· The Vela Pulsar in Very High Energy γ-rays with H.E.S.S. II DISSERTATION...

The Vela Pulsar in Very High Energy γ-rays with H.E.S.S. II

D I S S ERTAT I ON

zur Erlangung des akademischen Grades

doctor rerum naturalium(Dr. rer. nat.)

im Fach Physik:

Spezialisierung:Astroteilchenphysik

eingereicht an derMathematisch-Naturwissenschaftlichen Fakultat

der Humboldt-Universitat zu Berlin

von

Michael David Gajdus MPhys (Oxon)

Prasident der Humboldt-Universitat zu Berlin:

Prof. Dr. Jan-Hendrik Olbertz

Dekan der Mathematisch-Naturwissenschaftlichen Fakultat:

Prof. Dr. Elmar Kulke

Vorsitzender: Prof. Dr. Marek Kowalski

Gutachter: Prof. Dr. Thomas Lohse

Gutachter: Prof. Dr. Christian Stegmann

Externgutachter: Prof. Dr. Nepomuk Otte

Mitglied: Prof. Dr. Ullrich Wolff

eingereicht am: 14. Oktober 2015

Tag der mundlichen Prufung: 27. April 2016

Kurzfassung

Die vorliegende Dissertation analysiert Beobachtungen des Vela-Pulsars mit dem H.E.S.S.-Observatorium und prasentiert damit den ersten Nachweis gepulster Strahlung mit H.E.S.S..Die signifikante Detektion (11.4 σ) oberhalb einer bis jetzt unerreichten Energieschwelle vonlediglich 15GeV wurde mittels von 24 h Beobachtungsdaten erzielt, die mit dem im Jahr 2012in Betrieb genommenen CT 5-Teleskop aufgezeichnet wurden. Nach dem Krebs-Pulsar istdamit der Vela-Pulsar der zweite Pulsar, von dem sehr hochenergetische Gammastrahlungnachgewiesen wurde.

Im Verlauf der Arbeit wird die Leistungsfahigkeit des fortgeschrittenen ImPACT-Rekonstruk-tionsverfahren untersucht und fur Gammastrahlunngsquellen mit stark abfallenden Spektrenim niedrigsten, mit CT 5 erreichbaren Energiebereich (E < 50GeV) optimiert. Die Anwendungdes optimierten Verfahrens gewahrleistet so maximale Detektionssignifikanz bei der Analysevon Gammastrahlungsquellen. Die Software wurde außerdem fur zwei weitere Quellklassen,die sich hinsichtlichtlich der Steilheit der Energiespektren unterscheiden, optimiert, um imRahmen der H.E.S.S.-Kollaboration bei der Analyse weitere Quellen verwendet zu werden.Somit stehen hiermit geeignete Analysekonfigurationen fur eine große Vielfalt an Quelltypenzur Verfugung.

Die Spektren von Pulsaren im Bereich der sehr hochenergetischen Gammastrahlung sindvon besonderem Interesse, da sie Einblicke in einen bisher fast komplett unerforschten Ener-giebereich erlauben. Der experimentelle Nachweis stellt eine besondere Herausforderung dar,da die Rate der Gammastrahlen gemaß einem Potenzgesetz mit der Energie abfallt und dieresultierenden Zahlraten klein sind. Daher werden neue gewichtete Testgroßen eingefuhrt,um die Sensitivitat fur den Nachweis schwacher Pulsationen fur zukunftige Beobachtungenvon Pulsaren zu verbessern. Die Testgroßen sind Erweiterungen von bereits existierendenungewichteten Verfahren und anwendbar auf beliebige diskrete Daten, in denen ein Pulsoder mehrere Pulse erwartet werden. Sie sind leistungsfahig fur den Fall unbekannter Phasen-profile und erzielen bei Standardgewichtung eine 10%-ige Steigerung der Detektionssignifikanz.

Die optimierte Analysekette wurde zur Detektion des Vela-Pulsars im Energiebereichvon 15GeV bis 125GeV eingesetzt. Der phasengemittelte integrierte Energiefluss betragt4.29 × 10−11 erg s−1cm−2 und wird durch einen einzelnen Puls verursacht. Das Profil desPulses kann durch eine asymmetrischen Lorentz-Funktion mit einer schmaleren abfallendenFlanke beschrieben werden. Der Photonfluß fallt gemaß einem Potenzgesetz mit Index −5.39mit der Energie ab. Dies ist in guter Ubereinstimmung mit einem theoretischen Annular-Gap-Emissionsmodell, entspricht jedoch einem starkeren Abfall der Emission als experimentele mitFermi -LAT nachgewiesen. Weiter wurde die Emissionseffizienz des Vela-Pulsars im Bereichder sehr hochenergetischen Gammastrahlung ermittelt. Sie betragt 0.0025% und ist damitvergleichbar mit dem Wert, der vom MAGIC-Teleskopsystem fur den Krebs-Pulsar gemessenwurde. Ferner wird das beobachtete Signal im Hinblick auf andere Pulsare, die moglicherweiseebenfalls sehr hochenergetische Gammastrahlung aussenden konnen, diskutiert. Im Rahmender Suche nach Verletzungen der Lorentz-Invarianz wird eine recht schwache untere Schrankeauf die Energieskala ELIV derartiger Phanomene als ELIV > 10−4EP berechnet, wobei EP

die Planck-Energie bezeichnet.

Abstract

In the presented thesis the analysis of the first pulsar to be detected with the H.E.S.S.array is described. The high significance detection at the 11.4σ level down to a hithertounachievable 15GeV energy threshold is a result of 24 h of observations on the Vela pulsarwith the new CT 5 telescope introduced into the array in 2012. This is only the second pulsarto be detected in the very high energy (VHE) γ-ray regime.

The capabilities of the advanced, template-based ImPACT reconstruction software areanalysed and optimised in order to perform best for sources with steeply falling spectra inthe lowest energy band (< 50GeV) accessible with CT 5. This ensures that the maximumdetection significance is obtained when analysing sources. The software was also optimisedfor two other source classes differing in the steepness of spectra to be used on other γ-raysources by the H.E.S.S. Collaboration, thus providing customised analysis configurations for abroad range of source types.

Of particular interest are the very high energy (VHE) spectra of pulsars as this is an almostcompletely unexplored energy domain. This does however entail smaller signals as the rate ofγ-rays generally drops according to a power law function. New weighted statistical tests areintroduced to improve the sensitivity to weak pulsations for use with future observations ofpulsars. These tests are modifications of currently used tests and are applicable to any discretedata in which a single or many pulses are expected; they are powerful when the phase profileis unknown and with a basic weighting provide up to a 10% boost in detection significance.

The optimised analysis chain contributed to the detection of the Vela pulsar with phase aver-aged energy flux in the energy range (15, 125)GeV of 4.29(+1.14

−1.02)stat(+5.50−3.31)sys×10−11 erg cm−2s−1.

The single pulse is characterised with an asymmetric Lorentzian function with a narrowertrailing edge. The photon flux falls as a power law with index −5.39 which is moderatelyconsistent with an Annular Gap emission model but represents a steeper drop in emissionthan that measured with the Fermi -LAT. The emission efficiency of the Vela pulsar in theVHE band is also evaluated as 0.0025%, which is comparable to that of the Crab pulsar inthe same energy regime measured with the MAGIC telescope array. Comparisons to othercandidate VHE pulsars are also drawn. A weak constraint is placed on the energy at whichLorentz invariance violation occurs in terms of the Planck energy as ELIV > 10−4 EP.

I dedicate this work to Sarah Isabella Jackson.

Contents

Introduction 1

1. Physics of Pulsars 51.1. Pulsar Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2. The Compact Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.1. Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.2. Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.3. Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.4. Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3. γ-ray Production Processes in a Pulsar’s Magnetosphere . . . . . . . . . . . . 231.3.1. Magnetic Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.3.2. Curvature Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.3.3. Inverse Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . 28

1.4. The Rotating Magnetized Conducting Sphere . . . . . . . . . . . . . . . . . . 291.5. Novel Emission Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.5.1. Cascade Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.5.2. Striped Wind Emission . . . . . . . . . . . . . . . . . . . . . . . . . . 341.5.3. Narrow Region Wind Emission . . . . . . . . . . . . . . . . . . . . . . 35

1.6. Magnetospheric Gap Pulsed Emission Models . . . . . . . . . . . . . . . . . . 351.6.1. Pulsar Death Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.6.2. Polar Cap Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.6.3. Slot Gap Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411.6.4. Outer Gap Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.7. Moving Forward with Observation . . . . . . . . . . . . . . . . . . . . . . . . 50

2. The H.E.S.S. II Experiment 512.1. The Tracking System and Pointing . . . . . . . . . . . . . . . . . . . . . . . . 542.2. Central Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.2.1. System Dead Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.2.2. Event Time Stamp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.3. Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.3.1. Photo-multiplier Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . 592.3.2. Camera Thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.4. Array Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.5. Calibration and DST Production . . . . . . . . . . . . . . . . . . . . . . . . . 62

3. Reconstruction & Analysis 653.1. Observation Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.2. Event Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.2.1. Hillas Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.2.2. Multivariate Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2.3. Image Pixel-wise fit for Atmospheric Cerenkov Telescopes . . . . . . 703.2.4. Semi-Empirical γ-ray Likelihood Model Reconstruction . . . . . . . . 71

3.3. Optimisation of the Analysis Configuration for Low Energies . . . . . . . . . 713.3.1. Optimisation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 723.3.2. Quality Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.3.3. Weighting the Monte Carlo Spectrum . . . . . . . . . . . . . . . . . . 743.3.4. Cut Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.3.5. Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.4. Pulsar Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.4.1. Calculation of the Solar System Barycentred Event Arrival Time . . . 813.4.2. Calculation of the Rotational Phase . . . . . . . . . . . . . . . . . . . 843.4.3. Timing Irregularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.4.4. Interface and Use of Tempo2 . . . . . . . . . . . . . . . . . . . . . . . 853.4.5. Background Estimation & Significance . . . . . . . . . . . . . . . . . . 86

3.5. Pulsed Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.5.1. Weighted Pulsed Statistics . . . . . . . . . . . . . . . . . . . . . . . . 91

3.6. Clarification of Phasing Software Using the H.E.S.S. Optical Crab Data . . . 973.7. Data Quality Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4. Analsysis of H.E.S.S. Data on the Vela Pulsar 1034.1. Region Of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.2. Timing Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.3. ImPACT Analysis of the H.E.S.S. II Data set . . . . . . . . . . . . . . . . . . 107

4.3.1. Signal Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.4. Sky Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.5. Phase-folded Event Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.5.1. Characterising the Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.5.2. Energy Resolved Peak Shape . . . . . . . . . . . . . . . . . . . . . . . 1184.5.3. Phase Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.6. Very High Energy γ-ray Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 1204.6.1. Energy Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.6.2. Spectral Calculation Method . . . . . . . . . . . . . . . . . . . . . . . 1224.6.3. Energy Binning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.6.4. Energy Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.6.5. Best Fitting Power Law Spectrum . . . . . . . . . . . . . . . . . . . . 1234.6.6. Exponential Cut Off Power Law . . . . . . . . . . . . . . . . . . . . . 1264.6.7. Spectral Upper Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.7. Spectrum Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.7.1. Instrument Response Function Bracketing . . . . . . . . . . . . . . . . 1284.7.2. Varying the Energy Binning and Threshold . . . . . . . . . . . . . . . 1294.7.3. Comparison of the Spectrum . . . . . . . . . . . . . . . . . . . . . . . 129

4.8. Phasing Cross Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5. Interpretation of the Observations 1335.1. Magnetospheric Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.1.1. Magnetospheric Gaps Models . . . . . . . . . . . . . . . . . . . . . . . 1335.2. γ-ray Emission Luminosity and Efficiency . . . . . . . . . . . . . . . . . . . . 140

5.2.1. Orientation of the Vela Pulsar . . . . . . . . . . . . . . . . . . . . . . 140

5.2.2. The Vela Pulsar Emission Efficiency . . . . . . . . . . . . . . . . . . . 1415.2.3. The Crab Pulsar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425.2.4. The Geminga Pulsar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435.2.5. PSR J0540−6919 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.3. Phase Profile Evolution with Energy . . . . . . . . . . . . . . . . . . . . . . . 1445.3.1. The Asymmetric Width of the P2 Peak . . . . . . . . . . . . . . . . . 146

5.4. Lorentz Invariance Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Conclusions 151

Appendices 155

A. Landau Quantisation 157

B. Analysis Configuration Optimisation 159B.1. Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159B.2. Loose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160B.3. Extra Loose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

C. Pulsed Statistics 163C.1. Clarification of the Implementation . . . . . . . . . . . . . . . . . . . . . . . . 163C.2. Weighted Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

C.2.1. Normalising the Harmonic test statistics . . . . . . . . . . . . . . . . . 168C.2.2. Weighted test statistic Null Distributions . . . . . . . . . . . . . . . . 171

Bibliography 173

List of Figures 192

List of Tables 193

Acronyms 195

Acknowledgements 199

Introduction

The first pulsar - later understood as a rapidly rotating compact stellar remnant - wasdiscovered with radio observations at 81.5MHz (Hewish et al. 1968) by Jocelyn Bell-

Burnell in 1968. Today, however, our understanding and technology has enabled pulsarsto be viewed not only in the radio, optical, and X-ray regions of the spectrum but also atfrequencies up to 1020MHz, otherwise know as very high energy γ-rays where energy scales arereferred to using the electron volt rather than with frequency. This thesis will go into detailabout the ways in which pulsars can be detected at such an extreme of the electromagneticspectrum from the highly sensitive instrument required to the analysis techniques used toselect the signal from the huge background. Theoretical discussion will also be includedto outline the reasons behind looking at pulsars and what observations could mean for theunderstanding of physics in extreme environments. Much is still disputed about the nature ofpulsars but certainly one key thing is clear: their environment is extreme. With magneticfields around 108T and temperatures above 106K, such extreme environments are not likeanything stable here on Earth and are suited to producing γ-rays.The original discovery of pulsars was dependent on the fact that radio telescopes are

sensitive enough to detect the emission from a single pulse from a pulsar. The state of the artimaging atmospheric Cerenkov telescopes are, however, quite the other end of the scale andrequire many hours worth of pulsations - which occur at ∼ 10Hz - to be summed together inorder to distinguish the γ-ray pulse from the background.

Very High Energy The term γ-ray refers to photos in a huge energy range from low energy- below 30MeV - up to extremely high energy - above 30PeV. The very high energy (VHE)regime lies in between the two: from 30GeV up to 30TeV (Aharonian 2004).

VHE photons are, in general, not produced by thermal processes. By considering Planck’s

law, a surface temperature of 1014K is required to get a detectable thermal signal of VHEγ-rays1. This sort of temperature is very difficult to achieve, especially for any length of time;a type II supernova stellar core may, for instance, only reach ∼ 1011K (Podsiadlowski 2013)and is considered amongst the hottest phenomena in the universe. This means that theremust be other processes going on to produce photons of such high energies; observations inthis regime entail the exploration of the non-thermal universe. The overall γ-ray spectrumis an approximate power law with a negative index meaning that the flux drops steeply athigher energies and flux in the VHE regime is comparatively tiny.Three approaches to detect γ-ray have emerged: pair-conversion satellite-based detectors

such as the Fermi Large Area Telescope (Fermi -LAT) which are most sensitive in the highenergy (HE; 30MeV ≤ E ≤ 30GeV (Aharonian 2004)) regime; water Cerenkov arrays whichare deployed at high altitude and are most sensitive at the very upper edge of the VHE γ-rayregime; imaging atmospheric Cerenkov telescopes (IACTs) such as MAGIC, VERITAS, andH.E.S.S. which perform best in the VHE regime. The former are limited by weight and size

11% of the integrated flux of the Crab Nebula (Aharonian et al. 2004a) above 0.1TeV was used as thesensitivity limit

1

Introduction

Figure 0.1.: All sky map in galactic coordinates with all known very high energy γ-ray sources(points) overlaid with the Fermi -LAT all sky map in high energy γ-rays (continuousimage). Taken from Wakely and Horan 2014 on 19/08/2015.

leading to a much smaller collection area and lower maximum measurable energy but have allsky coverage and a high duty cycle. The water Cerenkov arrays have a higher minimummeasurable energy ∼ 100GeV, large field of view, large collection area and high duty cyclebut suffer from a poor background rejection power. IACT arrays are limited in minimumenergy due to the absorption of Cerenkov light by the atmosphere to ∼ 30GeV and have asmall duty cycle due to restrictions on light levels but achieve a much larger collection areathan the satellite experiments. The three approaches have come to complement each otheras together they explore a huge γ-ray energy regime over seven orders of magnitude from10MeV to 100TeV.

The All Sky View Due to the large field of view of such satellite-borne instruments and thecomparatively high flux of the sources, their view of the sky is very different from that ofIACT arrays; Figure 0.1 shows this stark difference with an all sky view obtained with theFermi -LAT overlaid with the discrete sources seen in the VHE regime. Most of the VHEsources lie along the galactic plane (across the middle of the image) including the indicatedVela pulsar which is analysed in this work. The clear difference is the continuum of emission inthe HE band compared to the discrete sources (162 on 19/08/2015 (Wakely and Horan 2014))in VHE. The result of the current, third, generation of IACTs has been to greatly increase thenumber of points on this map. The VHE sources mostly consist of remnants of supernovaeand active galactic nuclei. The emission as a result of supernovae can be categorised asdirectly due to the explosion - from which γ-ray emission is thought to originate as a result ofthe supernova shock front interacting with the interstellar medium or electron acceleratedin magnetic fields that interact with local radiation fields - or from the remaining compactstar whereby a nebula and γ-ray emission are driven by the highly energetic leptonic stellarwind. This latter group, due to their driving force, are referred to as pulsar wind nebulaeand comprise the largest source class in VHE γ-ray astronomy. Active galactic nuclei arethought to be powered by central supermassive black holes where particles are acceleratedalong huge relativistic jets and γ-rays are emitted through various electromagnetic and strong

2

Introduction

force processes.Only two of these VHE source are, however, pulsars: the Crab pulsar (PSR J00534+2200)

and the Vela pulsar (PSR J0835−4510)2. It is to the detection and analysis of the Vela pulsarwhich a large part of this work is devoted.

The chapters in this thesis are organised according to the following:

• Chapter 1 introduces the current understanding of the physics of pulsars from theirdetailed make-up to the mechanisms with which they are predicted to emit γ-rayradiation. The content of this chapter provides context for the importance of theinvestigation of these extreme astrophysical laboratories in later chapters.

• Chapter 2 describes the H.E.S.S. II IACT array which is used to measure the γ-rayemission from astrophysical sources and the ways in which the experimental physics isimplemented to obtain reliable shower images for analysis.

• Chapter 3 explains the software and methods used to reconstruct the γ-ray inducedCerenkov air showers and determine the properties of the γ-ray from the imagestaken with H.E.S.S. II. The process of optimising this reconstruction, particularly forpulsar analysis, and the software developed to produce higher level analysis results aredescribed. Statistical tests - including newly introduced weighted tests - that are usedto quantify the level of pulsation are discussed.

• Chapter 4 details the analysis of the H.E.S.S. II data set on the Vela pulsar showingthe results of the software and methods developed and described in Chapter 3. Thischapter reports on the first detection of a pulsed signal from a pulsar with H.E.S.S..

• Chapter 5 seeks to understand the analysis results reported in Chapter 4 in the contextof the physics of Chapter 1 to determine what can be gleaned from the data and whatconstraints can be put on physical models, properties of the Vela pulsar, or generalphysics.

2Neshpor et al. 2001 report on the detection of the Geminga pulsar (PSR J0633+1746) at a significance of4.4 σ in the ultra high energy regime (30TeV < E < 30PeV (Aharonian 2004))

3

1. Physics of Pulsars

Since their discovery in 1968 (Hewish et al. 1968) three classes of pulsar have emerged:

• Magnetars

• Accretion-powered pulsars

• Rotation-powered pulsars

Duncan and Thompson 1992 first proposed magnetar theory well after the observation of thefirst γ-ray burst (GRB; Strong, Klebesadel, and Olson 1974) in order to explain observationsof GRBs by the Burst And Transient Source Experiment (BATSE) aboard the ComptonGamma Ray Observatory (CGRO; Meegan et al. 1992). The theory suggests that the origin ofsuch bursts is a type of neutron star that has an extremely high magnetic field 1014···15G whichcan experience magnetic disruptions (quakes). More recently, observations of other objects,such as soft gamma repeaters (SGRs) & anomalous X-ray pulsars (AXPs), have also beenlinked to magnetars. Magnetars are, therefore, associated with bursts of γ-rays at irregularintervals (an active topic in very high energy (VHE) astronomy, see, for example, Aharonianet al. 2009a) and young, relatively long-period, isolated pulsars (2 s < P < 12 s) (Olausen andKaspi 2014).

Accretion-powered pulsars are associated with pulsars in a binary system - with a less dense,non-compact star that has overflown its Roche lobe - that accrete their companion’s stellarmatter (Nagase 1989). The accreted matter is then channelled, by the pulsar’s magnetic field,to the poles where X-ray hot spots are formed due to the in-falling matter impacting withhigh kinetic energy obtained as a result of the great loss of gravitational potential energy.When the magnetic and rotational axes are not aligned then a stationary observer will detectregular X-ray pulses due to the rotation of the hot spot around the axis of rotation.

The final class - the rotation-powered pulsar - is the topic of this chapter. Only one otherVHE γ-ray pulsar (Aleksic et al. 2011; VERITAS Collaboration et al. 2011; Aleksic et al.2012) has so far been detected; the Crab pulsar (PSR J00534+2200; Staelin and Reifenstein1968), this work introduces the second: the Vela pulsar (PSR J0835−4510; Large, Vaughan,and Mills 1968). These pulsars are isolated and have unaligned magnetic and rotationalaxes which causes a periodic change in the photon emission. This chapter will begin with adiscussion of basic neutron star & pulsar physics and will continue with a description of theacceleration & emission models of rotation-powered pulsars. The scope of this chapter is todescribe the current understanding of isolated non-proto-neutron stars1. This limits the scopeto cover the physics of the Vela pulsar, which is the subject of a large part of this work, andprevents this chapter from becoming overwhelmingly long.

1Proto-neutron stars are those that have recently formed i.e. with an age < 100 yr

5

Chapter 1. Physics of Pulsars

1.1. Pulsar Properties

Hewish et al. 1968 discovered pulsars in the data from a radio antenna and, initially, due totheir precise periodicity, were viewed as an indication for extra-terrestrial life. This thoughtwas, however, quickly replaced once more such sources were discovered that were spatiallyseparated from the first (three other pulsating sources of radio were published along with thefirst: Hewish et al. 1968). Baade and Zwicky 1934 suggested the existence of neutron stars asstellar remnants with a density approaching that of atomic nuclei and consisting mainly ofneutrons. This theory was then advanced in the year prior to Bell-Burnell’s discovery byPacini 1967 and the year after independently by Gold 1968 who suggested that such a stellarremnant that rapidly rotated could power the Crab Nebula. Cocke, Disney, and Taylor 1969confirmed this suggestion with the discovery of pulsations coming from the direction of thenebula. The Crab pulsar’s 33ms period ruled out a white dwarf star as the origin of thisradio emission. This is based on the consideration of the acceleration exerted on a mass atthe surface of a spherical star of radius R, mass M spinning with an angular velocity Ω:

Ω2r =GM

R2. (1.1)

Rearranging this equation relates the mass density ρ = M43πR3 to the rotational period P = 2π

Ω

ρ =3π

GP 2. (1.2)

This means that for a white dwarf to be the origin of the Crab pulsations it would require∼nuclear density. This problem led the way for the neutron star to be considered as theremnant of a collapsed massive star and the source of the radio pulsations.

The angular momentum of a collapsing stellar core is, of course, conserved during collapse.Considering a red giant star’s core with radius 0.2R� and period 100 day (Goupil et al. 2013;Mosser et al. 2012) the pulsar period would be 45ms. This is very near that of the youngCrab pulsar which has a 33ms period (Manchester et al. 2005). According to this simplecalculation the pulsar period can be related to the radius of its progenitor stellar core by theconservation of angular momentum:

Ppulsar = 0.45ms

(Rpulsar

10 km

)2(Rcore

R�

)−2(Pstar

P�

)(1.3)

where the solar values are taken as R� = 695, 500 km (Emilio et al. 2012) and P� =25day (Wohl et al. 2010). Clearly, the progenitor core of the Crab pulsar was somewhatdifferent from the sun.Rotation-powered pulsars exist in two distinct groups: pulsars and millisecond pulsars

which are distinguished based on their period. Millisecond pulsars are much older, havingbeen spun-up by a binary partner that has overflown its Roche lobe, with a period ≤ 10ms,whereas those pulsars that have not had a period of spin-up tend to have a period > 10ms(see Figure 1.10 in Section 1.6.1). There is a period limit below which a pulsar would break-up.Applying the virial theorem (Clausius 1870) to a neutron star considered as a rotating sphere

6

1.1. Pulsar Properties

with moment of inertia I = 25MR2 gives:

Erot =1

2IΩ2 =

4π2I

2P 2< Ekin =

1

2Vgrav =

1

2

GM2

R(1.4)

and assuming the neutron star has a mass M = M� and radius R = 10 km

Pmin = 0.3410−5

(R

10 km

) 32(

M

M�

)− 12

ms. (1.5)

A limit can be derived on the mean mass density by requiring that the rotational accelerationRΩ2 does not exceed the gravitational acceleration GM

R2 which leads to the limit

ρ >3π

GP 2. (1.6)

This is a weak limit since as P becomes small a pulsar becomes oblate causing the centrifugaleffect at the equator to increase and the gravitational acceleration to decrease.Approximating a pulsar as a rotating dipole means it will loose energy due to magnetic

braking. The period P = 2πΩ , therefore, increases with time at a rate P and a proportion of

the pulsar’s rotational kinetic energy is converted to observable emission. Again, consideringthe pulsar as a sphere with moment of inertia I = 2

5MR2 it has rotational kinetic energyErot =

12IΩ

2. The loss is then the time differential of this quantity:

dErot

dt=

1

2

(IΩ2 + 2IΩΩ

)= −CΩn (1.7)

where a dot above the quantity indicates its differential with time and it is approximated thatM & R are constant with time. Here n is the level of multipolarity. The time differential ofthe angular frequency is

Ω = − C

I(Ω)

(1 +

Ω

2I(Ω)

dI

dΩ

)−1

Ωn (1.8)

If it is assumed that the moment of inertia does not vary with angular frequency then thissimplifies to the spin-down law

Ω = −KΩn (1.9)

where K = CI . The magnetic braking index

nbr ≡ΩΩ

Ω2(1.10)

is equal to n when Ω is small or I does not vary with the angular momentum otherwise

n(Ω) = n−3 dIdΩ + 1

Ω2d2IdΩ2

2I +Ω dIdΩ

. (1.11)

The efficiency of this emission process η =Lγ

Evaries with energy and can reach 10−(1...2) (Fermi-

LAT Collaboration, Abdo, and al. 2010) for γ-ray emission. nbr varies based on the mechanismsinvolved that may - for a pulsar - vary with time but for a rotating magnetic dipole it istaken as three (Johnston and Galloway 1999). Departures from this value could be caused

7


by unseen glitches or phase transitions in the core (Glendenning, Pei, and Weber 1997) (seeSection 1.2.2.2). Johnston and Galloway 1999 approach the definition of nbr differently andintegrate Equation 1.92 in the time interval t1 → t2 obtaining:

n = 1 +Ω1Ω2 − Ω2Ω1

Ω1Ω2(t2 − t1). (1.12)

This means that the measurement of Ω & Ω - and not Ω - within a time interval is sufficientto calculate the braking index.

Considering a pulsar has a magnetic dipole moment m = m0 exp−iωt (where m0 = BsurfR

3

and m = −ω2m) which therefore radiates based only on the perpendicular component of the

magnetic moment m⊥ with power Udipole =23m⊥2

c3. This means the radiated power can be

written also with respect to the inclination angle between the magnetic and rotation axes α:

Udipole =2

3c3(BR3 sinα

)2(2π

P

)4

. (1.13)

The radiation is emitted at the frequency ν = P−1 ≈ 1 kHz and cannot penetrate the ionisedinter-stellar medium but rather illuminates a nebula. For the canonical neutron star withmagnetic inclination angle α = 20 ◦ and period 30ms, Equation 1.13 gives Udipole = 1038W.Equating this with Equation 1.7 gives a spin down rate 10−13 ss−1, the Crab pulsar spin downrate is measured as 4.22765×10−13 ss−1 (Manchester et al. 2005). Equating Equation 1.13 and1.7 allows a lower limit to be placed on the perpendicular surface magnetic field B⊥ = B sin2 α:

B⊥ >

√3c3

8π2

I

R6PP (1.14)

For the canonical neutron star with I = 1038 kg m2 & R = 104m this is B⊥ > 3.2 ×1019

√Ps P G3.

A characteristic age can be given to a pulsar based on its periods and the rate at which theperiod is increasing with time (this, therefore, does not apply to pulsars that are being or havebeen ‘spun-up’ by a binary partner star). The torque slow down (torque acting on the pulsarduring magnetic dipole radiation) can be written in terms of the braking index T ∝ −Ωn.Assuming n is constant in time, equating it with the torque slow down of a magnetic dipoleT = IΩ = − 2

3c3B2

R6Ω2 (Manchester 1992), and rearranging using Equation 1.9 shows that

Pn−1P is constant in time. Integrating this expression from t = 0 to the current pulsar age τgives:

τ =P (τ)

(n− 1) ˙P (τ)

(1−(P (0)

P (τ)

)n−1). (1.15)

When the assumption is made that P (0) � P (τ) and the braking index follows the magneticdipole model n = 3 then τ can be written tc ≡ P

2P, which is know as the characteristic pulsar

age 4.There is a limit on the radius out to which a pulsar - or any rotating object - can co-rotate.

2Differentiating Equation 1.9 leads to the definition of nbr in Equation 1.103The unit Gauss = 10−4 T for magnetic field density is convention in pulsar physics4A millisecond pulsar’s age is difficult to determine and is estimated from models of its formation into abinary rather than using this method

8

1.2. The Compact Star

Considering the tangential velocity of a material in co-rotation with a sphere rotating withangular velocity Ω, a light cylinder can be defined as the distance from the rotation axis atwhich this velocity reaches the speed of light rlc sin θ = c

Ω for a polar angle θ (usually boundedabove the pole at r = Pc) (Gold 1968). More usually rLC is considered at the rotation equatorand defined as rLC = c

Ω . This, for the canonical neutron star mentioned above, is at a distanceof 2.6× 109m and for the Crab Pulsar at 107m from the axis of rotation. This limit has aneffect on the possible models of the pulsar environment since it is an absolute limit on theradius out to which co-rotation is possible. The exact processes and emission regions involvedwith pulsed radiation from rotating neutron stars are still unclear and a discussion of thecurrent theories follows. Firstly, however, the physics of the stellar remnant itself will besummarised.


The equation of state (EOS) is the relation between the state variables of a system. Eachcomposition model for the compact star has a different EOS which describes the way thestar’s radius varies with its mass; measurements of the mass and radius of compact stars canseverely constrain these models and greatly help understanding. Each EOS also predicts amaximum possible rotation frequency and a moment of inertia. Figure 1.2 displays some ofthese models - some, for example, the strange star, are so exotic they contain only a thin layerof neutrons at the surface.

A compact star’s core is studied so intensely because its physical conditions are irreproduciblein the laboratory. The EOS has two of the three requirements so far, the first is the Tolman-

Oppenheimer-Volkoff (TOV) equation. Tolman 1939 & Oppenheimer and Volkoff 1939consider non-relativistic hydrostatic equilibrium within general relativity and arrive at theTOV equation

dP(r)

dr= −G

r2

(ρ(r) +

P(r)

c2

)(M(r) + 4π r2

P(r)

c2

)(1− 2GM(r)

c2r

)−1

. (1.16)

It describes the dependence of pressure P on radius within the star but is, however, derived fornon-rotating bodies thus serving as an approximation for all but millisecond pulsars (for whichit is not accurate). The solutions have, as a parameter, the central density of the neutron starand require a stability condition stipulating that the mass must increase monotonically withdensity. The second part of the EOS is the relation of mass m with radius r: dm

dr = 4πr2ρ.These two provide two thirds of the ingredients of the EOS; a pressure density relation P(T, ρ)is, however, lacking. All modern models approximate pressure to be a function of only themass density since the Fermi energy 5 is far higher than the thermal kinetic energy allowingthe approximation T ≈ 0 to be made. Knowledge about the physics of the strong interactionbetween hadronic matter in such extreme conditions is what is needed to progress in the longterm.

The rotating star that comprises a pulsar is often referred to as a neutron star in line withthe reasoning explained in Section 1.1 but the composition is far from certain which leadsto such stars being labelled as compact stars. A compact star consists of several layers, so

5The Fermi energy/momentum is the energy/momentum the most energetic fermion has when it is part of apopulation in the ground state. This is a result of the Pauli exclusion principle requiring no two fermionsto be in the same quantum state.

9


Figure 1.1.: Simplified cross-section of a neutron star with characteristic sizes and densities.Taken from Y Potekhin 2010

much is generally agreed upon. The basic model, first suggested by Ginzburg 1971, envisageda core and an envelope each being subdivided into outer & inner as shown in Figure 1.1.Beyond this, the structure of the envelope - or at least the crust near the surface of the star- is also more in consensus since the pressure is considered low enough for separate nuclei.Figure 1.2 shows six of the main theories concerning the composition of compact stars all ofwhich agree upon the composition down to the third layer (labelled as the crust). Fartherinto the star the mass density increases and models begin to disagree when nucleons existfreely. Except for the strange star model, the figure shows agreement about the compositiondown to the outer core of neutrons (n), protons (p), electrons and muons (μ) surrounded by alayer of superconducting protons. It is clear from the number of possible compositions thatthe physics at such mass densities as those found in the inner core is not well understood.The acronyms used for the description of the quark-hybrid star and the strange star - 2SC,CFL, etc. - are explained in Section 1.2.2.2.

Formation Stars inevitably run out of the fuel that powers them. Nuclear fusion is no longerviable when the stellar core matter has become mostly iron - the most tightly bound productof fusion in a star - and energy can no longer be extracted by nuclear fusion. The stellarcore becomes layered with iron at its core and lighter elements surrounding it. Dependingon the core mass, at this point stellar core collapse can occur in what is known as a Type IIsupernova: the outer envelope of the star is blown off at a speed of ∼ 0.1c due to the release of∼ 1053 erg of gravitational potential energy to form a nebula - known as a supernova remnant(SNR) - and the core, no longer held by the pressure of the fusion reaction, collapses undergravity. During the formation of a compact star, the core experiences a collapsing energydue to gravity that can be estimated from Newtonian gravity as ∼ 1044 J of which around1% goes into accelerated particles that form a supernova remnant nebula and 0.01% intoelectromagnetic radiation (Blanchard and Signore 2006). The rotation and magnetic fieldas well as any accretion of matter have the greatest effect on the evolution of a compactstar. Neutrino emission from the core and electromagnetic emission from the surface, at firstquickly cool the ∼ 1010...11K (Dexheimer and Schramm 2008) proto-compact star. At the end

10


Figure 1.2.: Cross-section of a compact star for various theoretical models. Filled black arrowspoint to cross-sections for each of these different models. Taken from Weber 2005

11


of a compact star’s lifetime - after all of the magnetic and thermal energy has been used up -it simply fades away.

Density Estimates In terms of density, a compact star is akin to a 20 km diameter atomicnucleus and has a mass limit, according to the Tolman-Oppenheimer-Volkoff limit (Oppen-heimer and Volkoff 1939) (see later in this section), of ∼ 3 . . . 5M�. This limit is, however,highly dependent on a compact star’s make-up. A neutron star is the canonical object shownas the “Traditional neutron star” in Figure 1.2. The group of tightly bound stellar remnantswhich, with regard to density, lie between white dwarfs & black holes and that have a nominaldensity ∼atomic nuclear density can be labelled as compact stars. The neutron degeneracypressure that stabilises the canonical neutron star (see later in this section) originates similarlyto the electron degeneracy pressure of a white dwarf but is far stronger because neutrons

are ∼ 2000 times heavier than electrons. The ratio of the de Broglie wavelength, λ =h

γmv,

of both particles at a constant, very high, temperature (v ≈ c) is simply the particle massratio; a white dwarf is ∼ 104 km (Kippenhahn and Weigert 1990) in diameter and a canonicalneutron star ∼ 101 km the ratio of which is on the order of the mass ratio of the neutron andelectron.The mass of the core largely determines the form of the remnant, the presence of heavy

elements in different ratios means putting limits on the total mass of the star to form differentremnants is not helpful. If electron degeneracy pressure is enough to stop the collapse awhite dwarf star is formed, this occurs for cores with mass less than the Chandrasekhar

mass MC = 1.4M� (Chandrasekhar 1931b; Mazzali et al. 2007). Beyond this mass is theneutronisation limit, which depends on the neutron degeneracy pressure. It is stronger thanthe electron degeneracy pressure because the neutron is almost 2000 times heavier thanthe electron. The density at the neutronisation limit can be estimated by considering therelativistic momentum of particles in the electron-degenerate (Fermi) gas p = 1

c (E2 −m2

ec4)

12

and the pressure in such a relativistic degenerate gas (for a derivation see Kippenhahn andWeigert 1990, Section 15.2). The relation of the mass density of electrons ρ to the Fermi

momentum pF is then6

ρ = μemu8πm3

ec3

3h3

(pFmec

)3

. (1.17)

This applies to the electron density and therefore the factor μemu has been used as itrepresents the mean particle mass per free electron. Using a reduced mass7 for an equalnumber of protons and electrons (mu ≈ me) results in a density of 1.2× 1010 kgm−3 abovewhich neutronisation occurs in electron-proton plasmas held by electron degeneracy pressure.During stellar collapse the situation is more complicated due to the presence of heavy nucleithat absorb electrons through electron capture reactions N(A,Z) + e → N ′(A,Z − 1) + νe.

This causes the density to increase slower than that of an ideal Fermi gas (P ∝ ρ43 ) but also

leads to nuclear fission due to the increase in number of neutron-rich nuclei. These fissionreactions release free neutrons into the stellar core in a process known as the neutron drip.This regime is passed when the newly created neutrons start to contribute to the pressure

6This is the highest particle momentum when all particles in a population occupy the lowest possible state.7The reduced mass is simply defined as the inverse of the sum of the inverse of the component particle masses:m−1

u =∑

particles i m−1i but this is most often only for two-body situations.

12


in the system, beyond this, poorly understood neutron interactions become important. Thisneutron drip regime begins at around 2.2× 1014 kg m−3 (Haensel, Potekhin, and Yakovlev2007) which, due to the electron capture reactions, is achievable in the stellar core.

A crude way to estimate when the electron capture reaction becomes energetically favouredis to consider when the Fermi energy reaches the mass difference between the reactants andproducts (neglecting the mass of the neutrino which is small (Ahmad et al. 2001)) in aninverse beta decay reaction:

EF =h2

2m

(3π2 ρ

m

) 23= (M(A,Z − 1)−M(A,Z)−melectron)c

2 = Δmc2 (1.18)

ρ =1

3π2

(2mΔm

h2

) 32

. (1.19)

For a white dwarf star (primarily made up of carbon and oxygen) electron capture is favouredat a mass density of ∼ 1011...13 kg m−3 for different reactants (value taken from Chamel andFantina 2015, using a more complex method). This is well below the nominal neutron stardensity of∼ 1017 kg m−3 (see later in this section for a brief explanation of this value) and closerto, but above, the white dwarf density, which is limited to 2.162× 109 kg m−3 (Chandrasekhar1931a).

The Canonical Neutron Star The foundational model of compact stars - the neutron star- suggests that the degeneracy pressure of neutrons supports the star and prevents furthercollapse. The compactness parameter xg =

rgR (rg = 2GM

c2is the Schwarzschild radius

and R the radius of the star) parametrises the effects of general relativity. For the canonicalneutron star with radius 10 km and mass 1.4M� then xg = 0.413 whereas modern equations ofstate suggest, at this mass, the radius will be 12 km thus altering this parameter to xg = 0.344.This significant difference suggests that checks of general relativity can be made using pulsarsand that general relativity must be taken into account when considering equations of state.Beyond the TOV mass (∼ 2 . . . 3M� (Y Potekhin 2010)) - a mass limit derived for bodies thatobey the TOV equation - neutron degeneracy pressure can no longer equal the force of gravityand, inter alia, a black hole, hypernova or GRB may occur. Considering the neutron star asa sphere of nuclear density 8 ρ0 = 1.2× 1017 kg m−3 with a mass at the Chandrasekhar

limit (Chandrasekhar 1931b) M = 1.4M� (Cutnell and Johnson 1995) gives a radius fromρ = M

43π r3

of R = 1.2× 105m. The canonical value is taken as 10 km with mass 1.4M�. A

commonly associated magnetic field strength to pulsars is 1012G which can be obtained usingthe canonical values given here in Equation 1.14:

B = 3.2× 1019(PP) 1

2G (1.20)

which with fiducial values of P = 1 s & P = 10−15 (Figure 1.10 shows these values to bereasonable) gives the usual magnetic field strength.

8This was calculated using an approximate nuclear radius rn = 2.13 fm (Mohr, Taylor, and Newell 2012) andaverage nucleon mass equal to an atomic mass unit (amu) mamu = 1.67 × 10−27 kg (Taylor 2009) thenρ = ma.m.u×N

43πr3n

for three nucleons N = 3.

13


The Doppler Effect Due to a compact star’s extreme density, the intense gravity significantlychanges the rotational frequency of the emitted radiation. Considering a photon emitted fromthe surface with rotational frequency ω0 and detected at infinite distance with ωinf then theDoppler Effect can be characterised by zg:

zg =ω0

ωinf− 1 = (1− xg)

− 12 − 1 (1.21)

and related to the compactness parameter xg. This effect also shifts the effective surfaceblack-body temperature of the neutron star T inf

eff = T 0eff

√1− xg and the apparent radius Rinf

from the equatorial radius R to Rinf = R(1 + zg) thus changing the canonical neutron starradius to 13 km and the realistic one to 15 km. This also inevitably affects the γ-ray luminosity

since Linfγ ∝ R2

inf

(T infeff

)4which is linked to the compactness parameter as Linf

γ = (1− xg)L0γ .

This effect not only alters the absolute values but the light bending effect of the intensegravity can enable both magnetic poles to be viewed at the same time by a carefully placed,stationary observer.

1.2.1. Envelope

The density of the envelope, which includes the outer crust, inner crust, and the mantle layersshown in Figure 1.1, is low enough to allow separate atomic nuclei to exist and is comparativelywell understood. It has an impact on the electromagnetic spectrum emitted at the surfaceand can be used to study the core. Moving from the edge of the outer core outwards, the firstelement to be reached is the mantle that consists of the so-called ”nuclear pasta”. In contrastto the liquid drop model (Gamow 1928), where nuclei are modelled as spheres, it becomesenergetically favourable for the nuclei to change form. They go through different stages suchas flat (”lasagne”) or stretched (”spaghetti”) and act as a liquid crystal (Pethick and Potekhin1998). It is, however, unclear whether these states are formed at all in collapsing supernovacores (Watanabe et al. 2009).Beyond the mantle is the neutron super-fluid inner crust which exists above the neutron

drip density ρndrip ≈ 4 . . . 6 × 1014 kg m−3 (Y Potekhin 2010) (see earlier in this section)around a rigid lattice of nuclei (Negele and Vautherin 1973). Moving further outwards bringsa lack of free neutrons that characterises the beginning of the outer crust. It consists ofdegenerate electrons and completely ionised ions in either a Coulomb liquid or Coulomb

crystal depending on the interactions between the ions. The ocean, as shown in Figure 1.1, sitsatop the outer crust and is a mixture of ions (completely and partially ionised) and electronsin liquid form. The ocean smoothly transitions into the atmosphere, also shown in Figure 1.1,in all but those neutron stars with the highest B-field in which case only a surface crust exists.

Atmosphere The atmosphere is a plasma. As the final layer it is that which is observed andcontains information about the magnetic field, chemical composition, gravitational accelerationand surface temperature of the neutron star. Its size ranges from a few millimetres to tensof centimetres depending on the surface temperature. Its chemical composition affects theabsorption of photons passing through it; the bottom of the atmosphere is defined wherethe optical thickness reaches unity. Models of the atmosphere for a range of magnetic fieldstrengths B ∼ 1012...13G (above which quantum electrodynamic effects may become significant)and effective surface temperatures Teff ∼ 1 . . . 5×106K suggest the presence of carbon, oxygenand nitrogen (Mori and Ho 2007).

14


1.2.2. Core

The core of a compact star is perhaps its most intriguing aspect, this view is supported bythe great variety of models that address its make-up, some of which are shown in Figure 1.2.Ginzburg 1971 suggests a two-part core - shown in Figure 1.1 - both regions several kilometresin depth, the outer of which has a mass density of 0.5 . . . 2ρ0 and the inner of ρ ≥ 2ρ0 (ρ0 isthe nominal nuclear density).

1.2.2.1. Outer Core

The composition of the outer core is comparatively well understood since at that mass density(ρ ≤ 2ρ0) the Hamiltonian can be calculated (Haensel, Potekhin, and Yakovlev 2007). Itis a mixture of a neutron superfluid & superconducting protons amongst highly degenerateelectrons and muons (Haensel, Potekhin, and Yakovlev 2007). The energy density can beexpressed as separate contributions from the Fermi liquid of nucleons and the Fermi gas ofleptons (electrons and muons):

ε (nn, np, ne, nμ) = εN (nn, np) + εe(ne) + εμ(nμ) (1.22)

where ni are the number densities for each of the particle types: neutron, proton, electronand muon. For a fixed baryon number density nb = nn + np and with the electro-neutralitycondition np = ne + nμ minimising the energy density specifies the EOSs and particleconcentrations. This leads to the relations between the chemical potentials of each of theseparticles μn = μe + μp and μμ = μe (where μi =

∂E∂ni

for i = n, p, e, μ). This is based on thereactions that enable equilibrium between these particles and cool the star due to the loss ofthe neutrinos (� = e or μ):neutron decay

n → p+ �− + ν� (1.23)

and electron capture:p+ �− → n+ ν�. (1.24)

Only the proto-compact star is opaque to neutrinos, therefore, for most compact starsμν = 0 (Haensel, Potekhin, and Yakovlev 2007). At this density, electrons are ultra-

relativistic μe = cpFe ≈ 122.1(

ne0.05n0

) 13MeV (n0 = 0.16 fm−3 is the nuclear satura-

tion density (Tondeur, Berdichevsky, and Farine 1986)) and muons are fairly relativistic

μμ = mμc2

√1 +

p2Fμ

mμc2(Haensel, Potekhin, and Yakovlev 2007). If the equilibrium state is

known then so is the equation of state since P = n2bd(ε/nb)dnb

, this reduces the search for anequation of state to that for the energy density ε(np, nn) as a function of the componentnumber densities.

1.2.2.2. Inner Core

The inner core is the subject of most disagreement due to the lack of a mathematical descriptionof the interaction of the strong force at high densities. Similar to the outer core, it is expectedto be several kilometres in radius but with a mass density ρ ≥ 2ρ0. Almost all models allowfor the hyperonisation of the nucleonic matter which introduces strange quarks. Exotic modelsinclude theories of pion and kaon condensates, de-confined quark matter and even higher

15


quark states of tetra-, penta- and six-quark states such as the usus, and the Ω+. All of theproduction processes of these states require the presence of a spectator massive particle N tobalance the kinematics of the reactions.

Hyperonisation This is the process of introducing hyperons - hadrons containing strangequarks - into matter. If the energy density decreases - at a certain baryon density - withthe conversion from a nucleon into a hyperon then it is energetically favourable to do so andthe matter will be populated with hyperons. Current models expect hyperonisation to onsetat densities ρ ≥ ρ0 (Y Potekhin 2010) implying a significant population of hyperons in theinner core of a compact star. These hyperons would simply decay were it not for the extremedensities of the inner core. The chemical potential μ generally increases with mass density atdifferent rates for different particle species. The threshold for the production of the lightesthyperon Λ (an up, down and strange quark) is:

μΛ ≤ μn (1.25)

which occurs at some critical baryon density nΛb below which the Λ simply decays via the

exothermic interaction Λ+N → n+N . The slightly more massive Σ− hyperon actually appearsbefore the Λ because the neutrality condition loosens its production threshold (Salpeter 1960)

μΣ− ≤ μn + μe− . (1.26)

This makes Σ+ (two anti-down quarks, one anti-strange), and other positively chargedhyperons, unfavourable since their threshold conditions contain the opposite sign of μe− . Athigher densities nb > 5n0 as the equilibrium of Σ− baryons and electrons shifts towards Σ−,several models predict the emergence of many Σ+ baryons and the loss of leptons at a numberdensity around nb ≥ 1 fm−3 resulting in a baryon soup with mean strangeness of around minusone (see Y Potekhin 2010, and references therein). The main points of theory lacking here arethe interactions between hyperons and between nucleons and hyperons.

Meson Condensate Further to baryonic hyperonisation, strange mesons can also be producedin a condensate by high-momentum electrons close to the Fermi momentum at around2n0 (Weber, Negreiros, and Rosenfield 2007). It is the in-medium properties of the kaonthat favour its production over the lighter pion (Kaplan and Nelson 1986) which suffers frompion-nucleon repulsion. The condensation reaction

e− +N → N ′ +K− + νe (1.27)

must, of course, involve a spectator massive particle N . The energy advantage is gotten fromthe bosonic, rather than fermionic as with the electron, nature of the kaon. In neutron richmatter, the kaon mass m

K varies (see Weber, Negreiros, and Rosenfield 2007, and referencestherein) according to

mK−

mK−= 1− 0.2n

n0. (1.28)

Based on the density, the effective kaon mass can be reduced to around 200MeV - from avacuum rest mass 495MeV (Nakamura and Particle Data Group 2010) - which is achievablefor a core number density of around 3n0. This condensation accompanies a higher neutrinoluminosity - evident in Equation 1.27 from the neutrino being produced - and, therefore, rate

16


of neutrino cooling. The requirement for the creation of a kaon condensate in this way is thatbaryonic hyperonisation does not occur too prevalently replacing all of the electrons.

Nucleon Star Stars where the fraction of protons and neutrons equalises are known asnucleon stars. The reaction

n+N → N + p+K− (1.29)

could be responsible for their creation since it lowers the energy per baryon (Brown 1996).The nucleon star maximum mass Mmax = 1.5M� - suggested by Thorsson, Prakash, andLattimer 1994 - limits the possible interpretations of various observations of compact stars,including the Vela X-1 compact star MV elaX−1 = 1.86± 0.16M� Barziv et al. 2001, implyingthey should be low mass black holes. Uncertainties about the attraction between kaons andnucleons renders this interpretation uncertain.

De-confined Quark Matter Ivanenko and Kurdgelaidze 1969 first proposed that compactstars may have a de-confined quark core. The development of quantum chromo-dynamics(QCD) resulted in studies involving the non-interacting quark model and perturbation the-ory (Collins and Perry 1975); these are, however, only valid for energies above those realisticinside the core of a compact star ( 1GeV). A simple estimate for the onset of quarkde-confinement can be made by requiring nuclei to be packed closer than their nominal radiusrN ∼ 1 fm giving ndeconf = 1.5 n0, however, changing this slightly rN = 0.5 fm, clearly gives avery different result ndeconf = 12n0; the number density range in which hardrons begin todissolve can, therefore, be said to be nb ∼ 2 . . . 10 n0. Considering the Fermi energy at thisdensity and the strange quark mass ms = 95 MeV

c2(Nakamura and Particle Data Group 2010),

the up and down quarks will readily convert to strange quarks meaning that the de-confinedquark matter should consist of up, down and strange quarks. Extending this considerationto the other quark flavours shows that nb ∼ 100 n0 is required for the next more massivequark (the charm quark) which is unrealistic inside a neutron star core. A compact star witha de-confined quark core and a population of charm quarks would be unstable to so-calledr -mode rotational instabilities (Weber 2005). These instabilities - fundamental f or rotationalr modes - are a result of counter-rotating surface vibrational modes (Weber, Negreiros, andRosenfield 2007) which are damped by viscosity and driven by the gravitational radiationthat they reinforce. These modes set a more strict mass limit than rotation induced massshedding (see Section 1.1). Hyperonisation in compact stars with cores cooler than T = 109Kmay limit the effect of the r -mode instabilities due to the additional viscosity contributedby the strangeness (Weber, Negreiros, and Rosenfield 2007). It is thought that observationsof the timing of the milli-second pulsar population could be used with the dependence ofthe r -mode instabilities on the microscopic properties of the core in order to constrain itscomposition (Alford and Schwenzer 2014). These instabilities, however, remain unobserved.The possible observational consequences of a star undergoing a phase transition to a de-

confined quark core - without seeking to specify at which radius or mass this would occur -according to Glendenning, Pei, and Weber 1997 are:

• The braking index (see Section 1.1) has a value far from the canonical value n = 3,possibly by orders of magnitude.

• Anomalous braking indices can be observed over a long period since the pulsar spindown is slower when | I | is large.

17


• The pulsar may spin up.

• The beginning and end of the spin up era are characterised by a large Ω leading to aneasy-to-measure braking index.

• Approximately 1% of pulsars may be passing through a phase transition epoch.

This is based on the change of the moment of inertia I of the neutron star during thephase transitions (Glendenning, Pei, and Weber 1997) which alters the braking index duringthe change according to Equation 1.11. The compact star could undergo consecutive statetransitions in which each of the three lightest flavours of quark is de-confined at a density ofnb ∼ n0, nb ∼ 3 n0 and nb ∼ 7 . . . 11 n0 (Y Potekhin 2010) for down, up and strange quarksrespectively (Blaschke et al. 2009). All models are, however, flawed due to their treatmentof the quark and baryon phases in different frameworks making them non-self-consistent.Nevertheless, a mixed state can enable a first order phase transition to quark matter. Theoryremains as yet unable to make strong claims on the properties and existence of a deconfinedquark core. Theoretical predictions of deviations of the braking index from three could beused with measurements of the braking index as a method of determining the composition ofthe core.

Color Superconductivity With the concept of bulk deconfined quark matter comes thatof color superconductivity through the formation of color Cooper pairs formed due tothe strong interaction between quarks of different flavours. These pairs cannot be colorneutral so the condensate breaks local color symmetry (Weber, Negreiros, and Rosenfield2007). Furthermore, the phase diagram (temperature versus chemical potential in quarks)becomes fairly complex (see Alford 2001 for a detailed description) due to the fact that quarksposses different masses, flavours and colors. It should be noted that the bulk matter remainselectrically and color neutral as well as in chemical equilibrium under the flavour change ofthe weak interaction (Becker 2009). The color superconducting state could affect variousaspects of the compact star which, according to include:

• Cooling through neutrino emission.

• The pattern of the neutrino arrival times from a supernova.

• Magnetic field evolution.

• Rotational stellar instabilities.

• Glitches.

Much theoretical work is still lacking before any clear predictions on any of these effects canbe made.Blaschke, Sedrakian, and Shahabasyan 1999 show that the quarks exist in a Type II

superconducting state (see 1.2.4.1) that allows the magnetic field to both penetrate the bulkcolor superconducting quark matter and have such a large decay time (see Section 1.2.4). TheCooper pairing required for the Bardeen-Cooper-Schrieffer (BCS) phase can occur asisotropic (non-LOFF) pairing whereby two quarks of different flavours and equal & oppositemomenta combine near the Fermi surface. Larkin, A. I. and Ovchinnikov, Y. N. 1964;Fulde and Ferrell 1964, however, determined there may be an intermediate phase (LOFFphase; named after the four authors of the aforementioned works) between the normal and

18


BCS phases in which the Cooper pairing occurs between quark pairs with non-zero overallmomentum. The LOFF state is expected to break translational & rotational invariance and,in position space, it describes a condensate that varies as a plane wave which is expected tomanifest itself as a crystalline structure. A situation where all three quark flavours participatesymmetrically - the color-flavour locked (CFL) phase - is the QCD ground state, assuming thestrange quark mass ms → 0. If, however, the strange quark is considered as massive enoughto be ignored, the two-flavour superconducting (2SC) phase takes over (Weber, Negreiros,and Rosenfield 2007).

R-mode instabilities, as mentioned earlier in this section, have a great affect on such quarkstars (those compact stars composed in part by deconfined quark matter). Depending onthe energy gap Δ of the color superconducting state 9 Δ ∼ 100MeV can render all CFLphase cores unstable to r -mode instabilities (Madsen 1998). The CFL phase and the 2SCphase - although less definitively - are both ruled out by the lack of a rapid pulsar spindown (Madsen 2000). This is based on an exponential reduction in viscosity and due tothe r -mode instability in many low-mass X-ray binaries and rapidly spinning pulsars. Thepossibility of meta-stable quark matter in either the CFL or 2SC phase in a hybrid star could,however, be possible (Madsen 2000).

1.2.3. Cooling

X-ray black-body emission from the surface of the compact star driven by heat conductionfrom the core and neutrino emission from the entire body of the star causes the star tocool. A central process is the modified Urca (murca) process which involves the consecutiveinteractions

n+N → p+N + �+ ν� (1.30)

p+N + � → n+N + ν� (1.31)

where � = e or μ. The modified aspect of these interactions comes from the spectator nucleonN without which these interactions are referred to as the (direct) Urca (durca) process (Gamowand Schoenberg 1941). The value of the relative abundance of protons xp - with respect toa critical abundance parameter xc that depends on the muon abundance - determines theactive cooling processes (Haensel, Potekhin, and Yakovlev 2007). In order for energy andmomentum to be conserved in the low proton abundance regime xp < xc, the mUrca processis dominant. For xp > xc the powerful dUrca process takes over and a state of advancedcooling is achieved.Considering n, e, p, & μ matter, nucleon-nucleon (N -N) bremsstrahlung(� = e, μ, τ & NN ′ = nn, np, pp):

N +N ′ → N +N ′ + ν� + ν� (1.32)

is also important. The emissivity of each of these processes varies significantly. Considering anon-superfluid core, the dUrca process (or a similar process for pion/kaon condensate coresinvolving a quark flavour change) is up to seven orders of magnitude stronger than the mUrcaand N -N bremsstrahlung processes. Super-fluidity in the core has the effect of suppressingthese processes (Haensel, Potekhin, and Yakovlev 2007). If neutron superfluidity is includednew processes are started in the core and inner crust such as quasi-baryon annihilation of

9A superconducting state occurs due to the Cooper pairs being at a slightly different energy than thesurrounding particles which inhibits the sort of interaction that leads to resistivity. This energy differenceis usually described as an energy gap with symbol Δ.

19


broken Cooper pairs of neutrons n+ n → n+ n+ ν� + ν�,� = e, μ, τ . The excited quasi-neutrons are recombined back into the condensate (Flowers,Ruderman, and Sutherland 1976). This process can be three orders of magnitude strongerthan the mUrca process in non-superfluid cores (Haensel, Potekhin, and Yakovlev 2007).

For the first 100 y the compact star is non-isothermal, undergoing thermal relaxation. Forages up to ∼ 105 y, neutrino emission dominates over photon emission and for older compactstars > 105 y reheating mechanisms may occur from the Ohmic dissipation of the magneticfield (Haensel, Potekhin, and Yakovlev 2007), see Section 1.2.4.Cooling has a great effect on a compact star and changes its magnetic field as well as

its spin-down rate, such change is linked to star-quakes. These events are not very wellunderstood and are theorised to be (Y Potekhin 2010):

• A change of the crust shape.

• A phase transition in the core.

• Interaction between the normal and super-fluid parts of the core and crust.

These phase transitions are able to heat the star, meaning both cooling and heating can occurfrom within the star. Measurements of the age and surface temperature - that is linked tothe internal temperature after thermal relaxation - can be used to constrain compositionmodels since each model predicts a different cooling evolution (temperature as a functionof time) (Y Potekhin 2010). Compact star mass has a great effect on the cooling processes.The neutrino luminosity scales with the internal temperature Lν ∝ T 6...8 depending on thecomposition model and mass. Constraints on the composition can be derived by consideringfour different composition models for which the most important aspect for cooling differs:nucleon dUrca, pion condensate, kaon condensate and Cooper pairing of neutrons. Relativelylow mass stars undergo low cooling rates mainly due to N -N bremsstrahlung regardless ofcomposition (Yakovlev et al. 2005) making them poor for distinguishing core compositionbased on neutrino luminosity or surface temperature. High mass stars, however, have higherrates of cooling and the neutrino luminosity for the four models goes in descending order:Nucleon dUrca, pion condensate, kaon condensate, and Cooper pairing of neutrons.

Figure 1.3 demonstrates that measurements of a compact star’s mass, surface temperatureand age can constrain core composition models. The plot in the figure shows the evolution oftemperature with compact star age for a non-superfluid core. The uppermost line correspondsto low mass compact stars which cool at the same rate independent of the four consideredcompositions. The other four lines are for the maximum possible mass of a compact starconsidered. The shading shows the range in which intermediate mass compact stars mustlie. Clearly, old compact stars are desirable to constrain the front edge of the contours. Themeasurements displayed do not rule out any of the models but more precise measurements ofold pulsars could constrain understanding. A similar consideration for a core of superfluidprotons is also not constraining however adding superfluid neutrons reduces the maximumage for all models to below that of the Geminga pulsar (PSR J0633+1746; Kniffen et al. 1975;Halpern and Holt 1992) in Figure 1.3 thus seemingly ruling out that possibility.

1.2.4. Magnetic Field

The magnetic field of an isolated compact star is created as a result of several contributions (seeY Potekhin 2010, and references therein):

20


Figure 1.3.: Plot of the surface temperature as a function of compact star age in years for anon-superfluid core. Points with error bars are measurements made of compactstars. Figure taken from Yakovlev et al. 2005.

21


• Differential rotation,

• Convection,

• Magneto-rotational instabilities,

• Thermo-magneto effects from the supernova.

Those currents adding to the magnetic field due to differential rotation occur in the innercrust or core where the electrical conductivity is highest. The magnetic field evolves basedon the several factors listed here (Y Potekhin 2010, ; a proto-neutron star’s magnetic fieldchanges far more):

• Ohmic decay,

• Hall drift,

• Starquakes - can cause magnetic field line reconnection,

• Thermoelectric effects.

Ohmic decay of any magnetic field due to the electrical resistance of non-superconductingmaterials in which these current flow can be characterised by a time scale

τD ∼ 4π σ R2

c2(1.33)

for a scale of variation of the magnetic field R ∼ RNS (Baym, Pethick, and Pines 1969). Sincethe conduction proceeds through the highly degenerate electrons, the conductivity σ can bewritten with respect to the electron number density ne and electron Fermi wavenumber kF :

σ ≈ nee2τtrc

h kF. (1.34)

The transport relaxation time τtr can be written in terms of the proton Fermi temperature Tp

and proton Fermi-Thomas wavenumber kFT =√

4kFmpe2

π h2 :

τ−1tr ≈ π2

12

(e2

h c

)2(T

Tp

)2 ck2fkFT

. (1.35)

For approximate values of the proton density ρp = 2× 1016 kg m−3, temperature T = 108K,and neutron star radius R = 10 km - which correspond to kF = 0.7 × 1015m−1, Tp =1.2× 1011K and τtr = 6× 10−14 s - the Ohmic decay time scale as given in Equation 1.33 isτD ∼ 1022 s = 23 × 103tuniv (Bennett et al. 2013 determined the age of the universe to be13.772± 0.059Gyr, 4.33× 1017 s). The magnetic field is locked in to the non-superconductingmatter of the inner neutron star with a decay rate that ensures very little change due toOhmic decay.

Cumming, Arras, and Zweibel 2004 study the relative contributions between Ohmic decayand the Hall drift in the crust of neutron stars. The relative contributions can be characterisedwith respect to the product of two parameters: Ψ - the electron cyclotron frequency for aneffective electron mass m =

EFc2

- and τ - the electron collision time. A large value of Ψ× τmeans that the Hall effect overshadows the Ohmic decay - tHall << tohm. The level of

22

1.3. γ-ray Production Processes in a Pulsar’s Magnetosphere

impurity Q in the lattice structure of the crust (see Section 1.2.1), as well as the surfacetemperature, and magnetic field alter the relative effects of the two contributions. The density,however, has little effect and, therefore, the transition between which effect dominates occursat all parts of the compact star at the same time. The level of impurity and temperaturehave an effect since they distinguish between scattering being dominated by phonons (hightemperature) and impurities (high Q). This shows how accretion can have a great effect onthe evolution of a neutron star’s magnetic field.Thermoelectric effects relate the thermal evolution of the neutron star to that of its

magnetic field. Elements of the thermal and electrical conductivity tensors as well as theplasma thermoelectric coefficients depend on temperature and magnetic field. The change ofmagnetic (Abrikosov) vortices (see Section 1.2.4.1) depend on their interaction with theneutron superfluid (Feynman-Onsager) vortices and the crust surface (Baym, Pethick, andPines 1969) at the boundary of this region.

1.2.4.1. SuperConducting Matter

A Type I superconductor (Landau and Lifshitz 1960) exists in an entirely superconductingstate when below its critical temperature and applied magnetic field. This implies a completelyexcluded magnetic field due to the Meißner effect10 (Meissner and Ochsenfeld 1933). Such amagnetic field is not compatible with compact stars. Type II superconductors, however, existin a state of partial superconductivity with regions of non-zero electrical resistance. Here thematerial can contain magnetic (Abrikosov) vortices where the magnetic field is concentrateddue to its expulsion from the superconducting regions. The characteristic time τnucl to expelthe magnetic field from a macroscopic region with respect to the magnetic field at that pointB and thermodynamic critical field Hc is ∼ τD

B2

H2c∼ 1015 s (Baym, Pethick, and Pines 1969).

This suggests that the magnetic flux is not affected by the conducting state of the material atthat point and the superconducting state of the matter in the compact star has little effect onits magnetic field.The main topics concerning the properties of the spinning compact star have now been

covered. Key properties regarding the mechanisms and understanding of the emission ofradiation from a pulsar and its magnetosphere will now be considered. This topic still requiresmuch work as no satisfactory theoretical explanation supported by observational evidence hasbeen found for the more newly detected γ-ray emission nor the original radio detection ofpulsars in 1968.


γ-ray photons are, in general, not produced through thermal emission of a black body;according to Wien’s displacement law a temperature ∼ 1014K would be required for ablack-body spectrum to peak at ∼ 100GeV. They must, therefore, be emitted by dynamicalphysical processes that involve a great deal of energy probably in a pulsar’s magnetosphere.Electromagnetic fields accelerate electrically charged particles according to Lorentz’s law

�F = q(�E + �v × �B

)(1.36)

10The Meißner effect is the expulsion of the magnetic field during the phase change to a superconductor dueto surface, London currents. It is distinct from the lack of a magnetic field within a perfect conductorwhich occurs due to electromagnetic inductance.

23


Figure 1.4.: Left panel shows the process of magnetic pair production and the right showsphoton splitting in a magnetic field contributed by the solid black line at the baserepresenting a massive charged body.

that describes the force F on a particle with electrical charge q and velocity �v in an electromag-netic field �E and �B. A non-zero electric field directly accelerates electrically charged particlesalong its field lines and if �v× �B �= 0 the charged particles rotate around the magnetic field linesinducing photon emission based on the non-zero Poynting vector �E× �H . Simply consideringthe Poynting radiation emitted by a magnetic dipole using particle-in-cell simulations theenergy cut-off of photons emitted in the magnetosphere of young pulsars measured by theFermi -LAT can be fairly well explained (Gruzinov 2013). Sufficiently energetic accelerationprocesses in a magnetic field include synchrotron and curvature radiation as well as inverseCompton scattering (bremsstrahlung is discounted since the photons emitted would mainlybe in the X-ray energy band).

1.3.1. Magnetic Attenuation

The magnetic field strength decreases farther from the pulsar. This allows photons of higherenergy to escape without undergoing magnetic attenuation. Magnetic attenuation is a resultof two processes which occur in the presence of an electromagnetic field: pair-productionγ → e− + e+ & photon splitting γ → γ + γ, as shown in Figure 1.4.

The attenuation length for both processes decreases with the photon energy, however, pair-production is the stronger process in environments resembling a pulsar’s magnetosphere (Michel1991b).

Magnetic attenuation has been suggested as the reason behind the energy of the cut-offof the γ-ray spectrum observed in many pulsars (Abdo et al. 2013). Figure 1.5 shows theenergy at which this cut off occurs for several pulsars. Pair production limits the energy ofaccelerated charged particles by shielding the accelerating electric field. Because the pairs areaccelerated in opposite directions the accelerating region is shortened by the increase in theamount of charge above and decrease below any given accelerating electron. The maximumelectron Lorentz factor by pair production due to this limitation can be shown to be (see

24


Figure 1.5.: High energy cut-off for several different values of period, magnetic field andemission altitude above the polar cap. Figure taken from Harding 2001.

25


Michel 1991b, Chapter 2, Section 6b for a brief derivation):

γpp < 3.5× 105h0

102m

((R

104m

)(Ω

2π rad s−1

)(B

1012G

)) 12

(1.37)

where h0 is a characteristic accelerating gap length.

1.3.2. Curvature Radiation

A charged particle emits radiation when accelerated due to perturbations in the electric fieldin the particle’s rest frame11. When this acceleration is induced by an external magneticfield then the resulting radiation is variously named cyclotron, synchrotron, and curvatureradiation. The path of a charged particle is helical around the magnetic field line with adrift velocity that causes it to essentially travel along field line. If the curvature of the fieldline is small then the helical component of motion is important and only synchrotron andcyclotron radiation are significant (for relativistic and non-relativistic perpendicular velocitiesrespectively). Curvature radiation is dominant when the helical path is stretched and thedegree of magnetic field line curvature is large.Cyclotron radiation does not reach γ-rays but synchrotron radiation is an active γ-ray

production mechanism. Considering a relativistic charged particle gyrating around a magneticfield line, its minimum frequency of rotation and emission is ω0 = eB

γm which is Landau

quantised (see Appendix Section A). The frequency at which it maximally radiates is ωH = γ3ωc

where ωc =eBme

is the cyclotron angular frequency (in essence a “magnetic bremsstrahlung”spectrum). In the conditions expected around a pulsar the particle emits photons of minimumenergy hωc(γ = 1) = 11.5B12 keV (B12 ≡ B

1012 G) resulting in the rapid de-excitation of the

particle (possibly before it even completes one complete gyration of the magnetic field line)and a near-zero perpendicular velocity to the field line. Synchrotron radiation is, therefore,not associated with γ-ray emission from pulsars in the region near to the pulsar’s surface.

The energy loss due to curvature radiation can be written

˙ECR = −e2β4γ4c

6πε0ρ2c(1.38)

where ρc is the magnetic field line curvature usually far larger than the gyration curvature.Such a particle emits a curvature radiation spectrum similar to that of synchrotron radiation:at low energies ∝ ω

13 and at high ∝ ω

12 e−ω (Ginzburg and Syrovatskii 1965) which peaks at

Ep ≈hγ3c

ρc. (1.39)

The spectrum effectively cuts off at the critical energy (the point at which it drops to aninsignificant level compared to the peak emission (Jackson 1998))

Ec =3hc

2ρcγ3. (1.40)

This cut off can be approximated using an estimate of the magnetic field line curvature for a

11its non-intrinsic magnetic field is introduced when a Lorentz transformation is done to a non-co-movingframe

26


field line that extends to an equatorial radius of about the light cylinder radius RLC from thepulsar, given by

ρc =4

3(RRLC)

12 (1.41)

where R is the star’s radius and RLC ≈ 107m (Sturrock 1971). Using the fiducial stellarradius R = 104m gives ρc = 4× 105m. This results in a curvature radiation peak emissionenergy and cut-off energy (the point at which the intensity drops to a small proportion of itspeak) of

Ec = 3( ρc105m

)−1 ( γ

105

)3keV. (1.42)

&

Ep = 2( ρc105m

)−1 ( γ

105

)3keV. (1.43)

This is too low for the production of significant numbers of γ-rays and is rather immuneto slight differences in magnetic field configuration. Furthermore, when the energy gained

by acceleration in an electric field√

eRΩBmec2

(for a compact star of radius R, rotating at Ω

with magnetic field B; Michel 1991b) is compared to that emitted by curvature radiation,Equation 1.38, then a limit on the Lorentz factor of radiating electrons can be placed

γr < 3.7× 106((

R

104m

)(Ω

2π rad s−1

)(B

1012G

)) 18 ( ρc

104m

) 12. (1.44)

This is less limiting than magnetic attenuation from pair-production that shields the accel-erating electric field, Equation 1.37. However, it shows that the cut off energy at this limitwould be in the megaelectron volt (MeV) regime. In specific environments where these limitsdo not apply in which case to obtain peak emission in the gigaelectron volt (GeV) band thenthe Lorentz factor of the charged particles would need to be 107. An ensemble of emittingparticles with a power-law energy distribution N(E)dE = KEΓdE over a sufficient energyband produces a power-law curvature radiation spectrum (Ginzburg and Syrovatskii 1965;Lyne, Graham-Smith, and Graham-Smith 2006)

Iω ∝ ω1−Γ3 . (1.45)

The emission accordingly changes when the power law distribution of emitting particles breaksdown.

1.3.2.1. Synchro-Curvature Radiation

Kelner, Prosekin, and Aharonian 2015 consider the case of an intermediate regime where thedrift velocity of the charged particle and the helical path both contribute to the emissionspectrum of a charged particle in a strong magnetic field. The time-averaged intensity is

simply that of the curvature radiation ICR multiplied by a factor 1 +(v⊥vD

)2where v⊥ is the

charged particle’s velocity perpendicular to the magnetic field line and vD is its drift velocity(essentially along the magnetic field line when considering strong fields). The synchro-curvatureradiation case is applicable when v⊥ ∼ vD. The exhibited spectrum of synchro-curvatureradiation is harder12 than that of curvature radiation, which goes as e−x where x = ω

ω�and

12A harder spectrum displays comparatively more higher energy photons.

27


ω is the synchrotron radiation critical angular frequency. A synchro-curvature radiation

spectrum goes as e−x23 (Kelner, Prosekin, and Aharonian 2015).

1.3.3. Inverse Compton Scattering

inverse Compton scattering is the scattering of a high energy photon off a quasi-free chargedparticle travelling with a relativistic velocity. The high energy, Klein-Nishina (Klein andNishina 1929), regime occurs when hν ∼ mc2 and the low energy Thomson regime wherehν � mc2. inverse Compton scattering differs from Compton scattering because the chargedparticle is moving with a relativistic velocity which causes the photon energy to increaseby a Doppler boost factor γ2 when transferring back to the observer rest frame from thecharged particle rest frame. The maximum increase in photon energy corresponds to a head-oncollision resulting in a photon energy:

hωmax = hωγ2 (1 + β)2 ≈ 4γ2hω0. (1.46)

However, the mean energy of a scattered photon, derived from the total number of photonsscattered per unit time, σT cUrad

hω0, is given by:

hω =4

3γ2β2hω0 ≈

4

3γ2hω0. (1.47)

This equation shows that thermal X-rays from a pulsar’s surface can be up-scattered to γ-rayenergies by electrons with Lorentz factors of only around 100 . . . 1000 which should be easilyachievable in a pulsar’s magnetosphere. This acceleration mechanism is however only expectedto significantly contribute to emission above 100GeV (Takata et al. 2006).

1.3.3.1. Resonant Inverse Compton Scattering

In pulsar magnetospheres due to the high magnetic field resonant and non-resonant inverseCompton scattering (RICS & nRICS) can occur. nRICS is similar to ordinary inverseCompton scattering, with a similar cross section, where the incident photon frequency muchhigher than the Landau frequency (νL ≡ eB

2π mc ; You et al. 2003). For much lower frequenciesthe cross section drops to zero. RICS occurs when the photon frequency is around the Landaufrequency. This resonance occurs because of the electron energy quantisation in the magneticfield (see Appendix A). The scattering involves the absorption of the photon at around theLandau frequency by a relativistic electron that almost instantaneously re-emits the photon.Lyubarskii and Petrova 2000, however, suggest that RICS only results in a significant γ-raycomponent in the MeV energy range.

1.3.3.2. Synchrotron Self-Compton Scattering

Synchrotron self-Compton (SSC) scattering is simply the inverse Compton scattering ofsynchrotron photons emitted by the accelerated electrons with the accelerated electronsthemselves. This mechanism was key in explaining the MAGIC detection up to 100GeVand the VERITAS detection up to 400GeV of pulsations from the Crab pulsar (VERITASCollaboration et al. 2011; Aleksic et al. 2011). The maximum expected (Lyutikov, Otte, andMcCann 2012a) photon energy produced by inverse Compton scattering with the secondary

28

1.4. The Rotating Magnetized Conducting Sphere

plasma in the SSC model is

EmaxγICS = 150

( η

10−2

) 13√

ξ

(λ

102

)−1

GeV. (1.48)

This is dependent on the ratio of electric field strength to magnetic field strength η = EB ,

the number of electron-positron pairs created as a result of the inverse Compton scatteringinteraction λ, and a scaling parameter ξ. Cheng and Wei 1995 derive the inverse Compton

scattering spectrum based on a synchrotron (rather than a thermal X-ray) target photonspectrum making the expression applicable for an SSC induced emission (see Cheng and Wei1995, Equation 29). It has also been suggested that a cyclotron self-Compton componentmay be important in a pulsar magnetosphere due to inward travelling electrons having atransverse velocity that is non-relativistic (Lyutikov 2013).


A good physical picture of a pulsar is a rotating sphere which has a central static charge, amagnetic dipole, and is a good conductor (Michel 1991b). The origin of the magnetic fieldis discussed in Section 1.2.4. The pulsar is in essence a Faraday disc extended normal toits surface to become a sphere which has a magnetic dipole field and an induced electricquadrupole field.

The Goldreich-Julian Model The Goldreich-Julian model (Goldreich and Julian 1969)that treats an aligned rotating magnetic dipole is, although very commonly referenced inpulsar related works, flawed. Many assumptions made, such as that charges of both signare supplied from the surface of the star, are simply wrong. In the case of positive charges,the work function of the ions is too high for them to be liberated. Nevertheless, the modelhas been pervasive across pulsar theory for last 46 years. A main feature is a co-rotatingmagnetosphere filled with a charge-separated plasma; one charge sign above the poles andanother around the equator. The model describes the charge density required for no Lorentz

force to be acting on the charges

ρGJ =�Ω · �B2πc

1

1−(Ω rc

)2sin2 θ

(1.49)

and the wind from the magnetic poles that is a result of this co-rotating magnetosphere reachingthe light cylinder. This no-force condition is referred to as the magnetohydrodynamic (MHD)condition: �E · �B = 0 magnetic field lines are electric equipotentials which must also be thecase inside the star. In summary, no photon emission is theorised in the Goldreich-Julian

model.

Charge Distribution In order to maintain the MHD condition within the star a quadrupolecharge distribution in the bulk of the star and on the surface (similar to the Goldreich-

Julian charge density) is set up, as shown in Figure 1.6, along with a static charge in the

29


Figure 1.6.: Charge distribution of an aligned rotator with bulk and surface quadrupolecontributions as well as a central positive charge denoted but the larger + at thecentre. Taken from Michel 2004

centre of the star13 (Michel 1991b)

Q =8πε0R

3ΩB

6. (1.50)

The torus around the equator and domes at the poles are of opposite sign and unconnected.Each, therefore, has a space-charge discontinuity at the edge. A force-free surface (FFS)is also formed, as shown by the dashed lines in Figure 1.7, which traps one sign of chargeand repels those of the opposite sign since on this surface the electric field parallel to themagnetic field reverses sign. This trapping geometry - known in non-neutral plasma physicsas the rotating terrella - results in a dome and torus of plasma from the magnetic dipoleand electric quadrupole fields. Charged particles do not escape above the poles since beyondthese FFSs they are accelerated back to the polar caps14; this acceleration does not produce

13The central static charge is a result of the radial electric field and Gauss’ law. Alteration of this chargewould produce surface charges that could be lifted off and would form an equally but oppositely chargedmagnetosphere.

14The Polar Caps are defined as the regions at the magnetic poles out of which the magnetic field lines do notclose within the light cylinder.

30


Figure 1.7.: Trapping surfaces shown as dashed lines. FFS stands for force-free surface. Theelectric field parallel to the magnetic field reverses sign at these FFSs. Thesurfaces pervade both inside and outside the star if the magnetic field is from amagnetic dipole at a point in the centre of the star. Taken from Michel 2004

31


Figure 1.8.: Distribution of charge around an aligned rotator, different charge signs shown byblack and grey. Taken from Michel 2004

X-ray hot spots since the acceleration ceases once the particle returns to within the FFS andconsequently the acceleration region is small. Any charge that does escape compounds theproblem since the system charge is then increased making further particle loss of the samecharge require more energy. Indeed, the lack of charges lifted from the surface means that it isunclear whether a plasma does surround a pulsar (it does not for an ideal aligned magnetisedrotating conductor); great doubt is thrown on the idea of a plasma that extends as far as thelight cylinder (Michel and Li 1999).Figure 1.8 shows the distribution of charges if a magnetospheric plasma did surround the

rotator. Michel 1980; Krause-Polstorff and Michel 1985a; Krause-Polstorff and Michel 1985b;Spitkovsky and Arons 2002 show with simulations that the magnetosphere does not follow theGoldreich-Julian model of uniform charge surrounding the body but rather has a torus ofcharge around the equator and domes of opposite charge over the poles (originally Michel1980). Within these non-neutral plasmas the MHD condition is fulfilled and any deviation

32

1.5. Novel Emission Processes

is compensated for by accelerating fields that act to re-fulfil this condition. This region ofnon-neutral plasma is referred to as the electrosphere. This plasma distribution does, however,leave large regions away from these structures where the MHD condition is violated andparticle acceleration is, in principle, possible. Simulated pair production in these regionssimply results in either particles entering the star or, with an increased pair production rate,the torus and domes closing to form a Goldreich-Julian-like magnetosphere that breaksdown as soon as the pair-production ceases (Smith, Michel, and Thacker 2001). See Micheland Li 1999 for a detailed description of the aligned rotator’s fields. This model does, however,suffer from the problem of the origin of the positive charges in the electrosphere; the negativecharges - electrons - could be drawn from the pulsar surface by strong surface vacuum electricfields (Ruderman and Sutherland 1975). Bogovalov 1999 models the pulsar environment withthat of a field originating from a magnetic monopole with an inverted field direction at one ofthe poles and a current sheet at the equator showing that the solutions are equivalent to analigned split magnetic monopole.

Plasma Instabilities Petri 2009b carry out simulations on the effect of rotation with respectto the plasma structure in the pulsar electrosphere. It affects the configuration due tothe diocotron (the plasma analogue of the Kelvin-Helmholtz instability in fluids) andmagnetron instabilities causing the charges of the domes and torus to be somewhat spreadacross the magnetosphere (Petri, Heyvaerts, and Bonazzola 2002a). The charge in the torusevolves as super-cells which cross magnetic field lines, however, when charge pairs beingsupplied to the electrosphere are considered the electrosphere evolves smaller electrostaticturbulence. Both enable a charge flux across magnetic field lines (Petri, Heyvaerts, andBonazzola 2003). Petri 2008 consider the magnetron instability at the light cylinder allowingthe torus to diffuse to infinity, it is suggested, however, that the charges would simply loopback to the pulsar (Michel 2005; Mestel et al. 1985; Goodwin et al. 2004).

1.5. Novel Emission Processes

Pulsar emission models mostly seek to explain two types of emission from a pulsar: a windand a pulsed photon component. The wind is in the form of energetic charged particles thatescape the light cylinder and are assumed to then be free of the pulsar. The pulsed componentin the γ-ray band is the topic of the rest of this chapter. In regions where �E · �B �= 0 (i.e.the MHD condition is violated) acceleration of charged particles can occur. This is becausethe magnetic field is so large the particles become locked rotating around a certain field line,the perpendicular electric field is not sufficient to move the particle to other field lines asperpendicular velocity to the magnetic field line is quickly lost due to synchrotron radiation.Therefore, electric field induced acceleration parallel to this magnetic field line can result inparticles with such high energies that they produce γ-rays from mechanisms such as curvatureradiation or inverse Compton scattering.The emission from a pulsar almost certainly relies on the inclination of the magnetic axis

to the rotational axis, furthermore the aligned rotator may not even be physically realistic.The oblique, or inclined, rotator’s fields are determined by the superposition of the alignedand the orthogonal rotators’ fields which are described by Michel and Li 1999 and lead toFFSs that are simply distorted spheres. Radiation as a result of this oblique configuration isnow considered. The main observational features, as defined by Harding and Muslimov 1998b,that must be explained by any emission model are as follows:

33


• A double peak phase-folded event distribution (phase profile) where peaks are separatedby 0.4 . . . 0.5 in phase with bridge emission in between.

• A spectral break in the approximate energy region of a few to a few tens gigaelectronvolts.

• A variation of spectral hardness15 across the γ-ray phase profile.

• The spectral hardness of the phase-average spectrum tends to increase with spin-downage.

In most models the spectral break is generally described by the curvature radiation spectrum.

1.5.1. Cascade Emission

Pulsed emission is conventionally associated with the light house-like beaming from a pulsar.The violation of the MHD condition is key for particle acceleration in the magnetosphere andin the aforementioned torus-dome model there are large regions in-between these filled areaswhere this violation occurs. Michel 1991a suggests that charges inserted into these ‘empty’regions of the magnetosphere would induce bunched cascades from which both coherent andincoherent curvature radiation would be produced that can reach - in this simple model -MeV energies. The coherent radiation - lower in energy - corresponds to radiation from thebunch at wave lengths long compared to the bunch size and correspondingly the incoherentradiation from short, high energy, radiation. This is determined from a simple model of theinjection of charged particles into the outer magnetosphere considering the electromagneticfields described in Michel and Li 1999. This phenomenological emission model was proposedmore to explain the radio rather than high energy emission (Michel 1978) but the γ-rayspectrum is essentially from curvature radiation much like the magnetospheric gap modelsdescribed in Section 1.6. This model does require the cascades to bunch which could beattributed to the Abraham-Lorentz force16 (Harpaz and Soker 1999) acting on the surfaceof the bunch (Michel 1978).

1.5.2. Striped Wind Emission

Alternatively, the pulsed emission from the pulsar is suggested to originate as a result ofthe pulsar wind (Coroniti 1990; Michel 1994). As a result of the obliquity, a striped wind isformed where adjacent layers have opposite magnetic field directions. The pulsar’s periodicityis imprinted in the striped wind and therefore in the emission. The emission originatesfrom Thomson regime inverse Compton scattering between charged particles in the windand: the cosmic microwave background (CMB) photon field (Penzias and Wilson 1965;Planck Collaboration et al. 2015a) or Doppler boosted synchrotron radiation from electronsaccelerated by magnetic field line reconnections in the current sheets. This emission occursfrom beyond the light cylinder (Petri 2009a). Petri 2009a accurately reproduces the spectrumand phase profile above 100MeV of the Geminga pulsar measured with the Energetic GammaRay Experiment Telescope (EGRET; Nolan et al. 1992). The maximum Lorentz factor

15Spectral hardness refers to the relative flux at higher energies. A spectrum is hard if comparatively moreemission come from higher energies.

16This is the reaction force on an accelerating charged particle as a result of the emission of electromagneticradiation, it is also referred to as the radiation reaction force.

34

1.6. Magnetospheric Gap Pulsed Emission Models

achieved by the charged particles can be expressed as√

Ecut4ECMB

≈ 106.5 (Petri 2009a) in the

observers frame. This model does not depend on the precise magnetospheric configurationclose to the pulsar.

1.5.3. Narrow Region Wind Emission

Aharonian, Bogovalov, and Khangulyan 2012 proposed an alternative wind emission modelto explain the VHE emission from the Crab pulsar up to ∼ 400GeV in which the wind isaccelerated in a narrow region δR

R ≤ 1 at a distance of around 30 times the radial distance ofthe light cylinder from the axis of rotation. Here the electromagnetic energy of the wind -which is the dominant component of its energy - is largely converted to kinetic energy. InverseCompton scattering can then follow with the pulsar X-ray emission as the target photonfield. The reason behind the narrow emission region is the low observed flux in this energyregime which would be far higher were the wind to be accelerated from the light cylinderradius onwards. The predicted flux drops when the Klein-Nishina regime is reached as thecross section for the inverse Compton scattering process drops compared to that in the lowerenergy Thompson regime (Klein and Nishina 1929).

The next section will detail the conventional models of γ-ray pulsed emission from pulsars.They only differ in the location of the acceleration zone of the charged particles but are notbased on a rigorous treatment of the electromagnetic dynamics of the system but are rather‘custom’ gap models where the assumption of a filled magnetosphere to the Goldreich-

Julian charge density is made and accelerating gaps inserted where the MHD condition isviolated. This major assumption is contradictory to the configuration described in this sectionbased on the analysis of the electromagnetic fields and charges of the system.


Models of pulsed emission from gaps in the pulsar magnetosphere are grouped according tothe location of the main emission region where the MHD condition is somehow broken: PolarCap, Slot Gap & Outer Gap are the three main groups of models, the acceleration regions foreach are shown in Figure 1.9. A solution to the electric field using ∇ · �E = 4π (ρ− ρGJ) isgenerally sought in the zones that do not fulfil the MHD condition since it is assumed in thesemodels that a charge-separated plasma of the Goldreich-Julian charge density surroundssuch regions implying no overall electromagnetic force acts on the charged particles outside ofthese gaps.

Two special relativistic effects that come into play are of similar magnitude: the aberrationeffect and the time of flight delay both of which introduce a phase shift ≈ − rem

RLCbased on

the radial position of emission rem as a proportion of the radial position of the light cylinderRLC (see Section 1.1). The retardation of the magnetic field is also a factor. A compactstar of radius R > 3GM

c3≈ 6 km (this is, of course, below the canonical radius 10 km) does

not prohibit photons emitted at any upward angle from escaping its gravitational field. Itdoes, however, act to bend the light’s trajectory17. A photon emitted at an angle φ0 to thenormal of the star’s surface is viewed at an infinite distance to have been emitted at an angleφinf . For small deviations from the normal φinf =

2GMRc2

φ0 sinφ0 but for large values of φ0 the

17Of course light does not follow a curved trajectory rather the gravitational body bends space-time and thephoton follows a geodesic.

35


Figure 1.9.: Schematic of the possible VHE emission regions in a pulsar’s magnetosphereaccording to the main models, taken from (Aliu et al. 2008).

36


difference can be significant: φ0 ≈ 90 ◦ implies φinf ≈ 180 ◦ (Michel 1991b). This effect alsoacts to widen the beam.Pulsar emission theory is incomplete, agreement is somewhat prevalent that the aligned

rotator is inactive and not a realistic pulsar model but important questions remain unanswered:

• How is the magnetosphere supplied with charges?

• How are the accelerating gaps in the magnetosphere (where the MHD condition isviolated) formed?

• How can the average net current flow from the pulsar be kept to zero (the current closureproblem associated with the particle wind)?

The first two questions are the more pertinent to pulsed γ-ray emission from pulsars. Regardingthe first question, it is possible that pair production populates the plasma with both chargesigns since the work function of ions is too large for them to be liberated off the surfacemeaning only negative charge can be supplied from the surface. Such pair production wouldonly occur in the gaps between the torus and domes and would quickly act to fill these gaps(the magnetosphere would, however, in no way be filled up to the light cylinder as in theStandard Goldreich-Julian pulsar model) but with the shrinkage of the gap the productionrate decreases and the process turns off (Michel 1991b). Within the Goldreich-Julian

model no inertial or gravitational forces are considered and this omission means that near thestar’s surface (where the effect is strongest) the MHD condition should be violated since theparticles are prevented from falling into the star. This violation is however small and doesnot constitute a significant acceleration mechanism.

1.6.1. Pulsar Death Lines

Sorting the known pulsars according to their period and period derivative with respect totime, as in Figure 1.10, reveals two populations: normal rotation powered pulsars (in themiddle of the figure) and millisecond pulsars (in the bottom left). Also visible, however, is alack of long period pulsars that change very slowly (points in the lower right). As describedin Section 1.3 curvature radiation & inverse Compton scattering are the primary γ-rayproduction mechanisms in a pulsar magnetosphere. Significant emission is suggested to be aresult of primary electrons drawn from the surface that form18 pair forming fronts (PFFs)which are established at a certain altitude above the surface of the pulsar; see Section 1.6.2.The PFF altitude is determined by the acceleration length of the primary electrons and thepair-production attenuation length. The acceleration length is ∝ 1

E‖and the pair-production

attenuation length is ∝ 1B . As shown in Equation 1.14 the pulsar’s effective magnetic field is

closely linked to the period P and the period derivative with time P (Beff ∝(PP) 1

2). For

rotation-powered pulsars P is almost always positive and the magnetic field slowly decayswhich increases the altitude of the PFF. The magnetic field becomes too weak for PFFs ataltitudes greater than one stellar radius (∼ 10 km altitude) (Harding 2007). Considering thePolar Cap and Slot Gap models where negative charges can be liberated from the pulsar’ssurface, curvature radiation is only possible for young pulsars with characteristic age τ ≤ 107 yr.The point at which γ-ray curvature radiation is no longer significant is shown by the upperdeath line in Figure 1.10. Due to the nature of the mechanism, γ-ray emission from inverse

18Primary electrons are those that are drawn from the star’s surface.

37


Figure 1.10.: Graph showing the period derivative against the period of observed radio andhigh-energy pulsars. Characteristic magnetic field and age contours are shownas well as the curvature radiation and inverse Compton scattering death lines inpink and blue respectively. Taken from Harding, Muslimov, and Zhang 2002

38


Compton scattering is possible beyond this line but even this process stops when the magneticfield drops further; the inverse Compton scattering death line almost corresponds to the edgeof the points in Figure 1.10 - those millisecond pulsars beyond the line could still emit γ-raysfrom inverse Compton scattering with X-rays from a hot polar cap (Zhang and Qiao 1998).

1.6.2. Polar Cap Model

The Polar Cap model (introduced by Ruderman and Sutherland 1975) is based on theSturrock model (Sturrock 1970; Sturrock 1971) with the assumption that ions cannot besupplied to the magnetosphere above the magnetic poles due to the high ion binding energyand those particles that pass the light cylinder are not easily returned to the pulsar. It isassumed that plasma surrounds the pulsar other than directly above the magnetic poles. Itis suggested that the plasma lifts away from the surface due to the loss of positive chargethrough the light cylinder along the open field lines19. Within this gap the MHD conditionis violated and charged particle acceleration can take place parallel to the magnetic fieldlines. Two gap types are theorised: space charge limited flow gaps where charges liberatedfrom the stellar surface emit radiation which goes on to produce electron-positron (ep) pairs,and vacuum gaps where photons passing through the gap undergo magnetic pair productionthus forming the PFF. This model has, however, since been disputed (Asseo, Beaufils, andPellat 1984). In whichever way it is formed, it is observationally interesting to investigate theconsequences of an acceleration region so close to the magnetic pole.

γ-rays travelling within the gap at an angle θγB to the local magnetic field with E > 2mec2

sin(θγB)

can form an ep pair; the positron is accelerated away from - and the electron toward - thestellar surface. In both situations the pairs produced emit curvature radiation which in turnproduces more pairs resulting in an electromagnetic cascade. The point at which most of thispair production takes place is known as a pair formation front (analogous to the force-freesurfaces from Section 1.4). Figure 1.11 shows a schematic of the PFF above the polar cap.This front shields the electric field due to the positrons being accelerated towards the surfacethus correcting the charge deficit δρ = ρ − ρGJ and preventing acceleration beyond thePFF (Harding 2007) (at this point it is expected that the central charge of the pulsar attractsthe charge back to the surface, the observational consequences of this model will be lookedat nonetheless). The downward moving positrons form a second PFF at a lower altitudethan the first that partially screens the electric field (Harding and Muslimov 1998a). InverseCompton scattering controls pair-forming near the surface but near-surface acceleration isnot possible due to the scattering of the soft X-ray photons from the surface. Higher altitudeacceleration in the curvature radiation energy loss regime where there is “up-down symmetry”between the electron and positron PFFs is significant in this model.The gap is filled with a copious supply of X-rays due to the extreme temperature of the

neutron star surface, as explained in Harding 2007 the pulsar thermal spectrum peaks at X-rayenergies which means that the surface temperature is ∼ 1MK leading to a flux of photonswith sufficient energy for ep pair production in the polar gap region of ∼ 107m−2s−1sr−1.

1.6.2.1. Observational Features

Cascades initiated in the Polar Cap gap produce a spectrum with a distinct energy cut-offbased on magnetic pair production at Ec, which - if the emission is assumed to originate from

19Anti-pulsars are defined from �Ω · �B > 0 (parallel) and pulsars when �Ω · �B < 0 (anti-parallel). This determineswhat sign of charge should be ejected as the pulsar wind at the poles of the pulsar.

39


Figure 1.11.: Schematic of the pair formation front (labelled as “onset of e+e−” the extent ofwhich is specified by the horizontal solid blue lines) above a pulsar polar cap forspace-charge limited flow type magnetospheric gap. Figure taken from Harding2007.

40


the Polar Cap outer rim - is given by

Ec ∼ 2P12

( r

R

) 12 ×{0.1 if (B0,12)

−1 ( rR

)3< 0.1

(B0,12)−1 ( r

R

)3otherwise

GeV (1.51)

for an emission radius r, neutron star radius R and magnetic polar cap surface magnetic fieldstrength B0,12 in units of 1012G (Aharonian and Volk 2001). This cut-off is an unavoidablefeature of the polar cap model and is shown for various rotational periods, surface magneticfields and emission altitudes in Figure 1.5. The energy spectrum as a whole is approximatelygiven by Equation 1.52 (Becker 2009) in the form of a power law of index Γ with a hyperexponential cut-off

f(E) = AE−Γ exp

(−C1γ exp

(− E

E1γesc

)). (1.52)

The parameter C1γ = 0.2φλB′R2

ρ where φ is the inclination angle between the magnetic and

rotational axes of the neutron star, λ is the Compton wavelength divided by 2π, B′ is themagnetic field strength in units of the critical magnetic field strength20, R is the radius ofthe neutron star and ρ is the charge density. In the original polar cap model (Rudermanand Sutherland 1975) the integrated pulse profile is a superposition of smaller pulses as theelectromagnetic cascade, initiated by a single pair production event, precesses around themagnetic axis whilst pointing at the observer. The model only describes zero magnetic axisinclination angle but to explain both single and double peaked phase profiles different viewingangles are considered in combination with the thick cone of emission. Both the observerviewing angle ζ and the magnetic axis-rotational axis inclination angle α considerably alterthe predicted phase profile (Grenier and Harding 2006). Figure 1.12 shows the predictedhollow cone of emission from the polar cap. In order to explain the Vela pulsar spectrum andphase folded light curve measured with EGRET (Kanbach 1999) Daugherty and Harding 1996are forced to take the acceleration region up to three stellar radii with an artificial increase ofthe emission at the polar cap rim. There is difficulty in reproducing the characteristic doublepeaked phase profile with simulations for polar cap models without requiring a very smallinclination angle (Hirotani 2006). Observational evidence against the polar cap model camewhen, in 2008, Aliu et al. 2008 reported the detection of the Crab pulsar at energies above25GeV showing that the expected spectral cut-off at sub-tens GeV energies does not occurfor this pulsar. This was compounded when in 2011 Aleksic et al. 2011 observed pulsation upto 100GeV, later in 2011 VERITAS Collaboration et al. 2011 reported them above 100GeVup to 400GeV, and most remarkably de Ona Wilhelmi et al. 2015 report the detection ofpulsations up to 2TeV.

1.6.3. Slot Gap Model

The edge of the polar cap gap is assumed to be a perfect conductor. The electric field E‖parallel to the magnetic field, therefore, drops near the edge with the boundary condition thatit must be zero at the first closed field line 21. E‖ also varies with magnetic colatitude and,

20the Schwinger critical magnetic field limit for pair production and non-linearity is Bcr = m2c3

eh= 4.4 ×

1013 G (Sauter 1931).21The first closed field line is that which when moving to higher colatitude (latitude from the magnetic axis)

from the magnetic axis the first field line that does not cross the light cylinder

41


Figure 1.12.: Predicted phase plot and geometry according to the polar cap model. Phaseplots shown for inclination angle α = 10◦ and over all viewing angles (ζ, upperleft) and at ζ = 18◦ (lower left). The central part of the right hand panel iszoomed to show the gap size with respect to the neutron star radius. Figuretaken from Grenier and Harding 2006.

therefore, so does the altitude of the PFF. This is because the altitude of the PFF follows

hPFF = racc + ratt (1.53)

where hPFF is the height of the PFF, racc is the acceleration length and ratt the pair attenuationlength. At the edge of the polar cap E‖ is smaller than near the magnetic axis due to theboundary condition at the edge. This drop in E‖ means charges are not accelerated as muchand thus require a greater acceleration length in order to achieve an equivalent Lorentz factorto those near the magnetic pole. The altitude of the PFF at the edge of the Polar Cap is,therefore, higher that near the magnetic pole; this is illustrated in Figure 1.13. As the PFFforms further from the surface, the magnetic field density is inevitably lower, thus reducingthe magnetic attenuation effect on γ-rays allowing more higher energy γ-rays to escape theregion. The parallel electric field is also effectively unscreened (Harding 2007) in the slot gapwhich allows charges to be continuously accelerated for greater distances along the last openfield line. It is this gap - along the last open field line extending far from the star’s surface -that is named the Slot Gap. The Slot Gap is limited at small values of magnetic colatitudeby the plasma created due to pair cascades above the polar cap, see Figure 1.9. This model isincluded to analyse the observational consequences rather then because it has a strong basisin electromagnetic theory. The model assumes a plasma that extends to high altitudes abovethe polar cap which is not predicted by the electrodynamic simulations and calculations (seeSection 1.4).

The width of the Slot Gap hSlotGap is larger for young pulsars since they have a shorter periodand higher magnetic field (Muslimov and Harding 2003). The solid angle into which γ-raysare emitted from the Slot Gap region is small due to its small extent ΩSlotGap ∝ θ2PCrhSlotGap

(θPC is the opening angle of the polar cap) but despite this the flux can still be relatively

42


Figure 1.13.: Illustration of the geometry of the slot gap above the magnetic axis of a pulsar.Figure taken from Muslimov and Harding 2003.

43


high. The flux ΦSlotGap =LSlotGap

ΩSlotGapdepends on the strength of the magnetic field relative to

the critical magnetic field. This is because in each of the regimes the efficiency - which is afunction of the period & magnetic field and with which spin-down power can be convertedinto the energy of primary particles - is different (Muslimov and Harding 2003). This flux isgiven by

(1.54)

ΦSlotGap = εγ[0.123 cos2 α+ 0.51θ2PC sin2 α

]

×

⎧⎪⎨⎪⎩9× 1034

(Lsd

1035 ergs−1

) 37 ( P

0.1 s

) 57 , B < 0.1Bcr

2× 1034(

Lsd1035 ergs−1

) 47 ( P

0.1 s

) 97 , B > 0.1Bcr

erg s−1sr−1

for the period P , the spin-down luminosity Lsd, the magnetic axis inclination angle α, andthe efficiency εγ with which primary particles transfer their energy to high-energy emission.For pulsars below the curvature radiation death line the slot gap disappears since the parallelelectric field is no longer effectively screened (Harding 2007).

The Upper Slot Gap The Slot Gap model is extended with the inclusion that the electronsthat initiate cascades at low altitude continue to accelerate and emit curvature radiation &synchrotron radiation as well as undergoing inverse Compton scattering at higher altitudes inthe Slot Gap. The curvature radiation limits the Lorentz factor achievable by the electronsin the upper Slot Gap region according to

γCR =

(3

2

E‖ρ2Ce

) 14

∼ 3× 107. (1.55)

Lower levels of magnetic field line curvature (i.e. the radius of curvature ρC is large) lead tolower fluxes of curvature radiation per electron meaning these electrons achieve higher Lorentzfactors. The spectrum of γ-rays due to curvature radiation from these electrons will thereforepeak at ∼ 30GeV. At even higher altitudes, where the magnetic field ∼ 106...8G, resonantinverse Compton scattering can occur with radio photons that are at the electron’s cyclotronresonance in its rest frame. This pushes the electron into a higher Landau state enabling it,once more, to emit synchrotron radiation. The non-resonant inverse Compton scatteringin the Thomson regime of radio photons can also occur which produces an emitted γ-rayspectrum that peaks at ∼ fewGeV; this can produce significant γ-ray emission if the radioemission region is located at a high altitude (Harding 2007).


The two pole caustic model (Dyks and Rudak 2003) considers a very similar magnetosphericregion as the Slot Gap model but does not simulate the γ-ray energy. The region is assumedthin and with uniform emissivity from the Polar Cap to the edge of the light cylinder alongthe last open field line (essentially the Slot Gap region). Only the γ-ray emission in termsof the phase profiles for both magnetic poles rather than the acceleration mechanism isconsidered taking into account the caustics from both poles and the relativistic effects suchas aberration and time of flight differences. The predicted pulse-profile, shown in the lowestpanel of Figure 1.14, fulfils many of the specifications set out at the beginning of Section 1.6.These include a double peaked profile with correct separation and intermediate emission.

44


The upper panel of Figure 1.14 shows the caustics from this emission region with respectto the viewing angle ζobs. The lower panel shows a comparison between the model appliedto the conditions of the Vela pulsar and data of the Vela pulsar gathered with the EnergeticGamma Ray Experiment Telescope (EGRET) (Kanbach 1999). Overall, peak shape andrelative position appear accurate, however, it seems the model could not account for therelative peak heights; reproducing a 50% difference when a 10% difference is observed. Thesecaustics are greatly affected by the relativistic effects described in the opening of Section 1.6.The high-energy emission from the Slot Gap as a function of viewing angle can be seenin Figure 1.15 as well as a sample phase-profile for a viewing angle of 100 ◦ (Grenier andHarding 2006). The phase-profile features the characteristic double-pulse with relatively widepulses as well as significant inter-pulse emission and a similar relative pulse height. The rightpanel shows the approximate locations of the emission regions for the sample phase-profile.The high-energy flux from such a magnetospheric gap (Hirotani 2008b) assuming no photonabsorption can be given by

(νFν)peak ≈ 0.0954fh3mκE

d20.707

sin2 αMeVs−1cm−2 (1.56)

where Ed2

is the spin-down energy flux, f is a factor related to the magnetic field curvature

and close to unity, κ a parameter related to the charge density, and hm ≡ θmax� −θmin

�θmax�

is the

gap transfield thickness since θmax defines the colatitude of the upper boundary and θmin

the colatitude of the last open field line. The key dependence here is on the cube of the gaptransfield thickness hm. The spectrum is based on curvature radiation originating at a largerdistance than the polar cap region meaning a power law with simple exponential cut off isexpected (Harding et al. 2008). It has, however, been suggested (Hirotani 2008b) that the SlotGap model could only explain about 20% of the high-energy emission from the Crab pulsarin the approximate energy range between 1MeV . . . 1GeV due to the thinness of the region.

1.6.4. Outer Gap Model

The Outer Gap model (Cheng, Ho, and Ruderman 1986) was proposed before the VERITASdetection of VHE pulsed emission from the Crab pulsar (VERITAS Collaboration et al. 2011)up to 400GeV. The model suggests γ-rays can be produced at a significantly greater distancefrom the neutron star surface where the magnetic field is far weaker than in the polar capregion. This is significant since a γ-ray flux close to the stellar surface should be magneticallyattenuated resulting in an abrupt spectral cut-off.Photon-photon pair production22 can occur with thermal X-rays from the surface or non-

thermal X-rays from the electromagnetic cascades in the gap. It mainly takes place betweencurvature radiation from inwardly accelerated leptons (the charge sign depends on whetherthe magnetic and rotation axes reside in the same hemisphere) that almost head-on collidewith thermal X-rays from the surface.

Figure 1.9 shows a schematic of the magnetosphere around a pulsar with the OuterGap region labelled. The Outer Gap is bounded towards the magnetic axis by the chargelayer on open field lines and at higher colatitudes by the charge layer on closed field lines.Close to the star the region is limited by the null surface (defined as the surface in a,

22This is similar to magnetic pair production as shown in 1.4 but with two free photons rather than one beingthe quanta of a magnetic field.

45


Figure 1.14.: Upper panel shows the photon mapping done by Dyks and Rudak 2003 showndepending on viewing angle. Lower panel shows their model applied to the Velapulsar for a magnetic inclination angle α = 70 ◦ and viewing angle ζobs = 61 ◦ inb) and this compared to data obtained with the EGRET (Nolan et al. 1992) ina). Figure taken from Dyks and Rudak 2003.

46


Figure 1.15.: The right panel shows the emission regions of the slot gap model labelling thoseregions which are responsible for the first (P1 ) and second (P2 ) peaks in thephase profile. The left hand panel upper plot shows the phase profile as afunction of the viewing angle with the lower plot showing it for a viewing angleof 100 ◦. Figure taken from Grenier and Harding 2006.

Goldreich-Julian, charge separated plasma-filled magnetosphere where �Ω · �B = 0 i.e. wherethe bulk charge sign changes). Charged particle energy will still be limited by curvatureradiation within the gap (Cheng, Ho, and Ruderman 1986) from which a power law withsimple exponential cut off is expected but inverse Compton scattering and synchrotronself-Compton scattering components are expected to feature for very-high energies. Thesynchrotron self-Compton scattering component was used to explain the emission up to400GeV from the Crab pulsar (VERITAS Collaboration et al. 2011). It was suggested that theregion was sufficiently compact and supplied with synchrotron photons to reproduce the γ-rayenergy spectrum due to this synchrotron self-Compton scattering component. The OuterGap has an extent hm ≈ 0.11 which is over double that of the Slot Gap and, thus according

to Equation 1.56, this boosts the peak flux by a factor ofh3m,OG

h3m,SG

≈ 10 (Hirotani 2008b, and

references therein). The Outer Gap can, therefore, account for the entire energy flux from theCrab pulsar in the GeV energy regime when magnetospheric lepton pair production is takeninto account (Hirotani 2008b). The spectral energy peak is then given by:

(ν Fν) = 4.06× 10−4fκ

0.3

(hm0.14

)3 (μ30

3.8

)2MeVs−1cm−2 (1.57)

which agrees with the numerical results found by Hirotani 2008a.

47


Figure 1.16.: The effect of rotation on the photon propagation direction. The left hand panelshows leading edge emission and the right panel trailing edge emission. Figuretaken from Hirotani 2006.


Hirotani 2011 analyses the curvature radiation produced by the accelerated leptons to determinethat the cut-off due to curvature radiation can be given by

Ec =

(3

2

)7

4hcρ

12c

(E‖e

) 34

. (1.58)

It is this dependence on the magnetic curvature ρc that causes the leading edge of the γ-raypulse to have a higher energy cut-off than the trailing edge since the leading edge particlemotion is bent towards the rotation; this is illustrated in Figure 1.16. The trailing edge alsocauses a pile up of photons and, therefore, a caustic. The time of flight delay effect, whichresults in pile up, and aberration effect, whereby the photon appears to have been emittedfrom a slightly shifted location, are both shown in Figure 1.16 for the leading and trailingedges of the pulsed emission.

The differential photon spectrum is given by a exponential cut-off power law:

dN

dE= KE−Γe−

EEc (1.59)

and has been measured for many γ-ray pulsars, see Abdo et al. 2013. Figure 1.17 shows theexpected phase-profile from an Outer Gap model emitter accounting for such caustic effects.The peaks (P1 & P2 ) in the phase-profile originate from the regions labelled in the righthand panel; the inter-pulse emission (I ) comes from the same region as the second peak. Afeature of the Outer Gap model is that there is very little inter-pulse emission, the pulses areparticularly narrow, but the second pulse displays a visible “shoulder” - a clear increase inthe emission immediately before the second peak in phase.

As is clear from the description of these models, none accurately and precisely describethe measurements made. Pulsar are certainly not all identical and several variables must beaccounted for in any model e.g. period, obliquity angle, etc. Much theoretical investigation is

48


Figure 1.17.: The right panel shows the emission regions of the outer gap model labellingthose which are responsible for the first (P1 ) and second (P2 ) peaks in the phaseprofile. The left hand panel upper plot shows the phase profile as a functionof the viewing angle with the lower plot showing it for a viewing angle of 80 ◦.Figure taken from Grenier and Harding 2006.

49


put into interpreting and modelling emission from the aforementioned ‘custom’ gap modelsinstead of continuing to explore possible emission scenarios from the more rigorous treatmentof the plasma configuration around the pulsar. Integrating these two trains of research will bekey in the future to unlocking the nature of the γ-ray emission from pulsars.

1.7. Moving Forward with Observation

Observations can be used to set limits on the properties of a compact star. They must,however, be more constraining that those available from theory. The radius is limited by theSchwarzschild radius R > rS = 2GM

c2but more so by the requirement that the speed of

sound is less than that of light vs < c, based on relativistic arguments (Haensel, Potekhin,and Yakovlev 2007), limiting the radius to R > 1.412rS . The centrifugal force must notexceed that of gravity which involves limiting the rotational velocity based on the overall

density of the star ω <√

3G4π ρ, for a canonical neutron star M = 1.4M� & R = 10 km

this limit is P > 0.5ms - this can also be reached using the virial theorem (Clausius 1870).Limits have been placed on the maximum mass (Rhoades and Ruffini 1974; Hartle 1978)and minimum period & radius (Koranda, Stergioulas, and Friedman 1997) of neutron starsbased on general relativistic and causal considerations. This mass-radius relation can beconstrained by observation. Looking in the X-ray part of the electromagnetic spectrumcan be used to determine the size of the pulsar itself as long as the magnetosphere hasall but disappeared. Observations have seemingly ruled out the possibility that, at least,PSR J1748−2446ad (Hessels et al. 2006) contains a canonical neutron core since it spins at toohigh a frequency (716Hz) for that composition model. Other possibilities to glean informationabout the compact star from observations include:

• Relation of surface temperature to core temperature.

• The surface temperature determines the black-body X-ray spectrum which is, in somemodels, the target field for inverse Compton scattering γ-ray emission.

• Use compact star cooling curves to constrain the equation of state.

A key goal in the future will be to link the observations of pulsars - in any wave band - toits structure and make up. A measurement of the polarisation of the radiation from pulsarsprovides further information and has been used in the optical and radio bands to determinethe pulsars’ obliquity. X-rays coming from the cooling surface of the pulsar may providefurther information about the structure and geometry, this property should be measurablein the next generation X-ray telescopes (Taverna et al. 2015). The nature of this chapteron pulsar physics shows that it is still a subject under intense research where the fledglingdisciple of γ-ray astronomy is just beginning to contribute in order to shed light on the physicsof these extreme objects. Chapter 4 describes the discovery of the first pulsar detected by theH.E.S.S. experiment and Chapter 5 seeks to interpret the observations in the context of thetheory of pulsars detailed in this chapter.

50

2. The H.E.S.S. II Experiment

This chapter describes the various sub-systems of the H.E.S.S. array required to operate CT 5and obtain Cerenkov data on astrophysical sources that can be used to further scientificunderstanding. Starting with details of how the telescope moves precisely & accurately, goingon to describe the triggering set-up, properties of the Cerenkov camera, operation of thesub-systems as a whole and finally describing the steps taken to convert the raw data into aformat that can be analysed to obtain scientific results such as sky-maps and fluxes.

The Imaging Atmospheric Cerenkov Technique The imaging atmospheric Cerenkov

technique can be used to detect localised γ-ray emission in the sky because a photon’strajectory, unlike that of an electrically charged particle, is not altered by interstellar andintergalactic magnetic fields. This means that cosmic protons are observed almost isotropicallyacross the sky but γ-rays can be observed as localised sources. Electromagnetic showers aretherefore of far more interest to ground-based imaging atmospheric Cerenkov telescopes(IACTs) than hadronic showers. The over-simplified Heitler model (Heitler 1954) for thedevelopment of electromagnetic showers in the atmosphere describes that in one radiationlength a photon will undergo pair production and an e± will produce a photon throughbremsstrahlung. This process is self-repeating and therefore results in a cascade of photons,electrons, and positrons1. Differences in air showers produced by primary γ-rays and protonsallow ground-based IACTs to distinguish between primary photons and protons based on theCerenkov emission from the shower.The particle showers that very high energy (VHE) particles create in the atmosphere

produce a large amount of Cerenkov light that manifests as a flash of optical light lastinga few nanoseconds. A Cerenkov telescope consists of an optical reflector and a camerawith pixels (often photo-multiplier tubes) whose exposure time is on the order of tens ofnanoseconds and which are sensitive enough to detect these brief Cerenkov flashes. Thelatest generation of IACT cameras can take hundreds to thousands of images per second.

Extensive Air Showers VHE charged particles have such a high kinetic energy that theytravel faster than the speed of light in the Earth’s atmosphere. This results in the well-knownCerenkov effect, which involves atmospheric molecules emitting radiation at the anglecos θc =

1n(ω)β . The angle is dependent on the refractive index of the atmosphere n(ω) - which

is in turn dependent on the angular frequency of the light travelling through it - and the speedβ = v

c of the particle travelling in the medium. Cosmic rays (consisting mainly of protons)are examples of such charged particles which induce the Cerenkov effect while travellingthrough the Earth’s atmosphere. VHE photons, γ-rays, are also incident on the atmospherebut do not induce Cerenkov light once within it since they are not electrically charged.Both the γ-rays and cosmic rays are, however, so energetic that they quickly interact resulting

1For high energy showers hadron production can sometimes take place but it is suppressed compared toelectromagnetic processes in γ-ray induced air showers

51

Chapter 2. The H.E.S.S. II Experiment

in particle cascades which differ depending on the initial (primary) particle. The chargedparticles in such a shower induce Cerenkov radiation.

Electromagnetic air showers are induced by photons and electrons. In the VHE regime, thedominant interactions are:

• Bremsstrahlung of electrons and positrons (e±s) in the Coulomb field of atmosphericnuclei.

• e± pair-production by photons in the Coulomb field of atmospheric nuclei.

• Ionisation and atomic excitation by e± which induces a swift shower extinction at thecritical energy, defined as the point at which e± bremsstrahlung losses equal ionisationlosses. This occurs at 83MeV in air (de Naurois 2013).

These showers generally have a negative charge excess due to the annihilation of positrons withatmospheric electrons 2. An air shower induced by a hadron usually involves electromagneticsub-showers, the fragmentation of atmospheric nuclei, and other hadronic interactions whichmainly include:

• Hadron production by interaction with an atmospheric nucleus producing mainly pions,kaons and nucleons.

• Charged meson decay producing muons.

• Electromagnetic sub-components from neutral pion decay into a photon pair.

Unseen neutrinos are produced in the charged meson decay and by any muons that decay.

The Newest Imaging Atmospheric Cerenkov Telescope The H.E.S.S. experiment wasoriginally an array of four equally sized IACTs (CT 1-4). This changed in 2012, when afifth telescope (CT 5), shown in Figure 2.1, was inaugurated. The telescopes are located inNamibia near the Gamsberg mountain (23°16′18.4′′S, 16°30′0.8′′E at 1800± 20m above sealevel). CT 5 is located in the centre of the array, has a dish size of 33× 24m2, and a focallength of 36m. The largest of its kind, CT 5 is able to detect lower energy air showers thanany other Cerenkov telescope. Working from a primary particle with such a low energy,10GeV say, an estimate of the required mirror area can be made. This estimate uses theFrank-Tamm formula (Frank and Tamm 1937) for the amount of energy dE radiated in theangular frequency range dω by a medium with magnetic permeability μ(ω) and refractiveindex n(ω) being induced to emit Cerenkov radiation by a particle with non-zero charge qtravelling a distance dx with speed v:

d2E

dldω=

e2

4π

(1− 1

β2n2

)μ(ω)ω. (2.1)

This equation holds when the Cerenkov condition is satisfied: the charged particle istravelling faster than light in the medium v > c

n(ω) . Considering that this energy is carriedaway by N photons, each with an energy of hν, then

d2N

dldν=

2πα

c

(1− 1

β2n2

)(2.2)

2The electro-production of hardrons can also occur for high energy showers.

52

where

α =e2

4πε0hc

is the fine structure constant. The assumptions for this calculation are:

• The shower consists of 10 charged particles with (±) unit charge.

• These particles are all created at the shower maximum at 10 km altitude.

• The particles travel for one radiation length ∼ 900m emitting Cerenkov light andthen stop emitting.

The assumptions are based on Monte Carlo simulations of Cerenkov showers indicating thatfor a shower induced by a 10GeV γ-ray the shower maximum occurs at around 10 km andcontains ∼ 10 charged particles (de Naurois 2013). The path length � ≈ 920m is equivalentto one radiation length in air 36.7 gcm−2 (de Naurois 2013) divided by the mass density of airat an altitude of 10 km, which is 0.4 kgm−3 (Picone et al. 2002). The considered radiationband is roughly the optical region of the spectrum since this is the range detectable with CT5. The expression for number of photons emitted per particle is then:

N =2πα

c

(1− 1

β2n2

)Δν|Δh| (2.3)

where Δh ≈ 920m is the distance travelled in the atmosphere, β = vc is approximated to

unity, Δν is around 400THz for the optical regime, and n is taken as the value at 10 km

altitude using the approximation: n(ω) = 1 + n0 exp− h

h0 ≈ 1 + 0.0003 ∗ exp− h8500m (Penndorf

1957)3. Including the dependence of n on height across the ≈ 900m path length is a relativelysmall correction. The number of photons produced in such a shower is ∼ 55, 000, spread overa light pool on the Earth’s surface of around 150m radius. Assuming a detector efficiency ofaround 10% then roughly 50 photons are measured with a detector with a collection area of650m2. CT 5 was designed in order to detect exactly these low energy showers. The CT 5reflector is a 614.5m2 (Bolmont et al. 2014) parabolic tessellated mirror that reflects opticallight onto a Cerenkov camera with 2048 photo-multiplier tube (PMT) pixels fitted withWinston cones to minimise loss of light in between pixels each of which has a field of view of0.067 ◦ resulting in a total camera field-of-view of 3.2 ◦ (Bolmont et al. 2014). The tessellatedmirrors of CT 1-4 are of the Davies-Cotton type (see Davies and Cotton 1957, for moredetails). The nature of the Davies-Cotton design means near-edge aberrations are lowerthan those produced by a parabolic mirror. However, the mirror is not isochronous, having aspread of arrival times: ΔT = d2

8fc (Bernlohr et al. 2003) dependent on the focal length f anddish size d. For CT 1-4, with 13m diameter mirrors, ΔT ≈ 5 ns. This is acceptable as it iscomparable to the intrinsic spread of arrival times of the Cerenkov light (∼ few ns (Bernlohret al. 2003)). This is, however, not acceptable for a 28m effective reflector diameter witha 36m focal length (Bolmont et al. 2014), which results in a spread of the arrival times of≈ 10 ns. The CT 5 parabolic mirror is used to avoid this problem and only suffers from smallanisochronisities due to the spherical shape of each of the 875 hexagonal mirror facets and

3Scale height of the Earth’s atmosphere is calculated by considering the ideal gas law with the dependenceof pressure on altitude in the atmosphere using reasonable numbers for the mean mass of atmosphericparticles 29×mH = 29 ∗ 1.67× 10−27 kg and temperature 288K.

53


slightly larger aberrations 4.

The H.E.S.S. array is operated in line with the phases of the moon; the year is split intoshifts of around 28 d separated by several days. Each shift is then broken up into nightlyperiods of data-taking, which last the duration of that night’s astronomical dark time 5.Observation data is taken only during astronomical dark time. Usage of the telescopes - forcalibration, observation, alignment etc. - is split into runs. These vary in length but usuallylast 1680 s when observing astrophysical targets tracked across the sky by the telescopes.

2.1. The Tracking System and Pointing

The CT 5 telescope has an altitude-azimuth mount and can track celestial objects with aprecision of 0.001° (Hofverberg et al. 2013). This precision is achieved using LEDs in the sameplane as the PMT pixels are fixed, with a CCD camera pointed at the camera lid (LidCCD)and mounted in the centre of the dish. During data processing the images from this CCDcamera are used to account for the mechanical hysteresis of the telescope and achieve thispointing precision. An antenna control unit (ACU), on a low level, is used to control four24 kW servo motors that power the movement of the telescope. The telescope is moved inthe azimuthal direction on six bogies, each with two wheels, that travel along a 36m circularmetal rail at the base of the telescope; and in the altitudinal direction by two gear driveunits, one on either side of the dish. These drives can achieve an acceleration of 0.5 deg

s

and maximum speed of 200 degs in azimuth & 100 deg

s in altitude, allowing the telescope toeasily track astrophysical sources in the sky and quickly move to any time-critical targets ofopportunity. The azimuth drive has a 540 ◦ range and the altitude drive a range from −32 ◦ to180 ◦, which means that it is capable of going beyond the vertical. This is classed as reversetracking and significantly reduces the time taken to slew across the sky.

To precisely reconstruct the position of events within the camera with respect to astrophysicalsources in the sky a so-called pointing model of the telescope is made using dedicated Pointingruns to analyse the hysteresis of the telescope structure. The exact position of the camerarelative to known optical astrophysical sources is determined using the LidCCD and anotherCCD pointed at the sky (SkyCCD) in the direction that the telescope is pointing. Thetelescope is pointed towards optically bright stars that are then photographed by the Sky-CCD and Lid-CCD (as a reflection from the telescope’s mirrors onto the Cerenkov camera’slid) the two images of which can be compared in order to determine the position of the camerawith respect to the telescope optical axis (the telescope structure bends dependent on theenvironmental conditions therefore many of these runs - the order of 100 targets - are carriedout each month).

2.2. Central Trigger

During the observation of an astrophysical target with H.E.S.S. I when two or more telescopessee an event in a small time window (80 ns (Funk et al. 2004)) a centralised hardware triggersystem signals to those telescopes to send their data to the central data acquisition system

4This is because the focal length varies as a function of the offset from the optical axis in both designs.5Astronomical dark time starts when the sun is ≥ 18 ◦ below the horizon and the moon is simply below thehorizon.

54

2.2. Central Trigger

Figure 2.1.: The fifth, and newest, H.E.S.S. telescope

55


(DAQ). This Central Trigger determines whether information from the camera will be stored.The system (and its name) were designed around a stereoscopic principle meaning duringphase I all operational telescopes would observe the same astrophysical target down to aminimum of two. With the introduction of CT 5, however, events are recorded even whenonly CT 5 is triggered (so-called mono mode). The array can now be operated very flexibly,for example, if a minimum of two telescopes are triggered on any one event or only CT 5 istriggered, this is the so-called Mono+Hybrid mode. Splitting the array to perform differenttasks or observations is also possible e.g. CT 5 observing a different source than CT 1-4. Thismeans that if either two or more telescopes trigger or only CT 5 triggers then the CentralTrigger signals that the event must be sent to a DAQ data processing node. The data analysedin this thesis come from CT 5 only. If, in the analysis, CT 1-4 are triggered on the sameevent as CT 5, their data is ignored (a hybrid analysis, where this data is also considered,is currently under investigation in the H.E.S.S. Collaboration). The Central Trigger alsogives each event a system-wide unique identifier using two numbers: Bunch Number & EventNumber within the bunch. This information, along with the DAQ node’s IP address, is passedto the telescope via the Local Module - see Figure 2.3 - and assigned to the event.

2.2.1. System Dead Time

The system dead time for H.E.S.S. II is dominated by that of the H.E.S.S. I cameras: 460μsas opposed to 15μs for the CT 5 camera (Bolmont et al. 2014). This results in the situationthat the H.E.S.S. I cameras partake in a minority of events when using the Mono+HybridCentral Trigger set-up that has been the operational standard since the introduction of CT 5.For a paralyzable system like monoscopic observations with CT 5, the dead time adds up andaffects the recorded rate thus

R = D exp (−DT ) (2.4)

where R and D are the recorded and delivered rates respectively and T is the dead time.A typical recorded CT 5 mono trigger rate is 1.5 kHz but the design specifies a maximumof 5 kHz which suggests that, due to dead time, only 2.25% and 7.81% of delivered eventsare not recorded at the operation and design rates respectively. The CT 5 dead time can bedetermined experimentally by looking at the time difference between active triggers. Figure 2.2was produced using CT 5 data and shows this clear step and the value of the dead time∼ 0.02ms. This dead time is taken into account when carrying out analysis of the data byreducing the exposure time. The events in the bins at lower than the dead time representevents that are triggered but contain no data, these events are useful when determining thedead time.

2.2.2. Event Time Stamp

The time stamp of an event is of particular importance to the analysis of variable sources suchas pulsars. In H.E.S.S. this is provided by the Central Trigger using a Rubidium-87 atomicclock synchronised to the global positioning satellite (GPS) time via a GPS 167 MeinbergGPS receiver (Meinberg Funkuhren GmbH & Co. KG 2015). The clock has 1Hz and 10MHzsignal channels, which are synchronised to the UTC second with a long term accuracy of< 2μs (Funk et al. 2004). Observations of the Crab pulsar using an optical camera mountedonto the lid of one of the smaller telescope’s Cerenkov cameras (Hinton et al. 2006), as wellas the use of two clocks simultaneously, have been used to verify the precision of the timing.

56

2.3. Camera

Figure 2.2.: Graph displaying the time difference between active triggers for some CT 5 data.

2.3. Camera

The Cerenkov camera of the CT 5 telescope weighs almost 3 t and is mounted onto thetelescope using a fully automated loading system. First light for the camera came in July 2012,a few months before the official inauguration of the telescope itself. The CT 5 camera is morecomplicated and uses newer technology than those of CT 1-4, each of which use a single centralprocessing unit (CPU) to trigger on, build, and send events, whereas the CT 5 Cerenkov

camera uses five CPUs. Due to the higher threshold of CT 1-4 ∼ 100GeV (Aharonian et al.2004b) and therefore lower event rate, they required a smaller amount of computing power tobuild the events. With CT 5, the ∼ 5 kHz maximum trigger rate and larger event size (7.4 kBas opposed to ∼ 2.4 kB with CT 1-4) mean that two CPUs operating in a load sharing modewere required to build the events. The process of event triggering, building and processingis shown in Figure 2.3. As can be seen in this figure, two data CPUs work in a flip-flopmode where one receives events for a short time period whilst the other has time to completebuilding the events from the previous time period. When an Accept flag is given from theL2AR trigger, the Trigger CPU passes the event on to the local interface module in thecamera, which informs the Central Trigger of the event having been triggered. If the CentralTrigger sends the confirm flag back, the data CPUs are instructed to send the events to theDAQ data processing node. Otherwise a reject flag is sent and the data in the Data FIFO isdiscarded. The LAN IP address of the DAQ data processing node to be used is passed to thecamera’s data building CPUs via the camera trigger CPU. The CT 5 camera can be used aspart of a minimally stereoscopic array and therefore uses the confirm signal from the CentralTrigger even if it is operating in monoscopic mode.

57


Figure 2.3.: Schematic of the procedure of the triggering and processing of a CT 5 event.

58

2.3. Camera

Figure 2.4.: Schematic of a photo-multiplier tube

2.3.1. Photo-multiplier Tubes

When a Cerenkov photon hits one of the camera’s PMTs during operation, it convertsthe optical signal into a measurable electrical signal. This process is shown in Figure 2.4.The photon hits the photo-cathode, a metal plate with a small work function and negativepotential, which, due to the photo-electric effect, results in the emission of an electron withkinetic energy determined by the photons energy and the work function. This is however smallcompared to the energy associated with the accelerating electric field and is not measured.The electron is attracted to a dynode farther into the PMT, which has a more positive electricpotential. The collision of this accelerated electron with the dynode results in the emissionof lower energy electrons which are in turn accelerated towards the next dynode. In a PMTthere are many of these dynodes in series, with increasingly positive electric potentials. Thisprocess ends at the anode. The electrical signal at the anode is then passed to a 12-bitanalogue-to-digital converter (ADC) with a sampling frequency of 20MSs−1, which takes thecontinuous voltage signal and converts it into a digital number proportional to the signal.This multiplication means that single photons can be detected using PMTs. The incidentphoton may not induce the emission of an electron and if it does the electron may not inducea cascade by, for instance, not colliding with a dynode. This contributes to the measurementefficiency of PMTs. The CT 5 camera uses Photonis XP − 29600 (Bolmont et al. 2014) PMTswith an operating voltage of 3000V and quantum efficiency of 30% (Bolmont et al. 2014). Ifthe signal from a PMT exceeds a pre-set threshold then the voltage is lowered to a safe levelto prevent overload and damage. The central camera will then repeatedly attempt to rampup the voltage which will remain at operational level if the signal has reduced to below thethreshold.

2.3.2. Camera Thresholds

During the commissioning of the CT 5 camera, the pixel and sector thresholds were calibratedto optimise the trigger rate whilst limiting the amount of night sky background (NSB) thattriggers the camera. These sectors are overlapping groups of 64 adjoining pixels in which acertain number of pixels must exceed the pixel trigger threshold for a sector to be triggered.Figure 2.5 shows the trigger rate of the CT 5 camera as a function of the two camera parametersthat vary the threshold of the camera trigger. The pixel threshold determines the signal atwhich individual pixels are labelled as containing a signal and the pixel multiplicity definesclusters of pixels required for a trigger. The NSB wall has been roughly defined in both plotsby a dashed red line and is due to residual light from human activity, star light, etc. The rate

59


Figure 2.5.: Plots showing the relationship between the CT 5 camera trigger rate against thecamera thresholds at 3 ◦ zenith angle. The left hand side shows the multiplicitywith the threshold set at 3 and the right hand side shows the pixel thresholdbeing varied with the pixel multiplicity set at 4.5. Two dashed lines show therough limit of the night-sky background (dominant at low threshold values) andthe astrophysical signal (dominant at higher threshold values).

is so high that these thresholds are critical in preventing the telescope from triggering only onNSB and missing the majority of the Cerenkov showers. The dead time of the camera is15μs, which therefore fundamentally limits the trigger rate to 67 kHz. However, limitations inthe data acquisition system (mainly network bandwidth) mean that the realistically achievablevalue is closer to a maximum of 5 kHz.

2.4. Array Control

The central H.E.S.S. DAQ consists of the processing and storage of slow control and Cerenkov

data as well as central operational control of the array. Concerning the Cerenkov data,the DAQ is involved when the data leaves the camera through optical fibre connections thatconnect to network switches using optical transceivers. These data are sent to one of sevencomputing nodes that process the data and then save it to a network mounted file systemin a ROOT framework-based (Brun and Rademakers 1997) H.E.S.S. data format. Only onenode is used at a time and is switched every four seconds distributing the processing over upto 20 processes on seven computing nodes. A node will buffer the data sent to it during thesefour seconds into a different data buffer for each telescope involved in the data taking. Onlyonce these four seconds are up and the Central Trigger has given the Cerenkov camerasanother IP address to which to send data, does the node begin to process the data. Thisprevents any possibility of added dead time of the system due to the DAQ. As can be seen inFigure 2.6 these four second bunches are then merged and saved to the H.E.S.S. data formatthat is used both online and offline. This means that a preliminary analysis can be done in arelatively short time and is available minutes after the end of the run.The right hand side of Figure 2.6 shows the three higher level processing tasks that allow

the crew operating the array to see, using a basic calibration (see section 2.5), the data beingtaken, the significance of any sources in the field of view, and details of the various systems’status. The various hardware components in the H.E.S.S. array are controlled by a central

60

2.4. Array Control

Figure 2.6.: Schematic of the procedure of the data processing of a CT 5 event in the dataacquisition system. One event buffer is used per telescope. Event numbersuniquely identify triggers in a data taking run and sash is the internal H.E.S.S.data format.

61


system which consists of over 100 processes all of which control the state of each hardwarecomponent, so-called state controllers. Depending on the complexity of the hardware therecan be many state controllers per device, for example, the CT 5 Cerenkov camera requireseleven. The DAQ state controllers allow the operation of the hardware devices to dependon other hardware devices. For instance, the Cerenkov camera’s high voltage (one statecontroller) should not be activated after the central trigger begins to accept events (anotherstate controller). The state controller also allows the system to be operated in an ordered wayto ensure that hardware can be brought to a safe state if a problem arises. Balzer, Fußling,Gajdus, et al. 2014 describe the state controller principle and the H.E.S.S. DAQ in moredetail; the DAQ was a significant portion of the work carried out in the thesis.

2.5. Calibration and DST Production

The calibration of the main observation data is done using data taken when the array is ina specific configuration. These calibration runs are used to determine the properties of theelectronic pedestal, single photo-electron peak position, and gain of the PMTs in order toaccurately translate from Analogue-to-digital Converter (ADC) counts to numbers of photo-electrons. The analogue-to-digital conversion is done using two channels: high gain (HG) andlow gain (LG). The HG is used when there is a relatively low signal (< 200 photo-electrons)and the LG when there are more photo-electrons, > 200. The equation to do this conversionis as follows:

NHGpe =

ADCHG − PHG

SADC,HG× FF (2.5)

NLGpe =

ADCLG − PLG

SADC,HG× (HG/LG)× FF (2.6)

where NHGpe & NLG

pe are the high and low gain number of photo-electrons, ADCHG & ADCLG

are the high and low gain ADC counts, PHG & PLG are the electronic pedestal values,SADC,HG is the single photo-electron value, HG/LG is the ratio of high to low gain and FFis the flat field coefficient. This calibration is carried out for each PMT individually and theruns needed are carried out every two days due to variation based on, amongst other factors,environmental conditions. The electronic pedestal is the signal from a PMT when it is notilluminated by Cerenkov light. This signal is sampled from many triggers and forms a peakcentred around a value that is used as PHG/LG; the width of this peak is dependent on theNSB level, which is relatively low and stable at the H.E.S.S. site. The electronic pedestalvalue is determined using dedicated runs by turning on the camera high voltage in a darkenvironment (i.e. camera lid closed in astronomical darkness) but is also be determined duringdata taking since the peak is visible in the signal from pixels that are not illuminated byCerenkov light during a Cerenkov event. The electronic pedestal value is visible as theabscissa position of the highest peak in Figure 2.7. The single photo-electron value (SADC,HG)is used in the denominator to convert the value into units of number of photo-electrons since itis the ADC count received when a single photon is detected by the PMT (Equation 2.6 requiresthe ratio of HG to LG since only the high gain channel can detect a single photo-electron).The SADC,HG value is determined by firing a very weak light source at the camera in a darkenvironment (the camera is moved to the closed shelter during astronomical dark time) andcan be seen by the abscissa position of the second highest peak in Figure 2.7. Finally, theflat fielding coefficient is used to equalise out the inhomogeneous PMT responses due to, for

62

2.5. Calibration and DST Production

Figure 2.7.: The displayed plot shows an electronic pedestal peak with the addition of aconvolution of the single, double, treble etc. photo-electron peaks for CT 5 pixel408. The electronic pedestal here is −11840.0± 0.1 count with a one sigma widthof 10.34± 0.07 count and the single photo-electron peak is such that 51.5 countsrepresents one photo-electron.

example, slight differences in gain, degradation due to age, different quantum efficiencies, etc.This is achieved by taking data whilst shining a uniform pulsed light source on the cameraand taking the mean PMT signal value. Figure 2.7 displays an example distribution for asingle CT 5 PMT.During observation runs, muons are detected by the camera and can be used to analyse

the losses in the detector and apply a correction factor to the amount of Cerenkov lightdetected in the camera, allowing for an absolute calibration of the event energy. The opticalefficiency losses come from several sources:

• Reflectivity of the mirrors ∼ 80%.

• Shadow of the camera and mounts ∼ 10%.

• PMT collection efficiency ∼ 70% (photon loss from mirror to PMT aperture).

• Wavelength integrated PMT quantum efficiency ∼ 20%.

Combined, these cause the optical efficiency to be ≈ 10% (Chalme-Calvet et al. 2014). Muonsinduce the emission of a definite amount of Cerenkov light which is related to the angleof emission, the path length, and the attenuation in the atmosphere (this last parameter isdetermined by simulation) (Vacanti et al. 1994). The process used in H.E.S.S. to determinethe correction factor applied to the Cerenkov signal due to this limited optical efficiency isgiven by Chalme-Calvet et al. 2014. This correction factor is required to reproduce correctly

63


normalised astrophysical source spectra. A further consideration for the calibration are thecommon modes of the camera. This is where the electronic pedestal of the PMTs vary asa result of using two separate power supplies for the upper and lower halves of the camera,described further by de Naurois 2013. This was first observed with the CT 1-4 cameras anddue to the recovery time-scale of the power supplies, on the order of 100μs, this effect is morepronounced with the CT 5 camera, which has a much lower dead time.

DST Production Data summary tapes (DSTs) are calibrated data files processed from theraw data and written to H.E.S.S. format ROOT files (Brun and Rademakers 1997) that canbe analysed to produce high-level results such as sky maps and spectra. One DST correspondsto one data taking run. The data taking process is as follows:

1. Raw data recorded onto hard drives on site in Namibia and merged into several ROOTfiles in chronological order.

2. Two copies of the merged raw data is transferred to LTO-4 magnetic tapes and shippedto two data centres used by the H.E.S.S. Collaboration in Europe.

3. Data, which mainly consists of intensity values for each pixel, is converted to number ofphoto-electrons by the calibration process, pointing corrections are also applied.

4. All images from a single run are saved into one ROOT file (DST) with other slow controlinformation, such as pointing information.

An image cleaning - intended to remove noise from the image - is also usually carried out.The cleaning considers an individual pixel and those that border it. The two most commoncleanings are 04/07 & 05/10 which refer to the number of photo-electrons recorded in thatpixel and its neighbours. For example, 05/10 cleaning would not set a pixel’s signal to zero ifits signal were either:

• 10 photo-electrons & any neighbour has a signal of 5 or more photo-electrons

• 5 photo-electrons & any neighbour has a signal of 10 or more photo-electrons

otherwise the pixel signal is set to zero. This effectively removes NSB in the image and leavesthe Cerenkov light to be analysed by reconstruction software chains which will be describedin detail in Chapter 3.

64

3. Reconstruction & Analysis

Calibrated air shower images contained within a DST (see Section 2.5) are analysed usingreconstruction software. A variety of methods exist but all seek to determine the energy,direction, and type of particle (primary particle) that induced the imaged air shower. Thebackground comes in two forms: non-air shower optical light known as night-sky background(NSB) from optical sources and an air shower background from cosmic rays that consist ofaround 1% electrons and positrons and 99% nuclei stripped of electrons, of which around79% are protons, 15% are alpha particles, and the rest are heavier nuclei (HZE ions) (Oliveand Particle Data Group 2014). The NSB manifests as a low flux due to, for example, lightfrom human activity in cities that is scattered in the atmosphere. This is mostly filtered outby the adjustment of the camera trigger thresholds (see Section 2.3.2) and the image cleaning(see Section 2.5). NSB also comes from optical sources in the field of view (FOV), however,as explained in Section 2.3.1, this is limited since pixels in the camera are turned off dueto bright stars. At the analysis level, areas of the sky in which there are stars of an opticalmagnitude less than four are excluded from the dataset to aid in the reduction of the NSB inthe data set and avoid image contamination.Hadronic cosmic ray air showers are less ordered than γ-ray air showers and do not

form the distinctive ellipse visible in the image of a γ-ray air shower. They produce a lessconcentrated patchy circle of light due to greater dispersion of the cascade particles based onthe relatively large transverse momentum transfer in hadronic interactions and features suchas electromagnetic sub-showers. A key difference between the images of air showers is thisgreater lateral spread. Figure 3.1 shows a simulated γ-ray and hadronic air shower where thisdifference in shape is clear. The identification of γ-ray induced air shower images becomesmore difficult at low energies (< 50GeV) because so few pixels are illuminated in the imageit closely resembles a low energy hadronic shower. When an IACT triggers on an event theimage recorded most often resembles one of the three panels in Figure 3.2. The image on theleft shows the distinctive muon Cerenkov light signature (the muon ring) which is only usedto calibrate the optical throughput of the detector, see Section 2.5. The image in the centre isan ellipse characteristic of a γ-ray air shower which has few illuminated pixels away from theellipse. On the right is the image of a hadron induced shower which has many illuminatedpixels away from two clear build ups.This chapter describes the analysis techniques used with the images from the Cerenkov

camera on an imaging atmospheric Cerenkov telescope (IACT). This includes the recon-struction, which is tuned to be sensitive to a particular type of source, and the high-levelresults. Only the data coming from the central, larger telescope of the H.E.S.S. array (CT 5)is used in the analysis (mono data), if any of the four smaller telescopes trigger their data isdisregarded.

65

Chapter 3. Reconstruction & Analysis

Figure 3.1.: The simulated particle trajectories of electromagnetic (left) and hadronic (right)air showers are shown. Taken from Volk and Bernlohr 2009

.

Figure 3.2.: Three camera images from CT 5, from left to right: a muon, a photon and ahadron (with electromagnetic sub shower) induced air shower. From left to righttaken from Chalme-Calvet et al. 2014; H.E.S.S. Collaboration 2012; Holler 2014respectively.

66

3.1. Observation Conditions

3.1. Observation Conditions

The calibration of the data takes into account the variation of the instrument (e.g. PMTsperforming differently) but the position in the sky in which the telescopes are pointed alsohas an effect. Each pointing position can be characterised by a zenith and an azimuthangle: directly upwards is a 0 ◦ zenith angle and polar north is a 0 ◦ azimuth angle whichincreases with easterly rotation. The zenith angle Z at which the telescopes are set affects theobservations in two main ways: the light from an air shower at altitude Aeff travels throughAeff

cosZ air and therefore dispersion and absorption effects reduce the intensity of light viewedfrom similar showers with increasing zenith angle. This has the effect of increasing the lowestenergy detectable air shower. The second effect is on the effective collection area Aeff whichgeometrically scales with zenith angle as 1

cosZ . This means that larger zenith angles decreasethe detectable flux threshold through an increase of Aeff thereby enabling the detection ofdimmer sources.The Earth’s magnetic field affects the movement of charged particles in the atmosphere

which has the effect of distorting the imaged ellipse. This effect varies based on azimuthalpointing direction due to the approximately north-south magnetic field lines, this effect isalso strongly affected by the position of the detector on the Earth’s surface. The maximumeffect occurs at a 45 ◦ azimuth angle - de Naurois 2013 describes a study into this effect; themagnetic field at the H.E.S.S. site has been measured to nanotesla precision. Monte Carloγ-rays are therefore simulated to occur from two azimuthal directions: north and south inorder to account for this effect in analyses.With respect to the pointing position in the camera frame of reference, the detector

acceptance falls off with increasing offset due to event truncation at the edge of the camera.This is corrected for when analysing the observation exposure. Generally, it is assumed thatthe camera acceptance is radially symmetric; this is not strictly the case since the offsetdepends on the zenith angle of the observations thus introducing a systematic error. Berge,Funk, and Hinton 2007 describe the details of this and other similar effects, however, asexplained in Section 3.4.5 the effect of offset is very limited in pulsed analyses as done in thiswork.

3.2. Event Reconstruction

The aim of the event reconstruction software is to determine the properties of the primaryparticle. These include:

• Energy.

• Direction.

• Primary particle type.

• First interaction depth.

• Impact point.

• Altitude at which the shower contains most particles.

The energy is related to the number of photo-electrons (p.e.s) detected in the camera asa result of the air shower. This relationship does, however, suffer from a degeneracy since

67


close-by showers of low intensity appear similar to distant, high-intensity showers meaning thatthe determination of the impact point is key. The impact point is the position on the Earth’ssurface through which the trajectory of the primary particle passes. The orientation of theimage in the camera is related to the direction of the primary γ-ray and the first interactiondepth is the altitude at which the primary particle first interacts in the atmosphere.The H.E.S.S. software used to analyse these images is fully integrated within the ROOT

software framework (Brun and Rademakers 1997). The reconstruction of showers in this workwas done using two software chains: the Image Pixel-wise fit for Atmospheric Cerenkov

Telescopes (ImPACT) analysis for the main work & the semi-empirical model γ-ray likelihoodreconstruction (Model++) software used independently to cross check results. They areprogrammed as part of completely independent data calibration chains in order to be a truecross check (see Section 2.5). The ImPACT analysis uses a multi-variate analysis (MVA) forbackground rejection and this MVA analysis in turn uses a parametrisation of the ellipsoidalimage called the Hillas parametrisation as input parameters and to remove clearly badevents.

3.2.1. Hillas Reconstruction

Images of the Cerenkov light produced by a γ-ray induced air shower show a distinct ellipse,as shown in Figure 3.2. This ellipse can be parametrised by the Hillas parameters (Hillas1985) describing the Hillas ellipse. These describe the orientation, shape, and signal containedwithin the ellipse and are used to reconstruct the primary γ-ray’s properties. Some of theparameters that are used within the H.E.S.S. Collaboration are shown in Figure 3.3. Theorientation of the Hillas ellipse as well as its width and length are determined by minimisingthe signal-weighted sums of the squared angular distance to each pixel. The sum of thesignals from all PMTs (i.e. number of photo-electrons) is known as the image amplitudeor size and is related to the energy of the primary γ-ray. Using stereoscopic observations,the direction can be determined by the intersection of the major axes of these ellipses. Thisis, however, not possible with monoscopic observations. Monte Carlo simulated γ-rays aretherefore used to identify the most likely direction using the skewness1 of the image becausethe image maximum is generally skewed away from the side of the ellipse from which theγ-ray originated (Murach 2012). To determine the first interaction depth and ImPACT point,Monte Carlo simulations are used to produce lookup tables with the Hillas parameters asinput. The shower maximum Xmax is determined as the brightest point in the ellipse.

3.2.2. Multivariate Analysis

Multi-layer perceptrons (MLPs) are a type of neural network that can be trained to classifydata, mapping an input to a certain type of output. The H.E.S.S. Mono MVA uses the Toolkitfor Multivariate Analysis (TMVA; Hoecker et al. 2007) with MLPs trained using the latestH.E.S.S. release of Monte Carlo simulated γ-ray shower images and off-source observationdata. Certain Hillas parameters are passed to the MLP that maps them to requested outputsuch as a binary or continuous value. The MVA principle and TMVA software library aredescribed by Hoecker et al. 2007.This MVA analysis uses one MLP for particle identification, one each for event direction

reconstruction and impact point reconstruction, and one MLP to determine the primary

1The image skewness is the third standardised moment of the image signal distribution.

68

3.2. Event Reconstruction

Figure 3.3.: Sketch of the Hillas parameters and their use from a stereoscopic view of the airshower in determining the direction of the original γ-ray. Taken from Aharonianet al. 2006

69


particle energy. The primary particle direction2 is in part determined by an MLP trainedusing the following input parameters:

• The Hillas ellipse width & length.

• The decimal logarithm of the Hillas ellipse amplitude.

• The decimal logarithm of the Hillas ellipse density log10amplitude

width×length .

• The skewness & kurtosis (the third and fourth standardised moments respectively) ofthe signal distribution in the Hillas ellipse.

The trained MLP returns the displacement from the Hillas ellipse’s centre of gravity (thesignal weighted centre of the ellipse) which has a two-fold ambiguity (the head-tail ambiguity).As with a simple Hillas reconstruction, this ambiguity is lifted with the use of the skewness,as described in Section 3.2.1. Another MLP trained with the same parameters is used toreconstruct the impact point.

The particle identification step uses a separate MLP trained with the above mentioned inputparameters except that the logarithm of the amplitude is replaced with another parameter

lengthlog10 amplitude . This extra parameter helps to distinguish the more circular images of hadronicair showers from the elliptical γ-ray air shower images. The density parameter particularlyhelps to flag any imaged muon rings as background since they occupy a large area of thecamera with a low density as opposed to high-density ellipses of imaged γ-ray air showers.The output of the trained MLP for particle identification is the continuous ζ parameter onwhich a cut is made to declare an image as a γ-ray or background.

Finally, the primary particle’s energy is determined using the regression mode of a singleMLP. This MLP is trained with the same parameters as those used with the direction MLP.Once trained, the MLP returns the decimal logarithm of the reconstructed energy based on ittraining with Monte Carlo gamma rays. This MVA is used directly after data taking on-sitein Namibia to obtain quick results for monoscopic data due to its good performance andspeed. During a normal analysis, however, it is often simply used as the seed to the fitting inthe ImPACT analysis.

3.2.3. Image Pixel-wise fit for Atmospheric Cerenkov Telescopes

The ImPACT analysis uses Monte Carlo simulated γ-rays to produce a template of the photondensity at ground level resulting from the extensive air shower from a certain direction andImPACT parameter with a certain primary γ-ray energy. This template translates into anexpected signal in each of the camera’s pixels. Signal from NSB is also simulated and includedwithin the template meaning that the uncleaned camera images are used which is in contrastto the MVA and Hillas reconstructions3. As described in Section 3.1, different observationzenith angles affect the observations and must be simulated. These templates are thereforegenerated in steps within this parameter space and due to this binning the image templatesmust be interpolated for the analysis of an event. The interpolated values are fitted to eachrecorded image from the Cerenkov camera. The shower and primary particle parametersare defined by the interpolated parameters of the best fitting interpolated image template.

2Note: the direction is a single point in the camera not a line3To reduce computational load only two extra rows of pixels in addition to the cleaned image are used.

70

3.3. Optimisation of the Analysis Configuration for Low Energies

The fitting procedure uses a likelihood method - based on that of de Naurois and Rolland2009 - and further described by Parsons and Hinton 2014. The likelihood for each pixel isdefined by

P (s|μ, σp, σγ) =∑n

μn exp−μ

n!√

2π(σ2p + nσ2

γ)exp

− (s−n)2

2(σ2p+nσ2

γ ) (3.1)

for a measured signal s and expected signal μ. The expression is a sum over the Poisson

distribution of the expected number of photo-electrons n convolved with the resolution of thepixel. The resolution of the pixel is taken as an approximate Gaussian of width consisting of asum of the electronic pedestal peak σp and the single photo-electron peak σγ , see Section 2.5for a short description of these two quantities. The log-likelihood is then given by

loge L = −2 loge P (s|μ, σp, σγ) (3.2)

and is approximately χ2 distributed. After summing over all significant pixels in the image,the likelihood of the event for a given set of shower parameters is given. This is minimisedaccording to a six dimensional fit of the primary particle properties: source position, impactpoint, Xmax, and energy. These parameters are initially set to those determined by the MVA(described in Section 3.2.2). The head-tail ambiguity is resolved by testing both startingpoints and using the one with the higher likelihood. The difficulty in fitting mono events (asthere is far less information when compared with multiple telescope observations) results indegeneracies in the fitted parameters and a relatively weakly constrained likelihood surface. Atwo-stage fitting procedure is therefore used: carry out the fit with Xmax fixed to an expectedvalue then repeat the fit keeping the source position fixed to its best fit values from the firststage of the fit.

3.2.4. Semi-Empirical γ-ray Likelihood Model Reconstruction

The analysis (Model++) is very similar to the ImPACT analysis and was the first to successfullyimplement this type of shower reconstruction within γ-ray astronomy. The Model++ analysisdiffers from the ImPACT analysis in the determination of the expected signal. Where theImPACT analysis uses Monte Carlo simulated γ-ray air showers to build a template ofthe photon density at the detector, the Model++ analysis uses a semi-empirical model todetermine this photon density which also includes signal from NSB. Holler 2014 adaptedthe Model++ analysis to monoscopic observations with CT 5. de Naurois and Rolland 2009fully describes the principles of this reconstruction technique. This reconstruction was simplyused as a cross-check of the results of the ImPACT reconstruction since it is adapted foranother calibration chain within the H.E.S.S. Collaboration. Holler 2014 optimised the cutconfiguration used in the analysis to be most effective for sources with a steep power lawspectrum with index −4.0.


With the introduction of the CT 5 telescope into the H.E.S.S. array came the first useof monoscopic data within the H.E.S.S. Collaboration. This required the development ormodification of reconstruction software as described in the previous sections of this chapter.The cut configurations of these reconstructions also needed to be optimised for general sourcetypes observed using CT 5, this optimisation procedure forms part of this work.

71


The analysis of data taken with Fermi -LAT has shown that almost all pulsars have aspectral cut off in the low gigaelectron-volt (GeV) energy regime (Abdo et al. 2013). IACTsbecome significantly less sensitive in the tens GeV energy regime as there is a steep declinein the effective area from the absorption of the Cerenkov light in the atmosphere and lowenergy showers emit significantly less light. It is desirable to optimise analysis performancein this regime not just for the analysis of pulsars but also for γ-ray bursts and other γ-raysources whose emission is expected to steeply decline in this energy band. For this reason anExtra Loose cut set was defined and optimised to perform best with sources whose spectrumsteeply declines with increasing energy. Loose and Standard cut configurations were alsodefined for less steeply declining source spectra for use with most other H.E.S.S. targets. Thedefinition of the Extra Loose configuration for ImPACT is described here as it is used inChapter 4 to analyse data on the Vela pulsar. The other five configurations (Extra Loose,Loose and Standard for MVA and Loose and Standard for ImPACT) defined as part of thiswork using the same procedure are described in Appendix B.

3.3.1. Optimisation Parameters

The optimisation procedure requires Monte Carlo simulations of γ-ray air showers alongwith off-source observation data (i.e. from areas of the sky containing no γ-ray nor clearastrophysical optical signal) to determine the best analysis cut configuration. The off-sourcedata are more effective than Monte Carlo simulations of the background as hadronic airshowers are difficult to model. These events will also match background data better thanMonte Carlo simulated background due to the accurate inclusion of electronic noise andartefacts.The optimisation centred on three parameters:

• The background rejection parameter ζ (zeta).

• The square of the angular distance from the event direction to the expected sourceposition in the sky θ2.

• The Hillas image size.

Pre-reconstruction cuts are applied to two other parameters but not optimised as theyare concerned with removing bad events: the number of pixels in the Hillas ellipse mustbe greater than 3 and the weighted centre of gravity of the Hillas ellipse must be within1.15 ◦ of the centre of the camera. The former removes events that are too small and difficultto classify and the latter removes events that are potentially truncated by the edge of thecamera. Both analysis chains (MVA and ImPACT) use the same cut parameters.

The ζ parameter is output by the particle identification MLP. The MLP is so trained thatγ-ray-like images have a ζ value close to one and background-like images a value close to zero.This is the principal signal-background discriminating parameter for both reconstructions. Theθ2 parameter is the squared angular distance from the primary particle reconstructed directionto the source position. Finally, the image size (hereafter simply size) is a pre-reconstructioncut that is used to filter small events that are difficult to classify and reconstruct. Higher cutvalues reduce the number of small, and therefore low energy, events that pass cuts which hasthe effect of reducing the effective collection area at low energies in the regime that the ExtraLoose cut set is designed to be most sensitive.

72


As aforementioned, the optimisation procedure involves two datasets: Monte Carlo generatedγ-ray shower images and off-source observation data. The data sets were analysed with thesame software and an extremely loose cut configuration. Once the reconstructed eventparameters are stored on file, all combinations of cuts on the three parameters are applied tothe data in the ranges: 0.0 ≤ ζ < 1.0 in steps of 0.01, 0.01 ◦ ≤ θ2 ≤ 0.5 ◦ in steps of 0.01 ◦,and 30 p.e. ≤ size ≤ 70 p.e. in steps of 10 p.e..

3.3.2. Quality Factor

The optimisation proceeded using a quality (Q−) factor, which is applicable for situationsof dominating background such as H.E.S.S. II observations, taken from Li and Ma 1983,Equation 10b:

S =NS

α√NOFF

(3.3)

Q =Safter cut

Sbefore cut=

NafterS

NbeforeS√Nafter

OFF

NbeforeOFF

=εS√εOFF

. (3.4)

Equation 3.3 expresses the approximate significance of an observation of NS signal events withNOFF background events with signal-background exposure ratio α. Equation 3.4 representsan approximate significance gain from the cut configuration being tested. The cut efficienciesεS/OFF for γ-rays and background respectively are simply the ratio of the number of eventsbefore and after the cuts have been applied. Equation 3.3 does not consider the statistical erroron the counts in the signal region NON (= NS + αNOFF ) and leads to an underestimationof the statistical error which in turn leads to an increase of the reported significance, seeSection 3.4.5 for the correct formula for the significance (Li and Ma 1983). During thisoptimisation the improvement of performance rather than the absolute signal is being testedmeaning this overestimation can safely be disregarded.This Q−factor was chosen as it is used to optimise the detection significance of a source.

This is often used at the beginning of experiments as source detection is, at first, desirable.Other parameters can be optimised such as the signal-to-noise ratio ( NON√

NOFF) which would

benefit the production of a spectrum as this is the critical quantity in each energy bin.The Monte Carlo γ-rays are simulated for a point-like source at different zenith, azimuth,

and offset values. The source is simulated with a power law spectrum:

d3N

dEdAdt= I0

(E

E0

)−Γ

(3.5)

of spectral index Γ = 2.0 with flux normalisation I0 and pivot energy E0. The spectral indexof the Vela pulsar was assessed using data from the Fermi -LAT (Abdo et al. 2010b). Anunpublished version of the spectrum is used on which to base the optimisation that onlyconsiders the emission from phase region [0.5, 0.6] above 10GeV (see Chapter 4 for the choiceof phase region and further information on the Fermi -LAT data analysis). The best fittingpower law has the following parameters:

φ0 = [1.8± 0.6]× 10−1MeV−1cm−2s−1

73


where

E0 = 1000MeV

Γ = 4.11± 0.10.

The Monte Carlo γ-ray data, which is simulated with a power law index of −2, must beweighted to simulate this steeper spectrum.

3.3.3. Weighting the Monte Carlo Spectrum

To simulate a power-law spectrum with index −Γ′ using Monte Carlo simulations with a powerlaw spectrum with index −Γ, the expression for the Q−factor is modified with a weighting.This weighting is based on the ratio of the two power laws

w =

(EE0

)−Γ′

(EE0

)−Γ. (3.6)

This weighted Q−factor uses an altered definition of the cut efficiency

ε =

∑i∈ afterwi∑j∈ beforewj

. (3.7)

For the Extra Loose cuts this was done with Γ = 2.0 and Γ′ = 4.0 using an energy pivot pointat E0 = 60GeV. Γ′ = −2.0 (unweighted) & Γ′ = −3.0 were used for the Loose and Standardcuts respectively.

3.3.4. Cut Optimisation

For the Extra Loose cut configuration the Q−factor was calculated for each set of tested cutparameters; weighting the γ-ray data to a power-law spectrum of index −4.0. For each cuton image size, the maximum Q−factor was determined in θ2 − ζ space. Figure 3.4 showsthe optimisation surface for image size > 40 p.e. for Extra Loose weighting with ImPACT.The Q−factor reaches a maximum at 2.11 where θ2cut = 0.045 deg2 & ζcut = 0.44. Figure 3.4displays a steady increase of the Q−factor with respect to the θ2 cut up to an optimal value.The optimum value appears to vary only slightly across the full range of ζ indicating stability.The ζ cut value however displays more structure. This can be understood from the distributionof ζ as shown in the left panel of Figure 3.5 for the data on which the cuts were optimised.The separation power of the ζ parameter is clear from the two peaks at zero and unity forbackground and γ-rays respectively. The middle peak represents events that are difficult toclassify. This distribution can be further understood when stretched over the size as shown inthe middle & right panels of Figure 3.5. It becomes clear that the central peak is made up oflow size events. Due to the positive correlation between size and primary γ-ray energy themiddle peak represents low energy events that the Extra Loose cut configuration is designedto include. The Q−factor displays the effect of the weighting to such a steep power lawspectrum as the cut configuration with the highest Q−factor retains the data in this peakdespite the large contamination with background data. It is normally inadvisable to placea cut on the peak of a distribution as the distribution will inevitably vary when used with

74


Figure 3.4.: Q−factor against the ζ and θ2 cut values in the scanned range for a size cutof 40 photo-electrons. Maximum Q−factor achieved is 2.11 for ζ = 0.44 andθ2 = 0.045 deg2. Red represents a higher Q−factor and blue a lower Q−factorvalue.

75


Figure 3.5.: Left panel: the distribution of the background discrimination parameter ζ witha size ≥ 40 photo-electrons cut applied for Monte Carlo simulated γ-rays shownwith a solid red line and off-source (background) data shown with a dashed blueline. The vertical dotted line indicates the optimal cut for the Extra Loose cutconfiguration for ImPACT. Middle panel: the distribution of ζ against size forMonte Carlo simulated γ-rays, right panel: the same for off-source data. Notethe logarithmic scale on the z axis.

different data. Such changes would not be tracked by the cut leading to a variation in the cutefficiency and different behaviour. The ζ cut was therefore restricted to a ζ value for whichthe distribution dropped to 15% of its central maximum.

The maximum Q−factor obtained for each size cut monotonically decreases with increasingsize cut and does not reach a maximum within the scanned range. Reducing the size cuthowever introduces smaller events that are difficult to reconstruct, furthermore, having apre-reconstruction cut on the number of pixels in the image > 3 together with 05/10 imagecleaning means that after cleaning the minimum size is 25 p.e.. Both the MVA and ImPACTanalyses poorly reconstruct images with a very low size and therefore the minimum sizepermissible was determined to be 35 p.e. for MVA and 40 p.e. for ImPACT as it requires theflexibility when performing the fit to the pixel values. This resulted in the definition of theExtra Loose cut configuration for the ImPACT reconstruction as the point on the surface forsize ≥ 40 p.e., ζ value that corresponds to 15% of the central peak and θ2 value correspondingto the maximum Q−factor with those conditions:

• ζ ≥ 0.44

• θ2 ≤ 0.045 deg2

• Image size ≥ 40 p.e.

along with some un-optimised pre-reconstruction cuts that are common with MVA:

• Number of pixels in the Hillas ellipse > 3

• The angular distance from the Hillas ellipse centre of gravity to the centre of thecamera ≤ 1.15 ◦.

Once the cut configuration is defined the corresponding instrument response function can bedetermined.

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3.3.5. Performance

The performance of an analysis can be characterised by several distributions: the energybias & resolution, the angular resolution, and the effective area together referred to as theinstrument response function (IRF). The energy bias and resolution are defined in simulatedenergy bins as the mean and width of the distribution: Ereco−Esim

Esimrespectively, where Esim is

the Monte Carlo simulated event energy and Ereco is the reconstructed event energy. Theangular resolution is defined by the value of θ at which 68% of simulated γ-rays from a pointsource originate (measured as an angle). The effective area is the product of the probabilitythat a simulated event reaches the final data set after analysis cuts with the area in which thesimulated shower impact points lie calculated with many simulated showers. It is best for theenergy bias and resolution as well as the angular resolution to be small implying an accurateenergy and direction reconstruction. It is also clear a large effective area is desirable as moreevents are recorded, however, not at the expense of the other quantities, an appropriatemedium is sought.

3.3.5.1. Improvements When Using Extra Loose Over Standard Cuts

A comparison of the IRFs from Standard cuts and Extra Loose cuts with ImPACT is shownin Figure 3.6. The boost in effective area at energies below 100GeV with Extra Loose cuts isclearly demonstrated in the upper left panel. This is a clear consequence of the definition ofeffective area and the Extra Loose cuts that allow more events through. The top right plotshows the point spread function (PSF) characteristic width. Clearly, the Extra Loose cutsperform worse over most of the energy range as they include small low energy events whosedirection is difficult to determine. The PSF width with Standard cuts only exceeds that of theExtra Loose cuts when the stricter Standard cuts begin to significantly limit the number ofevents in the energy bin. This can be seen on all of the plots in the figure and is characterisedby the drop in effective area for the Standard cuts at a higher energy. The two lower plotsare the energy bias and resolution. The energy bias shown in the bottom right is similar formost of the energy range but again the increased statistics comes into play below 100GeVand the energy bias for the Standard configuration increases. The energy resolution of theStandard configuration is clearly better for the overlapping energy range as a result of the lowerbackground contamination. At low energies (∼ 30GeV) the bias distribution is skewed andbegins to leave the Gaussian regime. This leads to a poorly understood reconstructed energyand corrected energy. The bias plot (simulated energy plotted against reconstructed energyalso known as the energy migration matrix) is used to correct the reconstructed energy toobtain a better estimate of the true energy of the γ-ray. This, however, results in a correctionerror based on the energy resolution at that reconstructed energy. Profiles in reconstructedenergy of the distribution from which the bias and resolution are derived Ereco−Esim

Esimshould be

Gaussian distributions as the bias should have an symmetric error and clear mean. Whenthis distribution becomes non-Gaussian - this often occurs at the lowest energies (∼ 30GeV)- the resolution sometimes appears to drop (this is the case for the MVA analysis at theseenergy but not the ImPACT analysis, see Appendix B). This is however a result of the skewednature of the distribution.

3.3.5.2. Comparing Extra Loose Between MVA and ImPACT

As Extra Loose cuts were defined for both MVA and ImPACT the advantage of the morecomputationally intensive ImPACT analysis can be assessed. The MVA Extra Loose cuts are

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Figure 3.6.: Performance plots for the ImPACT Standard and Extra Loose cut configurationsproduced using Monte Carlo simulated γ-rays and shown for the simulatedobservation conditions: zenith angle 20 ◦, azimuth angle 180 ◦, and offset angle0.5 ◦. All plots are binned in simulated primary particle energy. Bottom left &right panels show the energy bias and resolution respectively, top right shows the68% signal containment radius representing the angular resolution and the topleft the effective collection area.

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Figure 3.7.: Similar to Figure 3.6 but comparing the Extra Loose analysis configurations withMVA and ImPACT.

similar: ζ ≥ 0.55, θ2 ≤ 0.055 deg2 & size ≥ 35 p.e.. Pre-reconstruction cuts are also similarother than the removal of the number-of-pixels cut. As this MVA chain does not use animage fit, the image amplitude cut is sufficient to ensure reasonable sized events are presentand regardless of this cut, the image cleaning requires that all images contain a minimum oftwo pixels. The relaxation of the size cut is permissible as the MLP does not perform themulti-dimensional fit as used in the ImPACT reconstruction, which requires more flexibilityto distinguish between image templates. The ζ cut is different because the MLPs used byImPACT were trained on a steeper simulated γ-ray spectrum (power-law index of −4 ratherthan −2), which slightly changes the position of the central peak in ζ and results in a differentcut position.The comparison between the performances of these two methods using Extra Loose cuts

is shown in Figure 3.7. The effective area with ImPACT is very similar to that of MVA inthe energy range of interest E ≥ 30GeV and has a similar peak value. During an analysis, asafe energy range is often defined as the energy range in which the effective area exceeds 10%of its maximum and this can be used to compare the performance of different analyses anddifferent configurations. With this definition the two analyses have a very similar low safeenergy level which varies depending on the observing conditions but is around 29GeV for 20 ◦

zenith angle, 0.5 ◦ offset angle, and 180 ◦ azimuth angle. The angular resolution is also only

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marginally improved at the lowest energies with ImPACT. The energy bias and resolutionwith ImPACT are better across most of the energy range except at the lowest energies whereMVA beats ImPACT by almost one third in resolution. This is the effect of a departurefrom the Gaussian regime as noted in Section 3.3.5.1 which is undesirable and indicatesbetter performance when using ImPACT. Analyses on data on the Crab nebula and activegalactic nuclei PKS 2155− 304 - sources with both high and low background respectively -were performed to validate the configuration and are described in Appendix B. The resultswith the data on these non-pulsating sources is satisfactory, the performance of the ImPACTanalysis is clearly more reliable at low energies (< 100GeV) and therefore this chain was usedto analyse the data taken on the Vela pulsar; this is presented in Chapter 4.

3.4. Pulsar Software

A software module designed to analyse data to search for pulsations from rotation-poweredpulsars was written as part of this work to be used directly after the reconstruction stagedescribed above. This module (PulsarSearch) interfaces with a third-party software calledTempo2 (Hobbs, Edwards, and Manchester 2006) that calculates the pulsar rotational phase(hereafter simply phase) with a timing solution (ephemeris) and is the industry standardused by several scientific collaborations including the Fermi -LAT Collaboration (Abdo et al.2010b). PulsarSearch is used in part to determine the background rate within the sampledsky region using unpulsed regions of phase-space to produce pulsed energy spectra and skymaps. Statistical tests and peak fitting are also implemented to characterise the pulsation aswell as tests to determine systematics in the determination of the background.

PulsarSearch is also used to lookup and pass the ephemeris to Tempo2 for each runin the data set. The available ephemerides in the H.E.S.S. Collaboration were sorted into aConcurrent Versions System (Price 2015) repository based on their origin (catalogue).A memorandum of understanding was reached with the Fermi -LAT Collaboration to supplyephemerides to the H.E.S.S. Collaboration and some ephemerides, published on the internet,are included. For each DST analysed - given the name of the pulsar and catalogue in which aephemeris should be sought - a new ephemeris is automatically searched for by PulsarSearch.Once a valid ephemeris is found, PulsarSearch recovers the following event-wise propertiescalculated by Tempo2:

• The pulsar phase.

• The timing residual from the ephemeris.

• The event arrival time and its error in barycentric coordinate time (TCB).

• The time of emission in the pulsar reference frame retarded by the photon transitduration (pulsar emission time; PET).

As the distance to a pulsar cannot be determined to sufficient accuracy, the transformation tothe time in the pulsar frame of reference, carried out as part of the phasing, is always retardedby the light travel time to the pulsar. To calculate these event-wise values Tempo2 requiresthe position of the detector, the event arrival times, and the pulsar timing solution. WithinPulsarSearch, an interface with Tempo2 was implemented so that the afore-listed propertiesare calculated at the beginning of the analysis for all events in a DST. Tempo2 is sent bunchesof 10, 000 events at a time as it is more computationally practical and efficient than calling

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Tempo2 functions per event as that is resource intensive and therefore time consuming. Bydetermining the phase at the beginning of the analysis of each DST, the parameter can beused in event discrimination at any point in the analysis chain. PulsarSearch stores theevent-wise information as well as run-wise data including the ephemeris used.

3.4.1. Calculation of the Solar System Barycentred Event Arrival Time

Each event recorded with H.E.S.S. is assigned a time stamp (time of arrival: TOA) by aGPS167 Meinberg GPS receiver (Meinberg Funkuhren GmbH & Co. KG 2015) in coordinateduniversal time (UTC), see Section 2.2.2. UTC is a time system based on the Earth’s non-inertial reference frame and therefore the standard in astronomy is to convert this time tobarycentric coordinate time (TCB). TCB is a relativistic coordinate time scale that applies tothe solar system barycentre (SSB) reference frame (IAU 2006). Tempo2 transforms the eventTOA recorded at the detector tobsUTC in the UTC time system to the pulsar frame of referencetpsremm via the TCB time system (Edwards, Hobbs, and Manchester 2006):

tpsremm = tobsUTC −Δ� −ΔIS −ΔB. (3.8)

The applied corrections are grouped: Δ� describes the transformation of a detector TOA(UTC in the case of H.E.S.S.) to a TOA in the SSB frame (TCB), ΔIS accounts for thetransformation from the SSB frame to the pulsar frame of reference, and ΔB accounts for thetiming delays due to any orbital companion the pulsar has (not applicable for data analysedin this work). For γ-ray observations, several of the contributions to these transformationsare negligible. Tempo2 was originally designed for radio astronomy and as γ-ray photonsare at a frequency roughly 13 orders of magnitude higher any frequency dependent delay willlikely vary greatly between the two.The transformation from UTC to TCB (Edwards, Hobbs, and Manchester 2006) itself

involves several contributions:

Δ� = ΔA +ΔR� +Δp +ΔD� +ΔE� +ΔS� (3.9)

where

• ΔA represents the delay due to the passage of the detected photons through the Earth’satmosphere.

• ΔR� is the delay between the event TOA at the detector position and the TOA atthe position of the SSB, known as the Roemer delay with the value for a H.E.S.S.observation ≈ 462 s.

• Δp is the so-called parallax delay. This delay is the TOA difference due to sphericalwave fronts arriving at different times at the detector position and the end-point ofthe projection of the SSB-detector vector onto the SSB-pulsar vector (this correction isincluded in the estimate of the Roemer delay above).

• ΔD� is the delay due to the dispersion caused by the inter-planetary medium (mainlyelectrons and negligible for γ-ray observations as it is proportional to wave length).

• ΔE� accounts for the relativistic timing effects of spin, orbital acceleration, and gravita-tional potential of the non-inertial centre of Earth reference frame to transform to thequasi-inertial SSB frame; it is known as the Einstein delay.

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• ΔS� accounts for the time delay due to the passage of photons in the curved space-timeof the solar system (Tempo2 accounts for the Sun, Venus, Jupiter, Saturn, Uranus,and Neptune) and is known as the Shapiro delay with a approximate size ∼ 10−6 s.Tempo2 also accounts for the geometric increase in path length as a result of theeffective bending of the photon geodesic trajectory due to observations near the sun butno other solar system bodies.

The transformation from the SSB frame to that of the pulsar (i.e. considering the photonpropagation through interstellar space) is also split into components:

ΔIS = ΔV P +ΔISD +ΔFDD +ΔFDD +ΔES (3.10)

where

• ΔV P is the vacuum propagation time from the pulsar’s frame to the SSB frame. Thiscannot, in general, be calculated because the distance to the pulsar is not known to asufficient precision. All epochs are therefore delayed with the light propagation time.The time variation of this value is what is taken into account.

• ΔISD is the interstellar dispersion delay which, as it is dependent on the inverse squareof the frequency, is a negligible effect for γ-ray observations.

• ΔFDD accounts for any frequency dependent delays (no significant dispersive delayingeffects occur for γ-rays).

• ΔES is the delay caused by the relativistic time-dilation due to the relative velocityof the SSB and the pulsar (this correction is dominated by the errors on the pulsarvelocity).

The binary term ΔB is also composed of terms due to different effects but is not important forthe analysis in this work and is therefore neglected here. The transformation from the SSBframe to the pulsar’s does not consider the existence of gravitational waves along the photonpropagation path. This potentially allows for a direct measurement addressing the existenceof gravitational waves using pulsar timing. For this, however, timing must be accurate to∼ 100 ns, see Jenet et al. 2005 for an example of the method used.

Delays with Imaging Atmospheric Cerenkov Telescopes A correction of particular noteto IACTs is the delay due to the atmosphere ΔA. Observations of a radio or optical signalpropagate from source to detector in the form of photons and Tempo2 must account for thedelay due to the non-unity refractive index of air. Edwards, Hobbs, and Manchester 2006 notealso that the delay caused by the curvature of the atmosphere, which applied to all groundbased detectors, is < 1 ns and therefore insignificant. Signals detected with a Cerenkov

telescope on Earth, however, pass from a photon propagator to an air shower that inducesphotons which are then observed. Several aspects of this technique can introduce timingdelays with respect to a traditional photon detector:

• The particles in the shower do not travel at the same speed as photons in air.

• The time stamp assigned to a H.E.S.S. event assumes the event arrives at the centre ofthe camera i.e. the event travels parallel to the telescope pointing direction.

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• The Cerenkov light that is detected is emitted at a non-zero angle to the trajectory ofthe primary γ-ray.

The signal is propagated - for at least some time - through massive particles in the air showerthat travel at a velocity greater than that of light in the medium. This should be accountedfor when calculating temm

psr . An estimate of the maximum delay caused due to this effectcan be made assuming the γ-ray encounters the atmosphere and immediately creates anelectron-positron pair. This pair travels down to the altitude of the shower maximum andemits the Cerenkov light at that point. Estimating the first interaction depth to be atan altitude of 100 km and the shower maximum to be at 10 km (de Naurois 2013) with arefractive index for air (disregarding its decrease with altitude to give the largest travel timedifference) n ≈ 1.0003 (Penndorf 1957) gives the unrealistic maximum time lead to be

Δt ≤ 100− 10 km

c− 100− 10 km

cn

= 90ns. (3.11)

This is a negligible time lead for H.E.S.S. observations. H.E.S.S. observations are mostlycarried out with a pointing offset from the source position in order to get an estimate of thebackground rate from an area of the camera with the same offset from the camera centre. Thisalso creates a delay because the event time stamps are determined assuming zero pointingoffset. Considering this assumption the extra path that is traversed by the Cerenkov photonswith an angular pointing offset O for a direct distance to the shower of S is

Δpath = (1− cosO)S (3.12)

which leads to a delay, when assuming a typical pointing offset of 0.5 ◦, of

Δt = (1− cosO)nSc

= 1.3 ns. (3.13)

Again, this is a negligible time difference. The final consideration is that the Cerenkov lightis emitted at the Cerenkov angle θC = arccos 1

βn ≈ 1.5 ◦ which can also be characterised ina similar way (assuming β ≈ 1):

Δt =

(1− 1

βn

)nSc

= 10ns. (3.14)

The combination of the first time lead and the latter two time delays contribute to an omittedtime lead in Tempo2 for IACT observations of around 80 ns. This represents an insignificantfraction of both the period (around 100ms for the Vela pulsar, see Table 4.2.) and precisionwith which a pulse can be determined. The integration window used by the Cerenkov

camera to record events with CT 5 is 16 ns (Bolmont et al. 2014), the central trigger usedin H.E.S.S. to define the event TOAs has a limitation based on the TTL signal it receivesat 10MHz and a long term accuracy of < 2 μs. This means that, although this time leadis not accounted for, it is not a significant effect. In the future, if IACTs are to be used toaccurately time γ-ray pulsars to detect gravitational waves along the light trajectory thenthis would have to be accounted for (Manchester 2010).

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3.4.2. Calculation of the Rotational Phase

Using the PET, the pulsar rotational phase is calculated as a Taylor expansion in the photonfrequency ν:

φ(t) =∑n≥1

νn−1

n!(tpsremm − tP )

n + φ0 +∑

glitches

φg. (3.15)

A reference phase φ0 is used to allow for absolute phase alignment that is specific to theobservation frequency & location and is determined from a reference TOA given by the userwhich normally lies in the centre of the ephemeris validity range. tP is the epoch at which therotational frequency of the pulsar’s emission νPSR is equal ν and is again provided by theuser. Tempo2 uses the measured position of the H.E.S.S. detector as:

(X, Y, Z) = (5622462.3793m, 1665449.2317m, -2505096.8054m)

in geocentric coordinates determined using GPS triangulation (USAF 2008) - this is 1800±20mabove sea level. The precision with which this position is given introduces a source ofuncertainty on the transformation from UTC to TCB. The light travel time on this precision(0.0005m) is O

(10−12 s

)which is insignificant for observations with H.E.S.S.. The parameter

tP is an epoch set by the user and the terms φg are contributions to the phase as a result oftiming glitches.

3.4.3. Timing Irregularities

There are parameters that can be passed to Tempo2 to deal with irregular pulsar rotationwhich can be caused by, for example, glitches, unaccounted for planets, precession, and asyet unknown effects. Young pulsars particularly have extra structures in the residuals of thetiming (Hobbs, Lyne, and Kramer 2010) but generally fewer glitches (Espinoza et al. 2011).Glitches are modelled by adding glitch terms φg in Equation 3.15 which are non-zero onlyafter the glitch epochs tg:

(3.16)φg =

⎧⎪⎪⎪⎨⎪⎪⎪⎩0, if t < tg

Δφ+Δν (tpsremm − tg) +12Δν (tpsremm − tg)

2+(

1− exp

(−(tpsremm−tg)

τ

))Δνt (t

psremm − tg) , if t > tg

The glitch modelling consists of shifts in the phase Δφ, frequency Δν, and frequency derivativewith time Δν as well as an exponentially decreasing frequency shift Δνt with time constantτ . One set of these parameters must be passed to Tempo2 as part of the timing solution inorder to accurately calculate the phase during and after each glitch in the phasing epoch.Other contributions to timing noise manifest as a long term pseudo-sinusoid that can be

handled by whitening sinusoidal wave terms (Hobbs et al. 2004). These are subtracted fromthe timing residuals. The user defines the sinusoids in the timing solution passed to Tempo2,which then fits the timing residuals. The fitted sinusoids are subtracted from the timingresiduals to give whitened values that are free of this timing noise.

Timing Solutions - Ephemerides In order to calculate the pulsar’s rotational phase from anevent time stamp an ephemeris is required. This consists of the rotational frequency as wellas at least its first & second derivative with respect to time and a reference time from which

84


the phases are relative. The parameters that can be passed to Tempo2 are documented byHobbs et al. 2014, Chapter 3. These ephemerides are only valid in a certain time period -defined by start & finish given in modified Julian date4 (MJD) (Winkler 2015) - due topulsar glitches and poorly understood timing noise, see Section 3.4.3.

3.4.4. Interface and Use of Tempo2

Two interfaces with Tempo2 are implemented in PulsarSearch:

• The H.E.S.S. Tempo2 plugin.

• The direct integration within the H.E.S.S. analysis chain.

The H.E.S.S. Tempo2 plugin is not yet a standard part of the Tempo2 package. It is a toolavailable to the H.E.S.S. Collaboration and has a basic command line interface. The requiredinput is the name of the pulsar to be phased, the catalogue from which a timing solutionshould be found, and a list of the DSTs that contain the events to be phased. The output is aset of DSTs appended with an extra event-wise member that contains the event phase, arrivaltime in TCB, and PET. An average CT 5 DST contains around 2.5× 106 events for which theplugin requires around 62CPU min. These phased DSTs can then be used in further analysiswith PulsarSearch. This is, however, used as a quick access method since it duplicates thedata and is relatively inefficient.The direct integration method requires the Tempo2 library to be linked to the Pul-

sarSearch library. As Tempo2 is external software and PulsarSearch is a H.E.S.S.software module, a C++ preprocessor variable is used to toggle the inclusion of the Tempo2

library in the build of the H.E.S.S. software. The H.E.S.S. software therefore does not re-quire a working Tempo2 installation. The Tempo2 software, shipped to be built with theMake (FSF Inc. 2015) software building tool, was integrated into the SCons (Foundation2015) software building tool used by the H.E.S.S. Collaboration.

Once built and linked to Tempo2, PulsarSearch can be used within the H.E.S.S. AnalysisProgram (HAP) - a program written by the H.E.S.S. Collaboration to analyse H.E.S.S. data- to calculate the phase as part of a standard H.E.S.S. data analysis. The principle of theentire H.E.S.S. software is based around loading each event individually and carrying out alloperations before the next is loaded. This is not optimal for phasing with Tempo2 as callingthe Tempo2 functions per event are resource intensive. The procedure of a pulsed analysisfor each DST is, therefore, as follows:

1. PulsarSearch loops over all events in a DST and uses Tempo2 to calculate the phasefor each event.

2. HAP loops over events in the DST

a) Pre-reconstruction cuts are applied.

b) Reconstruction software determines the event properties.

c) Post-reconstruction cuts remove background-like events.

d) PulsarSearch loads the events phase from its storage and uses it to select thepulsed and background phase regions.

4The Julian date (JD) is a continuous count of the solar days starting at zero at noon on 1st January4713 BCE and was modified to reduce the absolute value by defining the modified Julian date (MJD) asMJD = JD − 2400000.5.

85


e) Sky maps and spectra are computed using phase information.

To calculate the phase, Tempo2 is provided with the timing solution and H.E.S.S. UTCTOAs of each event in bunches of 10, 000 events, this is a suggested limit hard coded intoTempo2. This process requires only around 20CPU min to calculate the phase for all eventsin a DST using the same machine with a similar load as used with the Tempo2 plugin.PulsarSearch outputs a PulsarLightcurve ROOT C++ object saved to a ROOTfile (Brun and Rademakers 1997) which uses the TTree ROOT class for data storage. Thisobject stores run-wise and event-wise information and many can easily be combined. Thisallows analysis to be done on each DST as a separate process, ensuring a far quicker analysis,the results of which can later be combined.

3.4.5. Background Estimation & Significance

The area in the sky in which the source is expected is designated in the configuration ofthe analysis and known as the ON region. A further background rejection stage beyond thepost-reconstruction cuts is usually carried out by designating areas of the sky from where noγ-ray nor optical source is expected, known as OFF regions. This allows an estimate to bemade of the number of background events that make it past post-reconstruction cuts and aredetected in the ON region. This background estimation method allows an excess NS to bedefined:

NS = NON − α×NOFF (3.17)

from the counts detected in the ON region NON and OFF region NOFF (Li and Ma 1983).Assuming all recorded events are from the background, the approximate error on the excess is

σ(NS) =√

α(NON +NOFF ) (3.18)

or if a signal is presentσ(NS) =

√NON + (α2 ×NOFF ). (3.19)

Here α is a normalisation factor defined as the ratio of the ON and OFF acceptance-correcteddetector exposures which is integrated over time and solid angle, see Berge, Funk, and Hinton2007, for more information. This is an important quantity and is discussed later in this section.Li and Ma 1983 use a maximum likelihood method to determine the significance of an excessNS as:

SLiMa =√2

(NON loge

[1 + α

α

NON

NON +NOFF

]+NOFF loge

[(1 + α)

NOFF

NON +NOFF

]) 12

(3.20)

which is henceforth referred to as the Li-Ma significance. This ignores the error on thenormalisation factor α, which, once taken into account (Spengler 2015), modifies the excessto:

Smod = sign(NS)

√S2LiMa +

(α − α

σα

)2

(3.21)

where α is the Gaussian distributed normalisation factor with mean α and error σα. Thisexpression clearly simplifies to SLiMa as α → α. The condition for which using Equation 3.21

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is necessary is given by Spengler 2015 as:

Ξ = σα

√NOFF

K

1

α (α+ 1)≥ 0.1 (3.22)

where the error on α is σα and K is the number of observations made (meaning NOFFK is the

mean number of OFF events per run). K can be seen as the number of data taking runscombined into a data set on a particular source. An analysis on a data set was carried outto estimate σα by calculating the distribution of Li-Ma significance across the sky whilstexcluding sources. This distribution should be standard normal, if the width exceeds onethen this is the effect of the omitted error on α. For those distributions with width above one,Equation 3.21 is used with different values of σα until the width is one. This was repeatedfor a 75 run off-source dataset made up of observations of a similar zenith (∼ 20 ◦), azimuth(∼ 180 ◦), and offset ∼ (0.5 ◦) angle range as the data set on the Vela pulsar analysed inChapter 4. This leads to a measurement σα = 0.0032± 0.0012 which can be combined withan estimate of the number of phase background events 4, 000 and alpha 1

3 to give Ξ = 0.3.Due to the large background, the parameters suggest that the Spengler significance shouldbe used but the conditions are in the transition region as 0.3 does not deviate greatly from0.1 and this constitutes a deviation of only around 5% from a standard normal distribution.The confidence level of a single observation is ξ = 1 − p where p = N(u = S; 0, 1) is the

probability that an event with significance ≥ S is from the standard normal backgrounddistribution N(0, 1). As a result of n trials the statistical significance is diminished to

pn =

n∑j=1

(n

j

)pj (1− p)n−j (3.23)

which can be approximated by restricting the sum to the first two terms

pn = np (1− p)n−1 +n(n− 1)

2p2 (1− p)n−2 (3.24)

and is applicable when p is small (p < 0.5)5. This post-trial probability must correspond to5σ significance in order for a source to be detected in γ-ray astronomy.The definition of the OFF data is particularly important not only because it determines

the excess - and therefore significance - but it also dictates the flux shown in the spectrumand counts in the sky-map. Berge, Funk, and Hinton 2007 describe the various methods usedto determine the rate of background in the ON region in γ-ray astronomy which tend to useseparate areas of the sky to estimate the background rate in the ON region. With the pulsarphase this is not required and the inherent systematic errors associated with, for example,variations in the camera between the ON and OFF region can be eliminated. Only data fromthe ON region is considered and a pulsed & an unpulsed region are defined. The unpulsedregion is used to estimate the level of background in the pulsed region. This also removesany unpulsed γ-ray emission from the measured excess which is classed as contaminated for astudy of the pulsed emission. The aforementioned α factor is simply the ratio of the size inphase of the pulsed phase region to the unpulsed phase region. Systematic variation in the

5Equation 3.24 comes from considering n trials of an experiment with the probability p of reaching signallevel S in a single trial. After n trials the probability of not reaching S is 1− pn = (1− p)n. This can thenbe rewritten as Equation 3.24.

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unpulsed region is assessed in PulsarSearch by fitting sinusoidal, linear, and polynomialsto ensure that no structure is evident in the phase of the unpulsed region. This is also doneper run to ensure stability in time. This method of background rejection is used in Chapter 4.

Sky Map & Spectrum The pulsed emission spectrum and sky map are produced usingthe excess counts as defined by the aforementioned background estimation method. Thereconstructed energy bias and resolution are both taken into account when producing thespectrum which is done using a forward folding method. This method involves using the IRFsto correct the reconstructed energy to better estimate the actual γ-ray energy using a lookuptable (energy migration matrix) filled using simulated γ-ray Monte Carlo events. This is atwo dimensional lookup table binned in simulated (abscissa) and reconstructed (ordinate)energy so that the relationship between the two from simulation can provide a correctionfor the reconstructed energy of measured events. Once the energies have been corrected thespectrum is produced by applying an a priori defined binning on the energy axis and simplycalculating the flux from the effective area, exposure corrected live time, and excess in eachcorrected energy bin. The sky map is produced integrated in energy space and therefore doesnot require the energy bias correction.

3.5. Pulsed Statistics

In order to determine whether or not there is a significant pulsation in a set of N phases φi in[0, 1), many pulsed test statistics can be used. The null hypothesis - uniformly distributedphases - can be written in terms of the (periodic) density function F(φ):

H0 : F0(φ) = 1. (3.25)

which, in the presence of a signal, is altered to include the relative signal function Fs(φ) witha proportion P of pulsed events

H1 : F(φ) = PFs(φ) + (1− P). (3.26)

The presence of a signal can then be characterised by a large value of ψ(F) (de Jager,Raubenheimer, and Swanepoel 1989):

ψ(F) ≡∫ 1

0(F(φ)− 1)2 dφ. (3.27)

The general Beran statistic (Beran 1969) is

ψ(F) = p2∫ 1

0(F0 − 1)2 dφ, (3.28)

it quantifies deviation of the signal distribution from a uniform distribution. If however - as isoften the case - the signal distribution Fs is unknown then estimators F are used. Pearson’sχ2 test is a powerful but simple test separating the phases into b bins:

χ2b−1 = Nψ(F) =

b∑j

(Xj − N

b

)2Nb

(3.29)

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where Xj is the content of the jth bin. This test statistic is clearly χ2 distributed with b− 1degrees of freedom. The main disadvantage of this test statistic is that it requires a binningof the phases. An alternative approach is to consider the Fourier coefficients of the phases:

αk =1

N

N∑i=1

cos (2πkφi) (3.30)

βk =1

N

N∑i=1

sin (2πkφi) (3.31)

for the kth harmonic. Both follow an asymptotic Gaussian distribution, see de Jager 1994for more details. The Cosine and Sine test statistics (de Jager 1994) require the phases tobe centred on an expected pulse maximum μ and therefore substitute φi − μ for φi and theyinclude a sum over the harmonics up to a maximum harmonic m:

Cm =

√2N

m

m∑k=1

αk =

√2

mN

m∑k=1

N∑i=1

cos(k(φi − μ)) (3.32)

Sm =

√2N

m

m∑k=1

βk =

√2

mN

m∑k=1

N∑i=1

sin(k(φi − μ)). (3.33)

Following from the Central Limit Theorem, the Sine and Cosine test statistics follow a standardnormal distribution in the absence of any signal in the data; their values therefore correspondto the significance of the signal. They are only sensitive to single peak pulse profiles andmost sensitive to symmetric pulses. If the pulse maximum is at the expected position μ theSine test statistic should contain only noise. This has been implemented in PulsarSearch

as a minimiser to determine the position of a pulse in the data in an unbinned way. Themethod most often used is to bin the data in a histogram and fit a functional form with aclear mean value. The optimal value of m is estimated as the inverse of the duty cycle of thephase profile (de Jager 1994). The Z2

m statistic is a combination of the Fourier coefficientssummed over all harmonics up to m:

Z2m = 2πNψ(F) = 2N

m∑k=1

(α2k + β2

k

)(3.34)

and is χ2 distributed with 2m degrees of freedom (Bendat and Piersol 2011). This is becausewhen the factor 2N is taken into the expressions for α & β they become standard-normaldistributed which then makes the sum of their squares χ2

2m distributed. Given Equation 3.28 itis clear that Z2

m should scale as p2N and p√N can be regarded as an approximate Gaussian

significance. This test statistic is most sensitive to narrow pulses when m is large and broadpulses for small values of m. The Rayleigh statistic is simply Z2

m=1 and is sensitive to broadsingle peaks (de Jager, Raubenheimer, and Swanepoel 1986).

The U2 test statistic (Watson 1961) is similar but with an extra factor in the sum, which istaken to infinity:

U2 = 2N∞∑k=1

1

2πk

(α2k + β2

k

). (3.35)

89


This test statistic is useful when the shape of the expected pulse is a priori unknown (Quesen-berry and Miller 1977). de Jager, Raubenheimer, and Swanepoel 1989 propose a rescaled Z2

m

test statistic using Hart’s rule (defined by Hart 1985, its use in this case is described by deJager, Raubenheimer, and Swanepoel 1989), the so-called H-test:

HM = max(1 ≤ m ≤ ∞)[Z2m − 4(m− 1)] = Z2

m=M − c(M − 1) (3.36)

suggesting that the maximum need only be taken for the range 1 ≤ m ≤ 20 and c = 4 tosuppress higher harmonics in the null case (de Jager, Raubenheimer, and Swanepoel 1989).This removes the arbitrary choice of m as is the case for the Z2

m test statistic. This test statisticis named the H-test after the use of Hart’s rule in its definition. de Jager, Raubenheimer,and Swanepoel 1989; de Jager and Busching 2010 analysed the distribution of Hm in theabsence of any signal using Monte Carlo simulations and concluded that the distributionfollowed e−0.4HM . Kerr 2011 however analytically determined the probability to observe avalue greater than Hm is given by

P(X > hM ) = e−HM2 ×

M−1∑n=0

GnIn(HM ) (3.37)

where

In = Πni=1

(∫ HM

−∞dxiθ(xi − xi−1 + c)

)

G =1

2e−

c2

which is then used to determine the confidence level of an Hm value and therefore itssignificance. A similar simple expression exp−0.398405Hm to that determined by de Jager,Raubenheimer, and Swanepoel 1989 is however a good approximation where m ≥ 10 (Kerr2011).

All of the aforementioned test statistics are implemented in the PulsarSearch modulealong with the Protheroe, Kuiper, and G test statistics (an interface is available withthe ROOT implementation (Brun and Rademakers 1997) of the Kolmogorov-Smirnovtest statistic; see Appendix C for definitions of all of the test statistics implemented). Asa clarification of implementation, a toy Monte Carlo was used to simulate 100, 000 sets of1, 000 phases. The phases were distributed uniformly in [0, 1) using the ROOT (Brun andRademakers 1997) framework implementation of the Mersenne Twistor pseudo-randomnumber generator (Matsumoto and Nishimura 1998) which has a period of 219937 − 1 andwhose seed is set to be unique in time and space. Each of the statistics were then calculatedand plotted to analyse their null distributions. Fitting an exponential to the distribution of theH test statistic obtained with PulsarSearch results in a best fit exponent of −0.397± 0.002which is within 0.74σ of the value (−0.4) reported by de Jager and Busching 20106. Thereduced-χ2 goodness-of-fit indicator 0.970 = 63

65 which was minimised in the range of thebinned data for which each bin had > 10 entries shows that the distribution is well fit bythe exponential. This validates the Hm test statistic implementation in PulsarSearch, seeAppendix C for the null distributions of the test statistics.

6de Jager and Busching 2010 report that the H test statistic is distributed as exp−0.398Hm but for H > 70 adownturn to an exponent of −0.4 is observed

90


3.5.1. Weighted Pulsed Statistics

An investigation was conducted into the use of weights to boost the performance of thesestatistical tests. The event-wise weight can include information beyond just the rotationalphase. Bickel, Kleijn, and Rice 2008 define a test statistic based on the likelihood of photonarrival times:

Q ≡ 2

T

∞∑k=1

((αkαwk)

2 + (βkβwk)2)

(3.38)

for the total integration time T and the weighted harmonic test statistics:

αwk ≡ IαN

N∑i=0

wi cos (πkφi) ; βwk ≡ IβN

N∑i=0

wi sin (2πkφi) . (3.39)

Kerr 2011 introduces a weighted-H test statistic based on the same αwk and βwk test statisticsusing a normalisation factor based only on the weights and N . Two approximations are madeon the functional form of the distribution shape in order to determine the normalisation. Herea similar weighted test statistic is introduced but using a different approach: the variance ofαwk and βwk (here after referred to as ψw) is required to remain unchanged compared to theunweighted test statistics using a normalisation Iψ which means Z2

m remains χ22m distributed.

The normalisation factor is

I2ψ =N∑N

i G2 −(∑N

i G2)2

N∑N

i w2iG

2 −(∑N

i wiG2)2 (3.40)

where G is C = cos(2πkφi) and S = sin(2πkφi) where appropriate based on ψ. This weighted-Z2m test statistic (Z2

wk) is then used for the weighted-H test statistic:

HwM = max(1 ≤ m ≤ M)[Z2wm − 4(m− 1)] (3.41)

where

Z2wm = 2N

m∑k=1

(α2wk + β2

wk

). (3.42)

See Appendix C for more details on the derivation of the normalisation. As Z2wm remains

asymptotically χ22m distributed the distribution of the weighted-H test statistic also follows

the same exponential distribution as the H test statistic.

Weighting In order to study the improvement that the weighting of the events gives, asignal was artificially embedded into the toy Monte Carlo data set. Ideal weighting would beaccording to the embedded signal phase distribution but this would introduce bias as it isnot an independent random variable. Weights are intended to be obtained from alternativeinformation about the event. To mimic a realistic weighting, which could be used in a H.E.S.S.data analysis, the PSF of an IACT was approximated by a King profile

K(θ) =A(

1 +(θ3

rc

)2)B. (3.43)

91


Here θ, as in previous sections, is the angular distance between an event’s reconstructeddirection and the target direction which was limited to a realistic ROI cut (θ2 ≤ 0.055 deg2) andnormalised to a unity integral in this range to be used as a probability density function (pdf).From the IRFs described in Section 3.3, a characteristic Monte Carlo generated θ2 histogramwas fitted with a King profile giving the following best fit parameters: rc = 0.0675± 0.0030,B = 0.107 ± 0.002, and A = 24438 ± 91. Two components were simulated: i) the pulsedcomponent from the pulsar which was assumed to follow the fitted King distribution and ii)the background (BG) component was approximated to follow a uniform distribution in θ2.The difference between the King and uniform distributions is the basis of the enhancementfrom the weighting. The value obtained from the King profile evaluated at the simulated θ2

value was therefore used as the weight:

wi =K(θ2i )∑Nj=1wj

. (3.44)

As the data are cyclic, a Von Mises distribution (Mardia and Jupp 2008) with unit period

P(x|μ, κ) = exp(κ cos

(x−μ2π

))2πI0(κ)

, (3.45)

where I0 is a modified Bessel function of the first kind and μ is the mean, was used toapproximate a wrapped standard normal distribution for the signal phase pdf. A uniform pdfwas used for the background. The concentration parameter κ is related to the duty cycle δ

(full width at half maximum of the approximated Gaussian) as δ ≈ arccos(1− 0.693κ )

π (de Jager,Raubenheimer, and Swanepoel 1989), this however only holds for κ > 1, in which case awrapped Gaussian was used where 10 periods, each consisting of a single Gaussian, wereincluded.

Data Set Size Scan To asses the stability of the weighted test statistics in the low statisticslimit i.e. N < 50, a toy Monte Carlo simulation with 10, 000 iterations was carried out for Nfrom 5 to 100 in steps of 10; these being the mean of the Poisson distributed N value. Nodata set was considered with number of events < 4 due to the sparsity of the data. The nulldistributions of all test statistics were binned and fitted to their expected asymptotic formsusing a χ2 minimiser. Taking the weighted H test statistic as an example, the goodness-of-fitχ2 value as a function of N is shown in Figure 3.8. Both versions of the weighted H teststatistic perform well down to low statistics, however it is clear that the weighted H teststatistic defined here more reliably maintains the asymptotic exponential distribution: themean and standard deviation of the ordinate values from the left hand plot are −0.402± 0.007for this work and −0.41 ± 0.02 for that by Kerr 2011. The right hand plot shows that assmaller data sets are reached the test statistic defined here deviates from the asymptoticdistribution significantly less than the other test statistic. The reduced χ2 value remainsbelow 2 for a data set with a mean of 10 events for the test statistic from this work but is at3.5 for the other test statistic. The weighted H test statistic defined here can therefore be saidto perform better in the low statistics limits. This is important when applying these weightedtest statistics at the highest energies accessible by a γ-ray observatory where statistics becomelow.

92


Figure 3.8.: The goodness-of-fit to the asymptotic distribution of the weightedH20 test statisticfor different data set size with Poisson distribution errors. Comparison betweenthe weighted H test statistic defined here and that defined by Kerr 2011. Lefthand plot shows the exponent of the best fitting exponential function to theweighted H test statistic null distribution for different data set sizes shown on theabscissa. Right hand plot shows the goodness-of-fit reduced χ2 value for a fit tothe asymptotic exponential function i.e. exponent fixed to −0.4 for different dataset sizes. The dashed black line shows an ideal fit where reduced χ2 = 1.

93


Figure 3.9.: Power of the test statistics shown in the inset legend to reject a false null hypothesis.The maximum harmonic used in all H tests displayed was 20. The errors on thepower are smaller than the markers and therefore not displayed.

Pulsed Proportion Scan The statistical power is quantified by the probability of correctlyrejecting the null hypothesis (not making a type II error; a false negative) and was calculatedwith the following phase-profile parameters: the mean was set to 0.5, the duty cycle 10%, andthe pulsed proportion (signal strength) varied from 0% to 20% in 0.5% steps of the 1, 000simulated phases per data set. The threshold probability above which a test statistic value isinterpreted as being from the null distribution is taken as 5%. This was done 10, 000 timesusing in a toy Monte Carlo simulation. The power of several test statistics for different signalstrengths are shown in Figure 3.9. The 5% threshold is evident from the value of the ordinate(0.05) for zero signal strength. It is clear that weighting improves each test statistic across thefull range of signal strengths. An improvement of 10% is observed for a 10% signal strengthby using the weighting with the H test statistic. The weighted H test statistic defined by Kerr2011 performs the same as the weighted H test statistic defined here.

Duty Cycle Scan The rejection power is dependent on the shape of the signal distributionso duty cycles from 1.5% to 50% were scanned over in 0.5% steps with a fixed 0.1 pulsedproportion to determine this corresponding dependence; this again was done 10, 000 times ina toy Monte Carlo and is shown in Figure 3.10. de Jager, Raubenheimer, and Swanepoel 1989

94


Figure 3.10.: Power of the test statistics shown in the inset legend to reject a false nullhypothesis as a function of the duty cycle of the peak with a fixed pulsedproportion of 10%. The maximum harmonic used in all H tests displayed was20. The errors on the power are smaller than the markers and therefore notdisplayed.

95


Figure 3.11.: Boost in detection significance (σwH−σHσH

where σ indicates detection significance)when using the weighted-H test statistic compared to the H test statistic as afunction of the H test statistic detection significance for a duty cycle of 10%and data set size of 300.

report that the H test statistic is relatively more powerful than other test statistic for narrowphase profiles, it is clear from the figure that the weighted-H test statistic is an improvementon the H test statistic across the entire range of duty cycles; the weighting proves effective forall three test statistics shown. Interestingly it seems that for moderate duty cycles the Z2

m=2

test statistic performs marginally better than the H test statistic but this starkly reversesbelow 10% duty cycle. The weighted H test statistic defined by Kerr 2011 performs the sameas that defined here.

Signal Significance Improvement Quantifying this boost with regard to detection signif-icance is shown in Figure 3.11 determined using the same toy Monte Carlo simulationsmentioned previously. The maximum boost observed with this modest weighting scheme is10% over the unweighted H test statistic. To generally compare the two: the H test statisticreturns a 5 σ significance in the presence of a 15.7% signal strength whereas only a 14.5%signal is required for the same with the weighted test statistic.

The method of background determination in H.E.S.S. is not suited to calculating a probabilitythat an event comes from the source distribution. Ideally, a background model and signal model

96

3.6. Clarification of Phasing Software Using the H.E.S.S. Optical Crab Data

are calculated from which the probability that an event comes from the signal distribution canbe determined. Within the H.E.S.S. software framework a rate of background is estimatedfrom off-source regions and subtracted from the on-source rate to determine the signal rateand significance. For pulsed analysis these regions are the same areas of the sky but differin pulsar rotational phase space. Several parameters can, however, be used to approximatea probability. The ζ and θ2 values described in Section 3.2 could be combined for such apurpose, this would however be most effective when no unpulsed underlying γ-ray signalwas present. An option is to use the gammalib software (Knodlseder 2012) which has beeninterfaced with the H.E.S.S. software in order to determine such a probability but testingremains to be done.

Here it was demonstrated that the weighted-H test more efficiently rejects the null hypothesisin the presence of a single Gaussian-like peak signal compared to the H test statistic andit performs better than the other weighted H test statistic defined by Kerr 2011 in the lowstatistics limit. These statistical tests are powerful when the phase profile is unknown howeverwhen knowledge of the pulse profile is available other tests, such as a likelihood ratio test, canperform better (Kerr 2011; de Jager, Raubenheimer, and Swanepoel 1989).

3.6. Clarification of Phasing Software Using the H.E.S.S. OpticalCrab Data

In 2003 a seven-pixel (one central and 6 outer pixels) optical camera was mounted ontoone of the smaller H.E.S.S. telescopes in order to verify the timing of the instrument. Thisinvolves the event time stamps set by the central trigger GPS clock and the use of them inthe software. The telescope was pointed at the Crab Pulsar. Hinton et al. 2006 describe theoriginal collection and analysis of the entire dataset. The dataset currently available consistsof 4 : 57min live time and 403, 450 signal events. A basic background rejection was carriedout whereby the six outer pixels were defined as background pixels and the central pixel thesignal pixel. The analogue to digital converter (ADC) counts for each pixel were plotted andfitted with a Gaussian. Equation 3.46 shows the conditions for acceptance of signal andrejection of background for each event based on this Gaussian functional fit:

Signal event | P(Xcentral pixel < ADC count) > 0.95 (3.46)

Background event | Πi = background pixels Pi(Xi < ADC count) < 0.05. (3.47)

The principle of this method is based on the optical point spread function being smaller than thepixel size so events with a large ADC count that were also detected in outer pixels are likely tobe background. The ephemeris used for both analyses was taken from the Jodrell Bank cata-logue (Lyne, Pritchard, and Graham-Smith 1993) at http://www.jb.man.ac.uk/pulsar/crab.html.PulsarSearch with a Tempo2 interface - discussed in Section 3.4 - was then used to

calculate the TCB event time stamp and rotational phase corresponding to each event timestamp. The phase-folded light curve is presented in the lower section of Figure 3.12. Ofparticular note is the phase offset of the peaks, which can be quantised by the fitted peakposition of the main, larger peak, around zero phase in the original data analysis. The peakwas fit with an asymmetric Lorentzian function which, as visible from the fitting residuals,well fits the data. This fit shows that a 0.2377(= 0.9987− 0.7610)7 phase offset exists. The

7The original paper used an empirical fit which therefore did not have a clear position of maximum; the one

97


Figure 3.12.: The phase-folded light curves from the optical data, upper, as published inHinton et al. 2006, lower, as processed by the current PulsarSearch softwareusing a smaller dataset with the best fitting asymmetric Lorentzian functionshown by the red dotted line and fitting residuals (χ = O−E

E where O is themeasured count in the bin and E is the value of the function at the bin centred)below.

98

3.7. Data Quality Checks

Figure 3.13.: The value of theHm=20 test statistic up to the event number given on the abscissa.The right hand plot was produced by another code in the H.E.S.S. Collaborationand is courtesy of Dr Gianluca Giavitto by private communication.

peak shapes and their relative position are however in agreement. This suggests that thereis a delay in the event timestamps - most likely in the transmission of the signal from thecamera to the central trigger - which was not taken into account in the data set currentlyavailable and causes a constant shift in the position in the peaks. This can be quantised giventhe period of the Crab pulsar (34.5ms according to the ephemeris used (Lyne, Pritchard, andGraham-Smith 1993)) to around 8.2ms (= 0.2377× 34.5ms). This is specific to the specialoptical set-up used to collect this data and is different when using the telescope to observeCerenkov air showers. Figure 3.13 shows the progression of the Hm test statistic - describedin Section 3.5 - against the number of events recorded. The figure shows a steady increase insignificance in both plots. Another software chain, used in the H.E.S.S. Collaboration for thedata calibrated within the Paris framework, was used to cross check the results, shown in theright hand plot. Both plots in Figure 3.13 reach a maximum of 2299.5 at the last event withthe same shape. The results were consistent both in phase-profile and calculation of the Hm

test statistic.


The Cerenkov data in the H.E.S.S. Collaboration is calibrated using two independent chains:Heidelberg (HD) & Paris. The calibration procedure is explained in more detail in Section 2.5.Once the calibration has been done DSTs are produced which occupy less memory than theraw data. For analysis, DSTs are selected based on quality criteria in order to remove anydata of bad quality that can corrupt analysis results. The newly defined selection criteria forCT 5 data are as follows:

• Broken pixel fraction < 6%.

used was estimated from the phase-folded light curve in the upper panel of Figure 3.12.

99


• Run duration > 5min.

• Mean CT 5 trigger rate > 1200Hz.

• CT 5 trigger rate dispersion < 10Hz.

• System trigger rate dispersion < 50Hz.

• Mean run zenith angle < 40 ◦.

• Homogeneous Hillas centre of gravity map using 05/10 image cleaning (see Section 2.5).

• No data taken during the commissioning phase of CT 5.

• No clouds during the runs.

A broken pixel is loosely defined as when a pixel produces either a persistently high signalor none at all during a run, the strict definition is slightly different between the HD andParis calibration chains. To reduce the probability that these problematic pixels contributeto a false positive (an image accepted by the reconstruction as that of a γ-ray when it isnot) no more than 6% of pixels are permitted to be broken in a run 8. The default durationof a H.E.S.S. data taking run is 28min meaning that a run of duration shorter than thecut value of 5min suggests a problem either with the data acquisition system, a sub-system,or bad weather forcing the data taking personnel to stop the run. The third criterion wasincluded to ensure the camera was performing as designed. As explained in Section 2.3.2,the camera trigger thresholds were adjusted to reduce the number of times the camera wastriggered by NSB but maximise the trigger rate on air showers. These settings mean that atrigger rate below 1200Hz indicates a problem with the camera triggering CPU or clouds inthe FOV, these runs are, therefore, discarded. The centre of gravity (CoG) maps (positionsin the camera of the weighted centre of the Hillas ellipse plotted in the camera frame ofreference over a single run) should be approximately homogeneous, an example of both goodand bad CoG maps are shown in Figure 3.14. There should also be no problems with the dataacquisition system nor the CT 5 camera machines. Finally, all data that was taken beforethe beginning of April 2012 was discarded due to the instability of the system during thecommissioning stages of CT 5.

8this was originally specified as 5% but was relaxed to 6% to included another run that otherwise appears ofgood quality within the Vela pulsar dataset used in Chapter 4.

100


Figure 3.14.: Camera displays showing the Hillas centre of gravity for each event in a datataking run, the left panel shows a bad (inhomogeneous) map and the right panela map that passes run selection.

101

4. Analsysis of H.E.S.S. Data on the VelaPulsar

The pulsar in the constellation Vela (PSR J0835−4510; the Vela pulsar) is associated withthe Vela supernova remnant (Large, Vaughan, and Mills 1968) which is seen in many partsof the electromagnetic spectrum: radio, optical, X-ray, and γ-ray (Large, Vaughan, andMills 1968; Wallace et al. 1977; Kellogg et al. 1973; Thompson et al. 1975). The Vela pulsaris an extremely powerful source of radio waves and γ-rays, the brightest pulsar at radiowavelengths and the brightest in the gigaelectronvolt (GeV) sky. It was used as a calibrationsource by the Fermi -LAT Collaboration (Abdo et al. 2009b) and was a key target of interestof the H.E.S.S. Collaboration after the commissioning of CT 5. It was the first pulsarobserved to glitch (Radhakrishnan and Manchester 1969) which led to the determinationthat pulsars are solid rotators and not oscillators. The Vela pulsar is contained within asynchrotron nebula (Frail, Bietenholz, and Markwardt 1997) and surrounded by an X-raywind nebula (Kellogg et al. 1973) as shown in the sky map in Figure 4.1. The synchrotronnebula, seen in both the radio and VHE γ-ray energy regimes, is show in the right imageof the figure and the X-ray wind nebula, in the left. As there is significant VHE γ-rayemission spatially coincident with the pulsar it is important to use the rotation phase of thepulsar to remove this emission component from the measured signal. The phase folded eventdistribution (phase profile) of the Vela pulse varies with energy, in the high energy regime(see the Introduction for its definition) there are three visible peaks: P1, P2, & P3 as labelledin Figure 4.2. The Vela pulsar has been studied since its discovery at the Molonglo Radio

Observatory in 1968 (Large, Vaughan, and Mills 1968), its properties have been determinedusing information from many parts of the electromagnetic spectrum and can be found in theATNF Pulsar Catalogue (ATNF 2015d; ATNF 2015a; Manchester et al. 2005); some ofthese properties are given in Table 4.1 alongside those for the only other pulsar detected withimaging atmospheric Cerenkov telescopes (IACTs): the Crab pulsar (PSR J00534+2200).The age, spin down and surface B-field are all quantities derived from measurements of otherquantities combined with assumptions about the pulsar system (see Chapter 1 for details).The Crab pulsar has a significantly larger E but is almost an order of magnitude farther away.Despite being relatively distant, the Crab pulsar has been observed across the electromagneticspectrum which can be linked to its extremely large spin-down luminosity (E); the highestof all detected γ-ray pulsars (Abdo et al. 2013). The Vela pulsar, however, has the highestintegrated energy flux in the range 0.1GeV ≤ E ≤ 100GeV which contributes to its rank asthe first pulsar target for H.E.S.S. II.This chapter will describe the analysis of the H.E.S.S. II Vela pulsar observations using

data from only the fifth telescope (CT 5) in the H.E.S.S. array. Data from the four smallerH.E.S.S. telescopes were simply discarded for this analysis. Using the quality selection criterialaid out in Chapter 3 the complete data set of 31.6 h was reduced to 24.3 h.

103

Chapter 4. Analsysis of H.E.S.S. Data on the Vela Pulsar

Figure 4.1.: Left: image of the Vela pulsar and surrounding X-ray wind nebula obtained withthe Chandra X-ray Observatory, taken from Durant and al. 2013. Right: im-age of the Vela X synchrotron nebula, taken from Abramowski et al. 2012,showing the radio sky map at 2.4GHz measured with the Parkes Observa-tory (Duncan et al. 1995) overlaid with very high energy γ-ray surface brightness(0.3, 0.6, 1.0, 1.6,&1.9)× 10−11 photons deg−1cm−2s−1 contours. The position ofthe Vela pulsar is shown with a white star. Note the difference in fields of view:left image 4.8 arcmin× 3.6 arcmin, right image 180 arcmin× 300 arcmin.

Figure 4.2.: Vela pulsar phase profile detected by Fermi -LAT. Upper panel for γ-ray energiesabove 0.02GeV with fit and peak labels. Lower panel for energies above 20GeV.Taken from Abdo et al. 2010b. The dashed magenta box denotes the P1 phaseregion, the dashed green box the P3 phase region, and the blue solid lined boxdenotes the P2 phase region.

104

4.1. Region Of Interest

PulsarName

Period[ms]

P[10−13 s s−1]

Distance[kpc]

Age [ky] E[1036 erg s−1]

SurfaceB-field[1012G]

Vela pul-sar

89.328 ±O(10−9

) 1.25 ±O(10−4

) 0.28 11.3 6.9 3.38

Crab pul-sar

33.392 ±O(10−9

) 1.25 ±O(10−5

) 2.0 1.26 450 3.79

Table 4.1.: Selection of measured and derived properties of the two detected very-high-energypulsars. Distance is given based on the association of the pulsars with otherastrophysical objects. Age is the spin-down characteristic age and E is the spin-down energy-loss rate. The surface B-field is derived from the relation given inEquation 1.14 using the canonical values for moment of inertia (1045 gcm2) andradius (106 cm). Data taken from Manchester et al. 2005

4.1. Region Of Interest

The extent of the region of interest (ROI) in the sky was defined during the optimisationof the cut configuration as a circle of radius 0.235 ◦ and centred on the Vela pulsar positionJ2000 Ra 08h 35m 20.65525 ± 0.00174s Dec −45◦ 10

′35.1545 ± 0.0114

′′as determined by

radio observations (Fey et al. 2004)1. The ROI in phase was defined based on the analysispublished by the Fermi -LAT Collaboration (Abdo et al. 2010b) and shown in Figure 4.2. Itis the same ROI as used by H.E.S.S. Collaboration 2015 to analyse the same H.E.S.S. data.It is clear from the figure that the emission contributing to the P1 & P3 pulses drops offwith increasing energy to the extent that P2 is dominant in the VHE regime. The phase ROIwas therefore taken as [0.5, 0.6]. The background phase region from which the backgroundrate in the phase ROI is estimated was also determined from the Fermi -LAT phase profile as[0.7, 1.0] for two reasons: the remnants of the P1 & P3 peaks are still visible in the range[0.0, 0.5] in the HE regime and a 0.1 phase buffer was taken away from the edge of the phaseROI. Using these phase regions during an analysis requires the signal to background exposureratio α to be 0.1

0.3 . The P1 and P3 ranges were similarly defined as the ranges [0.1, 0.2] and[0.2, 0.4] respectively according to those used by Abdo et al. 2010b.

4.2. Timing Solution

Two timing solutions (ephemerides) were used for the entire data set. These were obtainedthrough a memorandum of understanding with the Fermi -LAT Collaboration and producedusing data from the Parkes Observatory (Australia Telescope National Facility 2015). Theend of the second ephemeris’ validity range is two days before the end of the H.E.S.S. IIdata set at MJD epoch 56777. An analysis of the Fermi -LAT data on the Vela pulsar inthis time window was carried out in order to verify that the ephemeris still gives consistentresults. Figure 4.3 shows the results of this analysis: the event time stamps around theend of this validity range are plotted against the event phase. The over-densities near the

1This position is around 0.01 s in Ra and 0.27 ” in Dec away from the position given in the ephemeris usedin the analysis (see Section 4.2). This is far smaller than the systematic error on the pointing of CT 5∼ 2 ” (Hofverberg et al. 2013) and so is an insignificant difference for the analysis presented here.

105


Figure 4.3.: Event time stamp versus event phase for Fermi -LAT data on the Vela pulsar.The time stamps are reduced by the end of the H.E.S.S. II data set - MJD epoch56777 - so as to show one week before and after this date.

most prominent peaks P1 and P2 at approximately 0.13 and 0.55 respectively do not movein phase across the time range shown. This verifies that the ephemeris can be extendedby two days to cover the entire H.E.S.S. II data set. This check was also carried out onthe entire range of the H.E.S.S. dataset with consistent results. The ephemerides validityrange contains two glitches at MJD epochs 55408.8 d and 56555.8 d that are accountedfor with modifications to the rotational frequency and its derivative with time as wellas an exponentially decaying contribution, as described in Section 3.4.3. Second ordersinusoidal whitening terms are also included which reduce the amount of white noise in thetiming residuals, again see Section 3.4.3. The basic values in the ephemeris are given inTable 4.2. Furthermore, the proper motion of the Vela pulsar is also included in the ephemerisvRA = −(49.68± 0.06)mas yr−1, vDec = (29.9± 0.1)mas yr−1 (Dodson et al. 2003). This isdue to the Shklovskii effect (Shklovskii 1970): a secular change in the period derivativewith time due to Doppler effect resulting from the relative motion of the pulsar which isquantified as

PShk. =v2⊥Pdc

. (4.1)

106

4.3. ImPACT Analysis of the H.E.S.S. II Data set

Validity [MJD] Ra, Dec(J2000)

ν [Hz] ν [10−11Hz s−1] ν [10−21Hz s−2]

[56326.37,56492.91]

08h 35m 20.61s,−45◦ 10

′34.88

′′11.189 ±O(10−6

) −1.519± 0.004 −8.384± 1.065

[56595.83,56777.34]

08h 35m 20.61s,−45◦ 10

′34.88

′′11.189 ±O(10−7

) −1.873± 0.117 64.25± 23.28

Table 4.2.: Basic information from the timing solutions used in the analysis presented. Thefrequency ν and its derivatives with time ν & ν are given to a precision of 20significant figures in the actual timing solution.

This - when calculated with the distance to the Vela pulsar 287+19−17 pc (Dodson et al. 2003),

its transverse velocity (61± 2) km s−1 (Dodson et al. 2003), and using the period from theephemeris 89.37ms - is 1.25×10−19 s s−1 which is a O

(10−6

)effect on the frequency derivative

used in the ephemeris and therefore insignificant.


The Image Pixel-wise fit for Atmospheric Cerenkov Telescopes (ImPACT) reconstruction wasused with the Extra Loose cut configuration described in Chapter 3 to analyse and select theevents in order to obtain the lowest viable energy threshold. This threshold is quantified as theenergy at which the effective collection area of the instrument drops to 10% of its maximumvalue which occurs at 29.7GeV with an effective area of 8.7 × 103m2 for the observationsconditions: 20 ◦ zenith angle, 180 ◦ azimuth angle and 0.5 ◦ offset angle. This definition is alsoused for the maximum energy limit but considering a drop to 10% when increasing the energy.The total dataset was collected over two years in two distinct observation periods: one in 2013and the other in 20142. The 2013 data was taken at a time of greater instrument instability3

leading to a large proportion of data being discarded during the run quality selection cuts.The data calibrated in the Heidelberg calibration framework was used (see Section 3.7). Thedetails of the runs that passed run selection cuts for each of these data sets are given inTable 4.3. The 2013 data set is less than one sixth the size of that from 2014 so a comparisonof separately calculated analysis results is not useful. A more important distinction is made bysplitting the dataset into two in energy according to the number of excess events. The energyat which the 50% quantile in the number of excess events is reached is the point at which thedataset is split. Below this is labelled the Low Energy (LE) data subset and above which, theHigh Energy (HE); this point is at 34.4GeV in reconstructed energy. The number of eventsthat pass all analysis cuts as well as the excess counts in the phase ROI - corresponding to theP2 peak - and other basic statistics for all of these data sets are given in Table 4.4. It is clearfrom Column 6 of Table 4.4 that the rate of significance in 2013 is somewhat lower than in2014 indicating differences in the instrument performance. The drop in significance in the highenergy data set compared to the low energy is a clear consequence of the higher backgroundlevel in the HE data set. For the entire data set, the statistics were evaluated for the threephase ROIs (P1, P2, and P3 ) separately. The results are given in Table 4.5. Only integrated

2Although one data taking run from the 2014 data set was taken at the end of 20133This occurred during the commissioning of the CT 5 telescope

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Dataset

Timespan[MJD]

Numberof Runs

Livetime [h]

CT 5MeanTriggerRate[Hz]

ZenithAngleRange[◦]

MeanZenithAngle[◦]

Offset[◦]

EnergyThresh-old[GeV]

2013 [56366,56413]

12 3.45 2023 [23, 43] 32.6 0.5 -

2014 [56656,56774]

48 20.83 1534 [21, 32] 24.6 0.5 -

Total [56366,56774]

60 24.27 1636 [21, 43] 26.1 0.5 29.7

Table 4.3.: Information about the data obtained on the Vela pulsar using CT 5 over the 60data taking runs accepted after run quality selection.

Data set Counts inthe phaseROI

Counts inthe phasebackground

Excess Li-Ma Sig-nificance [σ]

SLi-Ma√T

[σ/√h]

2013 18, 567 54, 316 462± 157 2.96 1.62014 95, 374 274, 331 3, 930± 355 11.2 2.5

Low Energy 39, 546 112, 205 2, 144± 228 9.49 -High Energy 74, 395 216, 442 2, 248± 314 7.2 -

Table 4.4.: Information about the counts measured in the H.E.S.S. II data subsets, only statis-tical errors are shown for the excess measured.

SLi−Ma√T

is the Li-Ma significance

per square-root of the observation time.

108


P1 [0.1, 0.2] P2 [0.5, 0.6] P3 [0.2, 0.4]

ON Counts 109, 341 113, 941 219, 466OFF Counts 328, 647 328, 647 328, 647Excess −208± 382 4, 392± 388 368± 605SLi-Ma [σ] −0.54 11.4 0.61SLi-Ma√

T[σ h−

12 ] −0.11 2.31 0.12

I (Eth < E < Emax) [10−9 s−1cm−2] < 3.24 1.38(+0.33

−0.30)stat(+1.74−0.96)sys < 6.57

IE (Eth < E < Emax)[10−11 erg s−1cm−2]

< 57.2 4.29(+1.14−1.02)stat(

+5.50−3.31)sys < 116

Table 4.5.: Information about the counts and flux of the total data set for three regionscorresponding to each of the phase peaks observed with the Fermi -LAT (Abdoet al. 2010b). The energy threshold is defined in the main text and is set to 15GeV(see Section 4.6). The maximum energy is also discussed in the main text andis set to 125GeV. A power law spectral index of −4.8 and −5.33 is assumed tocalculate the integrated photon flux I and energy flux IE for the P1 and P3 phaseregions respectively according to that determined from the Fermi -LAT data byH.E.S.S. Collaboration 2015. The ON counts are those in the phase region ofinterest and the OFF counts those in the phase background region. SLi-Ma is theLi-Ma significance.

spectral upper limits are possible for the P1 and P3 peaks since neither were significantlydetected. However, a significant signal is obtained at the 11.4 σ level corresponding to theP2 peak with a signal-to-noise ratio of 13.3 (signal-to-background ratio 4%). The statistical& systematic errors given on the integrated photon and energy fluxes are derived from theerrors on the best fit power law function parameters given in Section 4.6.5. These errors aredetermined in Section 4.7 and propagated to I & IE through the integrals used to calculatethem I =

∫ Emax

EthFPL(E)dE and IE =

∫ Emax

EthEFPL(E)dE using the functional form of the

power law FPL(E) given in Section 4.6.2. As explained in Section 3.4.5, the error on the ratioof the ON and OFF exposures is negligible for analyses using the phase as a backgrounddiscriminator. In order to verify this, Equation 3.22 is evaluated for the total data set givingΞ = 0.36 from the average number of background events per data taking run (5, 477 events),α = 1

3 , and σα = 0.0032, as determined in Section 3.4.5. Spengler 2015 states that Ξ = 0.1 isthe limit for the reliable use of the Li-Ma significance i.e. neglecting the error on α. For thisdataset the value of Ξ lies close to this limit and according to Figure 2 of Spengler 2015 onlyrepresents a 5 − 6% variation on the width of the off-signal significance distribution. TheLi-Ma significance can thus be said to be appropriate for this data set. Furthermore to thisconsideration, only one trial was applied to the dataset with analysis cuts being refined usingother off-source data and Monte Carlo simulated γ-rays. The integral energy flux correspondsto 3.9 times the energy flux from P2 of the Crab pulsar in the same energy range4.

4.3.1. Signal Significance

The test statistics detailed in Chapter 3 were evaluated for each of the aforementioned datasubsets to test for the statistical significance of the periodic signal which is already clear from

4This value was determined from an extrapolation of the MAGIC power law spectrum (Aleksic et al. 2014).

109


Data set Low Energy High Energy Total

Number ofEvents

375, 627 723, 802 1, 099, 429

Significance [σ]

χ2 8.34 6.57 12.04Hm=50 11.90 9.12 15.15wHm=50 12.14 9.79 15.70Z22 5.71 4.96 7.83

Z210 11.87 7.56 14.08

Z220 11.92 8.94 15.10

wZ22 6.26 4.90 7.94

wZ210 12.02 7.84 14.38

wZ220 12.00 9.72 15.74

Table 4.6.: The statistical significance that the dataset of phases is not uniform in [0, 1] isshown as determined from the un-binned test statistics: H, weighted-H & Z2

m

and the binned χ2 test statistic; 201, uniform width bins were used and the Hstatistics used a maximum harmonic of 50.

the phase profile in Figure 4.7. A summary of the statistical significances obtained is givenin Table 4.6. A strong signal is evident in all three data sets. The sharpness of the peak isevident from the larger significance from Z2

m values with higher harmonics. Due to the sharppeak in the phase profile the highest harmonic used in the H-test was chosen as 50 ratherthan 20 in order to boost performance (Kerr 2011). The number of harmonics to use with theH-test was taken a priori as it was clear from the Fermi -LAT results(Abdo et al. 2010b) thatthe peak was narrowing with energy. When considering the data set binned in reconstructedenergy the H-test significance is 1.61 σ and 0.94 σ in the energy ranges (60, 125)GeV and(125, 200)GeV respectively.

The weighted test statistics shown in Table 4.6 were calculated using a weighting on thedata defined by the King profile described in Section 3.5.1 which is used to describe the γ-raypoint spread function (PSF) of CT 5 which was determined from Monte Carlo. Weighting isclearly not necessary with such a clear signal but is rather carried out as proof of principle.Applying this weighting to the two high energy bins evaluated with the H-test gives 1.98 σ and1.36 σ in the energy ranges (60, 125)GeV and (125, 200)GeV respectively. These significancesrepresent a boost when compared with the unweighted test statistic but reflect the similarlylow certainties obtained with the H-test and lie within an expected significance uncertaintyon the order of 1σ. This can be calculated from error propagation of equation 3.20.

The progression of the signal with time was assessed by evaluating the Li & Ma significanceand the H-test significance as a function of the number of background events, these progressionsare shown in Figure 4.4. The best fit square root function is shown based on the expectedbehaviour SLiMa ∝

√Nbg according to Equation 3.20 where Nbg = α×NOFF . At the beginning

of the data set it is clear that the function is a poor fit and suffers from low statistics leadingto a development that does not follow this trend. However, the progression of the significancefollows the expected trend with increasing statistics. The reduced χ2 values clearly sufferfrom the noise at low abscissa values (low statistics) but convey little information on goodnessof fit as the points are highly correlated. The statistic is simply used to clarify the generaltrend of the data. These progressions are another sign that the instrument and/or the Monte

110


Figure 4.4.: Left panel: The Li & Ma significance as a function of the number of backgroundevents. Right panel: The H test statistic significance as a function of the numberof background events. The best fitting square root functions (f = a

√x) are shown

with grey dashed lines; best fit statistics and parameters are also shown inset.

111


Carlo simulations of the instrument were not well understood during data taking and MonteCarlo production. Nonetheless, the progression is of acceptable quality to indicate an almostconstant acceptable data quality.

4.4. Sky Map

The directions of events in the phase ROI were binned in a sky map and subtracted by thesame counts from the unpulsed (background) phase region leaving an excess count sky map.This was smoothed according to the PSF of the instrument and is displayed in Figure 4.5(PSF shown in insert determined assuming a −4.0 spectral index). A single localised signalis visible centred on the position of the Vela pulsar. The source was fit as a point sourceconvolved with the instrument PSF (modelled by a triple Gaussian function) with a best fitlocation: J2000 Ra 08h 35m 21.24± 1.13s Dec −45◦ 10

′30± 3.6

′′, 0.33 σ and 1.43 σ standard

deviations respectively from the position used for the centre of the region of interest in thesky (determined from radio observations, see Section 4.1). This suggests that the source isconsistent with a point source at the location of the Vela pulsar being only 6.3 seconds of arcfrom the centre of the region of interest in the sky, this represents an insignificant deviation.Clearly the emission is well contained and resembles a point source. Demonstration thatthe background is well understood is clear from Figure 4.6. The histograms in the upperpanel are the distributions of θ2 values for the events in the phase ROI and unpulsed phaseregion. Away from the signal - at high θ2 values - the two histograms have similar bin contentsdemonstrating that the background in the signal region is well estimated from the backgroundregion. The histogram is not flat at high θ2 values due to the background γ-ray signal comingfrom the Vela-X nebula (Abramowski et al. 2012). The histogram in the lower panel showsthe distribution of Li-Ma significances corresponding to the sky map in Figure 4.5 both forthe full sky map and one where the sources of optical light and γ-rays have been excluded.The excluded distribution is well fit by a standard normal distribution further suggestingthat the estimate of the background contamination in the signal region is reliable and nounidentified signals are present in the data. However the visible shoulder on the distributionwhich is not well fit suggests that the exclusion region could be larger and hints at a largerPSF than expected from Monte Carlo simulations.

4.5. Phase-folded Event Distribution

The phases for all events that passed analysis cuts are plotted in the phase profile histogramin the upper panel of Figure 4.7 which shows a distinct single peak that corresponds to theP2 peak observed by Abdo et al. 2010b and the phase ROI defined in this work.

4.5.1. Characterising the Pulse

The histogram of the entire phase range was fit to a constant using a χ2 minimiser yielding a13.5 σ confidence level that the sampled distribution is not uniform in phase. The clear pulsein the phase on region was characterised by an asymmetric Lorentzian function with six

112


Figure 4.5.: Smoothed excess count sky map with a 4 ◦ field of view centred on the positionof the Vela pulsar (overlaid) shown in J2000 right ascension and declinationcoordinates. The region of interest in the sky is shown by the light green circlecentred on the position of the Vela pulsar. Inset is the instrument point spreadfunction which was used to smooth the raw excess count map.

113


Figure 4.6.: Theta squared (upper) and Li-Ma significance (lower) distributions. Thetasquared distribution shown for the region of interest in the sky and Li-Ma

significance distributions shown for the entire field of view where backgroundand signal are defined by regions in the pulsar phase. In the upper panel thesignal is displayed as filled green bars and background as empty black bars.Lower plot shows the significance distribution from the entire field of view in redand the significance distribution where areas of the sky that contain a γ-ray oroptical signal are excluded in blue. The best fitting standard normal Gaussian

distribution to the blue data is shown by the thicker black line. The fit parametersare given in the inset text.

114


Figure 4.7.: Phase-folded event distribution of the events that passed analysis cuts in theH.E.S.S. Vela data set binned into 201 bins. Upper panel: entire phase range,the blue dashed line shows the best fit to an asymmetric Lorentzian functionwith a symmetric constant and, below, the fitting residuals. The phase region ofinterest is indicated with a red shaded region and the phase background regionby a grey shaded area. The statistics box displays the various values quantifyingthe variation from a uniform phase distribution. Note: the χ2 value is relatedto a fit to a constant that is not displayed rather than for the peak fit shown.Lower panel: zoom of the phase region of interest with overlaid best fit functionas shown in the upper panel as well as fitting residuals below.

115


parameters:

f(x) =

⎧⎪⎪⎨⎪⎪⎩

I0

sLπ

(1+

(x−t)2

s2L

) + CL if x < t

I0A

sT π

(1+

(x−t)2

s2T

) + CT otherwise(4.2)

where

A = sTsL

+ πsT (CL−CT )I0

ensures the function is continuous at x = t,

I0 is the amplitude of the pulse,

t is the location of the pulse maximum,

sL/T are the leading and trailing widths of the pulse respectively,

CL/T are the level of the background leading and trailing the pulse respectively,

following the fit used by Abdo et al. 2010b on the Vela phase profile obtained from theFermi -LAT data set. In order to ensure that the asymmetric parameters improve the fitsufficiently to justify their inclusion, simpler models were also tested, all of those that wereassessed are as follows:

• Lorentzian with one constant (most nested) i.e. CL = CR & sL = sR leaving four freeparameters.

• Lorentzian with two constants, one each side of the mean i.e. sL = sR leaving fivefree parameters.

• Asymmetric Lorentzian with one constant i.e. CL = CR leaving five free parameters.

• Asymmetric Lorentzian with two constants, one each side of the mean having six freeparameters.

A χ2 minimisation was used to determine the best fitting parameters for each attempted fitall of which were done using the binned data set with 201 bins in the range [0.4, 1.0) to fitthe P2 peak. The range was chosen so as to avoid the upper edge of the P3 phase interval.A bootstrapping method with 1, 000 iterations was used by which to accurately obtain thestarting parameters for the fit. The contents of each bin in the phase profile was varied bya pseudo-random value drawn from a Gaussian distribution with zero mean and a widthdetermined by the error on the bin content, which follows Poisson statistics and is therefore√bin content, each time the data were refitted with the function. Starting parameters for

each of the bootstrapping fits were determined from fitting a straight line to the data for theconstant(s), using the centre of the signal phase region as the position of the peak and itswidth as the peak width. All fitting used the ROOT implementation of the MINUIT2 fittingpackage with a χ2 minimisation (James and Winkler 2004). The Fisher test statistic (F-test;Fisher 1922; Fisher 1925) was used to evaluate the benefit of the additional parameters used.The first three models are all nested5 because the parameters all exist in the complete model,

5Two models are nested if the more complex model (the one with the greater number of parameters) can betransformed into the other by constraining its parameters.

116


Complete Model Nested Model F-test Value F-test ConfidenceLevel [%]

Lorentzian withtwo constants

Lorentzian withone constant

10.61 99.9

AsymmetricLorentzian withone constant


16.34 99.99

AsymmetricLorentzian withtwo constants


9.524 99.99


Lorentzian withtwo constants

7.812 99.4


AsymmetricLorentzian withone constant

2.496 88.3

Table 4.7.: Comparison of the functional forms fitted to the phase-folded event distribution.

the asymmetric Lorentzian with two constants; this is a requirement to use the F-test. TheF-test is expressed as a ratio involving the residual sum of squares (RSS):

F =

RSSR−RSSC

npC−np

R

RSSC

N−1−npC

(4.3)

where

RSS =

N∑i=1

(yi − f(xi))2 (4.4)

for bin values yi, model values f(xi), and where npC & np

R are the number of parameters in thecomplete and restricted (nested) models respectively when fitting N bins. The distribution ofF has np

C − npR & N − 1− np

C degrees of freedom.This test therefore evaluates two points of interest: i) is the pulse narrower on one side? ii)

Is the level of the background higher on one side? The F-test was applied to all applicablecombinations of the four functional forms of the pulse the results of which are shown inTable 4.7.

The P2 peak is better fit by an asymmetric function at the > 99% confidence level butthere is less evidence to suggest that the constant level on either side of the mean of the peakis better described by different values: 88% confidence level. An asymmetric Lorentzian

function (with a single constant) was therefore determined to best fit the phase profile. Thereis no indication of a log-normal (P3) component - as observed in the Fermi -LAT phaseprofile (Abdo et al. 2010b) - both in the total and low energy data sets. Fitting was attemptedwith this additional component and no significant contribution from P3 nor improvement inthe goodness of fit - measured by the χ2 test statistic - was observed.

117


Parameter Low Energy High Energy Total

χ2/NDF 1.232 = 142.9/116 1.183 = 137.3/116 1.299 = 150.7/116

I0 18.67+2.76−2.63 21.87+4.86

−3.74 39.65+4.53−4.45

t 0.5616+0.0012−0.0015 0.5626+0.0018

−0.0019 0.5619+0.0010−0.0011

sL 0.01605+0.00336−0.00279 0.01479+0.00362

−0.00289 0.01515+0.00243−0.00209

sT 0.005442+0.001879−0.001636 0.002173+0.002069

−0.002026 0.003687+0.001385−0.001296

C 1854+4−4 3589+6

−6 5445+7−7

Table 4.8.: Parameters for the best fitting asymmetric Lorenztian function with a symmetricconstant background over the phase range [0.4, 1.0] with 201 bins using a χ2 method.Statistical fitting errors determine with the MINOS technique (see text) are shown.For the high energy data set, the upper error interval of t and lower error intervalfor sL could not be determined with the MINOS technique so the parabolic erroris shown.

4.5.2. Energy Resolved Peak Shape

Looking at the LE and HE datasets separately highlights the variation of the pulse profilewith energy. The best fit parameters for all three data sets are detailed in Table 4.8. TheMINOS technique was used to estimate the errors (see Eadie 1971 for a description of thetechnique which accounts for parameter correlations and non-linearities).

Peak Position The position of the peak is compatible across all of the data sets. Abdo et al.2010b determine the peak position from the best fit asymmetric Lorentzian function of theFermi -LAT data binned in energy. In the highest energy band (8GeV ≤ E ≤ 20GeV) theydetermine the peak position to be 0.5699 ± 0.0007 which is only compatible to within 6.6standard deviations from the result presented here for the total data set. This may be dueto the ephemeris used. Using the Fermi -LAT data to compare the ephemeris used in thiswork (Kerr ephemeris described in Section 4.2) to that used in Abdo et al. 2010b (GTPephemeris created using data from the Fermi -LAT (Abdo et al. 2010b)) shows that the GTPephemeris results in a peak shifted to higher phase by 1.31% (H.E.S.S. Collaboration 2015).The Fermi -LAT peak corrected to the Kerr ephemeris position is 0.5624± 0.0007 which iswithin 0.1 standard deviations of the H.E.S.S. result and leads to a combined measurement of0.5622± 0.0006.

Peak Width The leading and trailing edges of the peak vary between the LE and HE datasets at the 0.28 σ and 1.21 σ levels respectively. Comparing these values to those obtained inthe lowest energy bin ((0.02, 0.1)GeV) analysed by Abdo et al. 2010b: sLAT

L = 0.0263±0.0035and sLAT

T = 0.0170± 0.0027 yields agreement at the 2.4 σ for the leading edge but only at the4.4 σ level for the trailing edge strongly hinting at a decrease in width on the trailing side.

4.5.3. Phase Systematics

Systematics regarding phase could result from the variation of the background rate withinthe background phase region or the binning of the phase profile. These effects were analysedto quantify the systematic bias induced.

118


Energy Range [GeV] χ2 Value Number of Degreesof Freedom

Reduced χ2

[0, 10] 52.582 59 0.89123[10, 20] 65.62 59 1.1122[20, 30] 41.889 59 0.70998[30, 40] 67.621 59 1.1461[40, 50] 55.824 59 0.94617[50, 60] 62.26 59 1.0553[60, 70] 65.235 59 1.1057[70, 80] 56.168 59 0.952[80, 90] 59.569 59 1.0096[90, 100] 47.728 59 0.80895[100, 110] 51.849 59 0.8788[110, 120] 40.679 59 0.68947[120, 130] 55.354 59 0.9382

Table 4.9.: χ2 goodness of fit measure for fits to the background phase region [0.7, 1.0] indifferent energy ranges. The mean and standard deviation of the reduced χ2 valuesare 0.94182 and 0.14056 respectively.

Background Phase Region Systematics The phase region [0.7, 1.0], defined as the back-ground phase region, was fitted to a constant using a χ2 minimiser with reduced χ2 value ofχ2/NDF = 58.13

59 = 0.99. This is very close to unity - representing an ideal fit - and is lessdue to the noise in the background region. Furthermore no residual structures are evidentimplying a constant background phase region. To ensure this behaviour is constant withenergy the data was split in bins of 10GeV in the range 0-130GeV and fitted to a constantfor each energy bin. The results are shown in Table 4.9; each of the energy bins and phasebins contain sufficient statistics (N > 10) for fitting.The goodness of fit measure used was the reduced χ2 value for which an ideal fit with

infinite statistics yields 1.0. The most distant outliers from this values are obtained from thedata in the ranges (110, 120)GeV and (30, 40)GeV for downward and upwards fluctuationrespectively. Both are, however, within two standard deviations of the expected value andtherefore represent no significant deviation. The background phase region can therefore saidto be stable with energy and constant in phase.

Phase Profile Binning The binning of the phase profile was varied from 50 bins to 300 binsin steps of 1 bin in the phase range [0, 1] to analyse the variation of the best fit peak parameterswith binning. The width of the distribution of the parameters is considered as systematicvariation resulting from the binned nature of the phase profile. The peak position and peakwidths were distributed with mean and standard deviations as follows: tdistr. = 0.5631±0.0021,sdistr.L = 0.01822± 0.00243, and sdistr.T = 0.003819± 0.001016. These standard deviations canbe interpreted as systematic errors on the bets fit parameters given for the total data set inTable 4.8.

Kernel Density Estimator A kernel density estimator (KDE) with a Gaussian kernel ofwidth 0.05 in phase was used to characterise the distribution of phases as it has a reduced bias

119


from binning the data which is present in histograms. It was then compared to the best fittingasymmetric Lorentzian showing excellent agreement of the peak position: t = 0.5627±0.0016,this is only 0.3 standard deviations away from the combined measurement presented previously.The error on the KDE peak position was determined from a bootstrapping method as follows:

1. Produce a KDE for the data.

2. Sample the KDE function as a probability density function and produce 100 new datasets of equal size.

3. Produce a KDE for each of these new simulated data sets and determine the position ofthe peak.

4. Produce a histogram of the position of maximum corresponding to P2 from each dataset.

5. The width of the distribution in the histogram is the error on the position of the peak.

The position of the peak agrees with the binned fit (considering only statistical errors) of theH.E.S.S. data, deviating from it by only 0.42 σ. This technique also shows the fitting errorsgiven for the functional fits are approximately correct as statistical errors because they aresimilar to those determined by the bootstrapping method.

4.6. Very High Energy γ-ray Spectrum

A high level result - such as a spectrum - at such an early stage in any experiment, H.E.S.S. IIin this case, is difficult to obtain precisely and results are therefore subject to large systematicerrors. For IACTs, the simulation of the instrument must be accurate and precise to be ableto well understand the effective collection area of the instrument. With CT 5 this is not quitethe case and although this work uses the most up to date Monte Carlo simulation availableat the time of writing the spectrum is still preliminary. New releases of the simulation in thecoming months will correct small errors found in the current release.

4.6.1. Energy Distribution

The distribution of reconstructed energies for the excess events (excess = ON − α×OFF )calculated from the number of events in the phase ROI (ON) and the unpulsed phase region(OFF) given the exposure ratio α is shown in Figure 4.8. The errors on the points appearlarge - particularly in the mid-energy range - however they were calculated as described inSection 3.4.5 and cross checked with an alternative log-likelihood method of error estimationon the excess counts used in the H.E.S.S. Collaboration with consistent results. As can be seenfrom the figure there is a clear discrepancy between the measured excess and that predictedfrom simulation for several different simulated power law spectral indices (normalised tothe number of excess events). A power law spectral form was chosen based on a fit of theFermi -LAT results above 10GeV (H.E.S.S. Collaboration 2015). This is indicative of a MonteCarlo simulation that is not very compatible with the instrument; this will contribute tothe systematic error in the spectrum. The Kolmogorov-Smirnov (K-S) test was usedto compare the measured reconstructed energy distribution to the reconstructed energydistributions for many different simulated spectral indices. The best agreement was obtainedfor the simulated spectral index of −4.4. The K-S probability did, however, remain low, at

120


Figure 4.8.: Distribution of the reconstructed energy for the excess measured in the phaseregion of interest. The comparisons to theMonte Carlo simulated data for differentpower law indices are shown and detailed in the inset legend. Furthermore, theexpected excess distribution in simulated energy is also given for a power lawindex of −4.4.

121


just 24% indicating a poor agreement. This points to a mismatch between the simulation anddata that could arise from parameters describing the instrument being different in the MonteCarlo simulation as those used in the experiment. The energy distributions for the leadingedge and trailing edge of the P2 peak were also separately analysed. The split was done at themean of the best fitting distribution 0.5619 from Section 4.5. Due to the asymmetric natureof the peak - leading edge excess 3, 450 ± 290 and trailing edge excess 942 ± 219 - the twodistributions were normalised based on the bin with maximum content. The K-S test showsthat the compatibility of the two is at the 95.48% confidence level indicating no significantvariation in the energy distribution between the leading and trailing edges of the P2 peak.

4.6.2. Spectral Calculation Method

Production of a spectrum down to the naıve safe energy limit (the energy correspondingto 10% of the maximum effective area) of around 30GeV is optimistic and conveys anunderstanding of the instrument and reliability of the data calibration and simulation thathas not yet been achieved. This was however still attempted to asses the performance andagreement that the ImPACT analysis shows with the spectrum obtained with the Fermi -LATdata (H.E.S.S. Collaboration 2015). The H.E.S.S. spectrum was obtained using a maximumlikelihood fitting procedure that uses a forward folding technique to account for the instrumentresponse function (IRF), see Piron et al. 2001 for an explanation of the process. Two resultsare produced during each spectrum calculation: the best fitting function with 1 σ confidenceinterval band (the butterfly) and the spectral points. The butterfly is determined using afitting procedure to obtain the differential flux F (E) as a function of the fitted spectralparameters. A priori specified binning in energy, zenith angle, instrument efficiency, and offsetangle are used; the binning is based on an estimate of the scale of variation in each of theaforementioned parameters using standard observation targets. The excess counts in each ofthese bins is compared to an expected count from the tested model. A maximum likelihoodmethod is used to determine the model parameters that best describe the data. The secondresult is the group of spectral points which are calculated using the ratio of measured (Sm)and expected (Se) counts in each energy bin:

FP (E) = F (E)× Sm

Se. (4.5)

The points are shown in reconstructed energy whereas the butterfly, as the true result of thespectral calculation, is given in energy corrected by the energy reconstruction bias.

Two spectral forms were investigated: a power law

FPL = φpl0

(E

E0

)−Γpl

(4.6)

and a power law with exponential cut off

FECPL = φec0

(E

E0

)−Γec

exp

(−(

E

Ecut

)b). (4.7)

The naming of the second model is however dependent on the b parameter which clearlyaffects the steepness of the cut off. A simple exponential cut off power law is obtained whenb = 1, a sub-exponential power law when b < 1 and super exponential cut off power law when

122


b > 1.

4.6.3. Energy Binning

The bin with the highest content is at 42GeV with a value of 475± 104 counts which indicateswhere the excess is centred6. The lowest and highest energy bins in the energy excessdistribution with a positive excess and not compatible with zero at the 68% confidence levelbegin at 4GeV and 8TeV in reconstructed energy respectively. These represent statisticalvariation and cannot be used as bounds on the energy axis. The binning of the spectrum waschosen a priori to balance the number of points with the significance per point. The highdensity of excess around 42GeV meant that binning could be fine in that region but as theexcess drops off steeply to higher energies the same binning would be too fine. Therefore theoptimal binning for the low energy range was used and a minimum of 3 σ significance perspectral point was required to ensure validity at the higher energies where there is a lowerrate of excess counts. 27 bins were chosen based on this consideration of approximately 3 σsignificance bins around 30GeV. This choice is clearly dependent on the energy range whichwas simultaneously determined.

4.6.4. Energy Range

The safe lower bound of the energy range was chosen based on the inspection of the energyresolution and effective area achievable with ImPACT. At 20GeV the energy resolutionremains approximately in the Gaussian regime: the distribution mean represents a largereconstructed energy bias and the width a large energy resolution EReco

Esim= 1.6+0.6

−0.4. The

effective area is approximately 1, 000m2; this is 1% of its maximum value obtained at around1TeV. Calculation of the spectrum is however possible down to 15GeV where the effectivearea is around 20m2 and energy bias correction factor EReco

Esim= 1.9+0.7

−0.5; this energy bias is alsofather from the Gaussian regime than that at 20GeV. In principle, accurate knowledge of anon-zero effective area and a relatively small energy resolution are the necessary properties ofthe IRFs in order to produce a reliable spectrum. Any extension below 20GeV must thereforenot be relied upon for physical interpretation of the data but rather used to evaluate theinstrument. The maximum energy used for the spectrum was determined from assuming apower law index of −5.0 and taken as 125GeV by requiring that 10 excess counts be expectedin the highest energy bin. This is extending the spectrum to the very limit of what can beused, the spectrum above around 60GeV should, therefore, be treated with caution.

4.6.5. Best Fitting Power Law Spectrum

The data were forward-folded with the IRF in the energy range 15GeV to 125GeV with 27bins for which the best fitting power law function to the data is shown in Figure 4.9 with 3 σre-binned points. The spectrum is phase-averaged meaning the live time used is not reducedconsidering that the pulse has a duty cycle of 10%. The starting values for the parameters weretaken from the power law fit of the Fermi -LAT data φ0(20GeV) = 8.5× 10−8 cm−2s−1TeV−1

and Γ = −4.1 (H.E.S.S. Collaboration 2015); the reference energy used here (20GeV) wasalso selected based on the power law fits by H.E.S.S. Collaboration 2015 but makes littledifference to the resulting best fitting spectrum. The best fitting parameters are Γpl =

5.39 (+0.43− 0.38)stat (+0.82− 1.19)sys and φpl0 (20GeV) = 8.56 (+0.72− 0.74)stat (+10.18−

6This value was determined using a histogram with finer binning than that shown in Figure 4.8.

123


Figure 4.9.: Phase-averaged spectrum of the pulsed emission from the Vela pulsar in theenergy range 15GeV to 125GeV. 1 σ confidence interval band is shown in greenfor the best fitting power law shown by the blue solid line. 3 σ rebinned spectralpoints in reconstructed energy are shown with red filled circles with logarithmicbinning (unless expanded to make the 3 σ threshold). A 95% confidence intervalupper limit is shown for the energy range (125, 200)GeV. Spectral point residualsare shown in the lower plot with 1 σ error bars.

124


Figure 4.10.: Likelihood contour surface for the power law parameters showing the(1, 2, 3, 4,& 5) σ contours. The best fitting parameters are shown by the blackpoint with statistical errors determined with the MINOS technique. The redfilled circle corresponds to the best fitting power law parameters determinedfrom Fermi -LAT data by H.E.S.S. Collaboration 2015 shown with statisticalerrors only.

2.85)sys × 10−8 cm−2s−1TeV−1 with decorrelation energy of 19GeV. These statistical errorsdiffer from those in the figure as the MINOS technique (described by Eadie 1971) was used todetermined more accurate errors, the errors shown in the figure represent the fitting errorsprovided by the MINUIT2 fitting package (James and Winkler 2004). The systematic errorsare explained in Section 4.7. The 3 σ rebinned spectral points are shown; the last point - forthe energy range (58− 125)GeV - is however only significant to 1.89 σ. The 95% spectralupper limit, the 1 σ errors on the spectral points, and 1 σ errors on the residuals in theresidual plot were all calculated with the Rolke method (Rolke and Lopez 2001) implementedin the ROOT framework (Brun and Rademakers 1997)7. The residuals show an acceptablespread but some structure may be evident suggesting some correlation between the spectralpoints. The likelihood contour surface for the two parameters in the power law fit is given inFigure 4.10. There is a clear maximum at the position of the best fit parameters. The bestfitting Fermi -LAT power law above 10GeV determined by H.E.S.S. Collaboration 2015 isstatistically compatible in normalisation but not index. However, when considering systematicerrors on the H.E.S.S. measurement as well, the two are compatible to within 1 σ. Thismismatch could also be interpreted as a steepening of the spectrum. The shape of the contours

7In simple terms this method involves ordering the probabilities of a measurement given different parametervalues by the probability that the measured values arise given the null hypothesis. A threshold is thenused to accept or reject the null hypothesis. This hypothesis testing can then be inverted to determineconfidence regions and upper limits.

125


suggests some correlation between the parameters with a low normalisation accompanying ahigh index, however, as this correlation becomes strong far from the best fitting values andthe region around this best fit shows only small indication of correlation then the fit positionand errors can be said to be reliable. Pearson’s correlation coefficient (Pearson 1895) is−0.106 which represents a low degree of anti-correlation.

4.6.6. Exponential Cut Off Power Law

Abdo et al. 2010b use a sub-exponential cut-off power law to characterise the spectrum of theVela pulsar in the energy range (0.1, 100)GeV and Acero et al. 2015 report in the Fermi -LATThird Source Catalog (3FGL) that the phase-averaged Vela pulsar spectrum to be best fit witha sub-exponential cut off power law (Equation 1.59 with b < 1). Abdo et al. 2010b remarkthat this may be because the phase-averaged spectrum is a sum of several simple exponentialcut off power laws (b = 1 for P1, P2 & P3 ) in that energy range. The phase-resolved spectrawith exponential cut off power laws using 4 free parameters cannot be calculated with only oneyear of Fermi -LAT data so b is fixed and scanned over manually to test the improvement in fitobtained for the individual peaks (Abdo et al. 2010b). This is also the case with the currentH.E.S.S. data set on Vela. A scan was performed over the b parameter in Equation 1.59 toassess the improvement in fit obtained over a power law model. b was fixed so only one extraparameter compared to a power law function was used in the fitting.

The likelihood ratio test was used to determine if the extra parameter in this fit is statisticallyjustified. The quantity λ = −2 log10(LECPL/LPL)/footnoteSubscripts used here denote thepower law (PL), the exponential cut-off power law (ECPL) and super exponential cut-offpower law (SECPL) is used with Wilks’ theorem (Wilks 1938) for nested models - whichstates λ ∼ χ2(NDF = NDFSECPL − NDFPL) in the infinite statistics limit (a requiredapproximation when using this method) - to define the probability that the model with morefree parameters is a better fit. For the set of b values tested (b = 0.5, 1.0, 1.5, & 2.0) nominimum could be found for 1.5 & 2.0. For b = 1 the improvement p-value is P(X < x) = 0.923which corresponds to 2.2 σ, for b = 0.5 the p-value is 0.873 corresponding to 1.65 σ. The b = 1best fit represents the greater improvement and corresponds to a slightly better log-likelihoodvalue −16.70 compared to −16.30 for b = 0.5. When fitting with b fixed to unity, two minimaare found: the first is narrow and results in an extremely small error on the normalisationδφ0

φ0∼ O

(10−7

)and the other, a deeper minimum, essentially describes a power law in the

regime in which there is a measured signal with a cut off value that renders the exponentialterm ineffectual. Fitting the normalisation - whilst fixing the other parameters to thosedetermined by Abdo et al. 2010b using the Fermi -LAT data (Γec = 1.55, Ecut = 4.5GeV &b = −1.0) - results in convergence and agreement in normalisation at E0(= 20GeV). Withthese fixed parameters the power law model is no longer a nested model in the exponential cutoff model meaning Wilks’ theorem cannot be used. The Akaike information criterion (AIC;Akaike 1998) can, however, be applied to determine the relative quality of the models; acorrection must be applied based on the finite sample size (AICc; Burnham and Anderson2002):

AICc = 2k − 2 loge(L) +2k(k + 1)

n− k − 1(4.8)

where n is the sample size, k is the number of fitted parameters, and L is the maximum valueof the likelihood function for the model. The power law minimises the AICc which means thepower law model is favoured over the exponential cut off shape Abdo et al. 2010b measured

126

4.7. Spectrum Systematics

with the Fermi -LAT. The relative likelihood - defined as exp(AICcmin−AICci

2

)(Burnham and

Anderson 2002) for the ith model compared to the model that minimises the AICc- for theexponential cut off of 30%. The conclusions to be drawn from this are as follows:

• The simple cut off (b = 1) is favoured over the sub (b < 1) and super (b > 1) exponentialcut off models.

• The exponential cut off power law model is not significantly favoured over the powerlaw model.

In summary, The result is that the H.E.S.S. data alone are best fit by a power law butthis does not constrain the exponential cut off power law form as determined by Aceroet al. 2015; Leung et al. 2014. The energy cut off Ec as measured with the Fermi -LAT4.89±0.66GeV (Abdo et al. 2010b) should not be interpreted as a physical spectral attenuationbut rather a parameter in the model; γ-ray emission is reliably observed up to 58GeV with ahint of emission up to 125GeV. These data are insufficient to distinguish between a curved orpower law spectral shape in the H.E.S.S. energy regime; there is no statistical basis to modelthe H.E.S.S. data with an exponential cut off.

4.6.7. Spectral Upper Limits

The 95% upper limit on the photon flux and energy flux assuming a continuation of the best fitpower law from 125GeV up to 200GeV according to theRolkemethod (Rolke and Lopez 2001)are F (125−200GeV) < 1.89×10−9 cm−2s−1 andH(125−200GeV) < 2.27×10−10 erg cm−2s−1

respectively.The P1 & P3 phase regions were also analysed in order to determine the integrated flux

upper limits given in Table 4.5. The indices obtained for a power law spectral fit of the P1 & P3peaks using the Fermi -LAT data are −4.8± 0.4 and −5.3± 0.3 respectively (based on privatecommunication with H.E.S.S. Collaboration 2015, i.e. within the H.E.S.S. Collaboration).These were used to determine the pulsed upper limits for this work. The phase ranges [0.1, 0.2]and [0.2, 0.4], as given in Table 4.5, and the same forward-folding technique were used todetermine the 95% integrated spectral upper limits using the Rolke method (Rolke andLopez 2001).


As shown by the comparison of the Monte Carlo energy distributions with the measuredenergy distribution displayed in Figure 4.8 the agreement between Monte Carlo simulationsand the data is not good. This therefore warrants an investigation into the systematic errorsthat affect the result. In the determination of the spectrum a correction to the live time is nottaken into account which is based on the existence of empty events in the data set. Around5% of the raw data set consists of spurious events with trigger information but no cameraintensity data which contributes to the dead time calculation thus biasing the normalisationand implying that the normalisation should be 5% higher than is determined; this is anasymmetric systematic error. This effect is however small compared to uncertainties elsewhere.Primarily, the systematic variation in the fitted power law spectral parameters caused byuncertainties in the IRF were evaluated.

127


Source of Sys-tematic

Index Normalisation [10−8 cm−2s−1TeV−1]

Upward Fluctu-ation

DownwardFluctuation

Upward Fluctu-ation

DownwardFluctuation

Effective Area 5.39+0.43−0.38 5.39+0.43

−0.38 5.71+0.49−0.49 17.1+1.47

−1.47

Energy Resolu-tion

6.58+1.47−0.92 4.57+0.24

−0.22 14.1+2.59−2.59 10.1+0.94

−0.9.5

Table 4.10.: Best fitting power law spectral parameters with MINOS errors (Eadie 1971)obtained by bracketing the quantities given in the first column. The binning andreference energy were the same as that used for the main spectrum calculation.

4.7.1. Instrument Response Function Bracketing

Variation in three areas of the IRF were considered: effective area, energy correction, andinstrument efficiency. The effective area was bracketed upward and downward by 50%, thisvalue is based on comparing the effective area from two simulation packages being used inthe H.E.S.S. Collaboration. The energy bias - used to estimate the true energy from thereconstructed energy - was also bracketed by 45% to estimate the error associated withthe correction from reconstructed energy to estimated true energy. This value was basedon the energy resolution presented in the lower right panel of Figure 3.7. This is becausethe reconstructed energy is corrected to the estimated true energy by the the value of theenergy bias. As described in Section 3.3.5, the bias is the mean of the distribution Ereco−Esim

Esim

determined using Monte Carlo simulated γ-rays. The width of the distribution, the energyresolution, can be used to estimate the error on the correction from reconstructed energy toestimated true energy. The value used here can be considered an extreme upper estimateas this value is taken from the lower end of the energy scale where the bias is highest andthe energy resolution is the full width of the distribution whereas the half width would bea more accurate estimate of the error of the bias correction. Variations in the instrumentoptical efficiency - mainly based on the relative mirror reflectivity8 - are accounted for in thevariations of the other two quantities and are small compared to these variations. These testswere carried out separately, each time determining the best fitting power law parameters. Thevariation of the energy bias is energy dependent thus altering the normalisation and indexof the spectrum. The variations tend to have a larger effect on the flux normalisation. Theresults of this IRF bracketing are given in Table 4.10. The energy correction bracketing affectsboth the index and normalisation by a significant amount. This attests to the importance ofa small, well-understood energy resolution9, this is not the case for low energy (E < 50GeV)observations with CT 5 in monoscopic mode. The effective area when bracketed only affects thenormalisation but does so significantly. These variations are asymmetric and when combinedtogether in quadrature (therefore assuming they are uncorrelated) add a large uncertainty to thebest fit spectral parameters: σ+

index = 0.8, σ−index = 1.2, σ+

norm = 10.18× 10−8 cm−2s−1TeV−1,and σ−

norm = 2.85 × 10−8 cm−2s−1TeV−1. These errors are shown in Section 4.6.5 with thestatistical errors on the best fitting power law parameters.

8The mirror reflectivity affects the reconstructed energy of the event, see Section 2.5, and is relative to thedesign mirror reflectivity rather than perfect optical reflectivity.

9A large bias can be accounted for but if there is a large spread then the correction is not precise

128


Index Normalisation[10−8 cm−2s−1TeV−1]

Number of bins

25 5.40+0.45−0.38 8.58+0.73

−0.74

30 5.33+0.41−0.35 8.56+0.72

−0.72

35 5.22+0.40−0.34 8.35+0.72

−0.72

40 5.32+0.41−0.35 8.55+0.72

−0.72

Energy Threshold GeV

10 5.37+0.42−0.37 8.54+0.72

−0.72

15 5.39+0.43−0.38 8.56+0.72

−0.73

20 5.05+0.47−0.39 7.96+0.91

−0.91

25 4.91+0.52−0.44 7.52+1.15

−1.15

30 5.05+0.66−0.55 8.14+1.67

−1.91

35 5.73+1.13−0.83 10.8+2.5

−2.8

40 6.57+1.99−1.42 13.6+3.4

−4.3

Table 4.11.: Best fitting power law spectral parameters with MINOS errors (Eadie 1971)obtained by changing the number of bins and energy threshold according to thevalues in the first column.

4.7.2. Varying the Energy Binning and Threshold

Furthermore, the variations of these parameters based on changes to the number of bins inthe energy axis and energy threshold were also explored. Restricting the energy range is notstrictly a source of systematic error but will be investigated here nonetheless since at lowenergies the large energy bias may result in bad spectral reconstruction. Limiting the rangemay reduce this effect and more accurately allow the spectrum to be determined. The bestfitting power law parameters for each of the energy thresholds and binnings scanned over aregiven in Table 4.11. The normalisation is resilient to change in both threshold and binningand there is only a weak dependence of the index on the binning. The scan over the energythreshold introduces problems with limited statistics when too large as evident from the largechange of best fitting parameters when going from a 35GeV to 40GeV threshold. The meanand standard deviation for the index are 5.25± 0.28 which amounts to a spread less than thefitting error on the best fit index. The mean and standard deviation of the normalisationare (8.59± 1.05)× 10−8 cm−2s−1TeV−1 which represents a similar sized error as from fittingand a much smaller error than contributed by systematics from the IRF. It can therefore bestated that - within a reasonable range - the energy threshold and binning do not contributesignificantly to the systematic errors in the spectrum. The systematic errors quoted here mustalso be indicated as subject to further study as they are based on early Monte Carlo andsoftware distributions in the H.E.S.S. Collaboration.

4.7.3. Comparison of the Spectrum

H.E.S.S. Collaboration 2015 use the same H.E.S.S. II data and data from the Fermi -LATto determine the spectrum obtaining differing parameters, these are shown in Table 4.12.The normalisation at 20GeV obtained in this work is compatible within one statistical error

129


Data set Energy Range [GeV] Index Normalisation[10−8 cm−2s−1TeV−1]

H.E.S.S. II, thiswork

[15, 125] 5.39(+0.43−0.38)stat(

+0.82−1.19)sys 8.56(+0.72

−0.74)stat(+10.18−2.85 )sys

H.E.S.S. II [20, 120] 4.03± 0.20 6.43± 0.50Fermi -LAT [10, 100] 4.09± 0.10 8.50± 0.55

Table 4.12.: Best fitting power law parameters for three analyses: the analysis of H.E.S.S. IIdata in this work and the analysis of both the H.E.S.S. II data and Fermi -LATdata carried out by H.E.S.S. Collaboration 2015. The errors on the parametersdetermined in the other two analyses are purely statistical. The reference energyin all three analyses is 20GeV.

of that obtained with the Fermi -LAT data set and 2.9 σstat of the other H.E.S.S. II dataset (H.E.S.S. Collaboration 2015). The spectral index derived in this work clearly differsfrom the other two; by 2.26 and 2.28 statistical standard deviations from the H.E.S.S. IIdata and Fermi -LAT data derived spectra respectively. The large systematic error on theparameters determined in this work bring the parameters to within the 1 σ error interval.Leung et al. 2014 analyse 62months of Fermi -LAT data on the Vela pulsar concentrating onthe energy range > 50GeV obtaining a 4.2 σ significance measurement. The energy range(0.1, 300)GeV is fit with an exponential cut off power law with parameters Γec = 1.086±0.004,Ec = 383± 5MeV, and b = 0.510± 0.002 which is a result of the contributions from all threepeaks P1,P2 & P3. The values have very small errors and are likely to be only statisticalfitting errors. The best fit b value is also likely to result from this superposition contributionof three simple (b = 1) exponential cut off power laws being fit together (P1, P2, and P3 )as noted by Abdo et al. 2010b. Abdo et al. 2010b and this work both fix b = 1 for the P2spectrum so a comparison of parameters is not useful. A new release of the Fermi -LATreconstruction (Pass 8) which improves on almost all aspects of the analysis has recently beenbrought out and may change the Fermi -LAT spectrum. In future work the new H.E.S.S. dataset on the Vela pulsar should be compared to the improved Fermi -LAT reconstruction withthe aim of well constraining the spectrum around the 100GeV energy range.The systematic error derived here appears accurate given the discrepancy between the

results here compared to those obtained from the analysis carried out on the same databy H.E.S.S. Collaboration 2015. Inaccuracies and mismatches between the Monte Carlosimulations and the system itself are likely the cause of the distinctly steeper spectrumobtained in this work as compared to that shown by H.E.S.S. Collaboration 2015. As theanalysis is operating at its extreme threshold and due to the early nature of this analysis thespectrum itself must be labelled preliminary. More data is required to allow firm conclusionsto be drawn about the behaviour of the spectrum around 100GeV.

4.8. Phasing Cross Check

The pulsar specific analysis software, PulsarSearch, was adapted in order to be able to dealwith data calibrated using the Paris framework in H.E.S.S.. This enabled both sets of datato be used - those calibrated in Paris and in Heidelberg. The analysis configuration used wascomparatively sensitive at lower energies and was also optimised (Holler 2014) in a similar

130

4.8. Phasing Cross Check

fashion using the Q−factor as described in Chapter 3. The resulting analysis cuts do howeverhave different efficiencies as those used with ImPACT. The events of the same set of runs werephased and sky maps produced, however, due to issues concerning the Monte Carlo simulationin the Paris framework the spectrum was not cross-checked. The phases are stored as C++double-precision floating-point variables allowing 15 significant figures (“IEEE Standard forFloating-Point Arithmetic” 2008) to be recorded. For the Vela pulsar (using an approximateperiod of 89ms, see Table 4.1) this corresponds to O

(10−16 s

)precision which far exceeds that

of the H.E.S.S. event time stamp precision ∼ O(10−7 s

)(see Section 2.2.2). The event-wise

agreement in phase is to the level of the floating-point precision and indicates no time stampshift nor drift during the calibration of the data nor in the reconstruction. This thus confirmsthat as far as possible the calibration process does not introduce timing irregularities. The skymaps and, in particular, point-like nature of the source are also compatible when evaluatedwith the Model++ analysis chain (see Section 3.2.4).

131

5. Interpretation of the Observations

This chapter considers the results presented in Chapter 4 mainly in the context of the pulsartheory described in Chapter 1 in an attempt to narrow down the possible physical origins ofthe very high energy (VHE) pulsed emission observed but also with respect to the constraintsthat can be placed on the energy scale at which Lorentz invariance violation occurs.

5.1. Magnetospheric Emission

There are two schools of thought concerning pulsar γ-ray emission: the first describes theemission originating from within the light cylinder (see Section 1.1) and the other fromwithout. The conventional models are magnetospheric gap models. They occupy the firstschool whereby the environment around the pulsar extending to the light cylinder is filledwith corotating plasma. In this plasma �E · �B = 0 and no force acts on the charged particles,however, there exist gaps where �E · �B �= 0 and acceleration occurs along the charged particles’trajectory of the magnetic field lines. This enables curvature radiation to be emitted upto VHE energies. A less conventional but arguably more rigorous model (Krause-Polstorffand Michel 1985a; Petri, Heyvaerts, and Bonazzola 2002b) disputes this magnetosphericconfiguration in favour of a dome and torus of non-neutral, charge separated plasma (seeSection 1.4) around which large areas satisfy �E · �B �= 0 and injected charged particles causebunched cascades that emit curvature radiation (hereafter known as the cascade model).Incoherent curvature radiation is used to explain the emission up to the megaelectron volt(MeV) energy band however further detailed calculations and simulations are required to beable to make any statements about this mechanism in the context of emission from the Velapulsar.The second school of thought envisages emission from the wind region beyond the light

cylinder either through synchrotron radiation or inverse Compton scattering of the pulsarstriped wind (Coroniti 1990; Michel 1994) with X-ray or cosmic microwave background(CMB) photons (Petri 2009a; Aharonian, Bogovalov, and Khangulyan 2012). The emissionfrom the Crab pulsar observed up to 400GeV was originally explained using the OuterGap magnetospheric gap model (Aleksic et al. 2012). However, Aharonian, Bogovalov, andKhangulyan 2012 sought to explain the emission above 100GeV with an inverse Compton

scattering wind emission model. No predictions have however been made on the γ-ray emissionfrom the Vela pulsar using this model. The striped wind model does however reasonably wellfit the Fermi -LAT data on the Vela pulsar (Petri 2011).

Evaluation of the predictions of these models in the context of the results presented in thisthesis could prove informative on the nature of pulsed emission. Currently the magnetosphericgap models are most frequently used to predict the spectra and phase profiles of pulsars.

5.1.1. Magnetospheric Gaps Models

The three conventional gap models are distinguished by the location of the charged particleacceleration region: polar cap (PC; Sturrock 1971; Ruderman and Sutherland 1975), slot

133

Chapter 5. Interpretation of the Observations

gap/two pole caustic (SG/TPC; Muslimov and Harding 2003; Dyks and Rudak 2003), andouter gap (OG; Cheng, Ho, and Ruderman 1986). Pulsars are clearly not all identical:varying ages, obliquity, magnetic field strength, and initial angular momentum are some ofthe important differences between the more than 2, 500 discovered pulsars1. With the successof the Fermi -LAT, the number of pulsars discovered in the high energy (HE) regime (117)has enabled a catalogue to be compiled and population study to be carried out (Abdo et al.2013). This is however not yet possible with the two VHE pulsars so far detected, the Crabpulsar (Aliu et al. 2008) and the Vela pulsar. Particular constraint can be placed on emissionmodels based on the highest energy to which these VHE pulsars emit pulsed emission. Beyondthat, the study of individual pulsars using many energy bands of the electromagnetic spectrumis a second step toward understanding the nature of individual pulsars.

5.1.1.1. Characteristic Pulsar Emission

Figure 5.1 shows the emission from several pulsars - including Vela - over a large energy range.The radio emission as a result of synchrotron radiation is visible in all of the spectra, thethermal peak in the X-ray band is clear in only three (Vela, Geminga, and PSR B1055−52),this gives an excellent measurement of the surface temperature of the pulsar. In the other fourthis X-ray peak is hidden by the wide peak presumably due to curvature radiation from thehigh magnetic field in the magnetosphere. The extended tail down to low energies - particularlyprominent for the Crab pulsar - is due to γ-rays being absorbed in the magnetosphere andconverted into lower energy synchrotron photons. Figure 5.2 illustrates the various componentsvisible in the measured spectral energy distributions (SEDs) shown in Figure 5.1. The H.E.S.S.data are particularly key since they occupy the transition energy band between the curvatureradiation component and a possible TeV component from inverse Compton scattering.Observations of the Crab pulsar show that the tail of the curvature radiation peak appears toextend as a power law up to around 400GeV (VERITAS Collaboration et al. 2011; Aleksicet al. 2012) although it has also been suggested to be of an inverse Compton scatteringorigin (Lyutikov, Otte, and McCann 2012b). It is not clear whether the Vela pulsar also showsthis behaviour or it simply emits according to the tail of the curvature radiation peak.

The SED determined in this work and by H.E.S.S. Collaboration 2015 with the Fermi -LATdata are shown in Figure 5.3. The Vela pulsar energy spectrum presented in Figure 5.3 showsno indication of abrupt attenuation up to 60GeV with a hint of no cut off up to 125GeV. Adistinction between the power law and exponential cut off power law forms is however, notpossible given the statistics in the dataset.

5.1.1.2. Polar Cap Model

Applying Equation 1.51 - which is a prediction of the energy at which the PC model predictsa pulsar’s γ-ray emission should abruptly attenuate - to the Vela pulsar using the propertiesin Table 4.1 and considering the model predicts emission from between one and two stellarradii from the surface (Harding and Muslimov 1998b) gives an expected super exponentialcut off at around a few GeV. This is neither seen in the H.E.S.S. nor the Fermi -LAT data.The best fit spectrum with a cut off is a simple exponential cut off power law which is a lessabrupt cut off than that expected from the PC model (Daugherty and Harding 1996; Baring2004; de Jager 2002). The spectrum can be said to extend without an abrupt attenuation

1this number was taken from http://www.atnf.csiro.au/research/pulsar/psrcat according to the latest catalogue,version 1.53, compiled by Manchester et al. 2005.

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Figure 5.1.: Broadband emission spectra from the seven pre-Fermi -LAT high energy pulsarsobserved from radio to γ-rays. Taken from Thompson et al. 1999.

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Figure 5.2.: Plot to illustrate the components of the broadband pulsed emission from a youngVela-like pulsar. Sy corresponds to synchrotron radiation, kT to thermal radiation,CR to curvature radiation, CS to inverse Compton scattering and the dottedline is the mono-energetic curvature radiation spectrum with the same energy cutoff as for the CR curve. Taken from Romani 1996.

expected from magnetic attenuation of γ-rays near the surface of the pulsar and thus thisinterpretation can be ruled out. Abdo et al. 2010b also reject the interpretation of the Velapulsar data in the context of the PC model at the 16 σ level. The polar cap model cannotovercome the problem of this abrupt attenuation. The PC model is a lower magnetosphericgap model based on acceleration close to the pulsar surface and therefore inherently resultsin a steep cut off due to the magnetic attenuation of γ-rays in the large magnetic field atcomparably lower energies than the OG and SG models. It is therefore ruled out as a viablemodel for the emission from Vela measured in this work.

5.1.1.3. Upper Magnetospheric Gap Models

The upper magnetospheric models - the OG and SG/TPC models - are clear candidates toexplain the observed VHE emission as the emission region is modelled farther from the pulsarsurface where the magnetic field is weaker allowing a higher energy cut off. However, as thereis no theoretical indication that a magnetosphere extends to such distances from the pulsar itis difficult to attribute the observed emission to such models.

Synchrotron Self-Compton Scattering Model Synchrotron self-Compton emission (seeSection 1.3.3.2) could explain the emission from the Crab pulsar in the VHE domain but it isnot expected to make a significant contribution to the Vela spectrum either from primarycharged particles nor their secondaries2 (Harding and Kalapotharakos 2015). Harding andKalapotharakos 2015 suggest that the electrons and electron-positron pairs produced in

2Primary charged particles are those that are either pulled from the surface or the first to be created in pairproduction processes, secondaries are those charged particles that are created from pair production fromphotons emitted by primary charged particles.

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Figure 5.3.: Spectral energy distribution determined in this work for the power law model;best fit model shown by the solid red line and 95% confidence interval by the redshaded area. Fermi -LAT points and Fermi -LAT data fit above 10GeV for theP2 peak taken from H.E.S.S. Collaboration 2015. H.E.S.S. I upper limits for theP2 peak taken from Aharonian et al. 2007. The H.E.S.S. II upper limit from thiswork between 125GeV and 200GeV is not shown as it is far less constraining.

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cascades near the PC are accelerated and go on to emit primary and pair curvature radiationin the SG. Near and beyond the light cylinder, synchrotron photons are up-scattered bythe charged particles in synchrotron self-Compton scattering interactions to produce γ-rays. These synchrotron photons are produced due to the resonant cyclotron absorption ofsynchrotron radiation produced near the Polar Cap by the charged particles which gives thecharged particles a non-zero velocity perpendicular to the magnetic field line. Harding andKalapotharakos 2015 suggest that only Crab-like pulsars3 produce significant (detectable bythe current generation of γ-ray observing instruments) synchrotron self-Compton scatteringemission from both primaries and pairs. The VHE emission from Vela is suggested to originatefrom curvature radiation from the primary charged particles and not from synchrotron self-Compton scattering. This is because the middle-aged Vela has a longer period and thereforesmaller magnetic field at the light cylinder as well as lower levels of non-thermal X-ray whichare key for synchrotron self-Compton scattering emission (Harding and Kalapotharakos2015).

Outer Gap Model Wang, Takata, and Cheng 2011 consider a 3D OG model to explain theenergy dependent phase profile seen in the HE Fermi -LAT data (Abdo et al. 2010b) but alsoderive a spectrum which well describes the HE-VHE phase-averaged spectrum measured withFermi -LAT (Abdo et al. 2010b). The predicted P2 flux is, however, too low and does not wellmatch the spectral shape. Vigano and Torres 2015 similarly fit a model of synchro-curvatureradiation - an intermediate regime of synchrotron and curvature radiation (see Section 1.3.2.1)- to the Fermi -LAT data based on a general gap in the outer magnetosphere which couldeven be beyond the light cylinder. The model is however only extended to 10GeV where thepredicted flux is below that of the latest Fermi -LAT spectrum (H.E.S.S. Collaboration 2015)and this work4 so interpretation in the context of VHE emission is difficult; the emission doesappear to cut off at an energy too low to match the H.E.S.S. data. A closer fitting modelwhich was published when only the EGRET data was available (Hirotani 2007) is shown inFigure 5.3 (labelled: 2D Outer Gap Model). The agreement is poor at low energies since themodel is for the phase averaged spectrum whereas the Fermi -LAT data points are simply forP2 but as the fit extends to 100GeV P2 is the major contribution to the energy flux and theagreement between the data and the model improves. This is the most compelling agreementand is based on a curvature radiation spectrum with a negligible component due to inverseCompton scattering; a photon propagation direction from the rotation axis of < 107 ◦ isused.

Leung et al. 2014 analyse a more recent Fermi -LAT data set and report the continuation ofthe form of the previous Fermi -LAT exponential cut off spectrum (Abdo et al. 2010b) up toaround 80GeV. The spectral cut off is characterised by a cut off energy and b parameter (seeSection 4.6.2) determined to be Ec = (383 ± 5)MeV and b = 0.510 ± 0.002 which includescontributions from all three peaks whereas this work finds b = 1 best fits the H.E.S.S. datafor the P2 peak alone. Leung et al. 2014 model the VHE emission in a multi-contributionOG scenario. Various currents - possibly originating from the PC region - encounter manystationary gaps and combine to create a curvature radiation γ-ray spectrum. The currents aretweaked in order to reproduce the observed cut off behaviour. The best fitting model is shown

3Crab-like pulsars are young with high spin down energies, an optical pulsed component, giant pulsesetc. (Ferdman, Archibald, and Kaspi 2015)

4The model gives the isotropic luminosity (erg s−1) which when divided by 4π times the distance squared tothe Vela pulsar gives a flux that is too low, however, when tweaked it reproduces the spectrum well.

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in Figure 5.3 alongside the H.E.S.S. data; again shown as a phase averaged spectrum meaningthe flux at lower energies is higher than the blue Fermi -LAT points due to the contributionsfrom P1 & P3. The flux predicted by the model is clearly higher and the shape describesan exponential cut off rather than a power law as best fit to the data measured in this work.The steepness of the H.E.S.S. power law makes it difficult to distinguish an exponential cutoff from a power law and the uncertainty of the H.E.S.S. spectrum around 100GeV combinedwith the contributions from P1 and P3 mean this model is certainly not ruled out.

Aharonian et al. 2007 constrain a simple 1D OG model prediction of the inverse Compton

scattering emission from the Vela pulsar with H.E.S.S. I data. A 2D OG model does howeverpredict a much lower inverse Compton scattering flux from Vela which is well below theupper limits derived. These upper limits are shown in Figure 5.3 but do not constrain thecontinuation of the best fitting power law model derived in this work nor the continuation ofa power law fit to the Fermi -LAT data above 10GeV, also shown. However, if these limitswere included in any fit of the H.E.S.S. II data they may increase the relative likelihood ofthe exponential cut off shape as compared with a power law.

Annular Gap Model The Annular Gap model (Qiao et al. 2004) is an intermediate mag-netospheric model between the Polar Cap and Outer Gap models as it suggests the γ-rayemission originates from a part of the pulsar magnetosphere between the other two. Tworegions are important for the Annular Gap model: the core gap and the annular gap whichare separated by the critical field line above the magnetic pole of a pulsar. The critical fieldline is defined as the field line that crosses the light cylinder at the same point as the nullcharge surface5. The gap between the null charge surface and critical field line is named theannular gap, the core gap is the gap between the magnetic axis and the critical field line.The Annular Gap model is suggested to be important for pulsars with small periods (Duet al. 2011) and involves two emission components in the γ-ray regime: primary particles thatemit synchro-curvature radiation and secondaries that emit synchrotron radiation. Du et al.2011 accurately model both the HE γ-ray and radio SED as well as the phase profile for bothcomponents observed the Vela pulsar. This includes the phase difference between the radioand P1 γ-ray peaks which is due to the radio emission region being high in the magnetosphereclose to the light cylinder. The phase-averaged P2 component of the SED predicted by theAnnular Gap model is shown in Figure 5.3 and appears to best fit the SED measured withH.E.S.S.. The emission comes only from synchro-curvature radiation from primary chargedparticles and not from synchrotron radiation radiation from secondaries which falls steeplybelow 1GeV (Du et al. 2011). This model well fits the observed Fermi -LAT HE SED andseems to be able to explain the steep decline in emission measured in this work. This is ofcourse in slight disagreement with the Fermi -LAT points as the SED obtained in this work isclearly steeper than that observed by the Fermi -LAT.

Pulsar Wind Model Mochol and Petri 2015 fit the Fermi -LAT spectrum presented by Leunget al. 2014 using a synchrotron emission component originating beyond the light cylinder inthe wind region with very similar results. This makes it clear that differentiating modelsbased simply on the SED is not easily done. Neither models however reproduce the expectedlight cylinder of the Vela pulsar which is another observable discriminator. In the model,a synchrotron self-Compton scattering component is used to model the Crab pulsar SED

5The null charge surface is defined in Section 1.6.4 as the null surface: the surface at which the bulk plasmacharge changes sign in a Goldreich-Julian, charge separated plasma-filled magnetosphere.

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however a significant synchrotron self-Compton scattering component is not expected fromthe Vela pulsar. This is because in the model most of the energy stored near the Vela pulsaris in high energy particles that do not emit sufficient low energy synchrotron photons to beup-scattered to VHE.

The Annular Gap and Outer Gap models appear to best predict the SED of the Vela pulsarin the γ-ray band based on curvature radiation/synchro-curvature radiation; it is clear thatthe emission is not from the Polar Cap region as no abrupt attenuation of the spectrum ismeasured up to 60GeV. Both the multi-component and the 2D versions of the Outer Gapmodel reasonably well reproduce the form of the spectrum. The Annular Gap does howeverwell explain both the form of the SED and the observed γ-ray phase profile as well the radiopeak phase lag and shape. For this reason it appears the Annular Gap model is a betterdescription of the magnetospheric emission coming from the Vela pulsar. It is unlikely thatsynchrotron self-Compton scattering plays a large role in the emission. It is not absolutelyclear from the data whether simply the tail of a curvature radiation-type peak or anothercomponent - possibly arising from inverse Compton scattering - has been measured. Froman experimental point of view the power law is favoured because fewer free parameters arerequired and there is no statistical justification for the more complicated exponential cutoff power law form. From a theoretical point of view however, the inability to distinguishbetween the two spectral forms suggests that the simpler interpretation - the continuationof the curvature radiation-type tail as measured by Fermi -LAT (Abdo et al. 2010b) - isfavoured over the addition of another spectral component to justify the power law shape of thespectrum. If the information from the H.E.S.S. I upper limits were taken into account duringthe fit, the case for a steep power law/simple exponential cut off spectrum would becomestronger since the continuation of the Fermi -LAT power law spectrum above 10GeV is veryclose to the lowest upper limit. New data obtained with H.E.S.S. as well as the new Pass 8reconstruction introduced by the Fermi -LAT Collaboration (Atwood et al. 2013) should allowthe nature of the Vela pulsar SED around this energy regime to be resolved.

5.2. γ-ray Emission Luminosity and Efficiency

The luminosity and emission efficiency of the Vela pulsar in the VHE regime can be calculatedfrom the measured SED, this however also requires an understanding of the emission mechanism.Two models are considered: the SG/TPC model and the OG model following that done byAbdo et al. 2010b. The efficiency of the emission with regard to the spin-down power ofthe pulsar (see Section 1.1) is of interest and a direct result of the emission model and SED.Firstly, the orientation of the Vela pulsar must be understood.

5.2.1. Orientation of the Vela Pulsar

Two angles are of importance when studying a pulsar: the magnetic axis to rotation axisinclination angle α (obliquity) and the observer angle ζ to the axis of rotation. Watters et al.2009 simulate the beaming from many young pulsars to be able to predict the features ofobserved pulsars. The Earth’s observation angle to the Vela pulsar is determined to be 64 ◦

from both torus fits to X-ray images (Breed et al. 2015; Ng and Romani 2008) and beampattern & phase profile simulation of a populations of young pulsars (Watters et al. 2009).From simulations, Watters et al. 2009 determine that this observation angle implies differentmagnetic axis inclination angles depending on the emission model: the TPC model gives the

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range α = (62, 68) ◦ and the OG models gives α = 75 ◦. Du et al. 2011 use α = 70 ◦ andζ = 64 ◦ to recover the phase profile observed by Fermi -LAT from the caustics predicted bythe Annular Gap model.

5.2.2. The Vela Pulsar Emission Efficiency

In order to estimate the efficiency of the γ-ray emission η = LE, the luminosity L must also be

estimated from the measured integrated energy flux from the source Fobs and the distance tothe source D

L = 4πfΩFobsD2. (5.1)

Furthermore, a correction factor fΩ is required which is the ratio of the flux over the entiresky to that observed from Earth (Watters et al. 2009):

fΩ = fΩ (α, ζE) =

∫ ∫Fγ (α, ζ, φ) sin(ζ)dζdφ

2∫Fγ (α, ζE , φ) dφ

(5.2)

where α is the obliquity, ζE is the observer angle from Earth, and φ is the pulsar rotationalphase (hereafter simply phase). This correction factor is model dependent since the pulsedflux from the pulsar Fγ must be predicted by theories of the γ-ray pulsed emission. Thedetermination of the correction factor is dependent on the characteristic magnetospheric gapwidth w. Watters et al. 2009 use a dimensionless width which is assumed to be proportional tothe γ-ray efficiency because the more energy that goes into γ-ray emission the larger the gapsize must be. Watters et al. 2009 determine it to be 0.01 meaning the correction factor fΩ forboth models can be evaluated: fTPC

Ω = [1.1, 1.15] in the TPC model and fOGΩ = 1.0 in the OG

model. Watters et al. 2009 did not calculate fΩ for the Annular Gap model however as it is nota large factor it is a small effect; fΩ remains in the expressions for ease of use when consideringother models. The distance to the Vela pulsar is determined from 2.3GHz and 8.4GHzvery-long baseline (832 km) interferometry observations (Dodson et al. 2003) as 287+19

−17 pc(a symmetric error band of 18 pc was used). Given the energy flux from the Vela pulsarmeasured with CT 5 taken from Table 4.5 as (4.29(+1.14

−1.02)stat(+5.50−3.31)sys)× 10−11 erg s−1cm−2,

the luminosity of the Vela pulsar in the energy band (15, 125)GeV is

L = (4.228(+5.564−3.450)comb)

fΩ1.0

× 1032 erg s−1 (5.3)

where the combined statistical and systematic error is shown propagated from the error on themeasured integrated energy flux (systematic and statistical errors were added in quadrature)and distance. This leads to a measurement of the pulsed efficiency in the energy band(0.015, 0.125)TeV as ηV HE = L

E= 2.531 × 10−5 × fΩ where E = 1036.8 erg s−1 was taken

from Watters et al. 2009 (this agrees with that given in Table 4.1). This luminosity valueis two orders of magnitude less than that determined using Fermi -LAT observations (Abdoet al. 2010b) in the energy range (0.1, 100)GeV where it is calculated as 8.2+1.1

−0.9× 1034 erg s−1.This is unsurprising given the steep spectrum and higher energy band measured in this work.The efficiency calculated by Abdo et al. 2010b is (1 × fΩ)% whereas this analysis obtains(0.005 × fΩ)%. Figure 5.4 shows the general trend between luminosity and E for pulsarsfeatured in the Second Fermi -LAT Catlog of Gamma-ray Pulsars (Abdo et al. 2013). There isa weak proportional dependence of the two properties into which the Fermi -LAT measurementfits well. The point determined from this work is significantly lower as the Fermi -LAT point is

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Figure 5.4.: Luminosity versus spin down power for all pulsars in the Second Fermi -LATCatlog of Gamma-ray Pulsars (Abdo et al. 2013). The Vela pulsar, as determinedwith theFermi -LAT in the high energy band, is shown in red with a black circlearound it and as determined in this work in the very high energy band in greenwith combined statistical and systematic error bars.

primarily determined from the HE emission in the MeV regime whereas the point calculated inthis work is determined from the gigaelectron volt (GeV) regime. It is evident from Figure 5.3that to have a HE luminosity similar to that of the Vela pulsar in the VHE band a pulsarrequires a spin down power around two orders of magnitude lower than the Vela pulsar. TheCrab pulsar, being the only other pulsar measured in the VHE regime, is observed to emitbeyond that measured here from the Vela pulsar, it is therefore interesting to analyse itsγ-ray emission efficiency. Two other good candidate VHE pulsars (PSR J0633+1746 andPSR J0540−6919) are also considered.

5.2.3. The Crab Pulsar

Considering the emission in the same energy range ((15, 125)GeV) by extrapolating tolower energies from measurements made with the MAGIC array (Aleksic et al. 2014) in theenergy range (50, 400)GeV (including contributions from P1, P2, and the bridge emission)gives the energy flux 3.41 × 10−11 erg s−1cm−2. This combined with the distance to theCrab pulsar (2 ± 0.5) kpc determined using the proper motions of filaments in the Crabnebula and nearby objects (Trimble 1968; Kaplan et al. 2008), the luminosity of the Crabpulsar was determined to be 1.633fΩ × 1034 erg s−1. The spin down rate of the Crab pulsar

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as given in Table 4.1 is 450 × 1036 erg s−1 which means that the emission efficiency isηCrab = 3.63× fΩ × 10−5 = 0.00363× fΩ%. This is remarkably similar to that measured forthe Vela pulsar possibly suggesting that the same mechanism is at work for the very highenergy γ-ray emission from two pulsars and slightly hinting towards the emission from Velacontinuing as a power law as with the Crab pulsar. However, due to the differences between thepulsars, age for one, this interpretation is likely somewhat too far reaching. This is uncertainsince all three phase profile features are included - i.e. P1, P2, and the bridge emission -whereas for Vela only P2 contributes to the calculation of the emission. Considering only P2results in an energy flux of 1.11× 10−11 erg s−1cm−2, a luminosity 5.31fΩ × 1033 erg s−1, andan efficiency 0.00118fΩ% which is still close to that measured from Vela. If it is assumed thatthe emission efficiency for other VHE γ-ray pulsars is similar, say 3× 10−5, then predictionscan be made about their VHE γ-ray spectrum in the same energy range (15, 125)GeV. Thisgives an indication whether or not they are worth observing with the current generation ofIACT arrays.

5.2.4. The Geminga Pulsar

Geminga has been measured by the Fermi -LAT Collaboration to have an efficiency in the

HE regime of ηHEGeminga = 0.15× fΩ

(d

100 pc

)2(Fermi-LAT Collaboration, Abdo, and al. 2010)

suggesting fΩ is smaller than unity since the distance to the Geminga pulsar is 250+120−62 pc,

obtained from an optical parallax measurement (Faherty, Walter, and Anderson 2007). It wasalso detected above 35TeV by the Crimea Astrophysical Observatory IACT (Neshporet al. 2001) although not by the HEGRA IACT array (Aharonian et al. 1999).

The ATNF Pulsar Catalogue (ATNF 2015c; Manchester et al. 2005) gives the spin downpower to be 3.2× 1034 erg s−1 which when using the expected efficiency η = 3× 10−5 resultsin an integrated luminosity in the energy range (15, 125)GeV of 9.6fΩ × 1029 erg s−1. Thisgives the integrated energy flux 1.28

fΩ× 10−13 erg s−1cm−2 which is two orders of magnitude

lower than that measured from the Vela pulsar in this work and therefore not reasonablyobservable by the current generation of IACT arrays.Extrapolating the Fermi -LAT spectra determined in the range (10, 100)GeV (Aliu et

al. 2015) with a power law gives the energy flux in the range (15, 125)GeV to be 1.71 ×10−11 erg s−1cm−2, using the same distance as done previously gives a luminosity 1.28fΩ ×1032 erg s−1, and a VHE efficiency of ηVHE

Geminga = 0.399fΩ%. The extrapolation is not ruledout by measurements made with the VERITAS IACT array (Aliu et al. 2015). This efficiencyis two orders of magnitude higher than that determined for the Crab and Vela pulsars. In thissimplistic calculation, either the Geminga pulsar has a far higher emission efficiency to achievethis continuation and the emission processes are different or there is an extremely abrupt cutoff in the (100, 125)GeV energy range to compensate and reduce the emission efficiency tothe same level as the other two pulsars, which seems unlikely. Note that this is consideringthe phase averaged spectrum with two components P1 and P2 but as P2 is observed witha spectrum that is higher at higher energies (harder) the effect of P1 in the comparison issmall. The different emission mechanism may be a result of the higher characteristic age ofthe Geminga pulsar ∼ 340, 000 y (ATNF 2015c) as compared to 11, 300 y for the Vela pulsar(ATNF 2015d) and 1, 250 y for the Crab pulsar (ATNF 2015a). More likely, however, this issimply the effect of a too simplistic interpretation of the emission efficiency trend from onlytwo pulsars.

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5.2.5. PSR J0540−6919

Another promising Crab-like candidate VHE pulsar is PSR J0540−6919 located in the LargeMagellanic Cloud (LMC), discovered with the Einstein High Resolution Imager (Seward,Harnden, and Helfand 1984) and detected in HE with the Fermi -LAT (Marting and Guillemot2014). According to the ATNF Pulsar Catalogue (ATNF 2015b; Manchester et al. 2005)it has a spin down power of 1.5× 1038 erg s−1 which is only one third that of the Crab pulsarbut far higher than both the Vela and Geminga pulsars. Again assuming a VHE efficiencyof 3 × 10−5 gives a luminosity 4.5 × 1033 erg s−1. The distance to the LMC is estimatedas 48.1± 2.3stat ± 2.9sys kpc from variable Cepheid stars based on a tight period-luminosityrelationship and measured with the Hubble Space Telescope (HST) (Macri et al. 2006). Thisresults in a energy flux in the range (15, 125)GeV of 1.63

fΩ× 10−14 erg s−1cm−2. This is three

orders of magnitude lower than the energy flux from the Crab and Vela pulsars in the sameenergy range and again not detectable by the current generation of IACT arrays. Using thepreliminary result with the Fermi -LAT data (Marting and Guillemot 2014), fitting the range(5, 20)GeV (the Fermi -LAT spectrum is only specified up to 20GeV), and extrapolating tothe energy range (15, 125)GeV gives an energy flux in this range of 1.02× 10−13 erg s−1cm−2.This translates to a luminosity of 2.82fΩ × 1034 erg s−1 and efficiency of 0.0188fΩ%. Thisshows only one order of magnitude difference to the Crab-Vela efficiency trend but an energyflux two orders of magnitude lower than the Vela pulsar in this energy regime. The efficiencywould need to be ∼ 1% to produce emission at the same level as the Vela pulsar, which issimilar to the HE efficiency of Vela. Deep observations have been made of the LMC withH.E.S.S. I (H.E.S.S. Collaboration et al. 2015) however the energy threshold is so highthat only another, perhaps inverse Compton scattering, emission component similar to thatconstrained by observations with H.E.S.S. I of the Vela pulsar (Aharonian et al. 2007) wouldbe able to be constrained.

5.3. Phase Profile Evolution with Energy

The phase profile of the Vela pulsar varies across the many energy regimes over which it hasbeen measured. Figure 5.5 shows the phase profile as measured in VHE & HE γ-rays as wellas in radio, X-ray, and near ultra-violet wave lengths. It has also been measured in the opticalband with a similar structure to that of the near UV (Wallace et al. 1977; Manchester et al.1978). The three peaks (P1, P2 & P3 ) labelled in the plot (second row on the right) do notclearly correspond to any of the features in the radio component, however a signal is seen atthat phase in the near UV. This discrepancy with the radio phase profile suggests that theemission originates from a different location in the pulsar’s magnetosphere. Radio emission isoften associated with the PC region (Lyne, Graham-Smith, and Graham-Smith 2006) thuspointing to an SG/OG γ-ray emission region. The Annular Gap model explains the phaselag of the radio pulse based on it emission from high up in the magnetosphere near the lightcylinder (Du et al. 2011). Of particular note is the reduction of P1 and P3 to the point atwhich they are not seen with H.E.S.S. II - at energies > 20GeV. This decrease is also clearlyvisible in the phase profile measured by the Fermi -LAT at E > 10GeV and shown in thefigure, fourth row on the right. This sudden drop off with energy compared to P2 suggests thatthe origin of the emission from P1 & P3 is located closer to the pulsar surface where magneticattenuation limits the maximum energy photons that can escape. This also suggests that theorigin of P2 is not close to the pulsar surface. Abdo et al. 2010b note a hint that the P2 peak

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Figure 5.5.: The Vela pulsar phase profile shown for various energy bands. The top row showsthe γ-ray signal (energy range shown) as measured with H.E.S.S. II describedin this work. The following three rows show the phase profile as measured bythe Fermi -LAT at the γ-ray energies shown. In the bottom left is the phaseprofile measured at X-ray energies by the Rossi X-ray Timing Explorer (NationalAeronautics and Space Administration 2012) and bottom right in the near UVband measured by the Space Telescope Imaging Spectrograph (Woodgate et al.1998) aboard the Hubble Space Telescope (National Aeronautics and SpaceAdministration 2015). Along the bottom the phase profile measured in radio isshown in red. The part of the figure showing non-H.E.S.S. results was taken fromAbdo et al. 2009b.

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splits at GeV energies (not clear in Figure 5.5 but more apparent when more Fermi -LATdata is analysed) however there is no indication of that in the H.E.S.S. data, backgroundfluctuations make identifying such a feature difficult. The gap models - particularly theSG/TPC and the Annular Gap models - well reproduce the Vela phase profile in the MeV-GeVenergy band as measured by EGRET (Dyks and Rudak 2003) and Fermi -LAT (Wang, Takata,and Cheng 2011; Du et al. 2011) but the OG model has the inherent problem of being unableto directly explain the bridge emission (P3 ) observed between the two main peaks in the GeVband (Abdo et al. 2010b). Wang, Takata, and Cheng 2011 suggest that P3 is the result ofazimuthal structure in the shape of the OG. The TPC model fairly well predicts the relativeheights of the Vela γ-ray phase profile as a result of caustics originating from a region similarto the SG region (Dyks and Rudak 2003). Energetics are however not considered specificallyin the model and therefore no VHE γ-ray phase profile has been derived. The Annular Gapmodel considers both caustics and energetics to accurately model the evolution of the HEphase profile with energy as measured with the Fermi -LAT (Du et al. 2011).

5.3.1. The Asymmetric Width of the P2 Peak

The narrowing of the P2 peak with energy as observed by Abdo et al. 2010b with theFermi -LAT implies a reduction in the width of the emission zone with increasing energy.Leung et al. 2014, whose model is discussed in Section 5.1.1, predict the opposite and anincrease in the peak width with energy. The model put forward by Mochol and Petri 2015,also discussed in Section 5.1.1, on emission from the wind region includes the observednarrowing of the peak with energy. The Annular Gap model well predicts the narrowingof the peak (Du et al. 2011). The trend is shown in Figure 5.6 for both the leading andtrailing edge widths of P2 as defined from the best fit asymmetric Lorentzian functional fitfor both Fermi -LAT and H.E.S.S. data (see Section 4.5). Note that the H.E.S.S. points areshown for reconstructed energy which is subject to a large correction, see Section 3.3.5, and -particularly the lower energy point - should be expected to require shifting to lower energies.This effect is also likely to be the case - albeit a smaller correction - for Fermi -LAT pointsbeing most significant at the lowest energies (Wood et al. 2015). The figure clearly showsa drop in the peak width for both leading and trailing edges however the behaviour seemsdifferent between the two. Functional forms were fit to both: an exponential function ea+bE

with best fitting parameters a = −3.422+0.010−0.010 & b = −0.01699+0.00170

−0.00170 for the leading width

with goodness of fit measure χ2/NDF = 16/6 = 2.67 and a power law function c(

EE0

)−d

with E0 = 0.02GeV (fixed), c = 0.03802+0.00157−0.00151 & d = 0.3491+0.0110

−0.0110 for the trailing widthwith goodness of fit measure χ2/NDF = 58/6 = 9.67. The parameters are correlated withPearson correlation coefficients (Pearson 1895) of −0.50 and 0.95 for the exponential andpower law function parameters respectively. The data are from two experiments i.e. areindependent measurements and the errors on the points are purely statistical meaning thatalthough the large reduced χ2 values represent a poor fit they do not rule out the functionalforms as accurate models; note that in both cases only two free parameters are used. Thefit appears far more convincing when the lowest energy points from both instruments areconsidered erroneous due to being at the limits of the instruments, however, this approach mustbe treated with caution as these points are accepted in the spectrum and should, therefore, betreated as valid. The consistently wider leading edge may be the result of contributions fromthe same source as that of the bridge emission (P3 ) observed with the Fermi -LAT (Abdoet al. 2010b). The charged particles entering a magnetospheric gap may well have been

146


Figure 5.6.: Peak width (measured in phase with range [0, 1)) for the leading edge and trailingedge of the P2 peak. Fermi -LAT data taken from Abdo et al. 2010b.

accelerated in the PC/SG regions up to Lorentz factors of 107 and form a delta function orbump-like distribution in energy. It is expected that the electric field peaks in the centre ofthe gap and since higher electric fields are required for higher energy charged particles thegap decreases in size with energy. The fact that the widths do not drop at the same ratesuggests that another mechanism is at work. The caustic effect shown in Figure 1.16 meansthat observations of the leading edge are made up of contributions from many altitudes alongthe gap. This difference between the two could contribute to the observed trend. This couldalso be a result of an asymmetric electric field within the magnetospheric gap and could shedlight on the magnetospheric configuration within the gap as the radius of curvature of themagnetic field lines directly relate to the flux emitted in synchro-curvature radiation/curvatureradiation (see Section 1.3.2). In the cascade model, if inverse Compton scattering werea large contributor to the emission - the emission is normally attributed to curvature-typeradiation (see Section 1.5.1) - this behaviour could be explained by the greater beaming effectof higher energy emission as the Klein-Nishina regime is approached. The scattering angledrops with increasing energy according to the Klein-Nishina differential cross section

dσ

dΩ∝ F (Eγ , θ)

2(F (Eγ , θ) + F (Eγ , θ)

−1 − 1 + cos2(θ))

(5.4)

where

F =1

1 +Eγ

mec2(1− cos(θ))

, (5.5)

Eγ is the photon energy, Ω the solid angle, θ the scattering angle, and mec2 the rest mass

energy of the electron (Weinberg 1996). This causes a strong beaming effect that wouldmanifest as a narrowing pulse width with energy. This combined with a similar caustic effect

147


to differentiate the leading and trailing edges may be able to explain this trend howeverdetailed calculations are required for any conclusions to be drawn.

5.4. Lorentz Invariance Violation

In some theories of quantum gravity (QG; Amelino-Camelia et al. 1998; Smolin 2002) theprinciple of Lorentz invariance is permitted to be broken at around the Planck energyas a direct result of the quantum fluctuations in the vacuum that cause space-time to havea dynamical nature. These fluctuations would induce an energy dependent interaction forphotons travelling through the vacuum which would manifest as photons of different energieshaving slightly different speeds. This could be observed using the pulsed emission frompulsars (Kaaret 1999). If the photons of different energies were emitted simultaneously andtheir speeds were different but directions the same then the energy resolved phase profilewould display a relative offset assuming the photons were emitted at the same time. This canbe measured by determining the precise position of the peak in the Vela pulse profile observedwith H.E.S.S. and as measured with the Fermi -LAT. The modification is brought about by

adding energy dependent terms(

EELIV

)nto the speed of light c which relate to the energy

level ELIV at which the effect becomes significant (Jacob and Piran 2008):

c(E) = c

(1− s±

n+ 1

2

(E

ELIV

)n). (5.6)

where s± = ±1 to indicate a sub-/super-luminal speed. It should be noted that odd valuesof n violate CPT symmetry (Colladay and Kostelecky 1997), this must be considered wheninterpreting experimental results as it is currently understood to be preserved in nature (Kost-elecky 1998). As Eγ � EQG,n then c(E) is dominated by the n = 1, 2 terms. The arrival timedifference between two simultaneously emitted photons of different energies E1 & E2 from asource at a distance d is given by

Δt(n = 1) =d

c

|E1 − E2|ELIV

→ ELIV =d

c

|E1 − E2|Δt

(5.7)

for the linear term and

Δt(n = 2) =d

c

3

2

|E21 − E2

2 |E2

LIV

→ ELIV =

√d

c

|E21 − E2

2 |Δt

(5.8)

for the quadratic term based on the prescription given by Otte 2012. The position of theP2 peak - based on a best fit asymmetric Lorentzian function - from the Fermi -LAT datacorrected to the same timing solution as used in this work (see Section 4.5) in the lowestenergy range6 measured (0.02, 0.1)GeV is 0.5544 ± 0.0035. The asymmetric Lorentzian

fit described in Section 4.5 yielded a peak position 0.5619 ± 0.0010 in the higher energyrange (34.4, 125.0)GeV. These values are compatible to within 2.1 σ. Assuming a power lawaccording to that best fitting the data (index = −5.39) results in a mean energy in the highenergy band used in this work of 44.1GeV. The same was done for the Fermi -LAT data whichis best fit with a power law with exponential cut off (see Abdo et al. 2010b) yielding a meanvalue of 54.3MeV. The one-sided 95% confidence level upper limit on the time difference

6The lowest energy range was chosen to maximise |E1 − E2| and therefore ELIV .

148

5.4. Lorentz Invariance Violation

Instrument ELIV at the 95% confidence level [GeV ]Linear Term Quadratic Term

Fermi -LAT > 7.6× EP > 1.07× 10−8 × EP

Fermi -LAT & Fermi -GBM > 3.42× EP -H.E.S.S. > 0.059× EP > 1.15× 10−10 × EP

MAGIC > 0.017× EP > 2.13× 10−9 × EP

VERITAS > 0.016× EP -

This work (H.E.S.S. II) > 9.64× 10−5 × EP > 1.87× 10−11 × EP

Table 5.1.: 95% upper limits on the energy scale of Lorentz invariance violation as measured

by various experiments with respect to the Planck energy scale EP =√

hc5

G ≈1.22× 1019GeV.

between the two measured peaks is

Δt95% < 1.65ΔφP = 1.106ms (5.9)

where the errors on Δt are assumed to follow a standard normal distribution (P [X < 1.65] =95%). Using Δφ = 0.0075 & P = 89.372ms - taken from the timing solution for the Velapulsar used in this analysis - as well as the distance to the Vela pulsar 287+19

−17 pc (as used inSection 5.2) gives a lower limit on the Lorentz violating energy scale from the linear andquadratic terms:

E95%LIV linear = 1.176× 1015GeV (5.10)

E95%LIV quadratic = 2.278× 108GeV. (5.11)

This of course assumes that the peak positions at the time of emission are the same at differentenergies. These are weak limits mainly due to the proximity of the pulsar. Many other limitshave been placed on this energy scale using a similar method of assuming a vacuum dispersion:using gamma-ray burst signals measured with the Fermi -LAT (Vasileiou et al. 2013) &Fermi Gamma-ray Burst Monitor (Fermi -GBM) (Abdo et al. 2009a), using flaring activegalactic nuclei signals measured with H.E.S.S. (Aharonian et al. 2008) & MAGIC (MAGICCollaboration et al. 2008), and using the pulsed emission from the Crab pulsar measured withVERITAS (Zitzer and the VERITAS Collaboration 2013). The limits placed by these studiesalong with that derived here are given in Table 5.1. Theoretical models describing quantumgravity were seriously hindered by the limits derived using Fermi -GBM and Fermi -LATdata as they exceed the Planck energy scale at which point LIV was allowed to occur(see references in Abdo et al. 2009a & Vasileiou et al. 2013). The limits from this work areintrinsically hindered by the proximity of the Vela pulsar; being around 7.5 times closer thanthe Crab pulsar. The Vela pulsar results can however provide insight into Lorentz invarianceviolation (LIV) at different distance scales. Zitzer and the VERITAS Collaboration 2013showed a more advanced method using the Z2

m test statistic and testing different energy-dependent event-wise shifts in the event time stamps to find the shift that maximised the teststatistic. This method resulted in a limit of the same order of magnitude to that derived withEquations 5.7 & 5.8 for the Crab pulsar. Martınez and Errando 2009 introduced a method inwhich the probability density function (pdf) that an event with a certain energy was observed

149


at a certain time is made up of several contributions: the photon energy distribution at thesource; changes to the instrument collection area during the observation time; the energy biasof the instrument; the emission time distribution and the propagation delay at the source. Theterm that characterises the energy-dependent propagation delay of the photons is suggestedto take the form:

Ens − En

s,0

MnQGn

z

H0(5.12)

for source at a low red shift where Es & Es,0 are the photon energy at the source and detectorrespectively, MQGn is the QG effective energy scale, z is the source red shift, and H0 theHubble constant 67.8 ± 0.9 km s−1Mpc−1 (Planck Collaboration et al. 2015b). This wasevaluated with the same H.E.S.S. II data set used in this work by Chretien, Bolmont, andJacholkowska 2015 to be ELIV > 3.72 × 1015GeV and ELIV > 3.95 × 1015GeV for the suband super luminal cases respectively. These are the same order of magnitude as the linearcase derived in this work.

150

Conclusions and Outlook

Here was presented the first detection of pulsed emission using the H.E.S.S. array of imagingatmospheric Cerenkov telescopes (IACTs). Pulsed emission from the Vela pulsar wasdetected to a high confidence level and measured in the energy range 15GeV ≤ E ≤ 125GeVwhich both shows the capabilities of H.E.S.S. Phase II at the lowest energies and presentsnew findings that the Vela pulsar - along with the Crab pulsar - emits VHE γ-rays (this wasalso reported by Leung et al. 2014 using Fermi -LAT data). The first point bodes well forstudies of transients such as gamma-ray busts (GRBs) which are as yet undetected in thevery-high energy (VHE) domain and to which H.E.S.S.’s latest telescope (CT 5) is uniquelysuited with its huge 600m2 mirror area and quick reaction time ∼ 1min (Parsons et al. 2015).Discovery that the Vela pulsar continues to emit γ-rays into the VHE regime without a visiblecut off brings into question whether another emission component is required beyond that ofthe usual (synchro) curvature radiation. Using an optimised analysis configuration led to themeasurement of the Vela pulsar at the 11.4 σ level down to 15GeV which is an unprecedentedlow energy in ground-based γ-ray astronomy. Deeper observations with CT 5 have alreadybeen made on the Vela pulsar in the hope that the spectrum around 100GeV can be firmlystated, too late however to be included in this work. Furthermore, the new Pass 8 Fermi -LATreconstruction method will improve the sensitivity and further aid in the identification of theVela pulsar (and others) in the VHE regime.

Theoretical Implications This detection also tests current magnetospheric models on thecause of γ-ray emission effectively requiring that the emission zone be relatively distant fromthe pulsar surface. Findings here also provide insight into the size of emission region from thedistinct thinning of the pulse in the phase profile. This is suggestive of an emission zone thatbecomes smaller at higher energies given the assumption that the emission energy peaks at itscentre where the electric field is at a maximum. As has been repeated throughout, the generalassumption that the pulsars magnetosphere is filled with charge-separated plasma up to thelight cylinder (see Section 1.4) is difficult to justify when analysing the electromagnetic fieldsassociated with the system (Michel 1991b). This assumption is required when consideringmagnetospheric pulsed γ-ray emission models such as the Outer Gap, Slot Gap, and AnnularGap models. The Outer Gap model more so than the Slot Gap and Annular Gap modelsrequires the magnetosphere to be filled with plasma up to the light cylinder. Alternativetheory has however suggested that the pulsed emission may come from beyond the lightcylinder in the wind zone regardless of the magnetospheric configuration around the pulsar.While this seems to be able to justify emission at the energy levels observed from the Velapulsar it has not yet been used to explain the hint of pulsed emission reported by the MAGICCollaboration at E ∼ 2TeV from the Crab pulsar (de Ona Wilhelmi et al. 2015). A promisingtheory of cascade emission from charges injected into the empty magnetosphere also seeks toexplain pulsed emission based on the rigorous treatment of the configuration of the plasmain the magnetosphere. It has however not yet been developed sufficiently to predict theemission properties of specific pulsars but rather predicts some compelling general spectral

151


forms. Overall, the Annular Gap model best describes the spectral energy distribution andphase profile of the Vela pulsar reported in this work in VHE γ-rays, by the Fermi -LATCollaboration in high energy γ-rays as well as at radio frequencies.

Experimental Preparations Beyond the investigation into the second VHE pulsar, thisthesis presented work on the optimisation of the template-based analysis ImPACT and amulti-variate analysis (MVA) for monoscopic events from CT 5. These analyses reconstruct thedirection & energy and determine the particle type from imaged air showers to filter the γ-rayshowers from the huge background of hadronic showers. The optimisation was customisedto different source spectra in order to tune the analysis to be as sensitive as possible. TheStandard configuration was optimised considering a relatively shallow power law spectrumwith a −2 index and the Loose configuration similarly for a steeper spectrum with a −3 index.Both of these are intended for use with non-pulsed γ-ray signals for which the backgroundis sampled from separate region of the sky. The ExtraLoose configuration corresponds to areduced background rejection efficiency and is intended for use with periodically pulsed targetswith extremely steep spectra where the index is around −4. In this vein, a new backgroundmethod for H.E.S.S. was introduced whereby the pulsar phase is used to define the region ofinterest and off-source region thereby reducing the inherent systematic variation when usingother parts of the sky (and camera) to estimate the background rate.

Statistical Methods As part of the study of pulsed signals from pulsars, a new softwaremodule (PulsarSearch) was introduced into the H.E.S.S. Collaboration software used toanalyse the pulsed signal from pulsars. In PulsarSearch various test statistics to quantifythe measured signal were implemented as well as a new group of weighted test statistics whichboost the sensitivity to small signals. The most sensitive of which is the weighted-H teststatistic. The weighting allows any discriminator to be used to weight individual events toachieve a boost in sensitivity when compared to the best performing omnibus statistical testsused in such analyses, such as the H-test. The key was using an appropriate normalisationin order to preserve the null distribution of the harmonic components of the H-test. Thisincreased sensitivity was not required for use with the analysis of the Vela pulsar but at higherenergies where pulsed signals will be searched for in the future and statistics become muchlower such test statistics will become invaluable.

Outlook It is in this direction that IACT astronomy of pulsars is moving; higher energy(> 100GeV) detections in order to shed light on the γ-ray emission from these extreme objects.As is apparent from the discussion in Chapter 5, Fermi -LAT has been used to measurethe peak and subsequent tail of the emission from pulsars in the MeV-GeV energy domainwhich is most likely a result of (synchro) curvature radiation. In the future IACTs can be assuccessful in measuring the transition phase between the curvature radiation and possibleinverse Compton scattering or synchrotron self-Compton scattering components that mayreveal the true nature of the configuration of the pulsar environment. This IACT future isalmost entirely expected to come from the Cherenkov Telescope Array (CTA) which willconsist of a hybrid system of large, medium, and small sized telescopes to extend the sensitivityof the instrument to both lower and higher energies compared to the current generation ofIACT arrays. H.E.S.S. Phase II is extremely useful to this next generation since it is thefirst hybrid IACT array7 to be operated. The boost in sensitivity with CTA will directly

7An array of telescopes with more than one type and/or design.

152


translate into the detection of more VHE pulsars. Pulsars such as the Crab pulsar or theCrab-like pulsar PSR J0540−6919 (Seward, Harnden, and Helfand 1984) - located in theLarge Magellanic Cloud (LMC) and recently detected in the γ-ray regime by the Fermi -LATCollaboration (Marting and Guillemot 2014) - are excellent candidates for such studies sinceboth are bright and young. An array in both the Northern and Southern Hemisphereswill allow almost complete sky coverage. McCann 2015 perform a stacking analysis on 115Fermi -LAT γ-ray pulsars in the Second Fermi -LAT Catalog of Gamma-ray Pulsars (Abdoet al. 2013) with no significant excess detected above 50GeV and placed a limit on the flux of7% and 30% of the Crab pulsar flux in the energy ranges (56, 100)GeV and (100, 177)GeVrespectively. This does not bode well for the expansion of the VHE pulsar catalogue whichcurrently stands at two: Crab and Vela, however, no inverse Compton scattering componentwas considered during that study. This level of curvature radiation emission is however wellabove that of the sensitivity of CTA (CTA Consortium 2015) so the future of VHE pulsarastronomy remains fascinating. Further to IACT arrays, water Cerenkov arrays such as thenewly commissioned High Altitude Water Cerenkov (HAWC) observatory with their verylarge fields of view will also be able to contribute to the search (Alvarez Ochoa et al. 2015).

Only efforts to expand the catalogue of VHE pulsars combined with multi-wavelength studieswill yield an understanding of these objects since many variables are at play: obliquity, viewingangle, surface magnetic field strength, surface temperature, and possibly more. Fermi -LATachieved major success with the compilation of two HE γ-ray pulsar catalogues (Abdo et al.2010a; Abdo et al. 2013), this achievement convolved with the understanding of a far lowerflux limiting the number of detectable sources in the VHE regime should be a key sciencegoal with the next generation of VHE γ-ray detectors such as CTA and HAWC. Furthermore,such studies may provide more insight into the make up of pulsars in order to understand thebehaviour of matter at extremely high densities equivalent to that in the very early universewhich current observations have not yet managed.

153

Appendices

155

A. Landau Quantisation

Charged particles with a non-zero perpendicular velocity v⊥ to a magnetic field - such as thoseoutside a compact star - must be Landau quantised. A parameter b to determine the extentof relativistic effects on the charged particles and, therefore, the strength of the magnetic fieldcan be defined:

b =hωc

mec2=

B12

44.14(A.1)

where B12 = B1012 G

. For an extremely strong magnetic field b > 1 and certain quantumelectro-dynamical effects cause the vacuum to become birefringent. This affects the emittedelectromagnetic spectrum in the X-ray region of the spectrum (Ho and Lai 2003). Therelativistic Landau energy levels are given by:

EN = mec2(√

1 + 2bN − 1)

N ∈ N0. (A.2)

Considering b for ions of mass mi = Amamu (mamu is the atomic mass unit (Nakamura andParticle Data Group 2010)) and charge q = Ze in the magnetic field shows that it is smallbi = 0.68× 10−8 Z

A2B12 and relativistic effects can be safely ignored for ions. For electrons,however, with their smaller mass be = 0.0272B12 and relativistic effects are important.Electrons moving between Landau energy levels have been observed in the electro-magneticspectrum of compact star magnetospheres (Truemper et al. 1978), ion transitions remain tooweak to be detectable. With the fulfilment of the smallness condition, hωc kBT , mostelectrons are in the ground Landau level and the magnetic field is strongly quantising. This

occurs at a number density below nB ≡(π2

√2a3m

)−1(am =

(hceB

) 12 is the magnetic length for

characteristic electron transverse movement) and a mass density

ρB < 7× 103A

ZB

3212 g cm−3. (A.3)

As shown in Figure 1.1 on Page 10, this holds for the atmosphere and ocean but even at thehighest values of the magnetic field B ∼ 1018G quantisation does not occur in the core sinceρ ρB. This means the core equation of state is not affected by the magnetic field and theBohr-van Leeuwen theorem applies1.

1The theorem states that the thermal average of magnetisation is always zero

157

B. Analysis Configuration Optimisation

As briefly described in Section 3.3, the multi-variate analysis (MVA) and image-wise pixel fitfor Cerenkov telescopes (ImPACT ) reconstruction chains were both optimised for severalsource types. The cut values of the ζ, θ2, and Hillas image size (hereafter simply size)parameters were optimised for different source energy spectra. Monte Carlo simulated γ-rayshowers are generated for a source with a power law spectrum of index α = −2.0

d3N

dEdAdt= I0

(E

E0

)α

TeV−1cm−2s−1. (B.1)

This was used to define the Standard cut configuration for the MVA chain1 and was re-weightedto simulate sources with steeper power law spectra α: −3.0 and −4.0 for the Loose and ExtraLoose cut configurations respectively for both chains. As the observation conditions affect thesensitivity of the instrument (see Section 3.1), one set of conditions was chosen on which tooptimise. The γ-ray showers were simulated at zenith angle = 20 ◦, azimuth angle = 180 ◦,and offset angle = 0.5 ◦ and the off-source data used for the background estimation was limitedto the ranges:

• 15 ◦ ≤ zenith angle ≤ 25 ◦.

• 135 ◦ ≤ azimuth angle ≤ 215 ◦.

For each analysis extremely loose cuts were initially placed on the optimised parameters:

• ζ, no cut

• θ2 ≤ 0.5 deg2

• Size ≥ 30 photo-electrons (p.e.)

allowing almost the full parameter space to be scanned.

B.1. Standard

The Standard cuts are intended for use with most γ-ray sources and are aimed at being reliableand robust to variations in the data such as systematic changes based on the atmosphereor rate of night-sky background (NSB). The procedure described in Section 3.3 was used todetermine the cut configuration where the highest Q−factor was used as a guide to determinethe best analysis cuts for the MVA reconstruction described in Section 3.2.2. The StandardImPACT configuration had already been defined before this work.The Size was varied in steps of ten from 50 p.e. to 150 p.e. The maximum Q−factor for

each of these steps varied only slightly - changing by < 5% across the entire range - and wasachieved at ζ and θ2 values that also only varied by a few percent. The main aim of this

1The Standard ImPACT cut configuration was already defined before this work.

159

Appendix B. Analysis Configuration Optimisation

configuration is stability to variations in data and therefore for wide applicability the sizecut was chosen to be 60 p.e.. The maximum Q−factor for all configurations is 12.1 whichoccurs for a size cut of 130 p.e. but this only dropped to 11.7 for the cut of 60 p.e. chosen.The reason that the maximum Q−factor and the cuts on ζ and θ2 at which this is achieveddid not vary is because the size cut has little effect compared to the ζ cut which removes mostlow size events as they mainly occupy the range ζ < 0.9. The optimum cuts found are ζ ≥ 0.9& θ2 ≤ 0.016 deg2. Figure 3.5 shows the strong correlation between images of a low size anda ζ value around 0.5 meaning a cut on ζ at 0.9 eliminates the power of the low size cut. As itis not necessary to increase the size cut and the Q−factor does not vary significantly it isdesirable to have a looser size cut to retain low energy events where possible. The optimisationsurface - Q−factor versus the cuts on ζ and θ2 for a given size cut - showed an unambiguouspeak at the stated optimal cut values. The instrument response function (IRF) was calculatedfor this cut configuration and tested using quality selected H.E.S.S. data on the Crab Nebulaand PKS 2155− 304, two bright γ-ray sources. The Crab Nebula is in a region of high NSBand only visible at relatively high zenith angles > 45 ◦; the power-law spectral index has beenmeasured to be −2.6 (Aharonian et al. 2006). PKS 2155 − 304 is in a region of relativelylow NSB and has a steeper spectrum with power-law index −3.5 (Aharonian et al. 2009b).They are both well established sources whose data are used in the H.E.S.S. Collaboration tocompare analysis chains. The MVA Standard cut configuration performed as expected withdata on these sources and therefore the cut configuration is validated.

B.2. Loose

The Loose cuts are intended to be used for sources that roughly follow a power law spectrumwith an index of α = −3.0. This covers an important range of sources which are key tothe science possible with CT 5 and is aimed to be sensitive to an energy around 100GeV.The calculation of the Q−factor was done using the weighting described in Section 3.3.3.The MLPs used for ImPACT and MVA were trained slightly differently: MVA used MLPstrained on unweighted Monte Carlo simulated γ-rays (power-law index −2) whereas ImPACTused MLPs trained on Monte Carlo γ-ray with a power-law spectrum of index −4. This hasthe effect of altering the ζ distribution slightly. For similar reasons as with the Standardconfiguration, the same size cut was chosen for the Loose configuration. The surface ofQ−factor against ζ and θ2 for the chosen size cut of 60 p.e. using ImPACT is shown inFigure B.1; it does not vary significantly for the MVA reconstruction. The highest Q−factoris obtained at a very similar configuration as Standard cuts, therefore to relax the cuts relativeto Standard the peak visible in Figure B.1 at a lower ζ and higher θ2 value was selected.This results in a cut configuration ζ ≥ 0.78 & θ2 ≤ 0.021 deg2 for MVA and ζ ≥ 0.86 &θ2 ≤ 0.024 deg2 for ImPACT with a 8.5% and 8% γ-ray efficiency (number of γ-ray eventsafter analysis cuts divided by the number before) respectively.

Validation was once again performed using data on the Crab Nebula and PKS 2155− 304where results agreed within statistical errors to the published spectra thus validating theLoose cut configuration for both ImPACT and MVA.

B.3. Extra Loose

The optimisation of the Extra Loose configuration for ImPACT is described in Section 3.3.4and resulted in similar cuts for MVA. A different cut on ζ was obtained for the MVA chain

160

B.3. Extra Loose

Figure B.1.: Q−factor for ζ and θ2 cut values in the scanned range for a size cut of60 photo-electrons. Red indicates high Q−factor values and blue low Q−factorvalues.

161

Appendix B. Analysis Configuration Optimisation

because the neural networks used with ImPACT were trained using Monte Carlo simulatedγ-ray data weighted to a power-law energy spectrum of index −4.0 rather than unweightedas used with MVA (i.e. a power-law energy spectrum of index −2.0). The most significantchange is simply to the position of the middle peak in the ζ distribution (see Figure 3.5) whichis moved to lower values with the MLPs trained for use with ImPACT. The optimisationsurface is much the same - other than this slight shift in ζ - and is shown in Figure 3.4 forImPACT. Particularly for MVA, the maximum Q−factor corresponded to a cut on ζ locatedat the maximum of the middle peak in the ζ distribution. This choice of cut is prone tochange due to slight variations in the distribution of ζ for different data sets and thereforeas explained in Section 3.3.4 the ζ cut was restricted for both reconstructions to be withoutthe points of half maximum of the middle peak. The optimum value for θ2 was clear as theQ−factor simply increases to a clear maximum and then falls. The MVA cuts are thereforedefined as: ζ ≥ 0.55, θ2 ≤ 0.055 deg2, and size ≥ 35 p.e.. The ImPACT cuts are: ζ ≥ 0.44,θ2 ≤ 0.045 deg2, and size ≥ 40 p.e.. The difference in size cut is discussed in Section 3.3.4.The validation of the configuration was performed as for the Loose and Standard config-

urations. Here the spectrum showed distinct variation between adjacent spectral bins, thisis however expected since the pulsar phase is not being used for background discriminationleading to significant background contamination in the signal. The best fit to the spectralpoints agreed within statistical error with the published spectra. The Extra Loose configura-tion should not be used for sources where no pulsed-phase (or something similar) is used forbackground rejection.

162

C. Pulsed Statistics

The PulsarSearch software module is used to calculate the value of various pulsed teststatistics in order to determine if there is a pulse in a phase profile. Tables C.2 & C.1 show allof the test statistics that are implemented along with their definitions, weighted normalisationfactors and distributions. The following abbreviations are used in this appendix:

Cki ≡ cos (2πkφi) (C.1)

Ski ≡ sin (2πkφi) (C.2)

which are together referred to as the Gki . Furthermore, αk and βk are together referred to as

the ψk. Weighted statistics are indicated with a subscript w.The normalisations given for the Sinem and Cosinem test statistics contain Sk

i and Cki

which are written unchanged but should be modified with φi → φi − μ as with the definitionof the two test statistics.The H test statistic null distribution was originally determined from Monte Carlo simula-

tions (de Jager, Raubenheimer, and Swanepoel 1989) and prescribed to the expression givenin Table C.2, however later it was determined analytically, see Section 3.5 for the expression orKerr 2011 for the derivation. The Kuiper test statistic operates on the phases φi ordered inascending order and the χ2

b+1 & Gb+1 test statistics require the phases to be binned into b bins,the Kuiper test probability density function was obtained from Jetsu and Pelt 1996. TheKolmogorov test is also interfaced but the ROOT TMath library (Brun and Rademakers1997) implementation is used.

C.1. Clarification of the Implementation

Each of the above mentioned test statistics that were newly implemented in PulsarSearch

were checked using toy Monte Carlo simulations of uniformly distributed phases to confirm thenull hypothesis distribution reported in Tables C.1 & C.2. The toy Monte Carlo simulationconsisted of 10, 000 iterations of 5, 000 phases.

Alpha & Beta The ψ test statistics should be Gaussian distributed with zero mean due tothe central limit theorem (CLT; Billingsley 1979) as they are individually a sum of independentand identically distributed (i.i.d.) random variables (rvs). The distributions of these two teststatistics are shown in the top row of Figure C.1 and are shown to well fit their expecteddistributions being well fit by the Gaussian distributions displayed. The width of theGaussian can be normalised to unity using the square root of the factor preceding the sumof the Z2

m test,√2N .

Cosine & Sine The Cosine and Sine test statistics should follow a standard normal distribu-tion as, according to the CLT, they are the mean over k of the ψk test statistics respectively.

163

Appendix C. Pulsed Statistics

Test

Statistic

Defi

nitio

nNorm

alisa

tionFacto

rAsymptotic

NullDistrib

utio

n

αk

1N ∑Ni=

1C

ki

√N ∑

Ni(C

ki)2− ∑

NiC

ki

N ∑Ni(w

i Cki)2− ∑

Niw

i Cki

Gaussian

βk

1N ∑Ni=

1Ski

√N ∑

Ni(S

ki)2− ∑

NiSki

N ∑Ni(w

i Ski)2− ∑

Niw

i Ski

Gaussian

Cosin

em

√2

mN ∑

mk=1 ∑

Ni=1cos(2

πk(φ

i −μ))

Use

αwk

Standard

norm

al

Sinem

√2

mN ∑

mk=1 ∑

Ni=1sin

(2πk(φ

i −μ))

Use

βwk

Standard

norm

al

Z2m

2N ∑mk=1α2k+

β2k

Use

ψwk

χ2with

2m

degrees

offreed

om

U2

2N ∑∞k=1

12πk (α

2k+

β2k )

Use

ψwk

∑∞k=1 (−

1)k−

12×2k

2π2exp −

2k2π2φ

HM

max(1

≤m

≤M

)[Z2m−

4(m−

1)]

Use

Z2wm

exp −

0.4H

Table

C.1.:Pulsed

harm

onic

statisticsin

PulsarSearchwith

their

asymptotic

distrib

ution

intheinfinite

statisticlim

it.For

references

onthetest

statistics

seeSectio

n3.5.

164

C.1. Clarification of the Implementation

TestStatistic

Defi

nition

AsymptoticNullDistribution

χ2 b−

1

∑ b j(X

j−

N b)2

N b

χ2withb−

1degrees

offreedom

Gb−

12∑ b−

number

empty

bins

j[X

j∗loge(X

j/N b)]

approxim

atelyχ2withb−

1degrees

offreedom

(Sokal

and

Rohlf1969)

Protheroe

2n(n

−1)

∑ n−1

i=1

∑ n j=i+

1[(0.5−||p

i−

pj|−

0.5|+1/n)−

1]]

Determined

from

Monte

Carlo(P

rotheroe1985)

Kuiper

VN

=max(φ)[FN(φ)−

F(φ)]+max(φ)[F(φ)−

FN(φ)]

where

F(φ)=

φ&

FN

=

⎧ ⎪ ⎨ ⎪ ⎩0,ifφi<

φ1

i N,ifφ1≤

φi&

1≤

i<=

N−

1

1,ifφi≥

φN

P(N

1 2VN

≥z)=

∞ ∑ i=1

2( 4i2

z2−1) ex

p−2

k2z2

−8z 3

n−

1 2

∞ ∑ i=1

k2( 4k

2z2−3) ex

p−2

k2z2

+O( N−

1)

Table

C.2.:Further

pulsed

statisticsin

PulsarSearch

165


Figure C.1.: The null distributions for the following harmonic test statistics starting from thetop left: Alpha (k = 1), Beta (k = 1), Cosine (m = 1), Sine (m = 1), and Z2

m

for m = 1, 2, 10,& 20. The goodnesses-of-fit to the asymptotic distributions areshown for each test statistic in the inset statistic boxes.

166

C.2. Weighted Statistics

The distributions of these test statistics are shown in the second row of Figure C.1, bothshown with well fitting standard normal distributions.

Z2m & U2 The Z2

m test statistic should be χ2 distributed with 2m degrees of freedom (Bendatand Piersol 2011) and the U2 test statistic should be distributed (Watson 1961) according tothe function shown in Table C.1 but not shown in Figure C.1. The χ2

2m distribution arises asthe Z2

m test statistic is the sum of two i.i.d. rvs which are distributed according to a Gaussian

distribution with zero mean and unit variance (Abramowitz and Stegun 1964) due to thenormalisation factor 2N applied to the Z2

m test statistic. The correct χ22m distributions are

shown to well fit the toy Monte Carlo distributions shown in the last two rows of Figure C.1.


With attempts to boost sensitivity, weighted statistics are considered for each of the mainharmonic test statistics considered in Table C.1.

• Alpha statistic

• Beta statistic

• Sine statistic

• Cosine statistic

• Rayleigh statistic (simply 12Z

2m=1)

• Z2m statistic

• H statistic.

The normalisation of a weighted statistic W that corresponds to the unweighted statistic S,which is considered over bins of the dependent variable with content θi which is Poisson

distributed, can be determined with the following method. The variance of the weightedstatistic σ2

W can be written using the chain rule:

σ2W =

(∂W (θ)

∂θ1

)2

δθ21 +

(∂W (θ)

∂θ2

)2

δθ22 +

(∂W (θ)

∂θ3

)2

δθ23 + . . . (C.3)

where each term is for a bin for which the variance of the content is θi due to the Poisson

distribution. This can be written in a more compact form

σ2W =

∑θ′

(∂W (θ)

∂θ′

)2

δθ′2. (C.4)

By requiring that the weighted and unweighted test statistics have the same variance -as calculated with Equation C.4 - a normalisation factor for W can be determined. This ishenceforth known as the Lohse method.

167


C.2.1. Normalising the Harmonic test statistics

The Lohse method is now applied to the ψk test statistics with the intention to use them forweighted versions of the other harmonic test statistics of which they are part of the definition.Weighting the ψ test statistics alters their distribution therefore applying the Lohse methodensures at least the second central moment of the weighted distribution matches that of theunweighted statistic. Starting with the consideration of bins in rotational phase φ (hereaftersimply phase) indexed with q and bins in event weight w indexed p each with contents n, theψwk can be written:

αwk =Iα∑

p,q np,q

∑p,q

wpCq,knp,q (C.5)

βwk =Iα∑

p,q np,q

∑p,q

wpSq,knp,q. (C.6)

The normalisation factors Iψ (for clarity henceforth simply referred to as I) in Equations C.5& C.6 are considered to be a function of both m and N which means their contribution tothe variance of the test statistic must be taken into account in the calculation. The relativevariance of the ψwk according to Equation C.4 is given by

(C.7)σ2ψwk

ψ2wk

=∑p′,q′

(∂ loge ψwk

∂np′,q′

)2

np′,q′ .

where∂ loge ψwk

∂np′,q′=

1

I

∂I

∂N+

1

ψ′∂ψ′

∂np′,q′− 1

N(C.8)

using the abbreviation ψ′ =∑

p,q wpGq,knp,q and the chain rule: ∂ loge ψwk

∂np′,q′= ∂ loge ψwk

∂N since

∂N∂np′,q′

= 1 from N =∑

p,q np,q. Inserting this into Equation C.7 and expanding the bracket

gives

(C.9)σ2ψwk

ψ2mk

=∑p′,q′

⎡⎣⎛⎝( I

I

)2

+

(ψ′

ψ′

)2

+1

N2+ 2

(+Iψ′

Iψ′ −I

NI− ψ′

Nψ′

)⎞⎠np′,q′

⎤⎦

where a dot above a quantity indicates its differential with respect to np′,q′ (in the case of Ithis is identical to the differential with respect to N). The differential generator function forψwk is:

∂ψwk

∂np′,q′= wp′Gq′,k. (C.10)

Considering the terms in Equation C.9 separately:

Term 1: ∑p′,q′

(I

I

)2

np′,q′ = N

(I

I

)2

(C.11)

I and I can be moved out of the sum as they are functions of N which is already summedover np′,q′ .

168



(ψ′

ψ′

)2

np′,q′ =1

ψ′2∑p′,q′

ψ′2np′,q′ (C.12)

ψ′ can be moved out of the sum as it is already summed over np,q. The remaining sumcan now be written in terms of a sum over events (indexed with i) rather than bins as

Ψ ≡∑p′,q′ ψ′2np′,q′ =

∑Ni w2

iG2k where Ψ is used for brevity.


np′,q′

N2=

1

N. (C.13)

This is clear from the form of N =∑

p,q np,q.


2Iψ′

Iψ′ np′,q′ =2I

Iψ′∑p′,q′

ψ′np′,q′ = 2I

I. (C.14)

This is because∑

p′,q′ ψ′np′,q′ = ψ′ which is clear from Equation C.10 and the definition of ψ′.


− 2

NIInp′,q′ = −2

I

I, (C.15)

for similar reasons as those previously stated. This is cancelled by Term 4.


− 2

Nψ′ ψ′np′,q′ = − 2

N, (C.16)

again for similar reasons as those previously stated. This term can be combined with Term 3.This all leads to an expression for the relative variance:

σ2ψwk

ψ2wk

= N

(I

I

)2

+Ψ

ψ′2 +NI

I− 1

N. (C.17)

Using ψ′ ≡ NI Zmw which, unlike ψwk, is neither a function of I nor N (although it contains

a sum up to N it contains no terms with N) allows the variance to be defined:

σ2W =

ψ′2

NI2 +

I2

N2Ψ− I2ψ′2

N3. (C.18)

This equation can be made applicable to the unweighted statistic simply by considering thecase I = 1 and using the unweighted versions of ψ′ and Ψ:

σ2 =Ψ

N2− ψ′2

N3. (C.19)

Equating this with the weighted version is used to determine I (now using subscripts w for

169


the weighted versions of expressions):

Ψ

N2− ψ′2

N3=

ψ′2w

NI2 +

I2

N2Ψw − I2ψ′2

w

N3. (C.20)

This is a non-linear partial differential equation in I and N which is insoluble. The approx-imation is therefore made that I is small and the first term on the right hand side can beneglected. Given this approximation the equation can be rearranged to give the result for thenormalisation factor for ψwk:

I =

√NΨ− ψ′2

NΨw − ψ′2w

(C.21)

which is simple the square root of the ratio of the unweighted test statistic variance to theweighted test statistic variance. This can be used in the expression for ψwk:

ψwk =I

N

N∑i

wiGik (C.22)

to enable the weighted Z2m and weighted H tests to be defined as:

Z2wm ≡ 2N

m∑k=1

(α2wk + β2

wk

)(C.23)

andHwM ≡ max(1 ≤ m ≤ M)[Z2

wm − 4(m− 1)]. (C.24)

As ψwk is distributed according to a Gaussian with zero mean and unchanged variancecompared to ψk then Z2

wm and HwM retain their asymptotic null distributions as given inTable C.1.

The assumption that I is small and indeed that I is not a function of N is clear whenconsidering the infinite statistics limit i.e. N → ∞. Looking at the numerator of Equation C.21:

NΨ− ψ′2 = N

N∑i

G2i,k −

(N∑i

Gi,k

)2

(C.25)

using the definitions of Ψ and ψ′ shows that a factor of N is present. In the asymptotic limithowever

N∑i

G2i,k → N

⟨G2⟩

(C.26)

andN∑i

Gi,k → N 〈G〉 (C.27)

based on the law of large numbers (Bernoulli and Sylla 2006) whereby, in the asymptoticlimit, the sample mean tends to the population mean. This also applies to the denominator

170


and enables the factors of N to be cancelled, leaving:

I∞ =

√〈G2〉 − 〈G〉〈G2

w〉 − 〈Gw〉. (C.28)

This is clear as it is the definition of the variance of the test statistics. See Section 3.5.1 foran analysis of the low statistic limit.

C.2.2. Weighted test statistic Null Distributions

To ensure the correct implementation of these weighted test statistics, which should haveconsistent null distributions as the unweighted versions shown in Figure C.1, the null distribu-tions for the corresponding weighted test statistics were produced using the same toy MonteCarlo and are shown in Figure C.2. The Sine and Cosine test statistics have been left out andinstead the H and weighted H test statistics are shown. It is clear that all of the expectedasymptotic distributions are obtained thus clarifying the implementations and normalisationfactors used for the weighted test statistics.

171


Figure C.2.: The null distributions for the following harmonic test statistics starting fromthe top left: weighted Alpha (k = 1), weighted Beta (k = 1), weighted Z2

m form = 1, 2, 10,& 20, unweighted H (m = 20), and weighted H (m = 20). Thegoodnesses-of-fit to the asymptotic distributions are shown for each test statisticin the inset statistic boxes.

172

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List of Figures

0.1. All Sky Map in High and Very High Energy γ-rays . . . . . . . . . . . . . . . 2

1.1. Ginzburg Neutron Star Cross Section . . . . . . . . . . . . . . . . . . . . . . 101.2. Compact Star Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3. Temperature Versus Age for a Compact Star . . . . . . . . . . . . . . . . . . 211.4. Magnetic Attenuation Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 241.5. Polar Cap Model Spectral Energy Cut-off . . . . . . . . . . . . . . . . . . . . 251.6. Charge Distribution of an Aligned Rotator . . . . . . . . . . . . . . . . . . . . 301.7. Trapping Regions Above the Poles . . . . . . . . . . . . . . . . . . . . . . . . 311.8. Pulsar Magnetospheric Charge Distribution . . . . . . . . . . . . . . . . . . . 321.9. Possible VHE Emission Regions in a Pulsar’s Magnetosphere . . . . . . . . . 361.10. P − P Diagram for Observed Pulsars . . . . . . . . . . . . . . . . . . . . . . . 381.11. Polar Cap Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.12. Phase Profile Predicted by the Polar Cap Model . . . . . . . . . . . . . . . . 421.13. Width of the Slot Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.14. Two Pole Caustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461.15. Slot Gap Phase Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471.16. Emission Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481.17. Outer Gap Model Phase Profile . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.1. Diagram of CT 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.2. Time Separation Between CT 5 Event Triggers . . . . . . . . . . . . . . . . . 572.3. CT 5 Trigger Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.4. Photo-multiplier Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.5. Night Sky Background Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.6. CT 5 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.7. CT 5 Pixel signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.1. Simulated Air Showers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.2. CT 5 Images for Different Particle Types . . . . . . . . . . . . . . . . . . . . 663.3. Hillas Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.4. Q−factor Optimisation Surface . . . . . . . . . . . . . . . . . . . . . . . . . . 753.5. ζ (zeta) Distribution for γ-rays and Background . . . . . . . . . . . . . . . . 763.6. Performance Plots Comparing the ImPACT Cut Configurations . . . . . . . . 783.7. Comparison of the Instrument Response Functions for the MVA and ImPACT

Reconstructions with Extra Loose Cuts . . . . . . . . . . . . . . . . . . . . . . 793.8. Affect of Data Set Size on Weighted H Test Statistic Null Distribution . . . . 933.9. Weighted-H and H Test Statistic Power Versus Pulsed Proportion . . . . . . . 943.10. Weighted-H and H Test Statistic Power Versus Duty Cycle . . . . . . . . . . 953.11. Detection Significance Boost with the Weighted-H test statistic . . . . . . . . 963.12. Phase Profile of the H.E.S.S. Optical Crab Pulsar Data . . . . . . . . . . . . 98

3.13. H.E.S.S. Optical Crab Pulsar Data H test statistic Evolution . . . . . . . . . 993.14. Camera Image Centre of Gravity for a Single CT 5 Run . . . . . . . . . . . . 101

4.1. Image of the Vela Region in Radio, X-ray, and VHE γ-rays . . . . . . . . . . 1044.2. Phase Profile Measured with the Fermi -LAT . . . . . . . . . . . . . . . . . . 1044.3. Waterfall Diagram from Fermi -LAT Data . . . . . . . . . . . . . . . . . . . . 1064.4. Li & Ma and H Test Statistic Significance Versus Number of Background Events1114.5. Smoothed Excess Sky Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.6. Theta Squared and Significance Distributions . . . . . . . . . . . . . . . . . . 1144.7. Phase Profile of the Vela Pulsar from H.E.S.S. Data . . . . . . . . . . . . . . 1154.8. Reconstructed Energy Distribution . . . . . . . . . . . . . . . . . . . . . . . . 1214.9. Vela Pulsar Phase-averaged Spectrum . . . . . . . . . . . . . . . . . . . . . . 1244.10. Likelihood Contour Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.1. Multi-wavelength Spectral Energy Distribution . . . . . . . . . . . . . . . . . 1355.2. Pulsar Spectral Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.3. Spectral Energy Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.4. Luminosity Versus Spin-down Power . . . . . . . . . . . . . . . . . . . . . . . 1425.5. Phase Profile of the Vela Pulsar at Different Energies . . . . . . . . . . . . . . 1455.6. Evolution of Peak Width with Energy . . . . . . . . . . . . . . . . . . . . . . 147

B.1. Q−factor Surface Against Optimised Parameters for the Loose Cut Configuration161

C.1. Null Distributions for Harmonic Test Statistics . . . . . . . . . . . . . . . . . 166C.2. Null Distributions for Weighted Harmonic Test Statistics . . . . . . . . . . . 172

List of Tables

4.1. Pulsar Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.2. Timing Solution Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.3. Year-wise Data Set Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.4. Basic Observation and Analysis Statistics . . . . . . . . . . . . . . . . . . . . 1084.5. Phase Resolved Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.6. Energy Resolved Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.7. Comparison of Peak Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.8. Energy Resolved Best Fit Peak Parameters . . . . . . . . . . . . . . . . . . . 1184.9. Phase Background Variation with Energy . . . . . . . . . . . . . . . . . . . . 1194.10. Systematic Variation of the Spectral Power Law Fit . . . . . . . . . . . . . . 1284.11. Systematic Variation of the Spectrum Based on Binning and Energy Threshold 1294.12. Spectral Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.1. Comparison of Lorentz Invariance Violation Energy Scales . . . . . . . . . . 149

C.1. Harmonic Test Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164C.2. Further Test Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Acronyms

CT 1-4 Cerenkov telescopes 1− 4

CT 5 Cerenkov telescope 5ep Electron-positron

2SC Two-flavour superconducting3FGL Third Fermi Large Area Telescope source catalogue

ACU Antenna control unitADC Analogue-to-digital converterAIC Akaike information criterionAICc Corrected Akaike information criterionamu Atomic mass unitAXP Anomalous X-ray pulsar

BATSE Burst And Transient Source ExperimentBCE Before common eraBCS Bardeen-Cooper-Schrieffer

BG Background

CCD Charged coupled deviceCFL Color flavour locked

CGRO Compton Gamma Ray ObservatoryCLT Central limit theoremCMB Cosmic Microwave BackgroundCoG Centre of gravityCPU Central processing unitCTA Cherenkov Telescope Array

DAQ Data acquisition systemDST Data summary tapedurca Direct Urca

EGRET Energetic Gamma Ray Experiment TelescopeEOS Equation of state

FFS Force-free surfaceFOV Field of view

GPS Global Positioning SystemGRB γ-ray burst

H.E.S.S. High Energy Stereoscopic SystemHAP H.E.S.S. analysis program

HAWC High Altitude Water Cherenkov (observatory)HD HeidelbergHE High energyHG High gainHST Hubble Space TelescopeHZE High atomic number (Z) and energy

i.i.d. Independent and identically distributed

IACT Imaging atmospheric Cerenkov telescope

ImPACT Image pixel-wise fit for atmospheric Cerenkov telescopesIRF Instrument response function

JD Julian day

K-S Kolmogorov-Smirnov

KDE Kernel density estimator

LG Low gainLIV Lorentz invariance violation

LMC Large Magellanic Cloud

MAGIC Major Atmospheric Gamma-ray Imaging Cerenkov TelescopesMHD MagnetohydrodynamicMJD Modified Julian dayMLP Multi-layer perceptronmurca Modified UrcaMVA Multivariate analysis

nRICS Non-resonant inverse Compton scatteringNSB Night sky background

p.e. Photo-electronpdf Probability density function

PET Pulsar emission timePFF Pair-forming frontPMT Photo-multiplier tubePSF Point spread function

QCD Quantum chromodynamicsQG Quantum gravity

RICS Resonant inverse Compton scatteringROI Region of interestRSS Residual sum of squaresrv Random variable

SGR Soft γ repeaterSIMBAD Set of Identifications, Measurements, and Bibliography for Astronomical Data

SNR Supernova remnantSSB Solar system barycentreSSC Synchrotron self-Compton

TCB Barycentric Coordinate TimeTMVA Toolkit for Multivariate AnalysisTOA Time of arrivalTOV Tolman-Oppenheimer-Volkoff

TTL Transistor-transistor logic

UTC Coordinated Universal Time

VERITAS Very Energetic Radiation Imaging Telescope Array SystemVHE Very high energy

Acknowledgements

I thank Professor Thomas Lohse and Doctor Ullrich Schwanke for giving me the opportunityof pursuing my research and supporting me academically and financially. I thank DoctorMatthias Fußling for providing support from the very beginning of my work in the physicsof pulsars, analysis techniques, and with array control. Doctor Arnim Balzer for his vastknowledge of coding and the H.E.S.S. experiment from which I profited greatly. Doctor GerritSpengler for many thought provoking conversations on things that he believed were incorrector insufficient. Doctor Markus Holler for ensuring that I neither tripped up on my γ-rayanalysis nor on the ski slope and for his fine translation skills with the abstract and scientificsummary of this work. Doctor Robert Parsons for ensuring that I was more knowledgeableafter every time we spoke and for providing an impressive reconstruction. Doctor VincentMarandon for excellent data and unending valuable advice and help. Doctor Francois Brunfor Namibia. Professor Mathieu de Naurois for showing me how high the bar is set. DoctorMichael Meyer for his understanding and amiability with my endless questions and ourextended discussions. Thomas Murach for ensuring that I thought about what I said. I thankthe H.E.S.S. Collaboration in which it was my pleasure and honour to work. I thank the restof the Humboldt-Universitat experimental physics group for the work environment. VeronikaSchneider for being affable and helpful during my entire time at the Humboldt-Universitat.I thank also the Clarkson family - Andrew, Eva, Fabian, and Charlotte - for tolerating

me during my years of research and, in particular, my partner Joanna Clarkson for herunderstanding and fortitude.

I thank my mother and father, Alison and Bohdan Gajdus, for their limitless and valuablesupport. Finally, I thank my brother, Captain John Gajdus and my grandmother, SarahBriggs.This research made use of the Astrophysics Data System (NASA 2014) and the SIMBAD

database, operated at CDS, Strasbourg, France (Wenger et al. 2000).

Kenntniserklarung

Ich erklare, dass ich uber Kenntnis der dem angestrebten Verfahren zugrunde liegendenPromotionsordnung verfuge.

Berlin, den 14.10.2015 Michael David Gajdus

Selbststandigkeitserklarung

Ich erklare, dass ich die vorliegende Arbeit selbstandig und nur unter Verwendung derangegebenen Literatur und Hilfsmittel angefertigt habe.


Declaration

I declare that I prepared and completed this work myself and only with reference to theauthor(s) and works cited included the work of others.


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