax distributions, which can then be compared tosimulations. Telescope Array does not apply fiducial cuts, therefore producing biased Xmaxdistributions. These biased Xmax distributions are then compared to detector simulations ofdiffering composition and interaction models, that simulate the detector bias. A comparisonhas been made by comparing Telescope array data to a composition mix compatible with thePierre Auger Xmax distributions. This comparison shows Telescope array data is consistentwith the Pierre Auger data [48].Using the Xmax distributions from simulated showers, it is possible to determine 〈lnA〉 andσ2(lnA), from 〈Xmax〉 and σ(Xmax). This has been done for three different interaction models.102 Pierre Auger ObservatoryThe results are shown in figure 2.42. As expected, the results for 〈lnA〉 show the lightestcomposition at E0, with heavier composition at lower and higher energies for all models,though the three models are offset with respect to each other. All of the models also indicatea σ2(lnA) decreasing with energy, indicating a transition from a mixed composition to amore pure composition. The QGSJETII-04 model has a σ2(lnA) below zero at the highestenergies, an unphysical result. The correlation between S38 and Xmax has been examinedFig. 2.42. lnA moments determined from the Auger Xmax distributions, using three differentinteraction models. From [117].in the energy range from 1018.5 to 1019 eV, to determine the purity of the composition mix.with results shown in figure 2.43.[VEM]38*S0 20 40 60 80 100]-2[g cmmax*X6008001000Auger19.0−18.5=/eV)Elg(0.017±0.069 −= Gr2652 eventspreliminary[VEM]38*S0 20 40 60 80 100]-2[g cmmax*X600800100019.0−18.5=/eV)Elg( LHC-EPOS0.04=Grproton, 0.12=Griron, Fig. 2.43. (left) Xmax vs S38 for events with energy between 1018.5 and 1019 eV. (right)predictions for iron and proton primaries. From [117].2.7 Current results 103The correlation found using a ranking correlation coefficient is rG = −0.069± 0.017,suggesting a mixed mass composition. Since any mixture of p and He is also predicted toshow a positive correlation, this means that the composition mix must also include nucleiheavier than helium. This result has been compared against simulations spanning all possiblefractions of p, He, O, and Fe, in steps of 0.1 (see figure 2.44), suggesting a value for σ(lnA)between 0.85 and 1.6.)A(ln σ0.0 0.5 1.0 1.5 2.0)38* S, max*X(Gr0.5−0.4−0.3−0.2−0.1−0.00.10.20.3QGSJET II.04Data01234〉Aln 〈LHC-EPOSAugerpreliminary19.0−18.5=/eV)Elg()A(ln σ0.0 0.5 1.0 1.5 2.0)38* S, max*X(Gr0.5−0.4−0.3−0.2−0.1−0.00.10.20.3QGSJET II.04Data01234〉Aln 〈2.3cSibyllAugerpreliminary19.0−18.5=/eV)Elg(Fig. 2.44. Correlation coefficient rG vs σ(lnA) for different mixes of p, He, O and Fe. From[117].The correlation between S38 and Xmax has also been examined in different energy bins.Results of this are shown in figure 2.45, compared to predictions for pure and mixed com-positions. The results suggest a more mixed composition at energies up to 1018.7 eV, withcomposition becoming more pure at higher energies. This is consistent with the resultsobtained from the Xmax distribution measured by the FD.Fig. 2.45. Correlation coefficient rG vs energy, with model predictions for protons, iron, anda 50/50 mix of proton and iron. From [117].104 Pierre Auger ObservatoryRisetimeRisetime data is available for 125,005 events, with 237 events above 50 EeV. This provides thelargest data set available for studying composition at the highest energies. When comparingthe data obtained with detector simulations, the observed ⟨∆s⟩ indicates a compositionbecoming heavier with increasing energy (see figure 2.46). The ⟨∆s⟩ data can also beconverted to ⟨lnA⟩ for a given interaction model for comparison with other compositionmeasurements (see figure 2.47). When comparing the data with the Xmax data from the FD,there is a disagreement between the two. This indicates that the interaction models used insimulations do not adequately reproduce the physics of extensive air showers.Fig. 2.46. Results for the ∆ method for the 1500m array, with lines indicating the interactionmodel predictions. The inset plot is a zoom in of the results near the benchmark energy.From [105].Since ∆s is dependent on the depth of maximum in shower development, it is expectedthat ∆s will be correlated to Xmax observed by the fluorescence detector. This is shown for thehybrid data in figure 2.48. It is therefore possible to use ∆s to get an estimate of Xmax. Dueto the larger exposure of the SD, this can allow Xmax data to be extended to higher energies.The smaller hybrid dataset allows for a calibration using ⟨XFDmax⟩, to calculate ⟨XDeltamax ⟩. Thefollowing relation is used to calculate ⟨XDeltamax ⟩:⟨XDeltamax ⟩= a+b · ⟨∆s⟩+ c · log10(ESD/eV) (2.41)A plot of ⟨XDeltamax ⟩ against energy, along with the values of ⟨XFDmax⟩, are shown in figure2.49. As expected, the values obtained by the SD show a good agreement with the FD data,and show the trend of increasing mass with energy continues up to 1020 eV.2.7 Current results 105Fig. 2.47. ⟨lnA⟩ deduced from the delta method, compared with ⟨lnA⟩ deduced from the FDXmax. From [105]Fig. 2.48. Plot showing correlation of ∆s to XFDmax. From [105]Muon production depthThe muon production depth has been determined for over 2000 events, with energies above1.5×1019 eV, and zenith angles between 45◦ and 65◦ [70]. Some bias is introduced in theX µmax reconstruction, which is dependent on the energy, as well as composition and interactionmodel. This bias could be corrected for but it would introduce a large systematic uncertaintydue to the mass and model dependence. Due to this, results are given as X µmax folded with thereconstruction bias, X∗µmax. The results are given in figure 2.50.106 Pierre Auger ObservatoryFig. 2.49. 〈XDeltamax 〉, with model predictions for proton and iron. 〈XFDmax〉 is also shown forcomparison. From [105]Fig. 2.50. (left) 〈X∗µmax〉 as a function of energy. (right) σ(X∗µmax). The error bars show thestatistical uncertainties, while the shaded area shows the systematic uncertainties. From [70]The results for 〈X∗µmax〉 suggest increasing mass as energy increases, in line with othercomposition measurements. When comparing 〈X∗µmax〉 with the model expectations, resultsare highly inconsistent with all reasonable mass values for EPOS-LHC, and somewhatinconsistent at higher energies for QGSJetII-04. Better compatibility is observed withσ(X∗µmax), with a smaller dependence on hadronic interaction models. These results arecompared with Xmax results from the FD in figure 2.51, for 〈lnA〉 derived from 〈X∗µmax〉. TheXmax and X∗µmax results are highly incompatible, highlighting the inadequacies in currenthadronic interaction models.2.7 Current results 107Fig. 2.51. 〈lnA〉 derived from 〈X∗µmax〉, with the 〈XFDmax〉 result shown for comparison. Theerror bars show statistical errors, while the square brackets show systematic uncertainties.From [70]Muon densityThe engineering array of AMIGA has gathered data on the muon density of showers withenergy above 1017.5 eV. Muon densities, measured in conjunction with energy, can be anindicator of composition, and also a test for hadronic interaction models. With the 750 mspacing of detectors, the muon density at 450 m, ρ450 is used for a muon density measure[87]. The analysis is done for zenith angles below 45◦. Similar to the SD array, the CICmethod is used, giving a muon density corrected for zenith angle effects, normalised to 35◦,ρ35. The results for ρ35 vs energy are shown in figure 2.52. The measured muon densitiessuggest muon densities slightly above the prediction for iron primaries. This result is highlyincompatible with Xmax results, which suggest a lighter composition.Fig. 2.52. (left) Muon density ρ35 measured by AMIGA vs energy, with the insert showingthe normalised residuals. (right) comparison of the muon densities to the interaction modelpredictions, with error bars showing the statistical uncertainty, and square brackets showingsystematic uncertainty. From [87].108 Pierre Auger ObservatoryThe muon density can also be measured using the 1500 m SD array with HAS (horizontalair showers). This gives a measurement at higher energies with a much larger exposure.To compare the AMIGA and HAS density results with FD Xmax observations, a z-factor isintroduced:z =〈lnρ35〉−〈lnρ35〉p〈lnρ35〉Fe −〈lnρ35〉p(2.42)This can also be similarly applied to N19, measured with HAS, as well as for Xmax, measuredwith the FD. A plot of z vs energy, for AMIGA, HAS, and Xmax is shown in figure 2.53.Fig. 2.53. z-factor vs energy, for AMIGA, HAS, and Xmax measurements, using EPOS-LHC(left), and QGSJetII-04 (right). From [87].Muon density measurements at ultra high energies indicate a much higher muon densitywhen compared to the hadronic interaction models. The z-factor obtained by both HAS andAMIGA show a good agreement with this discrepancy, though the energy ranges do notoverlap. The trends in z-factor with energy obtained by HAS and AMIGA show a reasonableagreement with the Xmax data. The excess in muon densities is also shown in figure 2.54,showing expected muon densities vs 〈Xmax〉 for energies of 1017.5 eV and 1018 eV. To matchobservations, the simulated muon density would have to be increased by 38% at both energiesfor EPOS-LHC, while it would have to be increased by 50% at 1017.5 eV and 53% at 1018 eVfor QGSJetII-04 [87]. This indicates that current hadronic interaction models show a largedeficit in muon density, consistent with numerous other experiments at the highest energies[29].2.7 Current results 109Fig. 2.54. ρ35 vs ⟨Xmax⟩, with model predictions for EPOS-LHC and QGSJetII-04, atenergies of 1017.5 eV (left) and 1018 eV (right). From [87].Chapter 3FD telescope alignment studyTelescope alignment: the methodsTo accurately determine the arrival direction of cosmic rays, the pointing directions of thefluorescence telescopes must be known. Uncertainties in the elevation angles of the telescopealso affect the energy and Xmax reconstruction. It is desirable to determine the telescopepointing directions to within 0.1◦. The first section of this chapter describes the methodsused by others to determine the pointing directions of the fluorescence telescopes. The nexttwo sections describe my method for finding the pointing directions.3.1 Previous methods3.1.1 Using star trailsThe directions of stars in the sky are known very accurately. The ultraviolet light from starsis bright enough to be detected by the Auger telescope PMTs. To detect only cosmic rayshowers, the signal is high-pass filtered to subtract any background signal, making it difficultto detect stars. However, because the PMTs are counting photons, there is a variance in thesignal proportional to n, where n is the number of photoelectrons in a given time bin. Theamount of light entering a pixel can then be estimated as being proportional to variance. As astar passes across a pixel (see figure 3.1(a)), the variance rises as the light spot is entering thepixel, then remains constant while the spot is completely inside the pixel, then lowers backto the background level (see figure 3.1(b)). A star will cause a variance increase across a trailof PMTs (see figure 3.2). The timing of the increases in variance from stars can be used todetermine the pointing directions of the telescopes. Studies have been done by the Augercollaborators from the Milan group [38] and the Czech group [85]. The Milan method found112 FD telescope alignment studythe pixel offsets individually, comparing the time at the centre of the signal plateau with theexpected time. The Czech method performed a fit over the whole telescope to minimise thedifference between the observed and expected signal. There is some disagreement of up to0.3 degrees in derived telescope pointing directions between the two studies.Ð:Ë ÖTÙ ïPÙá`ËÌÏÀyÄÌ¾ÂÃ|Ó,ÂÁ¥¾¿ªÚ,½ÍkðgÃ¤Öw¾<¿Áª½wÇ¥ÇªËÌÆTÖÂªÈTÃ1ÂÃ|Ä Ã|Çª¿½×oÃ1ìR½ÍîO½wÇnîOÃ¤½wÆyÃ|Ç¤ÙO¼ÈTÃ¯×YË¨Ê`Ã|ÄÌÇÆPÀyÏ¯ÕÃ¤ÁÇ¾Á¥Ã1¾ÄÔÇ½<ÇªÈT½±ÉÑÆiÙâVÃ|Ó&×YË¨Ê`Ã|ÄÌÇÏÃ|¾Æ&ÂªÈy¾Â.¾ÇªË ÖwÆy¾Ä?É¾ÇnÍ½wÀYÆyÓ&ËÌÆkÂªÈTÃ1á(¾Á¥ËÔ¾Æy¿ÃÓy¾Âª¾`ÙOñnÁªÃ¤Ã|Æ×YË¨Ê`Ã|ÄÌÇ1ÏÃ|¾ÆÂªÈy¾ÂeÂ¾Ä ÂªÈT½wÀTÖwÈÂªÈTÃ<ÇÂª¾ÁR¿Áª½wÇªÇÃ|ÇÂªÈyÃ<×YË¨Ê`Ã|Ä}Â?Æy½&Ç¥Ë ÖwÆy¾ÄÉ¾ÇÍ½wÀyÆyÓiÙTsimSpotStar trackTimeR = 0.75 degr = 0.25 degÐ:Ë ÖTÙyìTÙá`¿¥ÈTÃ|Ï¾ÂªËÌ¿ØÂÁ¥¾¿¥Ú<½Í?¾«ÇªÂª¾ÁV¿Áª½wÇªÇªËÔÆTÖ«¾«¿¤Ë Á¥¿¤ÀYÄÌ¾Á'×YË¨Ê`Ã|ÄÙ/®Kò ó {9zô±uv|´c{9xR{¥³gy}²+w5Ú ÄHÅÇÆ¥ÈSÉfÊ ±xºÚ ÄHÅÇë ®dõ wty,wtutzZ|}x±y|o{9xQ{¥³ô|¬öTwt¶·´ô+{9|}x+y|}x/µ9´ýnþnþÿ$$¥þe#! gS#%IP-+#þÿ¯þ'IP$#1!*. þ'#&(HI÷%#K#6&+#dþe##$ZtnT-1#&©$±þ'DT7]OÚ ÄÅÇÆ¥ÈSÉfÊ O!5Ôþÿ þÿ&þ'IPa&M&§HIø#[$¥þe#Rþ''#%eN$I:1K#dþ'¹Ú ÄÅÇë 7 À Zù Ú ÄHÅÇÆ¥ÈSÉfÊ íúÚ ÄÅÇë ù2û¿@LÐÑoÒ2ÏMO!*#$$IPkþÿ+#Zþþÿa$#1"!*(a) Spot from a star crossing a PMT.VarianceTendTsignalTstartSignal plateauTimeBaselineÐ:Ë ÖTÙÎíÙtâDÓTÃ|¾ÄÎÇªË ÖwÆy¾ÄgÇªÈY¾×ÃÙ0ÁVÍÂÃ¤Á-ËÌÓTÃ|ÆZÂªË ÍÅPËÔÆTÖ<¾=ÇÂª¾Á|ßàÇ¯ÇªË ÖwÆy¾ÄgËÌÆ&ÂªÈyÃ«á(¾ÁËÌ¾Æy¿ÃÓy¾Âª¾Âc¾ÂªËÌÏ«ÃÃ+ÄÅÇÆ¥ÈÉÊOËÌÇ¾Ç¥Ç½P¿¤ËÔ¾ÂÃ|Ó=Â½Ë Â¤ÙþÿP`ÿTþþÿhI#[#%1hÑþÿP +#$1h$g&±þ'I&RR#þ'I:!5&c!ËDÌÍÎÏYý$Þþ%¹%##W&#Zþe#!5Ôþÿh`ÐÑoÒOÏTþ'P]gÐÑÒ2ÏMYþÿ+#Zþ"tnT&$G@Ó?HIþÿ`I#b +#%$1$W$&&[#a$þ'#$#1o7 À #þÿ$W!#Stþÿ` 0a#&þÿa&©þÿ($#1 #%a#$1RH&#$Rþÿa)+$þh#&%þÿa1K#$¥þ`-Tþ`rþÿ$+#1&#Zþe#Z$±þS78Øÿª$&ÔIPÞþÿ/&Õo¦¾P$#$þ%g&%#Zþ'MLI±þÿ/&7 ½ =#11K#ZþfþÿIyþg%ÑþÿZ#%K#%:&#Zþe# Þþ,!*rþ,! i-Tþ'$.$-+#%'#Zþ&R <QD(ÐÑÒ2Ï"H!*&©Pþi$&($þ'MR-0Tþ'$kþ§#S&Ö$b×+±þ'+#dþ'$ev7nýØ-$Ôþ'#&Y#=E#Zþ'M$þ?%#1(-þÿ$:ITþS2!5Ôþÿ[ÐÑoÒ2Ïe]cªÐÑÒ2ÏM$-&§þ'=þÿg1#&h#&þ'#1`&$5#Z$+#178Øÿ$fþ,!*RI±þÿ/&$P#Pþh;<M#%1TþhJþÿi$$§þÿ+#dþSU&-0&QJþÿ$+#1k$ÿ+#%-0h#&þÿZ +#%eN/&R #$1Oþÿt=&Zþg#1! #S/$6)+&þÿZ$'#IP$+#1$78Øÿ)+$¥þ*IPÞþÿ/&:! N<$" Þþþ'þÿ+#<þÿG$&PiþÿG +#$1G$ÿPÿ1%## 1 þW$!*$.HG$I#110$+#%1$7+j. E/$1M( Pþÿ!*Na).Hnÿ`ÿ?#&1S#A$+#1$7ZýÙ$¤ÿI#Zþ'5-1PþA%cþÿ-þ,!*I±þÿ/&$A$A$ÿ!5P327ÇD/7 À þ"$ÿ1& 0`Pþ'&þÿ+#Zþ! h#%:Zþ.Tþ'$þ'&[þÿZ$#1ÿPÿyþ6 þ.#þÿþIPþÿ$+#1k#¤ÿ$þÿ:IP&&1%'þÿZ$+#%1O-1#Zþ'# Ú ÄHÅÇÆ¥ÈSÉfÊ #$6&S#Zþ&=ª327]78Øÿ$Ñþ'IPW$+#1 $VþÿþIPW!.ÿ&þÿ6$þ'#$"1$$þnþþÿ6yþGgþÿ-MnT17/® ° |}zZs/¶M±y|o{9xQ{¥³6´tyo±ugyu±/reÛc´XUf`ÿTþIa#E[$¥þe#$`$$þÿP)+1& T!J3kj5lggþÿ(324 þ'1$-$7Ü c!5&þÿZ;<+#Zþ#1k/&+#Zþ$6*#($¥þe#P¾W`ÿTþ1ý$$q461+#Zþ $PþG&M¼1þþ'Z$I:1#Zþ'þ'$Vþ'#eN(,þÿ.1T#10T&+#Zþ'$/$þIÝXA1%#Zþý£I:þÿ dHIbþÿ 0-þ'1þÿ&Z%iþÿ`ÿTþ5\p%_7 ½ &þÿ$O$I:1K#dþ'!5þÿþ$&=þÿÞ0±þ6þÿ#Zþ'IP$-ÿ7«%7!*`&aPþ-þe#%N`#@(b) Variance signal increase from a star crossing aPMT.Fig. 3.1. Detection of a star crossing a PMT. From [38].Ð:Ë ÖTÙ ïPÙá`ËÌÏÀyÄÌ¾ÂÃ|Ó,ÂÁ¥¾¿ªÚ,½ÍkðgÃ¤Öw¾<¿Áª½wÇ¥ÇªËÌÆTÖÂªÈTÃ1ÂÃ|Ä Ã|Çª¿½×oÃ1ìR½ÍîO½wÇnîOÃ¤½wÆyÃ|Ç¤ÙO¼ÈTÃ¯×YË¨Ê`Ã|ÄÌÇÆPÀyÏ¯ÕÃ¤ÁÇ¾Á¥Ã1¾ÄÔÇ½<ÇªÈT½±ÉÑÆiÙâVÃ|Ó&×YË¨Ê`Ã|ÄÌÇÏÃ|¾Æ&ÂªÈy¾Â.¾ÇªË ÖwÆy¾Ä?É¾ÇnÍ½wÀYÆyÓ&ËÌÆkÂªÈTÃ1á(¾Á¥ËÔ¾Æy¿ÃÓy¾Âª¾`ÙOñnÁªÃ¤Ã|Æ×YË¨Ê`Ã|ÄÌÇ1ÏÃ|¾ÆÂªÈy¾ÂeÂ¾Ä ÂªÈT½wÀTÖwÈ ÂªÈTÃ<ÇÂª¾ÁR¿Áª½wÇªÇÃ|ÇÂªÈyÃ<×YË¨Ê`Ã|Ä}Â?Æy½&Ç¥Ë ÖwÆy¾ÄÉ¾ÇÍ½wÀyÆyÓiÙTsimSpotStar trackTimeR = 0.75 degr = 0.25 degÐ:Ë ÖTÙyìTÙá`¿¥ÈTÃ|Ï¾ÂªËÌ¿ØÂÁ¥¾¿¥Ú<½Í?¾«ÇªÂª¾ÁV¿Áª½wÇªÇªËÔÆTÖ«¾«¿¤Ë Á¥¿¤ÀYÄÌ¾Á'×YË¨Ê`Ã|ÄÙ/®Kò ó {9zô±uv|´c{9xR{¥³gy}²+w5Ú ÄHÅÇÆ¥ÈSÉfÊ ±xºÚ ÄHÅÇë ®dõ wty,wtutzZ|}x±y|o{9xQ{¥³ô|¬öTwt¶·´ô+{9|}x+y|}x/µ9´ýnþnþÿ$$¥þe#! gS#%IP-+#þÿ¯þ'IP$#1!*. þ'#&(HI÷%#K#6&+#dþe##$ZtnT-1#&©$±þ'DT7]OÚ ÄÅÇÆ¥ÈSÉfÊ O!5Ôþÿ þÿ&þ'IPa&M&§HIø#[$¥þe#Rþ''#%eN$I:1K#dþ'¹Ú ÄÅÇë 7 À Zù Ú ÄHÅÇÆ¥ÈSÉfÊ íúÚ ÄÅÇë ù2û¿@LÐÑoÒ2ÏMO!*#$$IPkþÿ+#Zþþÿa$#1"!*Fig. 3.2. Simulated track from Vega crossing telescope 4 of Los Leones. A signal was foundin the red pixels. No signal was found for green pixels but the star did cross the pixels [38].3.1 Previous methods 1133.1.2 Using CLF laser shotsNumerous studies have been done to test the alignment of the telescopes using the CentralLaser Facility (CLF). Most studies involve comparing the reconstructed SDP (defined in 2.4.2)to the direction of the CLF. Using the SDP allows for detection of telescope misalignmentperpendicular to the direction of the laser track. By using laser shots at different inclinationsit is possible to determine the pointing direction of the fluorescence telescopes.Patrick Younk and Brian Fick [115] used inclined CLF shots to find the pointing directionsof the telescopes for Los Leones, Los Morados and Coihueco. To do this accurately, tracksare required over a broad range of inclinations. This is only possible for the 1 or 2 telescopespointing towards the CLF. Figure 3.3 shows the tracks from the CLF shots used in one of theeyes.GAP-2006-087 5 Figure 3: The laser shot tracks in the three operational telescopes. Absolute telescope identification numbers are shown. The criteria for a shot to be included in the analysis of a telescope were the following: • Minimum of 20° degrees track length in the telescope. • The track is not within ~3° of a primary axis of symmetry (SDP vector zenith angle of 0°, 60°, or 120°). • The track does not meet the edge of the camera at an angle of less than 20°. Cuts were made to eliminate laser shots fired during poor atmospheric conditions such as clouds and haze. These cuts where: • Minimum of 10 pixels with pulses in the telescope. • Maximum of 100 pixels with pulses in the telescope. • The α parameter must be within +/- 2°. • The SDP vector zenith angle must be within 3° of the nominal value. • The SDP vector azimuth angle must be within 5° of the nominal value. 1 2 3 4 5 6 7 8 9 10 11 12 19 20 21 22 23 24 Fig. 3.3. CLF Tracks used to determine pointing directions of telescopes for Los Leones[115].The estimated precision of the technique was about 0.1 degrees for telescopes pointingtowards the CLF. The pointing directions agreed with the star trail methods to within error.For telescopes not pointing towards the CLF, the uncertainty of this method was too large.Misalignment of the telescopes can also be detected using the timing of the CLF lightarriving at the pixels. Timing is sensitive to misalignment in the direction of the laser track.This is most noticeable when the track crosses two telescopes and they have a differentalignment relative to each other. This will cause a discontinuity in the timing fit at thetransition between the two telescopes. Thomas Harrison, Jose Bellido and Bruce Dawson[49] used the timing to test the alignment of telescope 6 of Coihueco. Figure 3.4(b) showsthe time fit residuals for a CLF event which crosses into telescope 6. If the mirrors areproperly aligned, the residual would be flat. There is a clear discontinuity between telescope5 and 6 (the two leftmost telescopes) with the light arriving at the pixels in telescope 6 laterthan expected when compared to telescope 5. The timing in telescope 5 is consistent withtel s op 4. The pixels from Telescope 5 ave mostly been reje ted as outliers because of the114 FD telescope alignment studydiscontinuity. It appears that telescope 6 is misaligned. The timing discontinuity observedcorresponds to a misalignment of 0.7±0.2 degrees in azimuth and −0.2±0.3 degrees inelevation for telescope 6 [49].(a) Shower track for CLF event 4404871 (b) Time fit residuals for the event.Fig. 3.4. CLF event 4404871. The black pixels in (a) were rejected from the time fit. Therejected pixels are the uncoloured points in (b). From [49].3.2 Determining relative offsets with cosmic ray showersTo test if any of the telescopes are misaligned, the telescopes alignment can be checkedagainst adjacent telescopes (unless they are all misaligned by the same amount). This ispossible using CLF laser shots, however the CLF is in a fixed location and can only offer alimited range of track directions. Cosmic ray showers have random arrival directions & corelocations giving a wide range of track directions in all telescopes, so showers can be used todetermine relative offsets between any adjacent telescopes.3.2.1 TimingIf there is a relative pointing error between two telescopes, there will be a discontinuity inthe timing of the pixel pulses between the two telescopes. This will cause a discontinuity inthe timing fit, which is a plot of pixel time versus the χ angle defined in 2.4.2. A plot of thetime residuals (see figure 3.5(b)) can clearly show the discontinuity in the pixel timing.Since the timing of the pixel pulse is a function of the χ angle, only offsets which shiftthe χ angle of pixels will have an effect on the timing. Hence, this timing method will detectthe component of the offset in the direction along the shower track.3.2 Determining relative offsets with cosmic ray showers 115Eye 4 Run 2881 Event 4444azimuth [deg]elevation [deg]051015202530120130140150160170180 angle [deg]χ90 100 110 120 130time residuals to fit [100 ns]-8-6-4-202468]2slant depth [g/cm400 600 800 1000 1200 1400 1600 1800 2000 2200 2400)]2dE/dX [PeV/(g/cm-30-20-10010203040/Ndf= 413.6/5332χRun 2881 Event 4444time stamp: 912057069 s 906244949 nsTrigger: ’Physics - Int or L/R trigger’, ’Shower Candidate’hottest hybrid station: --- Mie attenuation: measured (h<4.3 km, VAOD at 3km: 0.0071)LIDAR: h(cloud)=4.0 km, 39%; CloudCam: no datain Coihueco mirror 5 6 ( in DAQ: 1 2 3 4 5 6 ) eV19 10× 0.06) ± 0.09 ±E = (2.47 2 254 g/cm±Xmax = 1883 )2 1.12 PeV/(g/cm±dEdXmax = 35.48 2157) g/cm±5,-6±,X0) = (34λ(Cherenkov-fraction = 3%1.2) deg±0.3, 94.0±)=(87.9φ,θ(12.14) km±3.93, -154.68±(x,y)=(10.42 0.28 km±dca to Eye=32.47 (a) Shower track for event 6872384Eye 4 Run 2881 Event 4444azimuth [deg]elevation [deg]051015202530120130140150160170180 angle [deg]χ90 100 110 120 130time residuals to fit [100 ns]-8-6-4-202468]2slant depth [g/cm400 600 800 1000 1200 1400 1600 1800 2000 2200 2400)]2dE/dX [PeV/(g/cm-30-20-10010203040/Ndf= 413.6/5332χRun 2881 Event 4444time sta p: 912057069 s 906244949 nsTrigger: ’Physics - Int or L/R trigger’, ’Shower Candidate’hottest hybrid station: --- Mie attenuation: measured (h<4.3 km, VAOD at 3km: 0.0071)LIDAR: h(cloud)=4.0 km, 39%; CloudCam: no datain Coihueco mirror 5 6 ( in DAQ: 1 2 3 4 5 6 ) eV19 10× 0.06) ± 0.09 ±E = (2.47 2 254 g/cm±Xmax = 1883 )2 1.12 PeV/(g/cm±dEdXmax = 35.48 2157) g/cm±5,-6±,X0) = (34λ(Cherenkov-fraction = 3%1.2) deg±0.3, 94.0±)=(87.9φ,θ(12.14) km±3.93, -154.68±(x,y)=(10.42 0.28 km±dca to Eye=32.47 (b) Residuals to the timing fit. uncoloured pixels wererejected from the time fit.Fig. 3.5. Event 68723824. The shower track crosses Co hueco telescopes 5 and 6.To find the the χ offset, ∆χ , from the timing offset, ∆t, we use the equation first describedin 2.4.2t(χ) = t0 +Rpctan(χ0 −χi2)(3.1)inverting this equation givesχ(t) = χ0 +2arctan((t0 − t)cRp)(3.2)Taking the derivative we getdχdt=2cRp11+((t0−t)cRp)2 (3.3)The χ offset can be approximated as∆χ =2cRp11+((t0−t)cRp)2 ∆t (3.4)Using multiple cosmic ray showers with tracks appearing in different orientations on thecamera, a fit can be applied to determine the relative pointing offset between two telescopes.If we define the relative pointing offset between the telescopes as−→S with magnitude |S|,direction β , and we define the direction of the cosmic ray shower track to be α , we get∆χ = |S|cos(α −β ) (3.5)These angles are defined in figure 3.6.116 FD telescope alignment studyazimuth [deg]elevation [deg]05101520253030405060708090 SFig. 3.6. Definition of shower and telescope offset directions. The black vector−→S (exagger-ated in magnitude) is the pointing offset of a telescope from its nominal direction, relative tothe offset of the adjacent telescope. Angles α and β are measured from the horizontal. Thehorizontal is the vector defined as the cross product of the direction of the shower spot at theboundary between the telescopes, and the zenith.Event SelectionThe method will be discussed below, but first a comment on selecting the shower events tobe used. An accurate time fit is needed in both telescopes. Events with a track length greaterthan 15 degrees in each of two telescopes were selected. Since the main source of error isthe error in the timing fit, we want events with a large value of ∆t∆χto maximize the timingsensitivity. ∆t∆χis large for large values of the shower impact parameter, Rp. Events with Rpgreater then 15000 m were selected. There was a total of 164 of these events. Consideringthat these events are divided across 20 telescope boundaries, this is not a very large numberof events. To get more events, an additional set was selected that had a minimum tracklength of 20 degrees in one telescope and 5 degrees in the other. For these events, instead ofcomparing the two time fits, the pixel times from the telescope with the short track would becompared to the time fit expectations from the telescope with the long track. 441 events metthese criteria (without meeting the first criterion). The first set of events will be referred to astype I events and the second set as type II events.3.2 Determining relative offsets with cosmic ray showers 117Determining OffsetsTiming fits had to be done individually in each telescope to find the relative pointing offsetbetween the two telescopes. To do this, events were reconstructed twice using Offline, oncewith the pixels in even numbered telescopes removed from the fit and again with the pixelsfrom odd numbered telescopes removed. The events had to be “mono” reconstructed (i.e.without surface detector information) because if the telescopes are misaligned the SD stationmay have the wrong time relative to the telescope pixel times and could affect the timing fit.For type I events, the two fits were compared to obtain an offset ∆χ . Figure 3.7 illustratestwo telescope time fits overlaid onto the residuals in the timing of each pixel’s pulse. Thetimes are displayed as residuals to the global fit (the fit using all of the pixels in the event).The most accurate way to determine the offset was to compare the two fits at the boundarybetween the two telescopes. The time when the shower was at the boundary was definedas the average of the arrival time of the latest pixel time from the first telescope and theearliest pixel time from the second telescope. Using the equation for χ(t) (equation 3.2), theχ angles for the two were found and the difference between the two was taken to get ∆χ .The two fits had slightly different SDPs which means the χ angle is defined differently foreach of the fits so the χ angles had to be translated to a common shower-detector plane.Fig. 3.7. Time fit for a type I event crossing Coihueco telescopes 5 and 6. The pixel timesare plotted as a residual to the Global time fit. The green points are pixels from telescope 6and the red points are from telescope 5. The two lines are the time fits for each individualtelescope relative to the global time fit.118 FD telescope alignment studyBecause the fits are mono, there is a strong correlation between errors in χ0 and Rp.This correlation must be taken into account when calculating an error in ∆χ to avoid a hugeoverestimation of the error. Offline calculates correlation coefficients between the errorsin Rp and χ0. For a function f (Xi), dependent on variables Xi, the error in f for correlatederrors is:σ( f ) = ∑i, jσiσ jd fdXid fdX jCorri j (3.6)where σi is the error in Xi and Corri j are the correlation coefficients between Xi and X j. Thecorrelation coefficients are between -1 and 1 with Corrii = 1.For type II events, only one of the fits is used. The fit from the telescope with the longtrack is compared to the pixel times in the telescope with the short track (see figure 3.8).The residual of each pixel’s χ angle is found to the χ(t) fit. The pixel’s error is a timinguncertainty so the χ uncertainty is approximated as:δ χi = δ tdχdt(3.7)= δ t2cRp11+((t−t0)cRp)2 (3.8)A chi-squared fit is then done on the pixel χ residuals in the telescope with the short trackto get the weighted mean of the residual to the χ(t) fit, ∆χ .3.2.2 SDPUsing cosmic ray shower SDP fits is another method capable of finding the relative mis-alignment between two telescopes. The shower-detector plane is sensitive to misalignmentsperpendicular to the track direction. With sensitivity in a different direction to the timingmethod above, the SDP method can improve the accuracy of the pointing offset determinationand can also be cross-checked against timing as this method is independent of timing.Event selectionThe SDP accuracy is typically not as good as the geometrical accuracy from timing fits with alarge Rp. Events had to be selected to minimize error in the SDP normal vector. The error inthe SDP normal is very different in different directions. The error in the SDP is best definedin the shower track coordinate system (see figure 3.9). The centroidal error is the error of theposition of the centroid perpendicular to the shower track. The angular error is the error in arotation of the track about the centroid. The error in the centroid of the track is less than the3.2 Determining relative offsets with cosmic ray showers 119Fig. 3.8. Time fit to a type II event. Instead of displaying residuals to t(χ), the plot isinverted to show the residuals to χ(t). The residuals are with respect to the fit to the pixelsin the telescope with the long track. The red points are the χ residuals from the PMTs intelescope 5 and the green pixels are from telescope 6.angular error in the SDP. The angular error is also a lot more dependent on the angular tracklength.The direction in which the telescope pointing offset is being measured is perpendicular tothe shower track, so the error in that measurement is roughly equal to the centroidal error. Aminimum track length of 20 degrees was chosen. At this track length, the typical centroidalerror is 0.07 degrees and the angular error is 0.44 degrees [116]. Since no timing is involved,there were no cuts on Rp.Determining offsetsThe relative offsets were found using the SDP normal vectors for each individual telescope.The events were reconstructed separately for each telescope using the same procedure as thetiming method. The relative offset between the telescopes in the direction perpendicular tothe shower track is the angular offset between the telescopes SDPs near the centroids. For acentroid in the x̂ direction, the offset will be:∆φ = (−→n2 −−→n1) · x̂ (3.9)120 FD telescope alignment studyGAP-2006-086 2 This convention should be applied immediately after the core location is estimated. The core is defined (for this convention only) as the intersection of the shower axis and the detector’s local horizontal plane. For a perfectly horizontal shower (no core defined), the convention is implemented as if the core is in front of the telescope building. For a perfectly horizontal shower which is also perfectly parallel to the back wall of the telescope building, the convention is implemented as if the core is to the right of the telescope building. The direction of the shower is not important in this convention. In other words, a vertical laser shot and a vertical shower at the same core location will have the same SDP vectors. 3. Natural Coordinates for Measuring SDP Vector Error Generally, an SDP vector is specified using the standard site coordinate system or an FD building coordinate system. However, neither of these coordinate systems is natural for specifying the errors on an SDP vector. For this purpose, we use the following coordinate system: Shower Track Coordinate System: The origin is the telescope building. The x-axis points toward the centroid of the shower track (CG of the illuminated pixels). The y-axis points in the direction of the actual SDP vector. The z-axis is determined by zyx ˆˆˆ =× . The azimuth, zenith, and elevation coordinates (φ, θ, Ω1, respectively) are defined as per the Auger Standard. See Figure 1A. An estimated SDP vector, Sestimate, can be described by an azimuth and zenith angle, Sφ and Sθ, in the shower track coordinate system. A natural way to report the rrors for the estimated SDP vector is shown in Figure 1B. The centroidal error is a translation of the estimated track’s centroid perpendicular to the actual track. The angular error is a rotation of the estimated track about the actual track’s centroid. The track centroidal error and angular error have values of (π/2 - Sφ) and (π/2 – Sθ), respectively. 1 Ω = π/2 - θ. z A B y Shower Track Sestimate Sθ Sactual x Sφ Centroidal Error, π/2 – Sφ Angular Error, π/2 - Sθ Actual Track Estimated Track Eye Track Centroid Figure 1: Diagram A shows the Shower Track coordinate system, a convenient system for reporting SDP vector errors. Diagram B shows the physical meaning of centroidal and angular SDP vector error. The shower track centroid is the CG of the illuminated pixels. Fig. 3.9. Diagram of the shower track coordinate system (left) and the definitions of angularand centroidal error (right). The centroid is the “centre of gravity” of the triggered pixelpositions, weighted by signal. The vector−→S in this diagram is the normal vector to theshower-detector plane, not to be confused with the relative telescope pointing offset inequation 3.10. From [116].where −→n2 and −→n1 are the SDP normal vectors for the fits in the two telescopes (see figure3.10). The vector x̂ can be anywhere on the SDP, however the error is minimized close tothe centroid where sensitivity to angular error is minimized. Since the error from both SDPnormals have to be taken into account, the error is minimized when x̂ is chosen to be betweenthe centroids.The errors in the SDP normal vectors are given in altitude and azimuth directions whichare not the same as the centroidal and angular directions, so the errors in altitude and azimuthwill be highly correlated. This correlation must be taken into account to avoid overestimatingerrors.For compatibility with the timing method above, we want the equation to fit the offsets tohave the same form as the timing method. Since the component of the offset being measuredis perpendicular to the shower track, the angle α is chosen to be an angle between thehorizontal and the direction perpendicular to the shower track (see figure 3.11). This is alsothe direction of the SDP normal vector. The component of the offset will be:∆φ = |S|cos(α −β ) (3.10)3.2 Determining relative offsets with cosmic ray showers 121Shower axisxnn12 Δ 𝜔 𝜙 Ω 𝜇 𝜌 𝜃 𝛿 𝜎 𝜆 𝜒 𝑋 𝑥 𝑇 𝑡 𝜏 Δ 𝜔 𝜙 Ω 𝜇 𝜌 𝜃 𝛿 𝜎 𝜆 𝜒 𝑋 𝑥 𝑇 𝑡 𝜏 EyeFig. 3.10. Diagram of the vectors used to find the pointing offset using SDP normal vectors.where−→S is the relative pointing offset in the direction β , and ∆φ is the relative SDP offset.azimuth [deg]elevation [deg]051015202530120130140150160170180S Fig. 3.11. Definition of the SDP offset geometry.122 FD telescope alignment study3.3 Determining absolute offsets with hybrid showersThe pointing directions of the telescopes can be checked by comparing the shower recon-structed in the telescope to the surface detector reconstruction. The surface detectors candetermine the position and timing of the shower core. Using hybrid cosmic ray showersis similar to using the CLF, except that the core location is not fixed at the centre of thearray and the core location and timing are not known quite as accurately. Because the corelocations are randomly scattered throughout the SD array, there will be a good variety oftrack directions for every telescope, unlike the case for the CLF tracks.Patrick Younk [113] has used golden hybrid showers (golden hybrids trigger 3 tanksso the shower has an SD reconstruction) to determine the azimuth of the telescopes bycomparing the SD core to the FD core (see figure 3.12). An azimuth offset in a telescopewould cause the angular core difference to have a non-zero mean. The longitudinal coredifferences could be used to find the elevation offsets but was not considered in the study.�� (��C�� >�#�� ?�� ?�� (�� ?�� ?�� ?�� ?�� !�� ?�� +� �� >�!�� 7�� ?��< �� !�� ?�� # �� ?�9�� ?�9�� 5�6��5�6� �7�� (�� 9�� ?�B�� &�� !��"��Fig. 3.12. Diagram of the angular and longitudinal differences between the hybrid and theSD core. From [113].S. Falk et al. [39] previously used the SDP reconstruction of Golden Hybrids to determineboth azimuth and elevation offsets. This was done by minimising the angular differencebetween the SDP and the SD core direction seen from the FD eye. For a single shower, thereis a degeneracy in azimuth and elevation offsets. Combining multiple showers with differentgeometry removes this degeneracy (see figure 3.13).3.3 Determining absolute offsets with hybrid showers 123azimuth [deg]elevation [deg]05101520253030354045505560 ]° [ϕ∆-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1]° [ϑ∆-1-0.8-0.6-0.4-0.200.20.40.60.81SDP, min2χ-SDP2χ110210]° [ϕ∆-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1]° [ϑ∆-1-0.8-0.6-0.4-0.200.20.40.60.81min2χ-2χ110210azimuth [deg]elevation [deg]051015202530120125130135140145150 ]° [ϕ∆-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1]° [ϑ∆-1-0.8-0.6-0.4-0.200.20.40.60.81SDP, min2χ-SDP2χ110210SDP ⇒ χ2Figure 2: (left) Two different SDPs projected onto the FD camera; (center) χ2SDP in parameter-space ∆ϕ-∆ϑ; (right) combined χ2 of the two events.telescope’s optical axis and a positive ∆ϑ denotes elevation that is larger than its nominal value.In practice, we are varying ∆ϕ and ∆ϑ untilχ2SDP =k ∑iqi( π2 − arccos(⇀vi (∆ϕ, ∆ϑ) · ⇀nref)σ)2(2)becomes minimal. Here the sum runs over all SDP pixels, qi denotes the signal in one pixel,σ = 0.35◦ is the average angular resolution of a pixel determined from Laser studies [6], and k =N∑i qiis a normalization factor that assures that χ2SDP/N ≈ 1 in case of a successful minimization.The SDP normal ⇀nref is fixed to the value obtained from the reference geometry and the pixeldirections ⇀vi are re-calculated in each minimization step using the current alignment corrections∆ϕ and ∆ϑ.As can be seen in Fig. 2 this χ2SDP does not have a well-defined minimum if one uses onlyone event. Obviously, all (∆ϕ, ∆ϑ)-pairs that result in a translation parallel to the shower de-tector plane, give a similar χ2. To solve this degeneracy, we combine the χ2s of two eventsthat have SDP orientations that differ by at least ∆θSDP > θmin. With the help of simulatedshowers we found that θmin = 24◦ is an optimum with respect to the number of available pairsand the precision of the resulting alignment constants. After the minimization we obtain thestatistical uncertainties σfit of ∆ϕ and ∆ϑ from TMinuit. The total uncertainty includes also theuncertainties of the reference geometry:σ2stat = σ2fit + σ2ref, (3)where σref is calculated using the NumericalErrorPropagation from Offline. For many suchpairs of events a weighted mean〈x〉 =∑ixiσ2i∑i1σ2i, (4)with standard deviationσ (〈x〉) =√√√√ 1∑i1σ2i(5)2Fig. 3.13. (left) Two different SDPs projected onto the FD camera; (center) χ2 SDP inparameter space ∆φ −∆θ ; (right) combined χ2 of the two events. ∆φ is the azimuth offsetan ∆θ is the elevation offset. From [39].The remainder of this section describes the use of hybrid showers to determine bothelevation and azimuth offsets of the fluorescence telescopes. In this study, two methods areused - one co paring the SD core with that predicted from the SDP fit, and one comparingthe core time predicted by the SD and FD fits.3.3.1 SDP methodEvent selectionEvents had to be golden hybrids with the SD axis and core reconstructed. For an accurateSDP, events with a 20 d gree track i one telescope w e selected. Although the SDPaccuracy does not depend on distance, the angular error from the SD core uncertainty does.SD core errors are typically around 100 m. Events with a large core to eye distance wereneeded. There were about 2000 events with a core distance of more than 15 km but at thatdistance, there are very few vertical shower tracks. This is because, at a 20 degree altitude,and a large core distance, the depth of the atmosphere is too shallow for the vertical showerto have developed and is not visible. Since a 20 degree track length is required for a goodtime fit, a core distance of greater then 10 km was chosen so that vertical tracks of sufficientangular length were included. Because there were so many events, an additional cut wasmade to only select showers that had triggered 25 or more pixels. Over 3000 events wereselected.124 FD telescope alignment studyDetermining the telescope pointing offsetThe component of the telescope pointing offset perpendicular to the shower track can bedetermined by finding the angular deviation of the SD core from the SDP. The angular offsetof the core from the SDP will be:∆φ = n̂ ·−−→Core (3.11)where n̂ is the SDP normal and−−→Core is the unit vector in the direction of the SD core fromthe eye. The core location was given in the site coordinate system and had to be convertedto a coordinate system specific to the eye. Events were reconstructed once with only pixelswith even numbered telescopes and once with odd numbered telescopes to avoid pixels fromother telescopes with different offsets from contaminating the SDP fit. Information frommany shower events were combined by fitting the equation:∆φ = |S|cos(α −β ) (3.12)to extract the absolute telescope pointing offset−→S . α is the angle between the horizontal andthe direction perpendicular to the shower track and β is the angle between the horizontal andthe direction of the telescope pointing offset−→S defined in Figure 3.14. So for each shower∆φ and α are known, and a fit of equation 3.12 for a sample of showers results in an estimateof |S| and β .3.3.2 Timing methodIn SD reconstruction, the arrival time of the shower core at the ground is calculated so thecore direction and time can be compared to those parameters from the FD time fit.Event selectionFor an accurate monocular FD time fit, a long angular track length is required as well as asmall value of dχdt . Events with a track length of 20 degrees and a core distance of at least 10km were selected. These are the same criteria as those for the SDP method so the same setwas used.Determining the offsetThe SD core time and location was converted into an FD time and χ angle. Light propagationtime must be taken into account when determining the time that the eye sees the showerarrive at the core. The SD-FD time offset must also be taken into account. This is a small3.3 Determining absolute offsets with hybrid showers 125azimuth [deg]elevation [deg]051015202530120125130135140145150 SFig. 3.14. Definition of shower and telescope offset directions. The black vector−→S(exaggerated in magnitude) is the offset of a telescope. Angles α and β are measured fromthe horizontal. The horizontal is the vector defined as the cross product of the direction ofthe shower core and the zenith.time offset beteen the clocks in the Flourescence detector and the Surface detector stations.This offset was found by using horizontal laser shots from the central laser facility [15]. TheSD-FD time offset used in this study is 300 ns for all eyes (this is discussed in 3.4.3). Thecomponent of the χ offset in the direction of the core will be:∆χ = χFD(tcore)−χcore (3.13)where χcore is the chi angle of the core derived using the SD fit and χFD(tcore) is the FD timefit’s χ angle at the core time (accounting for light propagation). The contributions to errorcome from both the FD and SD. The error contribution from the FD is predominantly fromthe timing fit, as described in section 3.2.1. The error contribution from the SD is not due toerror in the timing of the core, but rather the position of the core, as the position of the corewill affect propagation time to the FD, and uncertainties in core position are typically muchlarger than the timing uncertainties (multiplied by the speed of light).The offsets ∆χ are then fitted to determine the pointing offset−→S using the equation∆χ = |S|cos(α −β ) (3.14)126 FD telescope alignment studywhere β is the angle between the horizontal and the direction of the telescope offset and α isthe angle between the horizontal and the direction of the shower track defined in figure 3.15.Again ∆χ and α are known for each event, and estimates of |S| and β result for each fit.azimuth [deg]elevation [deg]051015202530120125130135140145150 SFig. 3.15. Definition of the timing offset geometry.3.3.3 Core bias effect on SDP methodIn inclined showers, there is a bias in the SD core position [39]. This is because at a givencore distance, a station on the upstream side of the shower will have a larger signal, asparticles heading to the downstream station have to traverse a greater atmospheric depth (seefigure 3.16). This results in a core position biased towards the upstream side of the shower,and due to a curved front being considered for axis fitting, the reconstructed axis direction isshifted to a smaller zenith angle. The bias in core position can be up to 100 m dependingon zenith angle and primary composition, as the difference in depth between upstream anddownstream stations is dependent on zenith angle, and the shower age and contribution ofthe electromagnetic component relative to the muon component are dependent on both zenithangle and composition.The effect of the SD core bias is to create an apparent telescope elevation offset. Dueto the dependence of the core bias on composition, a method for correcting the effect onalignment was chosen to be independent of the magnitude of the asymmetry. The angularsize of the SD core bias seen by an FD eye is inversely proportional to the core-eye distance.The apparent elevation offset ∆Y can be expressed as3.3 Determining absolute offsets with hybrid showers 127s1rupstreamshower frontradius points1 1, multiple copies of the same particle are injected into the detector. To avoida large signal spike from multiple particles arriving at the same time, the arrival times ofcloned particles are smeared using a log-normal distribution:τ′ = τeσG (4.9)where τ is the time delay of the particle from the shower front plane, and G is drawn from aGaussian distribution with mean of 0 and standard deviation of 1. σ is typically chosen to be0.1-0.2.Some biases will arise from using a large sampling area [45]. Most notably, the strongdependence of particle density on core distance. If in the area around the detector, densityis proportional to r−η , the relative excess in the density at the detector station will beη (η −1)δ 2/6. Since η is typically between 3-4, the excess is in the range of δ 2 to 2δ 2.This is much smaller than any natural fluctuations in detector signal, so it is not usually anissue for single events.4.3 Dethinning 163The density distribution of particles can also lead to a bias in particle arrival times at theground. Particles within the sampling area that are nearer the core are overrepresented relativeto particles farther from the core. Since particles nearer the core also arrive earlier (with asmaller time delay from the plane front), the distribution of arrival times at the detector isbiased towards particles arriving earlier. This effect is small, but can become noticeable forstudies which average over a large number of showers, such as the muon production depthestimates [26].Corrections can be applied to reduce the biases in particle number and timing caused bythe steep particle density distribution. A simple adjustment [19] is to correct the weights ofthe particles to compensate for the dependence of density with core distance:w′ = w(rprd)η−1(4.10)This correction does rely on prior knowledge of the lateral distribution to work, though thecorrection is fairly insensitive to the chosen value of η . Using a value of η = 3.5 gives goodresults for all shower types [45].Another de-thinning method has been developed by Stokes et al. [90] and is utilised forsimulations with Telescope Array. Instead of sampling over a large area for a detector, thismethod creates a number of copies of the particle on the ground, with the position and timingof each of the copies scattered depending on the original particle’s timing and geometry. Firsta vertex point is chosen along the trajectory of the particle at a distance D from the ground.Copies of the particle are then generated emerging from this vertex with a two dimensionalGaussian distribution centred on the original particle trajectory (see figure 4.4).There is a maximum distance Dmax that can be chosen to avoid the possibility of particlesarriving before the shower front (propagating at the speed of light from the time and place offirst interaction, t0 and x0). For a first interaction coordinates x0, t0 and particle coordinatesxi, ti with particle trajectory p̂i:Dmax =c2 (ti − t0)2 −|xi −x0|22(c(ti − t0)− (xi −x0) · p̂i)(4.11)The angle of the Gaussian cone must be carefully selected. It has to be small enoughto avoid introducing a bias in the particle distribution while being large enough to avoidcreating small areas of high particle density around higher weight particles. This angle isset to βd where d is the distance of the particle from the shower core. For a thinning ratiot f = 10−6 the value of β = 3deg /km for electromagnetic particles, and 1deg /km for muonsand hadrons. The energy of the copied particles is varied to avoid any spikes in the energy164 Simulation of extensive air showersOur method, as described in this paper, differs in two ways.First, as the dethinning process we take each CORSIKA output par-ticle with weight w and from it generate a swarm of w particles.We perform this step for the entire set of CORSIKA output particles.The details of this generation matter, and are described in this pa-per. Second, we compare the resulting dethinned showers withshowers of almost identical characteristics generated withoutusing the thinning approximation. This is a direct way of testingthe accuracy of the dethinning process. Since real surface detectorsfor ultrahigh energy cosmic rays sample showers coarsely, andmeasure only the time distribution of the number of particles thatstrike them, this is the important aspect of the comparison process.We present our comparisons between dethinned showers and non-thinned showers in this manner.2. Dethinning descriptionIn a thinned EAS simulation, particles are discarded from thesimulation in order to conserve computation time. In the case ofCORSIKA [3], for a given thinning level, eth, if the energy sum ofall j secondary particles falls below the thinning energyethE0 >PjEj ð1Þthen only one randomly assigned secondary particle survives withprobabilitypi ¼ Ei=PjEj: ð2ÞIf the energy sum is greater than the thinning energy, then second-ary particles with energy below the thinning energy survive withprobabilitypi ¼ Ei=ðethE0Þ: ð3ÞIn both cases, surviving particles have their weight multiplied by afactor of wi = 1/pi. Thus the weight of a particle reaching the end ofthe simulation after passing through k vertices iswi;tot ¼Qk1=pk: ð4ÞFor sufficiently low values of eth, it is clear that the thinned simula-tion output can be thought of as an accurate sample of secondaryparticle types, trajectories, and positions compared to a non-thinned simulation, for the observable parts of the shower. In thissituation, for a particle of weight, wi, the simulation, on average,removed wi 1 particles from a similar position in phase space.(Of course, if the value of eth is increased, this situation is no longervalid because the thinned simulation no longer has the same distri-bution of particle types, trajectories, and energies as the parentshower.) By comparing a dethinned shower with a similar non-thinned shower one can determine the accuracy of the sampling.The pivotal questions are then: (1) how can a thinned sample beused to reconstruct the full simulation? and (2) what is the maxi-mum value of eth for which the thinned sample accurately repre-sents the parent shower’s particle types, etc.?We address the first question by describing the process bywhich we dethin the showers. The original CORSIKA shower con-sists of a list of output particles (plus their weights, types, energies,positions, angles, and arrival time) that have struck the ground.Dethinning consists of adding particles to this list. For every CORS-IKA output particle of weight w we add w 1 particles to the list.When this is completed the weight of each particle is set to 1. Toinsert these particles we use the following procedure (see Fig. 1).1. Choose a vertex point on the trajectory of the weighted particle,in the way given in the next paragraph.2. Choose a point in a cone centered on the output particle’s tra-jectory, weighted by a two-dimensional Gaussian distributionwith a sigma of a few degrees (as described in Section 3). Thiswill be the inserted particle’s trajectory.3. Project the inserted particle to ground level, assign it a time andenergy (as described in Section 3), and add it to the particle listof the dethinned shower.4. Perform steps 2 and 3 w 1 times. For the case where w is notan integer, add one particle randomly based on the decimal partof w.There is a maximum distance from the ground that one canchoose for the vertex in item 1 above, which is set by the require-ment that no particle can have an arrival time that precedes the ar-rival of the shower front. A too-early arrival time occurs when thetotal time-of-flight from the point of first interaction, x0, to the ver-tex point and then to the position on the ground of the generatedparticle is less than the time-of-flight directly from x0 to final par-ticle position. This can be corrected by fixing the position of thevertex point to a position where the time-of-flight from x0 to theimaginary vertex and then onward to the final position of theweighted particle, xi is equal to the difference in the arrival timeof the weighted particle, ti, and the time of first interaction, t0. Thiscondition is: the distance along the weighted particle trajectory, p̂i,between xi and the vertex point isShowerCoreWeighted Particle Arbitrary VertexTrajectoriesSampledTrajectoryFig. 1. Geometry for a ‘‘Gaussian cone’’ with a vertex placed at arbitrary position onthe trajectory of the weighted particle.ox , tx t,pi^t’oi i1dt’d 2ShowerCoreWeighted Particle SwarmDmaxArbitrary VertexFig. 2. In order to ensure temporal consistency in the EAS simulation, we requireti t0 P dt01 þ dt02, where ti is the recorded arrival time for weighted particle and t0is the time of first interaction.760 B.T. Stokes et al. / Astroparticle Physics 35 (2012) 759–766Fig. 4.4. Diagram of the vertex used for dethinning. From [90].spectrum of particles at the ground. The energy is varied in a Gaussian distribution withstandard deviation of 10% in the original particle energy. In showers with a high zenith angle,copied particles can have much longer trajectories than the original particle and would beless likely to make it to the ground. To compensate for this, an acceptance factor for copiedparticles is introduced. The acceptance probability is:P = e−∆Xε (4.12)where ∆X is the difference in slant depth between the original particle and the copied particle,and ε is set to 50 gcm−2The height of the vertex is usually set at Dmax but this may be far too high for particlessuch as muons produced later in the shower’s development. This may give an excessivelywide distribution in the positions of copied particles. This is solved by limiting the vertexdistance to the smaller of Dmax and:D′ = |xi −x0|−X−1 (xi,x0,αh) (4.13)where h is the hadronic generation from which the particle originated, α = 30 gcm−2, andX−1 (xi,x0,αh) is the distance corresponding to a a slant depth of αh on a straight line fromx0 towards xi.4.4 Simulation of detector response for the Pierre Auger Observatory 165This method has been found to reproduce the original shower properties for a thinningratio t f = 10−6 at core distances of up to 4500m without introducing artificial fluctuations.4.4 Simulation of detector response for the Pierre AugerObservatoryAn accurate simulation of the detector response to simulated showers is essential to deducethe properties of observed cosmic ray showers. For this, the Offline software framework wasdesigned to handle both reconstruction of observed cosmic ray showers and the propagationof simulated showers into the detector (and subsequent reconstruction) [8]. The Offlineframework is modular in construction, where the series of modules run can be chosen by theuser to suit a specific task. Offline was developed in C++, with a framework that consists of3 main parts (also shown in figure 4.6):• The sequence of modules. These are assembled by the user to suit a specific task.Modules do not typically pass data to other modules directly but rather communicatethrough the event data, in a format common to all modules. This simplifies developingand updating modules individually for use in a long sequence of modules. The sequenceof modules is controlled by a run controller, with sequencing instructions specifiedusing an XML-based language. An example of the sequencing instructions is shown infigure 4.5.• Event data. This is the principal way in which modules relay data. It contains all raw,calibrated, reconstructed, and simulated data. The Event data is arranged in a hierarchystructure typically associated with observatory instruments. Modules are able to bothread from and write to the event data.• Detector description. This provides an interface to allow modules to access informationabout the detector such as the detector configuration, current detector performance,and atmospheric conditions.Event data is held in memory in a transient event. When data is to be written to a disk,it is stored as a persistent event via a file interface. File input/output in Offline is handledby a ROOT toolkit to store each event and read various file formats. These formats includeraw event and monitoring formats produced by the observatory, in addition to simulated dataproduced by programs such as AIRES, CORSIKA, CONEX, and SENECA.Another important component of Offline is the geometry package. Since the observatoryis spread over a large area, with multiple detector types, there are a number of different166 Simulation of extensive air showers EventFileReaderOG EventCheckerOG FdCalibratorOG SdQualityCutTaggerOG SdPMTQualityCheckerKG TriggerTimeCorrection SdCalibratorOG SdBadStationRejectorKG FdPulseFinderOG FdSDPFinderOG FdAxisFinderOG HybridGeometryFinderOG FdApertureLightOG FdProfileReconstructorKG RecDataWriterNG Fig. 4.5. An example of the module sequencing file for reconstruction of a hybrid event.preferred coordinate systems. These could include a system relative to a Fluorescencedetector station, or even a shower coordinate system dependent on the reconstructed showergeometry. The geometry package provides abstract geometrical objects such as points andvectors. The objects internally track the coordinate system used and allow for operations onthese objects to be written independent of the coordinate system used.4.4 Simulation of detector response for the Pierre Auger Observatory 167Fig. 4.6. Diagram of the general structure of the Offline framework. From [101]Chapter 5An improved dethinning method toreduce biases in detector simulations5.1 IntroductionSimulations of extensive air showers and the response of detectors to these showers play avital role in deducing the properties of the primary cosmic ray particle. A major problemwith this is that extensive air showers at the highest energy contain billions of particles. A fullsimulation of such a large number of particles is a computationally intensive process, makingthe simulation of large numbers of showers difficult. To simulate a large number of showers,a process known as thinning is employed, where only a fraction of particles are tracked ina simulation (details in 4.2). With appropriate weightings applied, overall properties of ashower can be preserved, with only small artificial fluctuations introduced by the statistics ofsampling a smaller number of particles. Simulation of a detector response is more difficult,requiring dethinning.5.1.1 DethinningIf thinning is employed in an air shower simulation, only a fraction of particles are trackedto the ground with each particle given a weighting which is inversely proportional to theprobability of that particle surviving the thinning procedure. Assigning this weight to particlesallows the average particle densities in a shower to be determined without bias. However,problems arise when attempting to simulate a detector of limited area. A detector simulationfrom a thinned shower without resampling would have fewer particles hit but with higherweights introducing artificial fluctuations in detector signals. This is corrected by resampling170 An improved dethinning method to reduce biases in detector simulationswhere particles are sampled from a larger area than the detector so that particles can beinjected into the detector with unity weight.The resampling method currently used in Offline defines a sampling region around eachdetector [19]. This sampling area is given a radial width as a fraction of detector core distanceδ and an azimuthal angle α in the shower plane (See figure 5.1). Particles can then be given aresampled weight wr = wAd/As, for a particle of weight w, a detector area Ad and a samplingarea As. The particle is then accepted into the detector n times where n is drawn from aPoisson distribution with n̄ = wr. For particles with wr << 1 this is equivalent to acceptingthe particle with a probability of wr. If n > 1, clones of the particle will be made and aregiven a time smearing to prevent spikes in detector signals.The values of δ and α must be chosen carefully. If δ and α are too large, particlesbeing sampled from regions of the shower far away from the detector may introduce biasesin particle density and arrival times. If δ and α are too small, larger resampled weightswill result in more particle cloning. If the particles injected into a detector are cloned, thevariance in the number of particles being injected is higher than the variance expected fromPoisson statistics, potentially introducing artificial fluctuations into detector signals. Artificialfluctuations are most noticeable when individual particles contribute a significant fractionof the total signal. Values of about δ = 0.1 & α = 0.15 rad are typically chosen for Augerdetector simulations. Sampling areas of this size typically have wr << 1 for lower weightparticles such as muons. Resampled weights are more likely to be larger than 1 for lowerenergy EM particles and/or at distances very close to the core, but in both of these cases theseparticles are very numerous and the artificial fluctuations introduced are not as significant.1 IntroductionThe CPU time required for the simulation of full (non-thinned) atmospheric air showers,on a large scale, is unmanageable. This problem is usually solved by the use of the thinningalgorithm. In this approximation, among the vast numb r of particles generated in thesimul tion, only a subsampl of them are followed up to the end. Statistical weights areassigned to the surviving particles in representation of those ones that, in similar regionsof phase space, have been removed along the procedure.A reliable simulation of any detector response depends on the quality of the estimationof the local density of particles at the position of the detector. Within this context, thesefluxes are obtained as averages over more or less extended areas around the location of inter-est. The module doing this in the Offline is the CachedShowerRegenerator. For the injectionof a realistic flux of particles in a specific surface detector, the resampling method definesa sampling-area (see Figure 1) which is defined by two parameters: a radial distance (δ)and an angular aperture (α) from the detector position (in the shower ref ren e frame). Theelection of the values of δ and α is delicate: they m st be small enough to guarantee thevalidity of the local averaging (times, energies, flux, etc.) and big enough to collect suffi-cient particles and avoid large artificial fluctuations (caused primarily by large remainingweights). Once this region is defined, all the particles within it are cloned according to theirweights, keeping their times and momenta. Thereby, the mean number of particles hittingone single detector is given by the sum of all the particle weights inside the sampling regionscaled by the ratio of the area of the detector and the area of the sampling region.Figure 1: Definition of the sampling region (shadow) at a given position (marked with apoint). This Figure has been taken from [1].The purpose of this note is to follow, in detail, the temporal structure of the simulatedair-showers before (at Corsika level) and after (at Offline level) the application of the re-sampling (or un-thinning) method summarized above. Even when we choose a samplingregion adequate for assuming a small variation of the curvature of the shower-front withinit, a time correction must be applied to preserve the intrinsic structure of the shower:t′ = tp + ~ns · (~rp − ~rs)/c (1)2Fig. 5.1. Definition of the sampling area [19]. The sampling area is an area in the showerplane with the detector at core distance rs and azimuth ζs. The sampling area extends out fora radial distance δ · rs and azimuth angle α .5.2 Biases caused by resampling 1715.2 Biases caused by resamplingFor the resampling procedure, it is assumed that changes in particle density over smalldistances are linear. Under this assumption, variations in particle density within a samplingregion would cancel out if the detector is in the center of the sampling region. This assumptionholds well azimuthally since variations in density are fairly small and roughly linear forvalues of α around 0.15. Radially however, the density varies much more steeply and itsnon-linear slope can have an effect on the density simulation at the typically used δ = 0.1.Biases in the timing of particle arrivals are also possible. The primary cause of timingbias is due to the steep LDF. Because of the steep LDF, there are more particles accepted fromcloser to the core within the sampling area. Due to the shape of the shower front, particlescloser to the core arrive earlier on average so particle timing has an early bias. Recent studiesin [45] have found a bias in the timing of muons in 60 degree showers that is independentof primary composition, energy, and hadronic model used (see figure 5.2). This bias is verystrongly dependent on the value of δ chosen. Corrections were suggested to reduce thisbias where the weighting of particles is scaled depending on their core distance relativeto the detector. If it is assumed that the particle density follows an r−η trend, weight isrescaled by a factor of (rp/rs)η−1 where rp & rs are the particle and detector core distances.A value of η = 3.5 is suggested. To compensate for biases caused by curvature in the showerfront, a correction can also be applied to the particle timing. This can be implemented asan adjustment to the time delay from the plane front, τ, to τ′ = (τ−τ f (rp))(rs/rp)+τ f (rs),where τ f (r) = r2/(2Rc) with R as the radius of curvature.The biases in timing and density are not likely to be noticeable for individual showersbut can introduce significant biases for studies averaging large numbers of showers. Recentwork on biases in X µmax (defined in section 2.5.2) has found that the timing bias caused byresampling does affect the simulated X µmax . Figure 5.3 shows the effect of correcting for thisbias on X µmax . The negative time bias caused by resampling will ultimately lead to shallowerreconstructed depths in the simulations. Since X µmax bias corrections are based on biases insimulations, this will lead to an overestimate of X µmax in the real data. Correcting for this biasin the simulation has ultimately led to a shallower X µmax in the real data.5.3 The new resampling methodWith biases in density and timing strongly dependent on the extent of the sampling area,it is highly desirable to minimise the sampling area. For a fixed sampling area, the areachosen must be relatively large to account for the higher weight particles to reduce artificial172 An improved dethinning method to reduce biases in detector simulationsFig. 5.2. Muon timing bias for δ = 0.1 in a 1019.5 eV 60 degree shower [45]. ∆ < τ> isthe bias in the mean muon arrival time.fluctuations, but this means that the majority of particles are being sampled from an unnec-essarily large area (see figure 5.4). This problem can be solved by introducing a variablesize sampling area for each particle, which is dependent on a particle’s weight. This has theadvantage of sampling lower weight particles from a small area but allowing a larger area forhigher weights, reducing both biases and artificial fluctuations.With the new resampling method, a sampling area is created around the detector for agiven particle with As = wAd . For very high weight particles, this sampling area may beexcessively large resulting in particles being sampled from too far away. To avoid this, a limitis placed on the size of the sampling area. The maximum allowable sampling area is definedas Amax = 4r2dδα where δ and α are the largest relative radial distance and azimuthal angleallowed. For particles which are overweight, i.e. wAd > Amax, cloning is required.For particles that are not overweight, the sampling area is given a radial and azimuthalsize δ ′ = δ√AsAmax, α ′ = α√AsAmax. If the particle is within the new sampling area, a singlecopy of the particle is accepted into the detector.Overweight particles are treated in a way similar to the original Offline method. Theparticle is cloned n times, where n is drawn from a Poisson distribution with n̄ = wAd/Amax.5.4 Performance comparisonsTo compare the performance of the two resampling methods, 7200 proton showers weregenerated in total at energies of 1019, 1019.5, and 1020 eV and zenith angles of 0, 38, and 605.4 Performance comparisons 173E [eV]1910×2 1910×3 1910×4 2010]2 [g/cm〉maxµX〈400450500550600198 122 92 42 27Epos-LHCQGSJetII-04protonironE [eV]1910×2 1910×3 1910×4 2010]2 [g/cm〉maxµX〈400450500550600198 122 92 42 27Epos-LHCQGSJetII-04protonironFigure 17: < Xµmax > as a function of energy before (left plot [1]) and after (right plot)the resampling correction. The prediction of different hadronic models for proton and ironare shown. Numbers indicate the statistics in each energy bin and brackets represent thesystematic uncertainty.13Fig. 5.3. Bias in the real X µmax estimate caused by resampling with the uncorrected (top) andcorrected (bottom) X µmax [26].degrees. The biases are found at the CORSIKA level by determining each particle’s meanacceptance < n> into a detector for a series of detector core distances rd averaged over allazimuths. Using the Offline resampling method, particles can be accepted into the detectorif the particle’s core distance rp, and azimuth ζp, obey the following criteria:∣∣rp − rd∣∣ < rdδ∣∣ζp −ζd∣∣ < α (5.1)For particles that fit both of these criteria, the mean number of copies accepted for detectorarea Ad is:n̄ =wAdAs(5.2)174 An improved dethinning method to reduce biases in detector simulationsr(m)0 500 1000 1500 2000 2500 3000 3500 4000weight10log0123456-310-210-110muon weight distributionmuon weight distribution(a)r(m)0 500 1000 1500 2000 2500 3000 3500 4000weight10log0123456-310-210-110EM weight distribution(b)Fig. 5.4. Contour plot of weight vs core distance for muons (left) and EM (right) for 1019.5eV showers. The black line is the weight where wAd = As for α = 0.15, δ = 0.1. Belowthis line particles could be sampled from a smaller area. Above the line particles requirecloning. Most muons are at, or slightly below, the hadronic weight limit (the maximumweight hadrons and muons are thinned to in CORSIKA). There is also a smaller number ofmuons above this limit which are created by pair production from photons, which inherit ahigher weight limit (the EM weight limit is a factor of 100 higher for an optimised simulation[64]). The EM component also has most of its particles close to the weight limit at smallercore distance. However a second peak appears in the weight distribution at larger coredistances. The second peak is likely muon halo particles (EM particles from decayed muons).If this is averaged over all detector azimuths, the mean number of copies accepted is:< n> = n̄2α2π=wAdAs× 2α2π=wAd4δαr2d× 2α2π=wAd4πr2δ(5.3)5.4 Performance comparisons 175With the new method if wAd < As the particle is accepted with n = 1 if the core distance iswithin:∣∣rp − rd∣∣ < δ ′i.e.∣∣rp − rd∣∣ < δ rd√wAdAsi.e.∣∣rp − rd∣∣ < δ rd√wAd4r2dδα(5.4)And if the azimuth is within:∣∣ζp −ζd∣∣ < α ′i.e.∣∣ζp −ζd∣∣ < α√wAd4r2dδα(5.5)The probability of a particle within the core distance criteria being accepted, and hence themean number accepted, when averaged over all azimuths is:< n> =2α ′2π=απ√wAd4r2dδα=1π√wAdα4r2dδ(5.6)Similarly to the Offline method , particles with wAd > As have a mean number of copiesaccepted:< n> =wAdAs× 2α2π=wAd4πr2δ(5.7)Figures 5.5-5.8 show biases for δ = 0.1, α = 0.15 for both resampling methods. Theplots show differences in density and timing against a reference value of δ = 0.02, whichis assumed to be the true density and time. Biases are given for both the muon and EMcomponents. For the EM component, particle contributions to density and timing wereweighted by their energy to better represent signals seen in a WCD.176 An improved dethinning method to reduce biases in detector simulationsr(m)0 1000 2000 3000 4000Relative density bias00.020.040.06° = 0θ eV 19E = 10r(m)0 1000 2000 3000 4000Relative density bias00.020.040.06° = 38θ eV 19E = 10r(m)0 1000 2000 3000 4000Relative density bias00.020.040.06° = 60θ eV 19E = 10r(m)0 1000 2000 3000 4000Relative density bias00.020.040.06° = 0θ eV 19.5E = 10r(m)0 1000 2000 3000 4000Relative density bias00.020.040.06° = 38θ eV 19.5E = 10r(m)0 1000 2000 3000 4000Relative density bias00.020.040.06° = 60θ eV 19.5E = 10r(m)0 1000 2000 3000 4000Relative density bias00.020.040.06° = 0θ eV 20E = 10r(m)0 1000 2000 3000 4000Relative density bias00.020.040.06° = 38θ eV 20E = 10r(m)0 1000 2000 3000 4000Relative density bias00.020.040.06° = 60θ eV 20E = 10Fig. 5.5. Biases in the muon density. The bias in these plots is the difference betweenthe resampled density and the reference density as a fraction of the reference density. Thered circular points are for the Offline method and the blue triangular points are for the newmethod. The energies from top to bottom are 1019, 1019.5, and 1020 eV. The zenith anglesfrom left to right are 0, 38 and 60 degrees.Biases in both particle density and timing exist for both EM and muons for the Offlinemethod. For the muons, the new resampling method shows almost no biases in both. Thisis because of the lower weights of muons and hence the smaller sampling distances withthe new method. The particle weights in the EM component are typically much larger thanthose for muons so at smaller core distances (. 1000 m). EM particles usually utilise themaximum allowable sampling area, so the new and old resampling methods are equivalent.For larger core distances the maximum sampling area is larger so EM particles are morelikely to utilise sampling areas smaller than the maximum, reducing biases. This effect isdependent on primary energy, with biases continuing out to larger core distances at higherenergy due to the average weight of particles being proportional to primary energy for agiven thinning level.5.5 Conclusions 177r(m)0 1000 2000 3000 4000Relative density bias00.020.040.06° = 0θ eV 19E = 10r(m)0 1000 2000 3000 4000Relative density bias00.020.040.06° = 38θ eV 19E = 10r(m)0 1000 2000 3000 4000Relative density bias00.020.040.06° = 60θ eV 19E = 10r(m)0 1000 2000 3000 4000Relative density bias00.020.040.06° = 0θ eV 19.5E = 10r(m)0 1000 2000 3000 4000Relative density bias00.020.040.06° = 38θ eV 19.5E = 10r(m)0 1000 2000 3000 4000Relative density bias00.020.040.06° = 60θ eV 19.5E = 10r(m)0 1000 2000 3000 4000Relative density bias00.020.040.06° = 0θ eV 20E = 10r(m)0 1000 2000 3000 4000Relative density bias00.020.040.06° = 38θ eV 20E = 10r(m)0 1000 2000 3000 4000Relative density bias00.020.040.06° = 60θ eV 20E = 10Fig. 5.6. Biases in the EM density. The bias in these plots is the difference between theresampled density and the reference density as a fraction of the reference density. The redcircular points are for the Offline method and the blue triangular points are for the newmethod. The energies from top to bottom are 1019, 1019.5, and 1020 eV. The zenith anglesfrom left to right are 0, 38 and 60 degrees.5.5 ConclusionsIt has been shown that this new resampling method can substantially reduce the biasesassociated with large sampling areas by sampling from the smallest possible area for eachindividual particle. This minimises the distance from the detector that particles are beingsampled from, but also allows for larger sampling areas for particles with higher weightsand/or smaller core distances. This minimises the artificial fluctuations caused by particlecloning. For showers thinned at the 10−6 level, biases in the muon density and arrival timeare almost completely removed because of the low average weight of muons. The EMcomponent still shows some bias at smaller core distances, reducing for distances & 1000 m.The EM bias worsens with increasing energy due to higher particle weights. The reducedbiases are especially useful for applications with a high sensitivity to particle timing, such as178 An improved dethinning method to reduce biases in detector simulationsr(m)0 1000 2000 3000 4000 > (ns)τ <∆-100-500° = 0θ eV 19E = 10r(m)0 1000 2000 3000 4000 > (ns)τ <∆-100-500° = 38θ eV 19E = 10r(m)0 1000 2000 3000 4000 > (ns)τ <∆-100-500° = 60θ eV 19E = 10r(m)0 1000 2000 3000 4000 > (ns)τ <∆-100-500° = 0θ eV 19.5E = 10r(m)0 1000 2000 3000 4000 > (ns)τ <∆-100-500° = 38θ eV 19.5E = 10r(m)0 1000 2000 3000 4000 > (ns)τ <∆-100-500° = 60θ eV 19.5E = 10r(m)0 1000 2000 3000 4000 > (ns)τ <∆-100-500° = 0θ eV 20E = 10r(m)0 1000 2000 3000 4000 > (ns)τ <∆-100-500° = 38θ eV 20E = 10r(m)0 1000 2000 3000 4000 > (ns)τ <∆-100-500° = 60θ eV 20E = 10Fig. 5.7. Biases in the muon mean arrival time. The red circular points are for the Offlinemethod and the blue triangular points are for the new method. The energies from top tobottom are 1019, 1019.5, and 1020 eV. The zenith angles from left to right are 0, 38 and 60degrees.the muon production depth (see section 2.5.2) analysis. The new resampling method is nowthe default method used in Offline for surface detector simulations.5.5 Conclusions 179r(m)0 1000 2000 3000 > (ns)τ <∆-100-500° = 0θ eV 19E = 10r(m)0 1000 2000 3000 > (ns)τ <∆-100-500° = 38θ eV 19E = 10r(m)0 1000 2000 3000 > (ns)τ <∆-100-500° = 60θ eV 19E = 10r(m)0 1000 2000 3000 > (ns)τ <∆-100-500° = 0θ eV 19.5E = 10r(m)0 1000 2000 3000 > (ns)τ <∆-100-500° = 38θ eV 19.5E = 10r(m)0 1000 2000 3000 > (ns)τ <∆-100-500° = 60θ eV 19.5E = 10r(m)0 1000 2000 3000 > (ns)τ <∆-100-500° = 0θ eV 20E = 10r(m)0 1000 2000 3000 > (ns)τ <∆-100-500° = 38θ eV 20E = 10r(m)0 1000 2000 3000 > (ns)τ <∆-100-500° = 60θ eV 20E = 10Fig. 5.8. Biases in the EM mean arrival time. The red circular points are for the Offlinemethod and the blue triangular points are for the new method. The energies from top tobottom are 1019, 1019.5, and 1020 eV. The zenith angles from left to right are 0, 38 and 60degrees.Chapter 6Muon shower front shape andcorrelation with first interaction depth,Xmax and XµmaxAbstractThe arrival time distribution of particles at the ground holds information about showerdevelopment and mass composition. This chapter focuses on the muon component of theshower. To study the muon front in high detail, showers were generated with CORSIKAusing very low (10−8) thinning. This study does not include the effects of a detector. Thepurpose of this study is to gain a better understanding of the physics of showers and look forhints of possible observables that could be correlated with Xmax, Xµmax, or the 1st interactiondepth. The possible effects of a detector will be briefly discussed towards the end of thechapter. One hundred 1019 eV showers were each generated at zenith angles of 0°and 60◦.Each zenith angle consisted of 60 proton and 40 iron showers. Showers were generated withCORSIKA version 6990 using QGSJET-II-03.6.1 IntroductionThe majority of hadrons produced in an air shower typically have a very small angulardeflection from the direction of their parent particle. This means that the hadronic componentof an air shower has a very narrow lateral distribution (<10m). For most purposes, includingin this study, it can be assumed that the hadronic component of a shower lies exactly on thecore. Muons are mostly produced as a result of decay of π+ & π−. Muons have a much larger182 Muon shower front shape and correlation with first interaction depth, Xmax and Xµmaxangular distribution than the hadronic component, due to the momentum of the recoilingneutrino from the pion decay. Muons produced predominantly by pions with an energy nearthe critical energy (when a charged pion’s interaction length is equal to its decay length) havean angular distribution large enough to be easily detected with an array of surface detectors.Since muons are produced close to the axis and travel in a straight line at close to thespeed of light, muons produced at a certain position on the axis will at a later time be along aspherical surface with the centre of curvature at the production point. From this it is possibleto determine the production depth of a muon from its timing if the geometry of the shower isknown. The area of the ground covered by muons produced at a certain depth does dependon the shower’s zenith angle. Nearly vertical showers have muons being produced relativelyclose to the ground. Because of this, the production profile of muons landing at a certainpoint on the ground will be largely determined by geometric effects. These effects include thedistance from the production point (favouring lower production distances), and the angulardistribution of muons produced (favouring higher production distances). These effects makeit much more difficult to study the muon production profile. Larger zenith angle showers,such as at 60◦, have the majority of muons produced far above the ground so they can covera core distance of a few km with a sufficient density to be detected. It is possible to study theproduction profile of muons at high zenith angles and this is the basis of the current MPDreconstructions [44].6.2 Muon timing studyFigure 6.1 shows the muon production height vs the arrival time at the ground for a verti-cal shower. There is a reasonable correlation between arrival time and production heightsuggesting that the SD time profile may approximate the muon production profile.Figure 6.2 shows the muon production height vs core distance for a vertical proton shower.For small core distances (<200m) the muon component is dominated by muons producednear the ground, as muons produced close to the ground will land with a high concentrationin a small area. This is because the ground close to the axis is within the muon productioncone along nearly all of the axis. For core distances larger than this, the muon productionheight peaks at a height increasing with core distance.If the muon production rate were uniform along the axis, the muon front would be in aconical shape. This is due to a combination of two effects. Due to the inverse square law,the muon component is expected to be dominated by muons produced closest to the ground.Muons are produced with a limited angular distribution, so at a given point on the ground,muons are only seen in large numbers from sufficiently large enough production heights for6.2 Muon timing study 183t (ns)0 100 200 300 400 500 600 700 800 900 1000production height(km)0510152025-810-710-610-510-410Zint vs tFig. 6.1. Contour plot of muon production height vs arrival time at the ground behind theplane front for muons with a core distance between 1000 and 1100 m. The shower is avertical 2×1018 eV proton. The contour variable has units of Nµm−2ns−1km−1.the point to be within the muon production cone. A combination of these two effects meansthat muons observed at a given core distance would be dominated by muons produced at aheight proportional to core distance, leading to a conical front. Because the muon productionis not uniform, the maximum in the muon production height profile at a specific core distancetends to be pushed towards the maximum in muon production. This effect is not large atsmall core distances, as the muon production rate does not vary greatly over a small range ofproduction heights near the ground, but is more relevant at large core distances.Figures 6.3 and 6.4 show a contour plot of the muon arrival time at the ground vs coredistance. The two black lines correspond to propagation at the speed of light from the 1stinteraction height and the maximum in total energy deposit, Xmax. The first muon arrivalscorrespond to the 1st interaction. However their density is extremely low and would bedifficult to measure even with a detector that covered 100% of the surface. For larger coredistances, most of the muons arrive between the corresponding times for 1st interaction andXmax. The muons at these core distances mostly come from above the maximum in muonproduction due to the angular distribution of muons produced. At smaller core distances, thereis a greater number of muons arriving later than the time corresponding to Xmax. The muonsproduced deeper than Xmax dominate at small core distances because they are concentratedover a smaller area around the core due to their limited angular distribution.184 Muon shower front shape and correlation with first interaction depth, Xmax and Xµmaxr (m)0 200 400 600 800 1000 1200 1400 1600 1800 2000productionheight(km)0510152025-810-710-610-510-410-310-210-1101Zint vs rFig. 6.2. Contour plot of muon production height vs core distance for a vertical 2×1018 eVproton. The contour lines indicate the density of muons as Nµm−2km−1.r (m)0 200 400 600 800 1000 1200 1400 1600 1800 2000t(ns)01002003004005006007008009001000-910-810-710-610-510-410-310-210-110t vs rFig. 6.3. Muon arrival time delay from plane front vs core distance contour plot for avertical 2×1018 eV proton. The contours are in units of Nµ m−2ns−1. The two black linescorrespond to propagation at the speed of light from the 1st interaction height and Xmax.6.2 Muon timing study 185r (m)0 200 400 600 800 1000 1200 1400 1600 1800 2000t(ns)01002003004005006007008009001000-910-810-710-610-510-410-310-210-110t vs rFig. 6.4. Muon arrival time delay from plane front vs core distance contour plot for a vertical2×1018 eV iron nucleus. The contours are in units of Nµ m−2ns−1. The two black linescorrespond to propagation at the speed of light from the 1st interaction height and Xmax.Different quantiles in the arrival time distribution of muons will most likely correspondto different stages of shower development. To study which quantiles are most useful, severalquantiles have been tested in this study. The shape of the shower front at different quantileswere fitted with numerous fitting functions to determine which function is the best fit. Onlytwo functions are shown in this chapter. Figure 6.5 shows the spherical and spherical + conefits to the shower front. Equation 6.1 is the fitting equation used for the spherical fit. R is theradius of curvature parameter and r is the core distance. Equation 6.2 is the fitting equationused for the spherical + cone fit. b is the cone slope parameter, R is the radius parameter andr is the core distance.t(r) =R−√R2 − r2c(6.1)t(r) =br−√R2 − r2 +Rc(6.2)For the vertical shower it is clear that the spherical fit is not a very good fit to the showerfront, especially for larger quantiles. The spherical + cone function fits reasonably well. For60◦, the spherical fits are fairly good and the spherical + cone fits are very similar, with onlya small conical component.186 Muon shower front shape and correlation with first interaction depth, Xmax and Xµmax0 500 1000 1500 2000r (m)0500100015002000t (ns) 9−108−107−106−105−104−103−102−101−101 spherical°00 500 1000 1500 2000r (m)0500100015002000t (ns) 9−108−107−106−105−104−103−102−101−101 spherical + cone°00 500 1000 1500 2000r (m)0500100015002000t (ns) 9−108−107−106−105−104−103−102−101−10 spherical°600 500 1000 1500 2000r (m)0500100015002000t (ns) 9−108−107−106−105−104−103−102−101−10 spherical + cone°60Fig. 6.5. Contour plot of the muon arrival time delay vs core distance for 1019 eV protonshowers. The contour units are Nµm−2ns−1. The points are the arrival time quantiles for eachdistance bin. The lines are the fits with a spherical function & a spherical + cone functionThe shape of muon arrival time distribution is a lot flatter in the 60◦ case, and is nearlyspherical in shape. This is because most of the muons are produced at higher altitudes, fartheraway from the ground.6.3 Muon time fitting studyTo see variations between showers and different compositions 100 showers were generated.The showers were 1019eV with 10−8 thinning. 60 proton and 40 iron showers were generated.The spherical + cone fitting function (equation 6.2) was applied from quantiles 0.1 to 0.9in intervals of 0.1. A graph could then be made of the fit parameters vs quantile whereeach shower is represented by a line to observe how a fit parameter correlates with itselfbetween different quantiles. Figures 6.6 shows the graphs of parameters vs quantile withproton showers colored red and iron showers colored blue.Both of the parameters show some separation between proton and iron showers. Theproton showers are more widely spread and overlap with the iron showers. For verticalshowers both the radius and the slope parameters separate compositions well but for 60◦ theslope does not. This is likely because the cone component is caused by muons produced6.3 Muon time fitting study 1870.2 0.4 0.6 0.8quantile0.050.10.15slope (m/m) slope°00.2 0.4 0.6 0.8quantile80001000012000r (m)radius°00.2 0.4 0.6 0.8quantile0.010.02slope (m/m) slope°600.2 0.4 0.6 0.8quantile1000015000200002500030000r (m) radius°60Fig. 6.6. The slope (parameter b, eqn. 6.2) and radius (R) as a function of quantile (detailsin text), to vertical and 60◦ 1019 eV showers. Proton showers are red and iron showers blue.near the ground. Only a very small number of muons are produced near the ground for a 60◦shower and those muons are produced in a very late stage of shower development which isless likely to correlate with primary particle composition.It is also useful to see how the fit parameters depend on first interaction height measuredvertically above ground level. Figures 6.7 shows the same plots but coloured by firstinteraction height. The colour gradient begins with red at 15 km above ground level,transitioning to yellow at 25 km, to green at 35 km, and blue at 45 km. Events withvery deep 1st interactions seem to be separated from shallower 1st interactions but all other1st interaction heights are mixed. This is clearer in figures 6.8-6.10.It is more useful to visualise the parameters as a scatter plot against first interactiondistance (measured along the shower axis from ground level), Xmax and Xµmax . Figures 6.8- 6.11 show scatter plots of the fit parameters. The slope parameter is not shown for 60◦showers because the slope is only very small and does not show very good correlation withother shower parameters.The spherical + cone radius R shows the best correlation with first interaction distance(figures 6.9 a & c). The correlation is also slightly better at the 0.1 quantile. The bettercorrelation for smaller quantiles is expected, as the muons at smaller quantiles correspondto muons produced at earlier stages of development. Larger core distances are dominatedby muons produced earlier due to their angular distribution whereas smaller core distances188 Muon shower front shape and correlation with first interaction depth, Xmax and Xµmax0.2 0.4 0.6 0.8quantile0.050.10.15slope (m/m) slope0°0.2 0.4 0.6 0.8quantile80001000012000r (m) radius°00.2 0.4 0.6 0.8quantile0.010.02slope (m/m) slope60°0.2 0.4 0.6 0.8quantile100002000030000radius (m) radius°60Fig. 6.7. The slope (parameter b in equation 6.2) and radius (R) as a function of quantile (seetext for details), for vertical (top) and 60◦ (bottom) 1019 eV showers. The colour gradientbegins with red at 15 km above ground level, transitioning to yellow at 25 km, to green at 35km, and blue at 45 km.1st interaction distance (km)10 20 30 40 50slope (m/m)00.050.10.150.2Spherical + cone slope vs 1st int height distance (km)maxX1 1.5 2 2.5 3 3.5 4slope (m/m)00.050.10.150.2Spherical + cone slope vs X_max heightFig. 6.8. The spherical + cone fit slope (parameter b in equation 6.2) as a function of the 1stinteraction distance from the core (left) and of the Xmax distance (right) for vertical 1019 eVshowers. The quantiles are 0.1 (lower clusters), 0.5 (middle clusters) and 0.9 (upper clusters).The red points are for proton showers and the blue points are for iron showers.are dominated by muons produced later in the shower’s development. Thus we expect thatthe radius R from the spherical + cone fit to be more sensitive to first interaction since theradius parameter has the largest contribution to the shape of the shower front at large core6.3 Muon time fitting study 1891st interaction distance (km)10 15 20 25 30 35 40 45 50R (m) 60008000100001200014000 R vs 1st int distance°0(a)Distance (km)max X1 1.5 2 2.5 3 3.5 4R (m) 60008000100001200014000 distancemax R vs X°0(b)1st interaction distance (km)50 100R (m) 100002000030000 R vs 1st int distance°60(c) distance (km)maxX12 13 14 15 16 17 18 19R (m) 100002000030000 distancemax R vs X°60(d)Fig. 6.9. The spherical + cone fit radius (parameter R in equation 6.2) as a function ofthe 1st interaction distance (left) and of the Xmax distance (right) for vertical (top) and 60◦(bottom) 1019 eV showers. The quantiles are 0.1 (upper clusters), 0.5 (middle clusters) and0.9 (lower clusters). The red points are for proton showers and the blue points are for ironshowers.distances. The slope parameter b from the spherical + cone fit shows some correlation forvertical showers and almost no correlation for 60◦ showers. This is expected as the slope ismostly dependent on muons produced near the ground. For 60◦ showers the muon productionhas stopped long before reaching the ground, so the slope component is negligible.The correlations of first interaction distance with the fit parameters are only present forsmaller first interaction distances (deeper first interaction depths). There is little correlationat larger first interaction distances. The likely reason for this is that for very shallowfirst interactions, second and subsequent interactions which are measurable at the surfaceoccur much deeper than the first interaction so the detectable properties of the shower areindependent of first interaction height. Measuring even earlier quantiles will probe earlierinteractions. Figure 6.10 shows the spherical + cone fit radius R for quantiles 0.1, 0.01, and0.001. Each of these quantiles represents a different hadronic generation though quantiles0.01 an 0.001 are impossible to measure with a detector that only covers a small fraction of190 Muon shower front shape and correlation with first interaction depth, Xmax and Xµmaxthe surface. The 0.001 quantile shows a better correlation for deeper first interactions butstill do not show good correlation for shallower first interactions.1st int height (km)15 20 25 30 35 40 45 50radius (m)100001200014000160001800020000220002400026000Fig. 6.10. Radius of the spherical + cone fit vs 1st interaction height for vertical showers.The squares are for the 0.1 quantile, the triangles for the 0.01 quantile and the circles forthe 0.001 quantile. The red points are for proton showers and the blue points are for ironshowers.A correlation of R with Xmax is apparent for both 0◦ and 60◦ showers. This correlationexists for all 3 quantiles, over the whole range of Xmax values. As expected, R increaseswith increasing distance to Xmax. There is also a small apparent separation between protonand iron showers in this plot, meaning that the radius R not only depends on Xmax, butalso composition. This could allow for a better composition determination if R could beaccurately determined from the SD in a hybrid shower. It is also not clear from these plotswhich quantile shows the best correlation with Xmax, so further analysis on this matter isrequired. The slope parameter b decreases with increasing Xmax distance. It is not entirelyclear why this happens. For showers with a smaller Xmax distance, most of the muons exceptthose very close to the core come from above Xmax where the shower is still developing. Thiscould mean that muons on average come from a smaller distance since that is where most ofthe muons are produced, increasing the cone component.The fit parameters can also be compared with the muon production profile. In this study,X µmax was obtained from the muon profile in the CORSIKA longitudinal file by calculatingthe rate of change in the number of muons with depth, and finding the maximum. Figure6.11 shows the radius R against the distance to X µmax along the shower axis. The FD canmeasure Xmax with good resolution, and if the SD could measure the Xµmax , figure 6.126.3 Muon time fitting study 191shows the proton/iron discrimination power that one could get if using Xmax and Xµmax . Thedistributions of Xmax and Xµmax for proton and iron show an overlap if the distributions aretaken separately. However, for showers with a given Xmax value have an Xµmax value that isdifferent depending on primary composition, so if both are known a much better compositionseparation is achievable. distance (km) maxmuX4 5 6 7 8R (m)80001000012000°0 distance (km) maxmuX18 20 22 24R (m) 100002000030000°60 distance (km) maxmuX18 20 22 24R (m) 16000180002000022000 0.5 quantile°60Fig. 6.11. Radius of the spherical + cone fit R vs X µmax distance for vertical (left) and 60◦(right) showers. The quantiles are 0.1 (lower cluster), 0.5 (middle clusters) and 0.9 (upperclusters). The bottom plot is a zoomed version of the 0.5 quantile for 60 degree showersThe fit parameters show a correlation with the distances to X µmax over the whole range ofsimulated X µmax. The radius of the spherical + cone fit increases with increasing Xµmax distanceas expected.When the radius R is plotted against distance to X µmax, vertical showers show a correlationsimilarly good to the correlation with radius against Xmax. For 60◦ showers, the correlationbetween radius and distance to X µmax is much higher than between radius and distance to Xmax.For both zenith angles, there is not as much of a separation between protons and iron asthere was with distance to Xmax . This means that the radius is measuring muon developmentinformation independently to primary composition. Additionally, the plots of Xmax vs Xµmaxshow a separation between proton and iron similar to the plots of radius vs distance to Xmax.192 Muon shower front shape and correlation with first interaction depth, Xmax and Xµmax)-2 (gcmmaxmuX400 450 500 550 600)-2(gcmmaxX700800900°0(a))-2 (gcmmaxmuX400 500 600)-2(gcmmaxX700800900°60(b)Fig. 6.12. Xmax vs Xµmax for vertical (left) and 60◦ (right) showers.6.3.1 Resolution of shower parameters using shower front shapeIn this section, the ability to resolve shower parameters (specifically Xmax, Xµmax, and 1stinteraction depth) using shower front shape is examined. A fit is performed of the showerparameters against spherical + cone fit R, and the RMS difference between the predictedvalue of the parameter from the true value is found. As the best quantile is not yet known,this is performed over a range of quantiles, to determine which quantile is best. For thismethod, the depth of 1st interaction, Xmax , and Xµmax are used instead of height as they varymore linearly with R, and they are easier to compare to each other. Although the plots shownpreviously for shower parameters against R show approximately linear trends, a quadraticfit is used to account for any slight non-linearity. Figures 6.13 and 6.14 show the RMSdifference vs quantile for the three parameters for vertical and 60◦ showers.As expected, for both vertical and 60◦ showers, the smallest quantiles are best fordetermining 1st interaction depth, though the dependence on quantile is not very strong,particularly in the 60◦ case. 1st interaction resolution is better for vertical showers, mostlikely for two reasons. First, due to the shorter distance from muon production to theground, radii of curvature are smaller for a given quantile, with more curvature in the showerfront for vertical showers, and any change in production depth will translate to a largerchange in arrival time at the ground. Secondly, due to geometric effects, muons produced atshallower depths are preferred at larger core distances, so muons from earlier in the shower’sdevelopment contribute more to the shower front shape.X µmax is best reconstructed for 60◦ showers. As geometric effects do not lead to a strongpreference for muons produced at shallower or deeper depths along the production profile,this is expected. This also explains why X µmax resolution is highest using the middle quantilesas these correspond to muons produced around X µmax. For vertical showers, there is a better6.3 Muon time fitting study 193quantile0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9)-2RMS error (gcm05101520253035maxXµmaxX0X Spherical + cone R parameter error vs quantile°0Fig. 6.13. RMS deviation of predicted parameters (1st interaction depth, Xmax, and Xµmax) vsquantile for vertical showers.quantile0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9)-2RMS error (gcm0510152025maxXµmaxX0X Spherical + cone R parameter error vs quantile°60Fig. 6.14. RMS deviation of predicted parameters (1st interaction depth, Xmax, and Xµmax) vsquantile for 60◦ showers.194 Muon shower front shape and correlation with first interaction depth, Xmax and Xµmaxresolution at earlier quantiles. It is not entirely clear why, but geometric effects likely play arole, as they are more relevant at greater depths, which would correspond to later quantiles.The Xmax resolution for both vertical and 60◦ showers follows a similar trend to theX µmax resolution though resolution is poorer in the case of Xmax. This is expected, as Xmaxis primarily defined by the electromagnetic component of the shower. As Xmax and Xµmaxare correlated, the best indicator of Xmax from the muon profile is likely to be Xµmax, so it isexpected that the Xmax resolution is related to the Xµmax resolution.Figures 6.15 - 6.18 show the distributions of Xmax and Xµmax for these 100 showers.)-2 (gcmmaxX650 700 750 800 850 900 950Number of events024681012141618 FeP distributionmax X°0Fig. 6.15. Xmax distribution for vertical showers6.3 Muon time fitting study 195)-2 (gcmmaxX650 700 750 800 850 900 950Number of events02468101214FeP distributionmax X°60Fig. 6.16. Xmax distribution for 60◦ showers)-2 (gcmmaxµX350 400 450 500 550 600 650Number of events02468101214161820 distributionmaxµ X°0Xmumax distributionXmumax distributionFePFig. 6.17. X µmax distribution for vertical showers196 Muon shower front shape and correlation with first interaction depth, Xmax and Xµmax)-2 (gcmmaxµX350 400 450 500 550 600 650Number of events0246810121416182022 distributionmaxµ X°60FePFig. 6.18. X µmax distribution for 60◦ showers6.3 Muon time fitting study 1976.3.2 Detector resolutionThis study has so far not considered the effect of a detector in the ability to resolve showerparameters with the shape of the muon front. Here detectability of the shower front shapewill be briefly considered.Detector densityThe limited sampling area of a detector array may have difficulties resolving the shower frontshape for a number of reasons including a limited sampling due to the sparse array, statisticaluncertainty due to the limited area of each detector, and the time resolution of detectors.A limited sampling due to a sparse array will have the effect of giving timing at a fewdifferent core distances. The number of stations will always be at least 3, as this many arerequired for an event to be reconstructed, with a larger number available at higher energies.This alone would not prevent a reconstruction of shower front shape, as this fit has twoparameters, if geometry is already known. If geometry is not known, a larger number ofdetectors may be needed, as geometry fitting will introduce 3 additional parameters usingtiming.Detector areaThe statistical uncertainty arising from the limited detector area is likely the greatest concernfor reconstructing shower front shape. It may be difficult to determine the uncertaintieswithout a full simulation study, but some conclusions may be drawn from results from theMPD analysis from the Auger collaboration, where a maximum in muon production is foundfor muons that land on the ground [70]. The method discussed in this chapter is likely togive similar results to the MPD analysis, which is done for showers with a high zenith angle.The MPD analysis could resolve the maximum in muon production depth to within 50 gcm−2 at 1.5×1019 eV, and 30 g cm−2 at 9×1019 eV. A similar resolution would be expectedwith this study for 60◦ showers. Vertical showers have densities of muons at the groundcomparable to that of 60◦ showers, for core distances used in this study, so X µmax resolutionis not expected to be worse for vertical events. Additionally, for the MPD study, there wasa minimum core distance of 1200 m due to the narrow time distribution of muons at smallcore distances, where errors due to kinematic delay (from muons having velocities slightlybelow c), and detector time resolution become dominant. Due to the wider time distributionof muon arrivals at the ground for vertical showers, smaller core distances may be used,where muon densities are greater, potentially allowing better X µmax resolution than the MPDanalysis.198 Muon shower front shape and correlation with first interaction depth, Xmax and XµmaxIt is reasonable given these factors, that the upgraded Pierre Auger Observatory surfacearray may have the ability to resolve a maximum in muon production profile sufficiently toprovide useful composition information from shower front shape over a wide range of zenithangles, for energies above ∼ 1019 eV.6.4 ConclusionsIt can be concluded that for both vertical and 60◦ showers, muon arrival times correlate wellwith production height (as seen in figure 6.1). In the case of 60◦ showers, due to the longdistances from muon production to the ground, muons can spread out to a spherical frontshape at the ground, with a radius of curvature tracing back to near the maximum in muonproduction, X µmax. For vertical showers, the limited angular distribution of muons leads tomuons produced at lower altitudes covering a smaller area, leading to shower fronts thatare between a conical and spherical shape. Despite the complexities in the shape of showerfronts for vertical showers, the shape of the shower front still closely correlates with X µmax.With detectors capable of distinguishing the muon component from the electromagneticcomponent, the determination of X µmax is still feasible for vertical showers. This may allow adataset similar to the MPD dataset over all zenith angles.Shower front shape for both vertical and 60◦ showers correlate with Xmax, but not as wellas for X µmax. Additionally, shower front shape did not depend significantly on compositionfor a given X µmax, but it did show a small dependence for a given Xmax. It may be possible forhybrid events to improve composition resolution using muon shower front shape information.Though it is theoretically possible to measure the distance to the first interaction usingthe leading edge of the shower front, it would require an extremely dense array of detectors,given the very low densities of muons in this leading edge.Chapter 7ConclusionsOver the course of this work a number of different aspects of UHECR detection have beenstudied. Firstly, misalignments in the Fluorescence Detector telescopes were discovered, andquantified by detecting inconsistencies in the reconstructed geometry of observed showers.Due to biases observed in simulated events due to resampling of thinned extensive air showersimulations, a new resampling method was developed, where the area used for resamplingwas allowed to vary to minimise the distance from which particles were sampled into adetector, reducing biases associated with sampling over a large area. Finally, with the SurfaceDetector array being upgraded with the ability to separate the muon component, simulationswere produced to study the shape of the muon shower front, to determine if the shape ofthe shower front could be used to reconstruct information on the longitudinal profile. It wasfound that X µmax correlates well with the shape of the shower front, and that it should befeasible at higher energies to determine shower front shape with the surface detector array ofthe Pierre Auger Observatory.Telescope Alignment• A discontinuity was observed in timing fits when crossing between Coihueco telescopes5 and 6. The most logical explanation for this is a pointing offset of one of the twotelescopes relative to the other.• A relationship between this time discontinuity and the shower geometry was derivedfor a given relative pointing offset. This relation then allowed for the relative pointingoffsets for every pair of adjacent telescopes to be determined. A similar relation wasalso determined for SDP fits, and provided additional data to fit relative pointingoffsets.200 Conclusions• Both timing and SDP data were consistent with relative pointing offsets, and could notbe explained by other factors, such as a fixed timing offset.• It was then necessary to find any offsets in the absolute pointing offsets of telescopes.Surface detector data from hybrid events, with core position and timing, could be usedto find any absolute pointing offsets.• complications arose in determining absolute pointing offsets due to biases in corepositions, and uncertainties in timing offset between the Fluorescence Detector and theSurface Detector. A method was developed to correct for these biases, allowing forindependent determination of pointing directions using both timing and SDP data fromobserved showers.• Though the different methods mostly agreed well, there were some discrepanciesbetween the timing and SDP results, as well as disagreements with other telescopepointing studies. Timing residuals from CLF shots, and geometry differences betweenFD and SD reconstructions showed that some telescope optics were misaligned in away that altered the telescopes field of view.• Physical inspection of the telescope optics showed that this misalignment was presentin telescope 4 of Loma Amarilla, which was then corrected.Resampling• Cosmic ray shower simulations at the highest energy are enormously computationallyintensive, containing hundreds of billions of particles. Because of this, a procedureknown as thinning is employed, where only a subset of particles are tracked, andreceive a weight based on the probability of being tracked.• Though thinning preserves the overall statistics of shower properties such as particledensities and time distributions when weighting is applied, the response of a particledetector would contain artificial fluctuations due to there being weighted particles atthe ground.• To prevent these artificial fluctuations, a method called resampling (also known asdethinning) is employed where weighted particles are sampled from a much larger areasurrounding the detector, and injected into the detector with unity weight.• In previously used resampling methods, biases were present in both the density andtiming of particles due to the large areas used for resampling, as the density of particlescan vary strongly with position.201• A new resampling method was developed, that drew particles from a much smallerarea where possible to do so while avoiding artificial fluctuations. This new resamplingmethod was shown to have significantly reduced biases in both particle density andarrival time.• This new resampling method is now the default method used in the Offline softwarepackage, for surface detector simulations.Shower front shape• The Pierre Auger Surface Detector is being upgraded with the ability to separate themuon and electromagnetic signals, to be able to resolve mass composition. In additionto densities of these components, timing may also be useful for this, especially themuon component.• To search for possible composition sensitive information in the shape of the muonfront, showers were simulated with a low thinning level, to obtain high statistics onindividual events.• Examining the shape of the shower front over a wide range of quantiles in the arrivaltime distribution, it was found that the shape is described very well by a sum of aspherical and conical function.• The radius of the spherical component of the shower front was found to show correla-tions with Xmax, Xµmax, and first interaction distance. This correlation was quantifiedover a range of quantiles, to determine what quantiles may be the most useful forresolving these shower parameters using shower front shape.• It was found that X µmax correlates well with the shape of the shower front over a broadrange of quantiles, for both vertical and 60◦ showers. First interaction depth could beresolved with extremely small quantiles, but these would be very difficult to detectwith a sparse detector array.• It is likely that the Pierre Auger surface detector will be able to resolve the shape ofthe shower front, and hence X µmax, well enough to gain useful composition informationfrom individual showers.References[1] Abbasi, R. U., Abu-Zayyad, T., Allen, M., et al. (2008). First Observation of theGreisen-Zatsepin-Kuzmin Suppression. Physical Review Letters, 100(10):101101.[2] Abu-Zayyad, T., Aida, R., Allen, M., et al. (2012). The surface detector array of theTelescope Array experiment. Nuclear Instruments and Methods in Physics Research A,689:87–97.[3] Adams, J. H., Ahmad, S., Albert, J. N., et al. (2013). An evaluation of the exposure innadir observation of the JEM-EUSO mission. Astroparticle Physics, 44:76–90.[4] Allard, D. (2012). Extragalactic propagation of ultrahigh energy cosmic-rays. 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