Search for Pulsed Very High Energy Gamma Ray Emission from the Millisecond Pulsar PSR J0437-4715 with H.E.S.S

Search for Pulsed Very High Energy Gamma Ray Emission from the Millisecond Pulsar PSR J0437-4715 with H.E.S.S.

Humboldt–Universit¨at zu Berlin Mathematisch–Naturwissenschaftliche Fakult¨at I Institut fur¨ Physik

Diplomarbeit

eingereicht von Till Eifert geboren am 15. September 1979 in Berlin

Gutachter: Prof. Dr. Thomas Lohse Prof. Dr. Hermann Kolanoski

9. Dezember 2005 Abstract

This work reports on the analysis of very high energy (VHE) gamma-rays and the search for pulsed emission from the millisecond pulsar PSR J0437-4715. This pulsar is an excellent candidate for pulsed VHE gamma-ray emission due to its close distance of ∼ 150 pc, a relatively low magnetic field of order 109 G, and a high spin-down luminosity of order 1033 erg s−1. Observations of PSR J0437-4715 were conducted in 2004 with the High Energy Stereoscopic System (H.E.S.S.). Located in Namibia and fully operational since December 2003, H.E.S.S. is currently the most sensitive system of Imaging Atmospheric Cherenkov Telescopes which collect Cherenkov light from extended air showers, created by interactions of VHE gamma-rays in the Earth’s atmosphere. This technique allows to detect cosmic gamma-rays with energies ranging from 100 GeV up to 100 TeV. By imaging the air showers onto a granulated camera, the energy and direction of incident gamma- rays can be reconstructed. The four 12 m-telescopes of the H.E.S.S. experiment provide an energy resolution better than 15% and an angular resolution below 0.1◦ for a single gamma-ray photon. In this work, methods for the search for periodicity in the arrival times of gamma-ray photons - pulsar timing analysis - and a procedure to optimise the analysis selection cuts were developed and implemented into the existing H.E.S.S. analysis software framework. The pulsar timing analysis consists of all necessary timing corrections, including two binary timing correction models, and statistical tests to calculate the significance of a possible pulsed signal within the dataset. All steps of the pulsar timing analysis were tested by extensive cross-checks, in particular with a standard timing analysis tool for pulsar radio astronomy and with a simulation of pulsed signals. The pulsar timing analysis was applied to 8.2 h of data taken on PSR J0437-4715 with the H.E.S.S. telescope system. No significant signal for pulsed emission from PSR J0437-4715 was found. From the data an upper limit at 99% C.L. on the integrated pulsed gamma-ray flux above 100 GeV of 1.0 · 10−12 cm−2 s−1 was derived. Zusammenfassung

Diese Arbeit beschreibt die Analyse hochenergetischer Gammastrahlung und die Su- che nach gepulster Strahlung vom Millisekunden-Pulsar PSR J0437-4715. Dieser Pulsar ist wegen seines geringen Abstandes von ∼ 150 pc, dem relativ niedrigen Magnetfeld in der Größenordnung von 109 G und der großen Leuchtkraft von ∼ 1033 erg s−1 ein aus- sichtsreicher Kandidat fur¨ die Suche nach gepulster hochenergetischer Gammastrahlung. Die Beobachtung wurde 2004 mit dem in Namibia stehenden High Energy Stereoscopic System (H.E.S.S.) durchgefuhrt.¨ H.E.S.S. wurde im Dezember 2003 in Betrieb genommen und ist momentan das empfindlichste System der Imaging Atmospheric Cherenkov Teles- copes. Bei der Wechselwirkung hochenergetischer Gammastrahlung in der Erdatmosphäre entstehen Luftschauer, die Cherenkov-Licht abstrahlen, welches von den Teleskopen des Systems nachgewiesen wird. Die Energie und Richtung der kosmischen Gammastrahlung kann dann durch die Abbildung der Luftschauer auf eine Kamera rekonstruiert werden. Die vier 12 m-Teleskope des H.E.S.S.-Experiments besitzen eine Energieauflösung von weniger als 15% und eine Winkelauflösung besser als 0.1◦ fur¨ ein einzelnes Gammastrahlungspho- ton. Im Rahmen dieser Arbeit wurden die Methoden zur Suche nach zeitlichen Periodi- zitäten in den Ankunftszeiten der Gammastrahlungsphotonen - eine Pulsar-Zeitanalyse - und eine Prozedur zur Optimierung der Standard-Analyse entworfen und in die be- stehende H.E.S.S.-Analyse-Software eingebaut. Die Pulsar-Zeitanalyse besteht aus den notwendigen Zeitkorrekturen mit zwei Binärmodellen und statistischen Tests, die die Wahrscheinlichkeit möglicher gepulster Signale in den Daten berechnen. Alle Schritte der Pulsar-Zeitanalyse wurden extensiv getestet. Dafur¨ wurde ein Vergleich mit dem Standard- Zeitanalyse-Programm fur¨ Pulsar-Radioastronomie durchgefuhrt.¨ Außerdem wurde die Pulsar-Zeitanalyse mit einer Simulation fur¨ gepulste Signale getestet. Mit der Pulsar- Zeitanalyse wurden die 8.2 h H.E.S.S. Daten von PSR J0437-4715 ausgewertet. Es wurde kein signifikantes Signal gepulster Strahlung gefunden. Aus den Daten wurde eine obe- re Grenze mit einem Vertrauensbereich von 99% fur¨ den gepulsten Fluss integriert ab 100 GeV zu 1.0 · 10−12 cm−2 s−1 berechnet. Contents

Introduction 1

1 Pulsars 3 1.1 Overview ...... 3 1.1.1 Discovery ...... 4 1.1.2 Identiﬁcation with Neutron Stars ...... 5 1.1.3 Pulsar Population ...... 6 1.2 High Energy Emission Models ...... 8 1.2.1 Magnetosphere ...... 8 1.2.2 Radiation Processes ...... 9 1.2.3 Polar Cap Model ...... 12 1.2.4 Outer Gap Model ...... 13 1.3 PSR J0437-4715 ...... 14 1.3.1 Detection in Radio and X-rays ...... 14 1.3.2 Theoretical Predictions ...... 15

2 H.E.S.S. Experiment 16 2.1 Imaging Atmospheric Cherenkov Technique ...... 17 2.1.1 Air Showers ...... 17 2.1.2 Cherenkov Radiation ...... 19 2.1.3 Detection Principle ...... 19 2.2 H.E.S.S. Telescope System ...... 21 2.2.1 Site and Telescopes ...... 21 2.2.2 Optics ...... 22 2.2.3 Camera ...... 23 2.2.4 Trigger ...... 24 2.2.5 Data Acquisition ...... 25 2.2.6 Monitoring ...... 25 2.3 Monte Carlo Simulations ...... 26

3 Methods and Algorithms 28 3.1 Standard Analysis ...... 28 3.1.1 Calibration and Preselection ...... 28 3.1.2 Geometrical Reconstruction ...... 30 3.1.3 Background Suppression ...... 30 3.1.4 1D Analysis Using 7 Background Regions ...... 31 3.1.5 Energy Estimation ...... 32 3.2 Timing Analysis ...... 34 3.2.1 Time of Arrival Corrections ...... 35 3.2.2 Clock and Frequency Corrections ...... 37 3.2.3 Solar System Corrections ...... 38 3.2.4 Relative Motion ...... 41 3.2.5 Binary Corrections ...... 42 3.2.6 Timing Model Parameters ...... 48 3.2.7 Statistical Tests for Periodicity Search ...... 51 3.3 Cross-Check of Timing Analysis ...... 55 3.3.1 Optical Crab Data ...... 56 3.3.2 Comparison with Standard Radio Timing Analysis Tool ...... 57 3.3.3 Simulation of a Pulsed Signal ...... 57 3.4 Optimisation of Hillas Cuts ...... 66 3.4.1 Method ...... 66 3.4.2 Results ...... 67

4 Analysis 70 4.1 Dataset and Analysis ...... 70 4.1.1 Quality Checks ...... 71 4.2 Results ...... 72 4.2.1 Low Energy Bin ...... 74 4.2.2 Low Zenith Angle Bin ...... 74 4.3 Background Dependence on Zenith Angle ...... 75 4.4 Flux Upper Limits ...... 79 4.5 Discussion ...... 80

Summary 83

Acknowledgments 91

iii List of Figures

1.1 Pulsar illustration ...... 3 1.2 Discovery of the ﬁrst pulsar ...... 4 1.3 Pulsar population ...... 7 1.4 Pulsar Magnetosphere ...... 10 1.5 Pulsar emission regions ...... 12 1.6 Spectrum of the Vela pulsar with model predictions ...... 13 1.7 PSR J0437-4715 phasograms in radio and X-ray wavelengths ...... 14 1.8 PSR J0437-4715 ﬂux prediction from Harding ...... 15

2.1 View on H.E.S.S. site ...... 16 2.2 Cherenkov detection principle ...... 20 2.3 H.E.S.S. telescope and night sky on site ...... 21 2.4 Technical drawing of one H.E.S.S. telescope ...... 22 2.5 Technical drawing of one H.E.S.S. mirror ...... 23 2.6 H.E.S.S. mirrors and camera ...... 24

3.1 Hillas parameters in camera display ...... 29 3.2 Intersecting shower ellipses ...... 30 3.3 Background estimation ...... 32 3.4 Leapseconds in UTC ...... 37 3.5 Illustration of the Roemer delay in the Solar System...... 38 3.6 Timing corrections due to the Roemer delay in the SSB ...... 39 3.7 Timing corrections due to the Shapiro delay in the SSB ...... 40 3.8 Timing corrections due to the Einstein delay in the SSB ...... 41 3.9 Timing corrections due to relative motion of the source ...... 42 3.10 Geometry of a Binary System ...... 43 3.11 Timing corrections in a binary system ...... 47 3.12 Timing phase residuals ...... 51 3.13 2D random walk in the Rayleigh-Test ...... 52 3.14 Illustration of the Kuiper-Test ...... 54 3.15 Optical Crab lightcurve ...... 56 3.16 SSB cross-check with TEMPO ...... 58 3.17 Binary models cross-check with TEMPO ...... 59 3.18 Simulated phasograms ...... 61 3.19 Simulation with S2B 0.0...... 62 3.20 Simulation of a pulsed signal with a S2B of 0.1 ...... 63 3.21 Simulation of a pulsed signal with a S2B of 0.15 ...... 64 3.22 Simulation of a pulsed signal with a S2B of 0.2 ...... 65 3.23 Effect of pulsar frequency shift ...... 66 3.24 Effective Areas for low energy and H.E.S.S. standard selection configurations 69

4.1 Cross-check of the central trigger GPS clock ...... 72 4.2 Theta squared plots for both cut configurations ...... 72 4.3 Phasograms for both cut configurations ...... 73 4.4 Phasograms in the low energy bin ...... 74 4.5 DC significance dependence on max zenith angle ...... 76 4.6 Phasograms in the low energy and low zenith angle bin ...... 76 4.7 Zenith angle distribution of the OFF runs relative to the target position . . 77 4.8 OFF run event and livetime Distribution ...... 78 4.9 Zenith Rate Distribution ...... 79 4.10 PSR J0437-4715 Pulsed Upper limits ...... 81

v List of Tables

3.1 Timing model parameters ...... 50 3.2 Simulation parameters ...... 59 3.3 Optimisation results for low energy events ...... 68 3.4 Standard H.E.S.S. selection cuts ...... 68

4.1 J0437-4715 observation parameters ...... 70 4.2 J0437-4715 timing model ephemeris ...... 71 4.3 Phasogram test statistics ...... 73 4.4 Phasogram test statistics and DC signiﬁcances in the low energy bin . . . . 75 4.5 Phasogram test statistics and DC signiﬁcances in the low energy and low zenith angle bin ...... 77 Introduction

Pulsars were discovered in 1967, when Hewish and Bell were studying interplanetary scin- tillations of radio waves. The name pulsar stands for pulsating source of radio which already unveils the most important characteristic: the intensity of the observed emission is regularly pulsing in time. This pulsation of the observed emission has a very high time stability, for some pulsars it is more stable than atomic clocks on Earth. The period of the variability is between a few milliseconds and a few seconds, depending on the specific pulsar. After many different speculations about the origin of these strange objects, little green men were also within speculations, the association with neutron stars was soon made. Neutron stars had already been postulated by theorists about 30 years earlier. The formation of a neutron star is assumed as follows: At the end of a star’s life, it collapses under its own weight. However, the material bounces back and is ejected at high speed, giving rise to an enormous explosion called a supernova. From the star’s core, a tiny dense neutron star is left behind. Next to black holes, neutron stars are the most compact objects known. Despite their very high surface temperature, neutron stars have been thought to be undetectable as they are extremely small. This was true until the discovery of pulsars. In a simple picture, the observed pulsation originates from the rotation of neutron stars. If only a small area of the pulsar surface emits radiation and the pulsar is spinning, an observer will detect a pulsed signal, as from a cosmic lighthouse. In the following decades, hundreds of pulsars were detected. The emitted radiation covers a large range of wavelengths: From radio waves, optical waves, X-rays, up to gamma-rays. However, there is still no detection of pulsed emission at energies above 20 GeV. This unexplored region may hold the key to questions on which theorists have disagreed for at least the last 20 years, in particular how and where high energy emission emerges from the pulsar and how it is related to radio emission. In the field of very high energy (VHE) gamma-ray astronomy, two different measurement approaches exist. Since VHE cosmic radiation is absorbed in the Earth’s atmosphere, a straightforward way is to use satellites. They are, however, limited by their small collection areas. In general, the VHE gamma-ray flux decreases with energy according to a power-law. Therefore, satellites are incapable of measuring gamma-rays with energies above the order of ten GeV in any reasonable time scale. The second approach is to use ground-based telescopes. When VHE gamma-rays enter the Earth’s atmosphere, they interact with nuclei producing an extended air shower. The relativistic secondary particles of these air showers in turn emit so-called Cherenkov light that can be detected by ground-based telescopes. This method is called Imaging Atmospheric Cherenkov Tech- 2 nique (IACT). The H.E.S.S. experiment located in Namibia is such an IACT system of the third generation. It consists of four 12 m telescopes and is fully operational since De- cember 2003. The H.E.S.S. telescopes can detect VHE gamma-rays ranging from 100 GeV up to 100 TeV. Despite successes in discovering many VHE gamma-ray sources, no pulsed VHE gamma-ray emission has been found so far. In this thesis, the first VHE gamma-ray analysis of the millisecond pulsar PSR J0437- 4715 is presented. This millisecond pulsar represents one of the most promising candidates for the search of pulsed VHE gamma-ray emission. PSR J0437-4715 was observed with the H.E.S.S. experiment in 2004. Chapter 1 gives an overview of pulsars and millisecond pulsars followed by a review about theoretical pulsar models for the emission of VHE gamma-rays. Subsequently, the characteristics of PSR J0437-4715 obtained from measurements in radio waves and X-rays are given. In chapter 2, the H.E.S.S. experiment is introduced. In addition to the Imaging At- mospheric Cherenkov Technique, all major hardware components are briefly described. Chapter 3 explains all methods that were necessary for the analysis in detail. This includes the standard H.E.S.S. analysis which is used for the reconstruction of observed gamma-rays and the background suppression. Special emphasis, however, is put on the timing analysis, i.e. the search for periodicities within the arrival times of the gamma-ray candidates. It comprises all the timing corrections that need to be applied to the arrival times, statistical tests for the quantitative search of pulsation, and a cross-check of all steps of the timing analysis. Furthermore, a procedure to optimise the analysis with respect to low gamma-ray energies is explained and tested in chapter 3. Finally, chapter 4 presents the results from the standard analysis and timing analysis of the millisecond pulsar PSR J0437-4715. Chapter 1

Pulsars

The millisecond pulsar PSR J0437-4715 is with its close distance of 150 pc and low magnetic field of order 109 G an excellent candidate among all existing pulsars for the search of pulsed very high energy (VHE) gamma-ray emission. It was discovered as a radio pulsar and soon later also found in X- rays. This chapter about the physics of pulsars provides a physical motivation for the VHE gamma- ray analysis of millisecond pulsars, with emphasis on PSR J0437-4715. In the first section, an introduction to pulsars is given. Beginning with the discovery of pulsars, the correct connection with neutron stars, physics of neutron stars, and the pulsar population including so-called normal and millisecond pulsars are presented. Subsequently, the second chapter gives an overview of the existing VHE emission models of pulsars. Therefore, the pulsar magnetosphere is introduced. In this context, the VHE radiation processes that occur in the magnetosphere like syn- Figure 1.1: Simplified pulsar illus- chrotron, curvature, and inverse Compton scatter- tration ing are shortly explained. In the final section, the millisecond pulsar PSR J0437-4715 is described with its physical properties that were obtained in the numerous radio and X-ray observations. Additionally, physical motivation for the VHE observation and theoretical predictions are presented.

1.1 Overview

Pulsars are fast spinning, highly magnetized neutron stars which emit a narrow radio beam along the magnetic dipole axis, see Fig. 1.1. In general, the magnetic axis of pulsars 4 Pulsars

Figure 1.2: Discovery of the first pulsar B1919+21 [3]. Left panel: First recording showing the observed interference. Right panel: More sensitive recording revealing the individual pulses. is inclined with respect to the rotational axis. Thus, for an observer with a line of sight close to the magnetic axis, the emission appears to be pulsating. Pulse periods are very stable over time, some very fast rotating pulsars, called millisecond pulsars, reach a time stability comparable or better than that achieved by the best atomic clocks. This is not very surprising when we consider a typical pulsar rotational energy of the order of 1043 − 1045 J and the comparably low rotational energy loss of the order of 1026 W. Since their discovery, about 1600 pulsars have been found. Seven pulsars are known to emit radiation with energies up to several GeV. The extreme physical conditions of pulsars together with their high pulse stability make them very interesting objects for a wide field of physics, in particular to probe general relativity beyond the weak-field limit of the solar system. Moreover pulsars provide valuable insights into the complex generation of neutron stars in supernova explosions and the subsequent evolution including binary systems.

1.1.1 Discovery

Pulsed emission from stellar origin was first discovered as a by-product, when Hewish work- ing with a research student, Jocelyn Bell, was investigating in interplanetary scintillation. In July 1967, Bell found large fluctuations of the signal, see left panel in Fig. 1.2. High precision follow-up observations confirmed these fluctuations. The result was a perfect regular pulse with a period of 1.337 s, shown in the right panel of Fig. 1.2. Remarkably, not long after this discovery a neutron star was already speculated to be the origin. Neu- tron stars were first proposed by Walter Baade and Fritz Zwicky [1] in 1934 and were not well known among astronomers in those days. Hewish received the Nobel Prize for this discovery. The question why Jocelyn Bell was not recognized can only be answered by the Nobel Committee. A good overview of pulsars and a nice review about the pulsar detection history can be found in [2]. 1.1 Overview 5

1.1.2 Identification with Neutron Stars Linking the observed pulsed radiation with fast rotating neutron stars was not straight forward, albeit the correct theory had been proposed already by Pacini [4] and indepen- dently by Gold [5]. Two other theories, trying to explain the pulsed emission, were more favored until they were ruled out by other pulsar detections. In the theory of oscillation of a condensed star, a periodicity of about 1 to 10 s for white dwarfs and 1 to 10 ms for neutron stars was predicted. The period is determined using gravity and elasticity to calculate the fundamental mode. When periods in between the two allowed regions (Vela 89 ms, Crab 33 ms) were discovered, the oscillation theory had to be abandoned. Addressing the pulsed emission to an orbiting binary systems, leads to other inescapable problems. A binary system emits gravitational waves due to its quadrupole moment. These gravitational waves carry away energy which in turn leads to a decrease in orbital period. Observations, however, clearly show a spin-down, i.e. an increase in period. Thus, it is clear that orbiting binary systems can not explain the observed pulsars. Nonetheless, binary theory found an application in explaining the emission of X-ray pulsars and was later applied to relativistic binaries. For the latter one, Hulse and Taylor received a Nobel Prize in 1993, demonstrating relativistic dynamics including the radiation of gravitational waves. As soon as the theories of oscillation and binary systems were ruled out, it became clear that pulsars are fast rotating neutron stars. Neutron stars and therewith pulsars are born in supernova explosions of massive stars (& 8 M ). Created in the collapse of the stars’ core, neutron stars are the most compact objects next to black holes. As a consequence of the conservation of angular momentum and magnetic flux of the progenitor star, pulsars gain their small rotational periods and huge magnetic fields. The outer layers of the progenitor are ejected with high velocities forming a supernova remnant. Striking evidence for the supernova-pulsar link theory was the association of more than 10 pulsars with supernova remnants. From timing measurements, pulsar masses were found to be in a narrow range of 1.35± 0.04 M . Modern calculations yield a size of about 10 km in radius which is quite similar to the very first calculations by Oppenheimer & Volkov. Acting as rotating magnets, pulsars emit magnetic dipole radiation which is the dominant effect for an increase in rotational period P , described by the spin-down P˙ . The power emitted by a rotating magnetic dipole is given by ˙ 8π 4 6 2 2 E = 3 Ω R B0 sin α, µ0c where Ω = 2π/P is the angular frequency, R denotes the pulsar radius, B0 is the magnetic field strength at a pole on the pulsar surface, and α denotes the angle between rotational and magnetic axes. This power can also be obtained from the angular kinetic energy of a rotating body: d 1/2 · IΩ2 = IΩΩ˙ , dt where I denotes the moment of inertia which has a typical value of I = 1045 g cm2 for pulsars. By equating the two power formulas, we can estimate the magnetic field strength 6 Pulsars at the pulsar surface: q 19 B0 ' 3.3 · 10 (P/s) · P˙ Gauss. Here, we assumed an orthogonal rotator (α = 90◦) with a radius of R = 10 km. Typical 12 14 values of B0 are of order 10 G, although field strengths up to 10 G have been observed. Millisecond pulsars have lower field strengths of the order of 108 to 1010 G which appear to be a result of their evolutionary history.

1.1.3 Pulsar Population A descriptive way of presenting the pulsar population is in terms of spin period P and spin- down P˙ because most characteristic pulsar properties depend on these two parameters. Fig. 1.3 displays a large sample of the pulsar population with logarithmic axes. Black small dots represent pulsars from which no gamma-ray emission has been observed, large red dots represent the seven high-confidence gamma-ray (∼ 10 GeV) pulsars, and the blue dots are the low-confidence gamma-ray pulsars. Lines of constant induced magnetic field (dashed blue lines), constant electric voltage (dotted red lines), and constant spin-down age (solid green lines) are also drawn into the plot. The spin-down age is estimated from P ν τ = = − 2P˙ 2ν ˙ using either period P or the spin frequency ν and their derivatives. The estimates for the age are obtained under the assumption that the initial spin period is much smaller than the present period and that the spin-down is fully determined by magnetic dipole braking. Initial spin periods are estimated to a wide range from 14 ms up to 140 ms. Pulsars are therefore born in the upper left area of Fig. 1.3 and move into the central part where they spend most of their lifetime. Inspecting the pulsars in Fig. 1.3, we clearly find two classes of pulsars.

Normal Pulsars Most of the ∼ 1600 known pulsars in total have spin periods in the range of 0.1 s to 1.0 s with period derivatives of typically P˙ = 10−15 s s−1. The longest period observed from a pulsar is 8.5 s. These so-called normal pulsars are thought to be observable for about 107 yr after the initial supernova explosion. Over this time they slow-down from their starting millisecond periods to some seconds and their magnetic fields become weaker. Subsequently, the energy output of the pulsar is diminished to a point where it no longer produces significant emission. Is is assumed, that the electric potential is not sufficient to produce the particle plasma which is required for the radiation processes. Thus, the pulsar is not observable any longer. This state seems to depend on a combination of P and P˙ and in Fig. 1.3, it corresponds to the lower right area.

Millisecond Pulsars About 100 pulsars located in the lower left part of Fig. 1.3 can not be explained by the above picture of the normal pulsar life. Instead these have both small periods (of the

1.1 Overview 7

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Figure 1.3: Period vs. period derivative for a large sample of pulsars. Small black dots: no gamma-ray emission. Large red dots: seven high-confidence gamma-ray pulsars. Large light blue dots: three lower-confidence gamma-ray pulsars. Solid lines: timing age. Dotted line: open field line voltage. Dashed line: surface magnetic fields. Taken from [6]. 8 Pulsars

˙ −18 −1 order of milliseconds) and small spin-downs P . 10 s s . The smallest known period is 1.56 ms. This population appears much older than ordinary pulsars. Indeed, the so-called millisecond pulsars represent the oldest population of pulsars with ages ∼ 1010 yr. It is assumed that millisecond pulsars emerge from normal pulsars situated in a binary system which was not disrupted during the supernova explosion. Such a spun-down normal pulsar in a binary system can spin-up again by accreting matter and therefore angular momentum at the expense of the orbital angular momentum of the binary system if the companion is suﬃciently massive and evolves into a red-giant overﬂowing its Roche lobe [7]. Thus, future millisecond pulsars enter the pulsar “graveyard” as normal pulsars in a binary systems. Then, due to the angular momentum transfer, these pulsars are “recycled”, i.e. they are spinning up and eventually become detectable as millisecond pulsars. During the accretion phase, X-rays are generated by the liberation of gravitational energy of the infalling matter onto the pulsar. These X-ray binaries relevant for the formation of millisecond pulsars can be divided into two classes, neutron stars with high-mass and with low-mass companions. High-mass companions are massive enough to explode in a second supernova. This can lead to a double neutron star binary, if the system survives this second supernova. In fact, such double neutron star binary systems have been discovered. In one case pulsed emission was observed from both neutron stars (PSR J0737-3039 A & B). Low-mass companions in X-ray binaries evolve and transfer mass onto the neutron star on a much longer time scale. This leads to very short rotational periods of a few ms. After the spin-up phase, the low-mass companion ejects its outer layers and becomes a white dwarf orbiting the fast spinning millisecond pulsar. The properties of millisecond pulsars and X-ray binaries are consistent with the described picture, which is illustrated by the fact that ∼ 90% of all millisecond pulsars are in a binary orbit while this is true for only less than 1% of the normal pulsars [8].

1.2 High Energy Emission Models

In this section an overview of VHE gamma-ray emission models for emission above 1 GeV is given. There are mainly two competing models which diﬀer in the assumed gamma-ray production mechanism and the location in the pulsar environment where the emission is produced. Polar cap models [9, 10] assume that particles are accelerated above the neutron star surface and that gamma-rays result from a curvature radiation or inverse Compton induced pair cascade in a strong magnetic ﬁeld. Outer gap models [11], on the other hand, assume that acceleration occurs along null charge surfaces in the outer magnetosphere and that gamma-rays result from cascade induced photon-photon pair production. Detailed reviews about the gamma-ray pulsar models can be found in [12, 13].

1.2.1 Magnetosphere Since pulsed gamma-ray emission was observed up to energies of 10 GeV by the Energetic Gamma Ray Experiment Telescope (EGRET) on board the orbiting Compton Gamma Ray Observatory [14], there is no dispute that particles are accelerated to extremely relativistic energies in the pulsar environment. It is also agreed that particles gain their very high 1.2 High Energy Emission Models 9

Lorentz factors in the range of at least 105 −107 by electric fields. These fields are induced by the rotating magnetized neutron star which is acting as a natural unipolar inductor generating huge electric fields (E~ ∝ ~v × B~ ). The electric fields are of such a strength that they can pull charges out of the star against the force of gravity (Goldreich & Julian [15]). In fact, the electric force is dominating over the gravitational force by a factor of about 1012. This becomes appreciable when we think of the enormous gravity which leads to a 11 gravitational acceleration of about 10 gEarth on the neutron star surface. Such a strong acceleration noticeably bends the emitted light, therefore we would “see” about 80% of the neutron star surface. Consequently, a resulting charge density (plasma) builds up in a neutron star magnetosphere, see Fig. 1.4. The magnetic field forces the plasma to corotate with the star. Therefore the magnetosphere can only extend up to a distance where the rotation velocity reaches the speed of light. This distance defines the so-called light cylinder which in turn separates the magnetic field into open and closed field lines. Plasma in the closed field lines is trapped into the magnetosphere and corotates forever, whereas plasma in the open field lines can be accelerated to highly relativistic velocities and leave the magnetosphere. This is thought to create the observed radio beam at a distance of order 10−100 km above the pulsar surface. Further, the plasma in the magnetosphere is able to cancel the electric field parallel to the magnetic field (allowing the field to corotate with the star) everywhere except at a few locations. These spots (where E~ · B~ 6= 0) are believed to exist above the surface at the polar caps and along the null charge surface, Ω~ · B~ = 0 where the corotation charge changes sign. These are the regions of particle acceleration and have given rise to the two classes of high energy emission models.

1.2.2 Radiation Processes The existence of rotationally induced potential drops, expected to exceed 1012 V, leads to the acceleration of charged particles (in particular e− and e+) to very high energies. This can lead to gamma-rays up to TeV energies as a result of a combination of curvature radiation (from electrons following curved field lines), synchrotron radiation (from electrons spiraling around field lines) and inverse Compton radiation (due to scattering of radio to soft X-ray photons by high energy electrons). However, the QED process of magnetic pair production [16] absorbs most of the VHE gamma-rays in ultra strong magnetic fields (leading to e−e+ pairs) before they can escape to the observer.

Synchrotron Radiation Charged particles moving in magnetic fields radiate energy. For non-relativistic velocities this is called cyclotron radiation, while at relativistic velocities it results in synchrotron radiation (SR). In magnetic fields, the induced motion of charged particles is simply uniform and circular around the field lines. Thus, particles with a non-zero velocity along the field lines move in a helical path along the field. The circular orbit can be nicely described 10 Pulsars

Figure 1.4: Illustration of the pulsar magnetosphere of an aligned rotator with open and closed magnetic field lines according to the model of Goldreich & Julian [15]. Charged particles can flow outwards along open field lines, whereas the plasma in the closed field lines is trapped forever. Taken from [7]. by its frequency, the so-called cyclotron frequency

eB ω = , γme given for an electron with mass me and velocity v in the magnetic ﬁeld B; γ denotes the usual Lorentz factor γ = 1 − β2−1/2 with β = v/c. This transverse acceleration leads to the emission of energy in the form of electromagnetic radiation. For non-relativistic particles, the emission frequency is simply 2πω and thus the spectrum consists of a single line. In the relativistic case, in contrast, the characteristic frequency of emission is the critical frequency 3γ2eB νcrit = , 2me given for an electron here. The overall SR spectrum consists of a large sum of many basic cyclotron harmonics and thus becomes quasi-continuous. Above νcrit, the spectrum is exponentially suppressed. Typical magnetic ﬁelds of neutron stars lead to a SR peak in X-rays. The SR power, i.e. its energy loss rate, is given by

e4B2β2γ2 PSR ∝ 2 . me

Electrons in the high magnetic ﬁelds of neutron stars, therefore immediately lose their transverse motion. Comprehensive reviews can be found in [17, 7]. 1.2 High Energy Emission Models 11

Curvature Radiation

In analogy to the SR, the curvature radiation (CR) emerges when charged particles move in (curved) magnetic ﬁelds. While SR is due to the transverse motion with respect to the magnetic ﬁeld lines, CR results from the parallel component. The characteristic frequency is obtained by replacing the radius of gyration in SR

βcγm r = e eB with the radius of curvature rcurvature of magnetic ﬁeld lines. Hence, we obtain νcrit ∼ 3 7 γ c/rcurvature. Assuming a dipole ﬁeld and an electron Lorentz factor of 10 corresponding to TeV energies, VHE gamma-rays with energies in the range of some GeV can be generated with this mechanism.

Inverse Compton Scattering

Inverse Compton scattering (ICS) is equivalent to the well-known Compton scattering process with a Lorentz boost. Now, the electron is moving and energy is transferred to the photon. The mean photon energy after the collision is found to increase with the squared electron Lorentz factor. Therefore, high frequency radio photons interacting with relativistic electrons of the order of γ = 103 − 104 are boosted up to X-ray energies. The boosting is limited by the incident electron energy. However, assuming very high electron Lorentz factors, it is easily possible to obtain VHE gamma-rays in the TeV energy range. A full treatment of the problem yields the Klein-Nishina formula for the scattering cross-section. This holds for all energies, while the Thompson cross-section can be applied 2 to photon energies below ≈ mec only. A complete review for ICS can be found in [18].

Pair Production

VHE gamma-rays generated in a combination of the discussed SR, CR, and ICS can undergo pair production. The induced electromagnetic cascade leads to a shift of the most energetic gamma-rays to lower energies. In the extremely high magnetic fields of pulsars, instead of the well-known QED pair production γ + γ −→ e+ + e−, the magnetic pair production γ −→B e+ + e− is the dominating process [16]. The mean free path lγB of gamma-rays with energy Eγ crossing magnetic field lines at an angle θ is exp((B sin θE )−1) l ∝ γ . γB B sin θ In fact, this leads to a sharp cutoff of the gamma-ray spectrum at a certain cutoff energy. Since the mean free path sensitively depends on the magnetic field which in turn depends on the distance from the pulsar surface, the acceleration region determines the energy cutoff. 12 Pulsars

Ω B

Light Cylinder

Ω . B = 0 POLAR GAP

OUTER GAP

Figure 1.5: Scheme of the VHE gamma-ray emission regions for the polar cap (red) and outer gap (blue) models in the pulsar magnetosphere.

1.2.3 Polar Cap Model

Polar cap models were first introduced by Sturrock (1971) [19] and Ruderman & Suther- land (1975) [20] who proposed particle acceleration and radiation near the pulsar surface at the magnetic poles. Meanwhile, a large variety of polar cap models have developed, mainly differing on the nature of particle emission from the stellar surface. This is still under discussion, since not much is known about the surface composition and physics. A certain subclass of models, based on free emission of particles of either sign, is called space-charge limited flow (SCLF) models. These assume that the surface temperature exceeds the iron and electron thermal emission temperatures. Although the electric field parallel to the magnetic field is zero at the surface for these models, the space-charge from the free emission falls below the corotation charge along open field lines. This is thought to be due to the curvature of the field or due to general relativistic inertial frame dragging. Therefore, a parallel electric field component is induced by the charge deficit. This electric field above the magnetic poles accelerates particles, which radiate ICS photons (γ ∼ 102 − 106) by resonant scattering of thermal X-rays from the pulsar surface and CR 6 photons (γ . 10 ). The photons in turn produce electron-positron pairs in the strong magnetic field. However, it is believed that the pairs cannot completely screen the parallel electric field. Thus, a stable acceleration region can form at 0.5 − 1.0 stellar radii above the surface, see Fig. 1.5. A super-exponential energy cutoff in the emission spectrum is predicted at several GeV due to pair production attenuation in the huge magnetic fields in the pulsar vicinity. 1.2 High Energy Emission Models 13

VELA PULSAR

EGRET -3 COMPTEL 10 OSSE RXTE CANGAROO 10-4 OPTICAL ROSAT

-1 PC Model s -5

-2 10 Daugherty & Harding 1996

10-6

MeV cm 10-7 Outer Gap Model Romani 1996 -8 Outer Gap Model 10 Hirotani 2000

10-9 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 106 Energy (MeV)

Figure 1.6: Spectrum of the Vela pulsar to demonstrate the two VHE gamma-ray emission models. The plot contains the observations ranging from the optical to the gamma-ray wavelengths and the two VHE model predictions, i.e. the polar cap (solid line) and outer gap (dashed line). Taken from [12].

1.2.4 Outer Gap Model

Based on the existence of a vacuum gap in the outer magnetosphere which may develop between the last open field line and the null charge surface (Ω~ ·B~ = 0), outer gap models, in contrast to polar cap models, predict an ICS peak at some TeV. The gap may arise due to charges escaping from the light cylinder along open field lines above the null charge surface which cannot be replenished from below. The outer gap acceleration region is illustrated in Fig. 1.5. First models developed by Cheng, Ho, and Ruderman (1986) [21] assumed emission from both gaps associated with the corresponding magnetic poles. More recent models by Romani [22] or Hirotani [11] assume the emission to take place from one pole only. These models can reproduce the observed spectra quite successfully, see Fig. 1.6. Outer gap models require electron-positron pairs to provide the current in order to accelerate particles in the outer gaps. These pairs are thought to be produced by photon- photon pair production. The photons in turn are generated by CR of primary particles or ICS of primary particles with infra-red photons. VHE gamma-ray spectra from outer gap models have energy cutoffs around 10 GeV and ICS contributions in the energy range of 100 GeV up to some TeV [11]. The presence of significant ICS contributions is mainly due to the much lower magnetic field in the outer gap compared to that at the pulsar surface. 14 Pulsars

1.3 PSR J0437-4715

At a distance of about 150 pc, PSR J0437-4715 is the closest and brightest millisecond pulsar known at both radio and X-ray wavelengths. It has a rotational period of 5.76 ms, 9 8 a characteristic age of 4.9·10 yr, a magnetic ﬁeld B0 ∼ 7·10 G, and a rotation energy loss rate E˙ ∼ 3.8 · 1033 erg s−1 = 3.8 · 1026 W. Furthermore, PSR J0437-4715 is in a 5.74-day binary orbit with a low-mass white dwarf companion of ∼ 0.2 M . The relatively small surface magnetic ﬁeld and the pulsar’s proximity to Earth make

PSR J0437-47150 a particular interesting pul- XMM−Newton 0.3 − 6 keV sar with respect to the VHE gamma-ray emission. As aforementioned, low magnetic fields effectively reduce the pair production probability for the highest gamma-ray energies. The distance d scales the observable flux as ∝ d−2. Chandra 0.1 − 10 keV

1.3.1 Detection in Radio and X-rays PSR J0437-4715 was discovered as a radio pulsar in 1993 during the Parkes survey of the southern sky for millisecond and low-luminosity ROSAT 0.1 − 2.4 keV pulsars [23]. The radio pulsation consists of one narrow pulse peak, see Fig. 1.7. PSR J0437- 4715 was the first millisecond pulsar detected in X-rays [24]. The X-ray satellite ROSAT observed the pulsar and discovered the X-ray pul- Parkes 1420 MHz sation in the 0.1 − 2.4 keV energy range. The X-ray pulsation is composed of one single broad pulse per period, with a pulsed fraction between 30% and 40%. The peak is at the same pulse phase as the radio peak, as can be seen in Fig. 1.7. The X-ray spectrum up to 7 keV from the combined data of ROSAT and Chandra, is incompatible with a simple blackbody model. It Figure 1.7: PSR J0437-4715 phaso- can, however, be described by two components, grams showing two phase cycles in radio a non-thermal power-law spectrum generated in (bottom) and X-ray (top) wavelengths. the pulsar magnetosphere with a photon index ≈ 2 and a thermal spectrum emitted by heated polar caps with a temperature of the order of 106 K [25]. The lack of any spectral features in the thermal component suggests that the neutron star surface is covered by a hydrogen atmosphere. XMM-Newton observations conducted in 2002 confirmed these results and moreover revealed that the pulsation shape and pulsed fraction are slightly energy dependent [26]. At ultraviolet (UV) wavelengths, constant (unpulsed) emission from PSR J0437-4715 was detected [27]. This was the first time that UV emission was observed from a mil- 1.3 PSR J0437-4715 15

10−5 PSR J0437−4715 10−6

−7 10 EGRET CR GLAST 10−8 s)

2 MAGIC 10−9

−10 10 H.E.S.S. GeV/(cm SR−prim 10−11

10−12 ICS 10−13

10−14 10−5 10−4 10−3 10−2 10−1 100 101 102 103 104 105 Energy (GeV)

Figure 1.8: Polar cap model predictions from Harding et al. [30] for PSR J0437-4715. Thin black lines: different predicted gamma-ray emission components. The top CR spectrum is the total emission along the field line, while the bottom CR spectrum is emission only above the radio emission altitude. Shaded lines: Sensitivities of the GLAST, MAGIC, and H.E.S.S. experiments. Black dots with downward arrows: EGRET upper limits. lisecond pulsar. The observed spectrum suggests thermal emission from the surface with a temperature of about 105 K. Furthermore, hints for a Hydrogen Lyα line were found. However, no statistically significant pulsation was found. Despite PSR J0437-4715’s close distance and very high spin-down flux, an upper limit for pulsed emission above 100 MeV at the level of 2.1·10−7 cm−2 s−1 was given by EGRET [28].

1.3.2 Theoretical Predictions The only outer gap model predictions were found from Zhang and Cheng [29]. The CR energy cutoff is in the range of 10 GeV. However, no ICS was considered and thus unfor- tunately no predictions for the emission of VHE gamma-rays were made. Harding et al. [30], using polar cap model calculations, predict a weak ICS component, see Fig. 1.8. In agreement with calculations from a 3D model polar cap incorporating variations of the general relativistic electric field (Venter, de Jager, and Tiplady) [31], the predicted CR energy cutoff is in the range of 1 − 20 GeV. This was also confirmed by calculations from Fr¸ackowiak and Rudak [32]. Bulik et al. claimed that PSR J0437-4715 is a promising target for VHE gamma-ray observations [33]. In their simulations, they extended polar-cap model calculations by including ICS. The thermal soft X-ray photons, which come either from the polar cap or from the surface, are Compton up-scattered to a very high energy domain and form a separate spectral component peaking at ∼ 1 TeV. Hence, PSR J0437-4715 would be within reach of high-sensitivity Cherenkov telescopes. The predicted flux above 100 GeV is between 8 · 10−12 and 200 · 10−12 cm−2 s−1. Chapter 2

H.E.S.S. Experiment

The High Energy Stereoscopic System (H.E.S.S.) is an array of four large Imaging At- mospheric Cherenkov Telescopes (IACT) situated in the Khomas Highland of Namibia, southern Africa. The name H.E.S.S. was also chosen in honor of Victor Hess, who received the Physics Nobel Prize in 1936 for his discovery of cosmic radiation. The H.E.S.S. experiment investigates in the ﬁeld of very high energy (VHE) cosmic gamma-ray astronomy, i.e. at photon energies above 100 GeV. Since December 2003, all four telescopes are fully operational and make the instrument the most sensitive IACT system nowadays. It can explore gamma-ray sources with intensities at the level of a few per mill of the ﬂux of the Crab nebula, which is the so-called standard candle in VHE gamma-ray physics. With this unprecedented power, H.E.S.S. succeeded in many important gamma-ray detections [34], in fact the number of known VHE gamma-ray sources was more than doubled in 2004 already. H.E.S.S. is operated by an international collaboration of about 100 physicists spread over more than 20 institutes in Europe and southern Africa. This chapter introduces the Imaging Atmospheric Cherenkov Technique and gives an overview of the experimental setup and some key components.

Figure 2.1: View on the H.E.S.S. site in the Khomas Highland of Namibia. 2.1 Imaging Atmospheric Cherenkov Technique 17

2.1 Imaging Atmospheric Cherenkov Technique

Apart from visible light and radio waves, all electromagnetic wavelengths are absorbed in the Earth’s atmosphere. Thus, the VHE gamma-rays of interest can be observed directly only outside the atmosphere. For this direct approach satellites are used, such as EGRET. Most VHE gamma-ray spectra, however, are described by a falling power-law, hence the gamma-ray rates become very small in the VHE domain. Thus, a large collection area is essential for such observations. Satellites with a typical area of 1 m2 consequently can not measure gamma-rays with energies above ∼ 10 GeV on any reasonable timescale. Ground based techniques, on the other hand, diluted by an indirect measurement namely the collection of Cherenkov light, avail of an enormous collection area. Detectable Cherenkov light is emitted when a charged particle traverses a transparent medium with a speed higher than that of light in this medium. Because both, relativistic cosmic protons and gamma-rays generate such charged particles in so-called extended air showers, IACTs suﬀer from a low signal-to-noise ratio. Basically, the technique of collecting Cherenkov light is nothing but utilizing a large fraction of the atmosphere above the telescopes as a homogeneous calorimeter. The ground based Cherenkov technique with its huge detector volume proved to be a very successful method to measure the highest energetic gamma- rays. In this section air showers and Cherenkov light are brieﬂy introduced followed by a short review of the detection method.

2.1.1 Air Showers Whenever a relativistic particle interacts with an atmosphere’s nuclei, secondary particles are produced which again can undergo interactions to produce more secondary particles and thus an extended particle air shower is created. Generally, it is useful to distinguish between electromagnetic and hadronic air showers. In the former one, only electromagnetic interactions appear, accordingly such an air shower is induced by photons, electrons, or positrons. In hadronic air showers also strong and weak interactions play a major role.

Electromagnetic Air Showers The development of electromagnetic air showers is dominated by a few well understood QED processes. Albeit many potential processes exist, in the high energy regime, electrons and equally positrons mainly lose energy via Bremsstrahlung in the Coulomb ﬁeld of air atoms. Photon interactions in turn produce mainly electron-positron pairs. Below a medium dependent critical energy, EC ' 86 MeV in air, the main source of electron energy loss is through collisions with atoms and molecules thus giving rise to ionization and thermal excitation; photons lose their energy below a certain limit through Compton scattering and the photoelectric eﬀect. Consequently, when a VHE gamma-ray enters the Earth’s atmosphere, it induces a cascade of of electrons, positrons and photons. This is called an extended electromagnetic air shower. Initially, the number of particles grows exponentially and simultaneously the 18 H.E.S.S. Experiment energy per particle degrades. Then, below the critical energy per particle, energy is mainly dissipated by ionization and hence not in the generation of further particles. The main electromagnetic shower properties can be described in terms of one parameter, the radiation length x0. The Earth atmosphere’s composition of mainly oxygen and −2 nitrogen translates into a radiation length of x0 = 36.66 g cm corresponding to a total thickness of ' 28 x0. The unit of the radiation length is given in atmospheric depth which is independent of the local medium density and can be transferred to a length l by dx = ρ (l) dl. One radiation length is the average distance an electron needs to reduce its energy by 1/e. Accordingly, the mean energy per particle in the air shower is given as a function of the passed distance x by

x − x hE (x)i = E0 e 0 where E0 indicates the incident particle energy. Similarly, due to pair production the 7 intensity of a photon beam is reduced to 1/e after traveling a distance of x = 9 x0. The shower maximum where most secondary particles are produced, is approximately located at a distance of E0 xmax ' x0 ln + 0.5 . EC Therefore, the air shower of an incident 1 TeV gamma-ray has a shower maximum at ' 8 km above sea level. Electromagnetic air showers are rather narrow, their transverse size mainly caused by multiple scattering of the electrons and positrons away from the shower axis. A good measurement of the air shower’s lateral extend integrated over the full shower depth is given by the Moli`ere radius (RM). It is a function of the radiation length and the critical energy, but almost independent of the incident energy. Roughly one RM represents the radius of a cylinder in which 90% of the energy are contained on average. In air, the −2 Moli`ere radius is RM ' 9 g cm corresponding to about 75 m.

Hadronic Air Showers In contrast to electromagnetic air showers, hadronic air showers induced by cosmic protons and nuclei, are more complex due to the multitude of eﬀects mostly caused by strong interactions. An incident hadronic particle hitting the Earth’s atmosphere mainly un- dergoes inelastic scattering or collision with air nuclei. Thereby, a signiﬁcant part of the primary energy is consumed in nuclear processes, such as excitation, nucleon evapora- tion, and spallation, resulting in low energetic particles at the MeV scale. Secondly, fast hadronic particles with large transverse momenta are produced, including protons, neu- trons, charged pions, and neutral pions. Because of the charge independence of hadronic interactions in each collision, on average one third of the pions produced will be π0’s. These pions will decay into two photons, π0 → γγ, before having a chance to reinteract hadronically. Naturally, these photons induce an electromagnetic sub-cascade, proceeding along its own laws of electromagnetic interactions as already discussed. Charged pions, + + on the other hand, decay into neutrinos and muons, π → µ νµ, the latter giving rise to small Cherenkov light cones (so-called muon-rings) when hitting a telescope. 2.1 Imaging Atmospheric Cherenkov Technique 19

As a consequence, hadronic air showers are more inhomogeneous and laterally extended than electromagnetic ones. A more complete discussion covering electromagnetic as well as hadronic showers can be found for instance in [35].

2.1.2 Cherenkov Radiation Highly relativistic particles traversing the air will deform the atom’s electron shell on its way. These excited atoms emit electromagnetic waves which superimpose coherently if the particle speed exceeds the local speed of light. As a result, a light cone namely the Cherenkov light, along the incident particle axis will emerge [36]. The half opening angle θc of this cone is given by c c 1 cos θ = n = = , (2.1) c v nv βn where v denotes the particle velocity and n is the refractive index in the local medium. Since cos θc ≤ 1, Eqn. 2.1 requires a minimum particle velocity βmin and hence an energy threshold, 2 m0c Emin = p1 − 1/n2 under which no Cherenkov photons will be emitted. Here, m0 is the particle rest mass. This formula points out the threshold dependence on the mass. As a consequence, Cherenkov light is mostly emitted by the lightest charged particles, i.e. electrons and positrons. The spectral distribution of the Cherenkov photons is proportional to λ−2 and has a maximum emission at a wavelength of λ ∼ 300 nm which corresponds to the ultraviolet band. Although some Cherenkov photons are absorbed or scattered by air atoms on their way, from an incident 1 TeV gamma-ray roughly 100 Cherenkov photons per m2 reach the ground on average. While the whole shower development proceeds on a time scale of some µs, the faint Cherenkov light front on the ground lasts for a few ns only since the emitting particles move with about the same speed as the photons. As a matter of fact, only the short duration of the Cherenkov light front generates an intensity high enough to make it detectable by very fast cameras.

2.1.3 Detection Principle Given the Cherenkov light’s uniform distribution on the ground over an area of ∼ π(100 m)2, the air shower can be detected through its Cherenkov emission from any suﬃciently sensitive light detector placed inside the light pool. An imaging instrument, that resolves the air shower morphology, is required to reconstruct the shower direction and also to suppress the hadronic background. Fig. 2.2 right side, illustrates the Cherenkov light cone leading to the light pool on the ground. As we have seen above, charged hadronic VHE cosmic-ray particles, like protons and nuclei, produce superﬁcially similar air showers to those of gamma-rays. To make things 20 H.E.S.S. Experiment

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Figure 2.2: Detection principle of Cherenkov light. Left: Sketch of the optical geometry, the air shower ellipsoid is projected by the reflector dish onto the camera located in the focal plane. Right: An air shower’s Cherenkov light cone illuminates the ground on which highly sensitive telescopes record this flash. worse, the cosmic proton background is dominating over the cosmic gamma-rays by a factor of about 104. Nonetheless, as aforementioned, a hadronically induced air shower is more inhomogeneous and wider on average which is reflected in the recorded image shape. This subtle difference between the cascades can be exploited to reject almost all of the hadronic background. Therefore, a finely granulated camera, in the form of pixels, is required to resolve the air shower shape and therewith deduce the incident particle identity to reject the proton background. Electromagnetic air showers induced by cosmic electrons, on the other hand, are obviously indistinguishable from those produced by cosmic gamma-rays. Due to the electron’s lower rate and steeper spectrum, this dilution is accepted as unavoidable background. In addition to background rejection, the camera pixels are required for the reconstruction of the primary particle’s properties, e.g. the shower direction and primary energy. On the left side of Fig. 2.2, the principle of optical projection is illustrated. The optical reflector on the dish projects the Cherenkov light from the air shower onto the camera mounted in the focal plane of the telescope. In the camera, the projected air shower is recorded. To extract physical properties of the primary particle, each recorded image (event) that passed some selection criteria is parametrized by an ellipse using a Hillas-type analysis [37]. Basically, the ellipse orientation determines the shower direction, which corresponds to a good approximation to that of the incident particle. Further, the image intensity corresponds roughly to the energy, in accordance with calorimeters. A more accurate 2.2 H.E.S.S. Telescope System 21

Figure 2.3: Left: One H.E.S.S. telescope with the camera hood. Right: Excellent optical conditions on the H.E.S.S. site; stars are visible down to the horizon. energy reconstruction can be achieved by taking into account the distance between shower axis and telescopes and utilizing detailed Monte Carlo simulations which simulate the air shower as well as the detector response. In stereoscopic observations, several telescopes record the same air shower from different angles. Thus, the telescopes have to be placed close enough to collect light from the same air shower. The benefit of stereoscopy is a significantly improved angular resolution, background rejection, and energy resolution. In particular, muon events can easily be rejected by requiring more than one telescope to trigger simultaneously. This is because Cherenkov light from muons is very faint and thus can only be detected if the muon hits one telescope directly.

2.2 H.E.S.S. Telescope System

In this section the experimental setup is shortly introduced, including telescopes, cameras, optics, and the trigger system.

2.2.1 Site and Telescopes The H.E.S.S. site is situated in the Khomas Highland of Namibia, near the tropic of Capricorn, about 1800 m above sea level. This southern hemisphere location oﬀers many advantages: excellent optical conditions (see Fig. 2.4 right side), the galactic center culmi- nates in zenith, mild climate, easy access, and good local support. IACTs require moonless nights for observations, of which 1600 hours are available per year. About 1000 hours of these are usable for observations, i.e. without any clouds [38]. On the site, the four telescopes are arranged in form of a square with a side length of 120 m, and the two diagonals oriented south-north and east-west. The telescope spacing is a balance between a large base length for good stereoscopic viewing and a small length in order to have two or more telescopes hit by the same Cherenkov light cone. Simulations 22 H.E.S.S. Experiment proved the chosen arrangement to be the most suitable one for observations of Cherenkov light originated in gamma-rays with energies between 100 GeV and 10 TeV. Each telescope is equipped with an altitude- azimuth mount to point the telescope at any source on the sky, see technical drawing in Fig. 2.4 and tele- Camera scope image in Fig. 2.3 left side. This tracking system is a friction drive on rails, with a maximum speed of ◦ −1 00 about 100 min , and an accuracy better than 10 . Mirror dish More details about the drive system can be found in [39]. One telescope has a total weight of roughly 60 tons, which includes the massive steel structure, the camera, the mirrors, and the drive system.

The key characteristics of the H.E.S.S. system are: Mount

• Energy threshold: 100 GeV Circular rail • Single shower angular resolution: 0.1◦

• Pointing accuracy: 2000

• Energy resolution: 15% Figure 2.4: Technical drawing of one H.E.S.S. telescope. In fact, the H.E.S.S. telescopes meet all design speciﬁcations. Only the pointing accuracy does not fulﬁll its expectations. Although the telescopes were constructed with emphasis put on high mechanical stability and rigidity of the mount and dish, the total systematic pointing error is of the order of 2000 and not 200 as aimed for. This inaccuracy is mainly caused by uncertainties in the exact positions of camera pixels and Winston cones. Deformations of the steel structure exist as well, but are well understood. Test measurements showed that 500 seems feasible for future observations [40]. A more detailed description on the whole H.E.S.S. system can be found in [41, 42].

2.2.2 Optics

The mirror of an IACT has to focus the Cherenkov light onto the camera. Therefore, it is crucial to have a big mirror area to catch enough of the faint light and to have a good image quality, i.e. a reflector point spread function smaller than a camera pixel. Each H.E.S.S. telescope consists of a 13 m diameter mirror dish. For cost effectiveness, it is segmented into 381 small round mirrors. Fig. 2.5 shows one of these 60 cm diameter round mirrors which is made of aluminized glass with a quartz coating, giving an initial reflectivity better than 80% (due to aging effects, the reflectivity is decreasing). Altogether, the mirror area is 107 m2 and has a focal length of 15 m. The mirror facets are arranged in a Davis-Cotton design (see Fig. 2.6 left site) to provide a high image quality also for off axis events. 2.2 H.E.S.S. Telescope System 23

The orientation of each facet is adjustable by motors, so that the mirrors can be remotely aligned via an automatic procedure.

Corresponding to an average air shower altitude, the Glas mirror mirrors are focused to an object distance of 10 km. The point spread function is over most of the ﬁeld of view well Support contained within a single pixel. frame Detailed information about the mirrors, the alignment and optical characteristics can be found in [43, 44]. Actuators and motors

2.2.3 Camera Figure 2.5: Technical draw- The camera has to capture and record the Cherenkov im- ing of one H.E.S.S. mirror ages. Thus, the requirements are: fast exposure of ns order, small pixel size to resolve image details, large ﬁeld of view for observations of extended sources and surveys, and a trigger to recognize Cherenkov images and at the same time reject night sky background. These requirements are met by the H.E.S.S. cameras, with the following features.

• Large ﬁeld of view of 5◦. For comparison, the moon has an angular diameter of 0.5◦ and the largest known VHE gamma-ray source, the shell-type supernova remnant Vela Junior, has an angular diameter of 2◦.

• 960 pixels, correspondingly each covers 0.16◦. The pixels are made of round photo- multiplier tubes (PMT) each equipped with a Winston cone of hexagonal shape to close the gaps between the PMTs. The mean quantum eﬃciency is about 25% [45].

• Modular structure: 16 PMTs are grouped together with their associated electronics into one drawer.

• The ampliﬁed PMT signals are sampled every ns using an analog ring buﬀer which has a capacity to store the last 128 ns.

• The camera trigger allows a maximum rate of 2.5 kHz. More details are given together with the central trigger below.

• All electronics, camera trigger, power supply, cooling etc. are integrated into the camera.

• Dimensions: 1.6 m diameter and 1.5 m length, weight about 800 kg.

Fig. 2.6 right side, displays a photography of a H.E.S.S. camera. Further information about the camera and its calibration is available in [46]. 24 H.E.S.S. Experiment

Figure 2.6: Left: Picture cut-out of the mirror dish showing the separate round mirrors. Right: Camera with the lid open disclosing the pixels.

2.2.4 Trigger A fast response to Cherenkov images and a good rejection of noise events are the key features of the trigger. The H.E.S.S. trigger consists of a component associated to each camera and a central trigger. Both are brieﬂy explained in the following.

Camera trigger The camera is divided into overlapping trigger sectors of about 64 pixels each. If a certain number of pixels in any trigger sector is above a threshold of about 5 photoelectrons within a time window of roughly 1.5 ns, the camera will be triggered. This is a simple way to reject uncorrelated PMT signals caused by photons of the night sky background. Once a camera triggers, a signal is sent to the central trigger and simultaneously the camera begins to digitize and readout all pixels.

Central trigger Requiring more than one telescope to trigger represents a very powerful way to suppress muon events, which appear only in one telescope, and moreover enables a signiﬁcantly improved image reconstruction since the same event is viewed from two or more angles. Alongside, the requirement reduces telescope dead-time and hence allows to reduce the camera trigger and energy thresholds. The hardware-level central trigger of the H.E.S.S. system searches for coincidences in the trigger signals received from the individual cameras in a time window of ∼ 80 ns. The system accounts for diﬀerent arrival times caused by the pointing angle. When a coincidence is found, the central trigger causes the telescopes to continue the readout of the pixel data. Otherwise, for non-coincident trigger signals, the telescope readout electronics is stopped after a few µs. 2.2 H.E.S.S. Telescope System 25

Using the central trigger, the system rate is of the order of 250 Hz in routine operation, depending on the weather conditions and also on the zenith angle. Coming back to the readout process, it is of course a matter of high interest for timing analysis to know when and where the event time stamps are generated. When a camera triggers, analog data spanning over the proper 16 ns is integrated and read out. At the same time, as aforementioned, a message is sent to the central trigger. In case the trigger is conﬁrmed due to coincident camera messages, an event including the current central trigger GPS time is sent back to the telescopes. Each telescope with data now adds its digitized pixel data and in addition its own GPS time stamp. Thus, the telescope time stamps are slightly delayed with respect to the central trigger one. All GPS devices have an intrinsic time error of less than 1 µs. A profound description of the H.E.S.S. trigger system can be found in [47].

2.2.5 Data Acquisition The data acquisition (DAQ) system collects and combines all data from the telescopes and the monitoring devices, stores the data, and performs a ﬁrst online analysis. A list of the key feature of the DAQ is given here:

• All devices and PCs are hooked-up using commercial technology, either a 100 MBit/s local area network or ﬁber optics;

• Processing is done using a Linux cluster;

• Data storage is secured with hardware RAID-10 systems and tape drives;

• Due to the lack of any high-speed Internet connection, the stored data is transmitted and distributed through tapes;

• During normal operation, the system data rate is about 4 MByte/s which amounts to more than 500 GBytes of raw data a month;

• The whole DAQ software is written in an object oriented way utilizing C++;

• Interprocess communication is implemented with omniORB which is based on the industry standard CORBA;

• For data analysis and storage, the well-known software framework ROOT is used.

A more complete documentation of the H.E.S.S. DAQ can be found in [48].

2.2.6 Monitoring Permanent monitoring is crucial especially during observations. Critical parameters from the cameras including currents, rates, and temperatures have to be known for good data quality and moreover in order to not damage any device. Atmospheric parameters, on the other hand, inﬂuence the number of Cherenkov photons reaching the ground and hence 26 H.E.S.S. Experiment the trigger and system rates. Therefore, good knowledge of the weather is helpful for the data analysis. In particular it is essential for determining the energy ﬂux level, since the event rate and therewith the rate of observed gamma-rays depends on the weather. The weather is monitored using:

• Radiometers on each telescope measuring the infrared emission of water molecules in the direction of pointing, eﬀectively detecting clouds in the ﬁeld of view.

• One central Ceilometer utilizing pulsed infrared light to determine the cloud density from the backscattered light. This device is capable to point in the same direction as the telescopes.

• One weather station recording the temperature, air pressure, humidity, wind speed and direction, and amount of rainfall.

• One all sky scanning radiometer providing an all sky survey of clouds.

2.3 Monte Carlo Simulations

In addition to the only calibration source available, the Crab nebula, detailed Monte Carlo (MC) simulations are mandatory for data analysis in ground based gamma-ray astronomy. In particular, MC simulations are used for energy calibration, to optimise the background suppression, and to study performance characteristics of the experiment such as the angular and energy resolution. First, the extended air showers are simulated by a program called CORSIKA [49]. In case of gamma induced air showers, i.e. electromagnetic air showers, the code relies on well known QED processes and cross sections. For hadronic air showers, on the other hand, phenomenological approaches have to be used since perturbative QCD is not applicable in the energy transfer domain of interest. Thus, the simulation becomes more inaccurate. CORSIKA incorporates the whole photon propagation down to the ground, taking into account environmental parameters such as absorption, scattering, and geomagnetic ﬁelds. In the end, the output of CORSIKA consists of a list of simulated photons each possessing a wavelength, direction, position, and emission time. Secondly, another piece of software, namely sim hessarray [50], simulates the detector response to these Cherenkov photons. Among many aspects, it considers:

• the mirror reﬂector layout and orientation with respect to the air shower,

• shading eﬀects caused by support structures,

• transmission of the Winston cones,

• PMTs with correct quantum eﬃciency,

• camera and trigger electronics response,

• analog signal shape. 2.3 Monte Carlo Simulations 27

Finally, the normal analysis chain is applied to the simulated data in order to compare with observational data. Chapter 3

Methods and Algorithms

Analyzing observational data requires a set of complex software which embodies the methods and algorithms that are needed to extract physical information from the raw data. This chapter is devoted to introduce and explain all methods which were applied within this thesis. In the ﬁrst section the standard H.E.S.S. analysis, as generally used within the H.E.S.S. collaboration, is shortly introduced including background suppression, a 1-dimensional signal extraction, and energy estimation. Special emphasis was put in the next section on explaining the whole timing analysis necessary for the search of time periodic signals of pulsars. First, corrections to the time of arrival and time of emission for binary pulsars, are explained in full detail. Then, statistical tests specialized on periodicity searches are described and compared. In the following section, cross-checks of all timing corrections with existing software, optical Crab data, and simulations of a pulsed signal, are described in detail. Since pulsar spectra are predicted by all theoretical models to have a cutoﬀ somewhere in the GeV energy range, the standard analysis selection cuts were optimised for very low energies. Methods and results of this optimisation are explained in the last section of this chapter.

3.1 Standard Analysis

This section describes the standard analysis which is performed to search for VHE gamma- ray sources. It does, however, not consider any time information. Typical H.E.S.S. observations are taken in 28 minute runs. Until the ﬁnal result, all runs have to pass a series of calibration, selection, reconstruction, and background rejection steps. All of these are explained here.

3.1.1 Calibration and Preselection At the very beginning, before analysis, every H.E.S.S. run has to pass several selection criteria. These criteria are supposed to remove runs taken at poor observing conditions 3.1 Standard Analysis 29

φ Figure 3.1: Hillas parameters of the shower ellipse in the camera online event display. Each pixel’s color corresponds to its intensity LocalDistance x in PEs.

width length

(weather) or affected by hardware malfunctions. To remove background noise and noisy pixels, the run images are calibrated [51] and cleaned. The image cleaning, also called tailcut, keeps only image pixels which either have a photoelectron (PE) signal above 10 and a neighboring pixel having a signal above 5 or the other way around, i.e. a pixel with a signal greater than 5 and a neighboring pixel with at least 10 PEs. Another not so conservative tailcut configuration is as above but with the requirement of 10 and 5 PEs replaced by 7 and 4 PEs. In this so-called 0407 tailcut, more events pass the image cleaning compared to the 0510 tailcut. This becomes an important issue for low energy events. The calibrated and cleaned images are parametrized by their first and second moments using a Hillas-type analysis [37]. Air shower images are of elliptical form and thus fairly well described by the parameters (comp. Fig. 3.1):

• width and length of the ellipse;

• φ, the inclination angle of the ellipse’s major axis with respect to the camera x-axis;

• LocalDistance, the distance from the ellipse’s center of gravity to the camera’s origin;

• size, the total image intensity in PEs.

Further, two image quality selection cuts are applied to ensure a proper functioning of the following analysis:

• In order to avoid truncated ellipses on the edge of the camera, the LocalDistance is required to be below a certain limit.

• A minimum total intensity (size) value ensures the images are well reconstructable. 30 Methods and Algorithms

For further event reconstruction, the image data of at least 2 telescopes has to pass the LocalDistance and size cuts.

3.1.2 Geometrical Reconstruction

Exploiting the stereoscopic observation mode, i.e. having the Hillas parameters of the shower ellipse of at least two telescopes, it is possible to reconstruct the shower direc- y tion of the primary particle and the shower core position on the ground on an event by event basis. This technique was pioneered by the HEGRA collaboration [52]. First, the intersection point from each pair of ellipse’s major axis is found in the ﬁeld of view (see Fig. 3.2). All intersection points are weighted taking into account the sine of x the angle between the two axes, the two image sizes, and Telescope 1 the ratio of width/length from both images. By projecting this average intersection point onto the plane of sky and plane of ground, the shower direction and core location Telescope 2 are obtained, respectively. For each event the typical angular resolution is 0.1◦ and the core position is on average reconstructed with ∼ 10 m accuracy Naturally, the shower direction is important to dis- Figure 3.2: Intersection of the tinguish whether a gamma-ray candidate belongs to an shower ellipses. assumed source or not (more details in section 3.1.4 on page 31) and also for spatially resolved analysis. The shower core position on ground is the point which the incident particle would have hit, if it had not been absorbed long before in the atmosphere. From this core position the impact parameter is determined for each telescope by the distance between core and telescope position. The impact parameter together with the total image amplitude is used in the energy estimation (see section 3.1.5 on page 32).

3.1.3 Background Suppression

As aforementioned, a powerful background reduction, i.e. a gamma hadron separation, is essential for the analysis due to the abundance of hadronic air showers. This is achieved by exploiting the subtle eﬀect of hadronic air showers being wider and longer on average compared to those initiated by VHE gamma-rays. Selection cuts on the Hillas parameters width and length have long been used to perform background rejection in VHE astronomy. They have, however, a poor acceptance at high energies. Additionally, they omit the information of the shower image recorded by multiple telescopes. For this purpose, an improved set of width and length parameters called mean scaled width (length) was developed by the HEGRA collaboration [52]. In the H.E.S.S. standard analysis, selection cuts on mean reduced scaled width (length) (MRSW,MRSL) are used: 3.1 Standard Analysis 31

Ntels 1 X widthi − hwidthi MRSW = i , (3.1) N σwidth tels i=1 i N 1 Xtels length − hlengthi MRSL = i i . (3.2) N length tels i=1 σi

Here, Ntels is the number of telescopes with data; widthi (lengthi) is the the Hillas width length parameter width (length) of telescope i; hwidthii, hlengthii and σi , σi are the Monte Carlo expectation values of telescope i for width (length) and their corresponding standard deviations. The expectation values are derived from gamma-ray simulations based on image intensity, reconstructed impact parameter, and zenith angle. Therefore, the MRSW (MRSL) represents the mean difference in standard deviations of the width (length) in the observed camera image from that expected from a gamma-ray simulation. PNtels In contrast to the HEGRA parameters MSW = 1/Ntels i=1 widthi/ hwidthii, the MRSW (MRSL) include the standard deviation of the expected values. Since the expected mean values hwidthi and hlengthi are not as well determined for some values of impact parameter and image size, it is reasonable to incorporate this effect with the standard deviation. The applied selection cuts are optimised a-priori to yield the maximum significance. Besides the standard H.E.S.S. selection cuts, a set of selection cuts optimised for low energy events was applied in the analysis of PSR J0437-4715, see section 3.4 on page 66.

3.1.4 1D Analysis Using 7 Background Regions After suppressing most of the background shower events, the remaining background of the signal region has to be determined. A simple approach is to use other sky regions without gamma-ray sources, apply exactly the same analysis for these so-called OFF regions as for the signal or ON region and therefrom obtain a background estimation. Yet, the system efficiency depends on many parameters, the most important being the observational zenith angle, the weather conditions, and the camera acceptance. Clearly, these parameters should be held constant for the background estimation. Therefore, a simple and also powerful choice for the OFF regions is to take neighboring regions in the same field of view of the camera (comp. Fig. 3.3). By pointing the camera center slightly next to the ON region (typically the offset is 0.5◦), it is possible to have the same radial distance from the ON and all OFF regions to the camera center. This observation method, called wobble mode, guarantees that the camera acceptance, which has rotational symmetry to a good approximation, is the same for all regions. Most other parameters the system efficiency depends on, are the same because the OFF data is taken at the same time and in almost the same sky region. Only the zenith angle differs, but this can be taken into account by using an alternating wobble offset mode, changing from +0.5◦ to −0.5◦ from run to run. Another aspect of background estimation is to have as much statistics as possible in order to reduce the statistical error. As can be seen in Fig. 3.3, there are seven OFF 32 Methods and Algorithms

Figure 3.3: Background estimation from OFF regions (blue) within the same ﬁeld of view (outer black circle) and with the same radial distance to the camera center as the ON region (red).

regions around the camera center. Depending on the region’s radius and the offset, it is possible to include many background regions in the field of view without risking to contaminate the regions with gamma-rays from the signal region. To select events from a specific region, the squared angular distance between the shower direction and the region center position is used. This squared angular distance is called θ2. Consequently, a selection cut of the order of 0.02 deg2 is applied to obtain events lying in the region. The total number of background events NOFF has to be normalized by the number 1 of OFF regions, NBackground = αNOFF. Here, α is the normalization, i.e. #OFF regions. Together with the total number of ON events NON, the signal significance is calculated by the following formula from Li and Ma [53]:

√ 1/2 1 + α NON NOFF S = 2 NON ln + NOFF ln (1 + α) . (3.3) α NON + NOFF NON + NOFF

3.1.5 Energy Estimation For each telescope image, the event energy is estimated using lookup tables containing the mean energy of simulated gamma-rays as a function of image size, impact parameter, zenith angle, and camera oﬀset. The estimated energy of the observed incident particle is then the mean of all telescopes with an energy resolution better than 15% for all energies. Within these 15% there is, however, a systematic error. For low energetic gamma-rays, the energy is estimated too large on average. For very high energy showers, on the other hand, the energy is estimated too low on average. The high energy bias is due to saturated PMTs and or the fact that shower images are too big to ﬁt on the camera so the image amplitude is underestimated. The method of using simulated gamma-ray events to calculate the energy implies that the energy of other cosmic ray particles as protons is not estimated correctly. 3.1 Standard Analysis 33

Effective Area For the determination of the flux, the number of collected gamma-ray events has to be divided by the collection area and the collection time. In the case of IACTs, the effective collection area, or just effective area, is located in the atmosphere and is characterized by the fact that showers from this effective area trigger the telescopes. Naturally, this quantity depends on the instrument. Moreover, the area is a function of the shower energy, the observation zenith angle, and the camera offset. Very energetic showers produce more Cherenkov photons and thus are detectable from farther away. High zenith angles, on the other hand, lead to an increased distance to the shower maximum. Therefore, low energetic showers can not be detected with high zenith angles. Whereas for high energetic showers, which produce a high number of Cherenkov photons and thus are detectable from far away, the effective area even increases with the zenith angle since a larger part of the sky is seen by the telescopes. The camera offset affects the camera acceptance and thus the effective area. To determine the effective areas, again Monte Carlo simulations are used. Because of the energy dependence, the effective areas also depend on the spectral shape. Therefore, in a full spectral analysis the effective areas have to be reproduced iteratively with the observed spectral shape until the assumed spectral shape of the effective area and the observed spectrum match.

Spectrum Flux Determination Knowledge of the event energy, the observation time and the effective area allows us to dN determine the energy spectrum. In most cases, the differential energy spectrum dE or 2 dN E dE is the quantity of interest. The binned differential energy flux in a given energy bin [E,E + ∆E) is given by

 E ∈∆E E ∈∆E  dN 1 iX 1 iX 1 =  − α  (3.4) dE ∆tlive∆E Aeﬀ (i) Aeﬀ (i) i=1...NON i=1...NOFF with

Aeff (i) = Aeff (Ei, ZA, offset) . (3.5)

Here, ∆tlive denotes the dead time corrected observation time; ∆E is the energy bin width; α represents the OFF normalization ; and Aeff (i) is the effective area for a given energy of event number i, the zenith angle (ZA), and camera offset.

Energy Threshold Intuitively, we might expect the energy threshold to be the lowest detectable energy. In- deed this definition is not used, since this quantity is difficult to determine and depends on the observation time. Instead, the energy threshold is the energy for which the differential gamma-ray rate is maximal. The differential energy rate is given by the energy spectrum 34 Methods and Algorithms

dN dE multiplied with the effective area Aeff : dR dN (E) = (E) A (E) . (3.6) dE dE eff As the effective area depends on the simulated spectrum (which should agree with the source spectrum), the energy threshold also depends on that spectrum. For convenience dN −2.6 the Crab spectrum is assumed, i.e. dE ∼ E , if not specified otherwise.

3.2 Timing Analysis

Search for pulsed VHE gamma-ray emission means in the first place to search for a time periodicity in the measured data. Naturally, first of all the standard analysis must be applied to the raw data in order to suppress hadronic background, select from the remaining events those in the signal region, reconstruct their energy, and estimate the background from neighboring regions. At this point, the timing analysis begins. Given the arrival timestamp of the gamma-ray candidates from the central trigger, it is the timing analysis’ responsibility to search for time periodicities and possibly determine the corresponding significances. Actually, the whole timing analysis can be done separately using external software tools. Although powerful and well tested tools like the radio astronomers program TEMPO [54] exist, there are mainly two reasons not to use such an external program. First, there are some subtle differences between radio pulsar observations and those in the VHE gamma-ray regime mostly caused by the fact that radio observatories measure continuously electromagnetic waves’ intensities whereas VHE observatories measure single photons and in general have much less statistics. Second, it is very desirable to have all analysis tools in a single framework for the purpose of usability as well as easier further developments. Therefore, the full timing analysis was implemented into the existing H.E.S.S. analysis framework. Part of the timing analysis is:

• Time of arrival corrections Assuming a periodic signal in the gamma-ray candidates has been found, a more precise analysis of a long exposure would quickly reveal that the periodicity is not completely constant over time. This phenomenon becomes comprehensible as soon as we admit that our observation frame is not inertial since we are using telescopes on a rotating Earth orbiting the Sun. This among other eﬀects, e.g. pulsars possess an intrinsic spin-down, make the time correction inevitable.

• Timing models To perform these time of arrival corrections, besides position information of all objects involved, a physical timing model is required for the calculations. In the case of binary pulsars, this becomes exceedingly true and simultaneously very complex since no relative position information is available and General Relativity eﬀects have to be taken into account. 3.2 Timing Analysis 35

• Statistical tests When all timing corrections are successfully applied on the gamma-ray candidates, statistical tests are performed to calculate the probability for the corrected timing data to be compatible with a flat distribution. This yields the final variability significance.

For a better understanding of the following text, a few terms must be introduced here. Rotational phase or phase: Corresponding to the rotation of a neutron star it is common to speak of a rotational phase or just the phase which denote the position of the light beam on the surface or equally on the pulsar waveform at a particular time of observation. Generally, the rotational phase and the phase are described by a number between 0 and 2π and between 0 and 1, respectively. Time of arrival (TOA): In radio pulsar observations, roughly five minutes of data are averaged with the predicted pulsar frequency and corrected for the Doppler offset at the observatory. This already produces a pulse profile, which is matched with a high signal-to-noise template to determine one effective TOA. Typically, this TOA for each sample is at the peak position. In VHE observations, each TOA corresponds to one single event timestamp. As a consequence, not all such TOAs are located at a special pulse position, in contrast to radio TOAs. Phasogram and light curve: Once the pulsar period is known, a compact way to gather and display the timing data is represented by the so-called phasogram or light curve. In this approach the data is simply folded with the rotational period producing one mean phase profile. Again, radio and optical pulsar observations differ from those at X-ray and Gamma- ray wavelengths in that the former measure a continuous intensity over time whereas the latter detect single photons. Hence, radio light curves depict the averaged intensity over one rotational period and show a smooth curve. In the case of single photon detection, each event is filled into a binned histogram, namely the phasogram, according to its phase derived from the event timestamp.

3.2.1 Time of Arrival Corrections Before deriving the phase, we ﬁrst have to correct the observed time of arrival (TOA) for several eﬀects, including

• pulsar speciﬁc behavior, mainly the spin down and glitches (if any),

• the acceleration of the observatory on the rotating Earth orbiting the Sun,

• dispersive delays in the interstellar medium (only relevant in radio frequencies),

• orbital acceleration of binary pulsars,

• and of course all sorts of clock related delays. 36 Methods and Algorithms

Apparently, a detailed timing model containing all these eﬀects is needed to correct the observed TOAs for the analysis of pulsar timing data. Aiming to express the pulsar rotation in a reference frame co-moving with the pulsar, we start with a Taylor expansion of the spin frequency

dν 1 d2ν ν (t) = ν + (t − t ) + (t − t )2 + ... (3.7) 0 dt 0 2 dt2 0 around a reference time t0, where ν0 = ν (t0) = 1/P0 with P0 being the pulsar period. For most time spans, both the ﬁrst and second time derivativeν ˙ andν ¨ can be approximated as constant. This is in particular true for millisecond pulsars which are very stable over time, thus have a very smallν ¨. By serially numbering the pulsar’s rotations with n and taking into account ν = dn/dt we ﬁnd 1 dν 1 d2ν n = n + ν (t − t ) + (t − t )2 + (t − t )3 + ... (3.8) 0 0 0 2 dt 0 6 dt2 0 where n0 is the pulse number at the reference time t0. From this, the phase is simply calculated as the residual with respect to the last integer value of n,

φ = n − bnc (3.9) where b...c denotes the ﬂoor function. With the accurate knowledge of the spin down parameters and the observed TOAs (of a pulsating source), we therefore expect from Eqn. 3.8 integer values of n or likewise φ = 0 corresponding to the pulses. This, however, holds only if the arrival times were observed in an inertial frame of the pulsar. Since the telescopes are located on the rotating Earth which in turn is orbiting the Sun, the observation frame is obviously not inertial. The problem is solved by transferring the topocentric TOAs, measured with the observatory clock, to the center of mass of our Solar System, the so-called Solar System Barycenter (SSB), as the best approximation to an inertial frame available. A similar approach is applied to the pulsar, if it is part of a binary system. Though this correction is more tricky due to the absence of accurate position and motion information. The timing model is concisely speciﬁed in the following equation yielding the corrected arrival time D t = t + ∆t − TOA clock f 2 +∆tRoemer, + ∆tShapiro, + ∆tEinstein, +∆tbinary (3.10) where tTOA, tclock, D, and f denote the observed TOA, the clock corrections, the dispersion measure, and the observing frequency, respectively. The second line contains all correction terms within the Solar System. Finally in the third line, ∆tbinary is the correction term for binary acceleration (if any). All terms are discussed in detail below. 3.2 Timing Analysis 37

Figure 3.4: Taken from International Earth Rotation Service [55]. UTC in blue follows TAI (horizontally) and approximates UT1 in red to 0.9 s.

3.2.2 Clock and Frequency Corrections

As for H.E.S.S., observatory times are usually obtained by clocks that run the Coordinated Universal Time (UTC) and are linked to the Global Positioning System (GPS) time. UTC is a compromise between the highly stable International Atomic Time (TAI) and the irregular Earth rotation embodied in the Universal Time (UT1). In contrast to TAI, UT1 utilizes the Earth rotation as a clock and hence maps 24 hours on one full rotation which is called a solar day. Due to the general but irregular slow down of the Earth’s rotation, the atomic time diverges from the Earth time. In other words, if standard time was based upon TAI, coincidence with the solar day could not be maintained. Therefore, the standard time is UTC which utilizes the high precision of TAI but introduces an additional time offset, namely the leapseconds, to maintain the difference TAI - UT1 to be less than 0.9 seconds. Fig. 3.4 illustrates the evolution of UTC and UT1 as a difference to atomic time. Now that we adequately understand UTC, as a matter of course the leapseconds are removed from the TOAs to gain the atomic time TAI. For the sake of completeness, the term D/f 2 in Eqn. 3.10 was presented. It is, obviously, for typical values of D of the order of one entirely negligible for high frequencies, as in the case of Gamma-rays. In radio observations, on the other hand, it describes the pulse delay due to dispersion in the interstellar medium and has to be taken into account. 38 Methods and Algorithms

Jupiter SSB ~r 3 ~n Source direction

Sun ~robs

~r2

Observatory ~r1 Earth

Figure 3.5: Illustration of the Roemer delay in the Solar System.

3.2.3 Solar System Corrections

All correction terms in the second line of Eqn. 3.10 represent time transfers and delays within our Solar System.

Roemer Delay

First, the classical Roemer delay is the transfer of the TOAs to the SSB. This time delay is simply the dot product between the unit vector towards the source ~n and the position vector of the observatory with respect to the SSB ~robs divided by the speed of light c:

1 ∆t = ~r · ~n. (3.11) Roemer, c obs

As illustrated in Fig. 3.5, the vector ~robs is obtained from the sum of the vectors: ~r1 that connects the observatory position on the Earth’s surface with the center of the Earth (Geocenter), ~r2 which spans from the center of the Sun to the Geocenter, and ~r3 from the SSB to the center of the Sun. Note, the SSB can be located even slightly outside of the Sun depending mainly on Jupiter’s position. Timing accuracies that satisfy the needs of millisecond pulsar analysis, require the inclusion of very precise positioning information of all major solar bodies. Further, deformation and polar movement of the Earth is needed in order to express ~r1 very accurately in the SSB reference frame. The ﬁrst task, high precision positions, is achieved by using so-called Solar System ephemerides, e.g. DE200 ﬁles published by the Jet Propulsion Laboratory (JPL) [56]. Secondly, Earth orientation parameters are made available as bulletin by the International Earth Rotation Service [57]. A typical Roemer delay can be seen in Fig. 3.5 3.2 Timing Analysis 39

Sample Roemer delay

[s] 200 SSB

t 150 ∆ 100 50 Figure 3.6: Timing corrections due to the 0 Roemer delay in the SSB. -50 -100 -150 -200 52000 52100 52200 52300 52400 Epoch [MJD]

Shapiro Delay The second term of the Solar System corrections is the relativistic Shapiro delay [58, 59]. This effect embodies the signal propagation through the curved space-time near the Sun. It can be as large as ∼ 120 µs for a signal passing the Sun’s limb. Since IACTs observe photons only during night time, it seems as if the Shapiro delay due to the Sun’s gravitational field is of no importance. Nonetheless, given that all time model parameters are obtained from radio observations, it is highly desirable to apply exactly the same corrections as it is done in radio astronomy. Additionally, there is a similar Shapiro delay in binary systems due to the partner’s gravitational field, which definitely has to be taken into account. Therefore, we will discuss the Shapiro delay here. Let us begin by taking a linear version of the Parametrized Post Newtonian (PPN) metric from [60],

−4 g00 = 1 − 2 U + O c , −3 g0k = O c , −4 gmn = −δmn (1 + 2 γ U) + O c , (3.12) in which c is the speed of light and U = U (~r) is the gravitational potential (dimensionless),

X G mb U (~r) = 2 . (3.13) |~r − ~rb| c b The sum is carried out over all major bodies b of the solar system with the SSB in the origin. The parameter γ is a measure for the space-curvature, in General Relativity γ is equal to one. Although the simplified metric neglects all higher order terms in c, it is sufficient to show the relativistic timing effects we are interested in. From Eqn. 3.12 we obtain the approximated line element:

ds2 = (1 − 2 U) c2dt2 − (1 + 2 γ U) dx2 + dy2 + dz2 . (3.14)

Emitted photons from the pulsar naturally followed the null trajectory, characterized by 2 ds = 0. Integrating over this null trajectory from the space-time point (tpsr, ~rpsr) of the 40 Methods and Algorithms

Sample Shapiro delay s] µ [ 0 Shapiro t

∆ -20 -40 Figure 3.7: Timing corrections due to the -60 Shapiro delay in the SSB.

-80

-100

-120 52000 52100 52200 52300 52400 Epoch [MJD]

pulsar to the space-time point of observation (tobs, ~robs) will result in an expression for the elapsed time, written ds2 = 0 r Z tobs Z ~robs 1 + 2 γ U ⇐⇒ cdt = (dx2 + dy2 + dz2) tpsr ~rpsr 1 − 2 U

⇐⇒ c (tobs − tpsr) = |~rpsr − ~robs| +

X Gmb ~n · (~robs − ~rb) + |~robs − ~rb| + (1 + γ) 2 ln . (3.15) c ~n · (~rpsr − ~rb) + |~rpsr − ~rb| b

In the last line only the two leading terms of a Taylor expansion were used and ~n and ~rb denote the unit vector towards the pulsar and the vector to the b-th solar body, respectively. In principle, all massive bodies lying on the way of light propagation have to be taken into account in the sum over b. Most objects, however, keep a constant distance to the line of sight to the pulsar over tens of years and hence contribute only a constant time delay. In the same manner, we can neglect the gravitational ﬁeld of the pulsar. Consequently, we only include the Sun, albeit a signal passing close by Jupiter may be delayed by as much as 200 ns. Note, the term |~rpsr − ~robs| in Eqn. 3.15, the length of path, is of course equivalent to the Roemer delay plus an additive constant. Further assuming ~n ∼ ~rpsr − ~r , the denominator becomes an additive constant and we ﬁnd ! GM ~robs − ~r ∆t = 2 · ln · (1 + cos θ) , (3.16) Shapiro, c3 AU where θ is the pulsar-Sun-Earth angle, AU is the astronomical unit, and γ was set to one. Fig. 3.7 shows a typical Shapiro timing correction over the period of one year.

Einstein Delay The last barycentric correction term is the Einstein delay [59]. It describes the combined eﬀect of gravitational redshift and time dilation of an atomic clock on the Earth. Simply 3.2 Timing Analysis 41

Sample Einstein delay

[ms] 1.5

Einstein 1 t ∆ 0.5 Figure 3.8: Timing corrections due to the 0 Einstein delay in the SSB. -0.5

-1

-1.5

52000 52100 52200 52300 52400 Epoch [MJD] speaking, the eﬀect arises from the variation of the gravitational ﬁeld when the atomic clock on the Earth follows the elliptical orbit around the Sun. In fact, the Einstein delay is nothing more but the transformation to a clock running at the SSB which again is the Barycentric Dynamical Time (TDB). Again, the PPN metric (Eqn. 3.12) serves as a well suited starting point, from which we derive the proper time equation: v2 dτ 2 = (1 − 2 U(~r)) dt2 − (1 + 2 γ U(~r)) dt2. (3.17) c2 Here, τ is the terrestrial clock, t is the coordinate time, ~r (t) and ~v are the vectors to the clock and the corresponding velocity with respect to the SSB, and U(~r) is the gravitational potential at the clock’s location. v2 −2 A Taylor expansion of the integrand around c2 ∼ 0 to order O c yields Z t 2 1 v 0 τ = t − U + 2 dt . (3.18) 0 2 c v2 Here, the terms U and c2 can be interpreted as the gravitational redshift and the Doppler shift. Thus, the Einstein delay amounts to an integral of the expression 2 d∆tEinstein, X G mb v = 2 + 2 − constant (3.19) dt c rb 2 c b where the sum is again taken over all major bodies of the Solar System excluding the Earth, rb is the distance between the Earth and body b, and the additive constant is chosen in such a way that the right-hand side cancels over a long time. A high precision semi-analytical model developed at the Bureau des Longitudes [61] for Eqn. 3.19 was used throughout this analysis. Fig. 3.8 shows the Einstein delay over one full year.

3.2.4 Relative Motion As long as there is no further motion or acceleration between the pulsar and the SSB, the Solar System correction terms discussed above are suﬃcient. If we now, however, consider 42 Methods and Algorithms

Sample Parallax correction Sample proper motion correction s]

µ 1.5

[ -7.2 [ms] -7.4 1 Parallax

t -7.6 ∆

Proper motion 0.5

-7.8 t ∆ -8 0 -8.2 -8.4 -0.5 -8.6 -1 -8.8 -1.5 -9 52000 52100 52200 52300 52400 52000 52500 53000 Epoch [MJD] Epoch[MJD]

Figure 3.9: Timing corrections due to relative motion of the source. Left: Parallax; Right: proper motion a pulsar having constant velocity with respect to the SSB, there will be some additional timing effects. The transverse component of the velocity vt will change the unit vector towards the pulsar ~n in Eqn. 3.11 and will therefore be observable as a proper motion µ. In contrast, the radial component vr will not be measurable, although it induces a constant Doppler effect. More timing correction effects can be derived from Eqn. 3.15 by adding a constant velocity to the pulsar position ~rpsr → ~rpsr + ~v · t. One is the so-called Shlovskii effect that arises from the increase of distance d to the pulsar when there is transverse motion, v2 ∆t = t t2, (3.20) Shlovskii 2 d c also known in classical astronomy as ”secular acceleration”. d denotes the distances to the source. Typically, tShlovskii is negligible due to the far distance to the pulsar. Another timing effect resulting from Eqn. 3.15 is the annual parallax, written: 1 ∆t = − (~r · ~n)2 − |~r |2 (3.21) π 2cd obs obs For a distance d = 1 kpc the effect amounts to a maximum change in the delay of . 2.0 µs and is therefore only observable for a few millisecond pulsars where it can provide an accurate distance estimate. In the timing correction model, the parallax is incorporated by one additional parameter. Typical parallax and proper motion timing corrections are shown in Fig. 3.9.

3.2.5 Binary Corrections If the pulsar is a member of a binary system, further timing corrections will be necessary due to additional acceleration eﬀects. In principle, the pulsar position vector ~rpsr has to be exchanged by ~rpsr = ~rbb + ~v · t + ~r1, (3.22) 3.2 Timing Analysis 43

00 auxiliary circle y z0, z00 z P 2 orbit pulsar ~r y00 1 y0 i center of periapsis y ellipse BB 00 1 E φ x ω ~r2 ~n x, x0 x00 binary companion observer

Figure 3.10: Geometry of a Binary System. On the left side, the two rotations from the plane of sky (perpendicular to the line of sight) to the plane of orbit are displayed. The right sketch illustrates the binary system in the center of mass frame including some angles and vectors (explained in the text).

where ~rbb points to the binary barycenter (BB), ~v is the constant velocity of the BB (compare Relative motion in 3.2.4), and ~r1 represents the time dependent position vector of the pulsar with respect to the BB. This leads to minor effects in the already discussed Solar System correction terms. However, there are more or less similar corrections within the binary system itself, which definitely have to be taken into account. In contrast to our Solar System, precise positioning information for the binary system is evidently not directly available. Therefore, a parameterized model describing the motion is needed. The first successful model was developed by Blandford and Teukolsky (1975,1976, hereafter BT) [62]. They assumed a slowly precessing Keplerian orbit in which relativistic effects are small perturbations. With an increasing level of accuracy in the observations, better models were demanded. Major progress was achieved by applying a solution of the post Newtonian two-body problem. This lead to the development of many precise models , e.g. Epstein-Haugan (1977,1979), and Damour-Deruelle (1986, hereafter DD) [63]. Since PSR J0437-4715 is in a binary system which was described with the BT and DD models, these two models will be explained in some detail. First, let us consider some geometrical relations with regard to the Keplerian ellipse (see Fig. 3.10). The plane of orbit is tilted by two angles with respect to the plane of the sky. Beginning from the coordinate system of the plane of sky (x, y, z), a rotation around the x-axis describes the inclination with the angle i. Then, a rotation around the new z0-axis with the periastron angle ω leads to the plane of orbit (x00, y00, z00). Both rotations are displayed in Fig. 3.10 on the left side. The source direction ~n viewed from the observatory, now pointing towards the BB, is also drawn. 44 Methods and Algorithms

In the plane of orbit, we assume the orbit to be of elliptical shape, having the BB identical to the right focus point in the origin, and with the major axis aligned on the x00-axis (see Fig. 3.10, right side). The pulsar and its companion are characterized by their masses m1, m2 and their spherical coordinates

~r1 = (r1, π/2, φ) , ~r2 = (r2, π/2, φ + π) , (3.23) with respect to the BB. Here, φ denotes the true anomaly, i.e. the angle between the pulsar position, the BB, and periapsis (point of least distance of the orbit to the BB). In addition, it is useful to deﬁne the eccentric anomaly as E = ∠ (periapsis, center of ellipse, P). P is the intersection point of an auxiliary circle around the ellipse center (with a radius equal to semimajor axis) and a line perpendicular to the major ellipse axis, that goes through the pulsar position (compare Fig. 3.10, right side). From Kepler’s equation, we ﬁnd the relation for the eccentric anomaly to be 2π E − e sin E = (t − T0) , (3.24) Pb

q b2 with e being the eccentricity, i.e. e = 1 − a2 where a and b are the semimajor and semiminor axes of the ellipse, respectively; T0 is the epoch of periastron (time of periapsis passage); t is the current time; and Pb is the binary system period.

Blandford-Teukolsky Model Blandford and Teukolsky [62] assumed in their model a classical Kepler ellipse for the binary orbit. So, the radii r1 and r2 in Eqn. 3.23 are given by m m a 1 − e2 r = 2 r, r = 1 r, r = = a (1 − e cos E) , (3.25) 1 M 2 M 1 + e cos φ with m1, m2, a, e, φ, and E as deﬁned above and M = m1 + m2. Further, a PPN metric similar to Eqn. 3.12 was assumed with the gravitational potential equal to

−G m1 G m2 U(~r, t) = U1(~r, t) + U2(~r, t) = 2 − 2 . (3.26) c |~r − ~r1 (t)| c |~r − ~r2 (t)| BT derived and discussed three time corrections in accordance with the Roemer delay, the Shapiro delay, and the Einstein delay in the Solar System. This time, the Einstein delay relates the proper pulsar time tp at the pulsar’s position ~r1 with the coordinate time t: 2 dt ~r˙ p = 1 + U(~r , t) − 1 + Oc−4 . (3.27) dt 1 2 c2 By applying 2 2 G m 2 1 ~r˙ = v2 = 2 − , (3.28) 1 1 M r a 3.2 Timing Analysis 45 which can be derived using Kepler’s energy relation G m m E = − 1 2 , (3.29) 2 a and dropping all constant terms, we ﬁnd: dt G m G m2 1 p = 1 − 2 − 2 . (3.30) dt c2 r c2 M r

Note that U1 is not inﬁnite, since the emission region is located somewhere beyond the pulsar surface. The integration is most easily done in terms of the eccentric anomaly E. Again, dropping constant terms and overall multiplicative constants, the integrated Eqn. 3.30 results in G m (m + M) P t = t − 2 2 b E p c2 a M 2π G m (m + M) P = t − 2 2 b e sin E. (3.31) c2 a M 2π Next, we discuss the Shapiro delay. Let us begin from the last term of Eqn. 3.15,

1 2 m2 ln (3.32) r − ~r · ~n from the point of view of the pulsar, the constant numerator already dropped, γ set to one, ~n still being the unit vector from the Earth to the pulsar, and ~r = ~r1 − ~r2. Applying all necessary rotations, we ﬁnd ~r · ~n = r sin i sin (ω + φ) . (3.33) Thus, the Shapiro delay amounts to 1 + e cos φ ∆t = 2 m ln . (3.34) Shapiro,bin 2 1 − sin i sin (ω + φ) BT now argue, that since the maximal variation of this delay is 14 µs m2/m for ◦ ◦ i = 0 and 89 µs m2/m for i = 89 , it would not be measurable and is therefore omitted from the binary corrections. Next, BT considered the analogon to the Roemer delay, written 1 ∆t = − ~n · ~r Roemer,bin c 1 2 a1 1 − e sin i sin (ω + φ) = − , (3.35) c (1 + e cos φ) where the coeﬃcient a · m2/M is absorbed in a1. Further, Kepler trigonometry relating the true anomaly φ with the eccentric anomaly E (tem) at the time of emission tem,

1/2 1 − e2 sin E cos E − e sin φ = , cos φ = (3.36) 1 − e cos E 1 − e cos E 46 Methods and Algorithms allows us to eliminate φ from Eqn. 3.35:

a 1/2 ∆t = − 1 sin i sin ω (cos E − e) + cos ω 1 − e2 sin E . (3.37) Roemer,bin c −2 Finally, BT incorporate the Einstein delay from Eqn. 3.31. Since tem = tp 1 + O c , we can write the complete BT binary correction as

tp = t − α (cos E − e) − (β + γ) sin E, (3.38) where 1/2 a α = χ sin ω, β = 1 − e2 χ cos ω, χ = 1 sin i, (3.39) c and 2 G m2 (m2 + M) Pb e γ = 2 2 . (3.40) c a1 M 2π Yet, Eqn. 3.38 is an implicit equation in that the eccentric anomaly E depends on tp 0 or equally tem. This, however, can be solved by deﬁning another eccentric anomaly E :

0 0 2π E − e sin E = (t + T0) . (3.41) Pb

Comparing Eqn. 3.41 with the corresponding one for E (tp), we ﬁnd P t − t ≈ b 1 − e cos E0 E − E0 . p 2π 0 On the other hand, E = E and tp = t to zeroth order. Thus, 0 0 tp − t ≈ −α cos E − e − (β + γ) sin E , and so 0 0 0 2π (cos E − e) + (β + γ) sin E −2 E − E = − 0 + O c . Pb (1 − e cos E ) Therewith, Eqn. 3.38 becomes the ﬁnal BT correction formula

0 0 ∆tBT,bin = − α cos E − e − (β + γ) sin E (α sin E0 − β cos E0) [2π (cos E0 − e) + (β + γ) sin E0] − 0 . (3.42) Pb (1 − e cos E ) Secular eﬀects can be accommodated by the replacements:

ω → ω +ω ˙ · t, χ → χ +χ ˙ · t,

e → e +e ˙ · t, Pb → Pb + P˙b · t.

Here,ω ˙ describes the precession of the longitude of periastron; the change of orbital period P˙b can be a damping eﬀect like tidal dissipation or the emission of gravitational waves; ande ˙,χ ˙ describe changes in the binary orbit. Fig. 3.11 shows typical timing corrections for the BT and DD model. 3.2 Timing Analysis 47

Sample BT model corrections Sample DD model corrections 4 4 [s] [s] 3 3 BT, bin DD, bin t t ∆ 2 ∆ 2

1 1

0 0

-1 -1

-2 -2

-3 -3

-4 -4 52000 52005 52010 52015 52000 52005 52010 52015 Epoch [MJD] Epoch [MJD]

Figure 3.11: Timing corrections in a binary system. Left: BT model; Right: DD model

Damour-Deruelle Model In 1986, Damour and Deruelle devised a more theory independent relativistic timing model [63] (DD model) which is based on an elegant new analytic solution for the post Newtonian two-body problem [64]. The model is valid under very general assumptions about the theory of gravity. It diﬀers from the BT model mainly in two aspects: The Shapiro delay is taken into account (important for large inclination angles, compare Eqn. 3.34) and the DD model allows for periodic eﬀects in the orbital motion. All binary timing corrections from the DD model can be summarized in

∆tDD,bin = ∆tRoemer,bin + ∆tEinstein,bin + ∆tShapiro,bin + ∆tAb,bin, (3.43) so that t = tTOA + ∆tSSB + ∆tDD,bin yields the overall corrected time in the DD model, similar to the initial timing formula in Eqn. 3.10. In the equation above, ∆tSSB is the sum of all SSB corrections, ∆tRoemer,bin denotes the Roemer delay in the binary system, ∆tEinstein,bin and ∆tShapiro,bin are the relativistic Einstein and Shapiro delays and ∆tAb,bin describes aberration timing delays caused by rotation of the pulsar. Going into some detail of these timing corrections, we first come back to the periodic effects in the orbital motion of the pulsar. In the BT model only a secular drift of the longitude of periastron is included, i.e. it allows forω ˙ , whereas the DD model contains both secular and quasi-periodic effects in ω given by

ω = ω0 + k Ae (u) andω ˙ = 2π k/Pb, (3.44) where Ae, u are the true and eccentric anomaly, respectively; and k is a parameter relating the two eﬀects. Here, the true anomaly is deﬁned by " # 1 + e1/2 u A (u) = 2 arctan tan , (3.45) e 1 − e 2 48 Methods and Algorithms and a generalized Kepler equation is used to relate the eccentric anomaly with the time " # t − T P˙ t − T 2 u − e sin u = 2π 0 − b 0 . (3.46) Pb 2 Pb

The post Keplerian Roemer delay is given by

∆tRoemer,bin = −χ sin ω [cos u − e (1 + δr)] 1/2 h 2 2i −χ 1 − e (1 + δθ) cos ω sin u, (3.47) in almost the same manner as in the BT model (compare Eqn. 3.37). The post Keplerian parameters δr and δθ represent post Newtonian variations in the orbital motion of the order of (v/c)2. Unaltered, the Einstein delay is deﬁned just as in the BT model, i.e.

∆tEinstein,bin = −γ sin u, (3.48) with the same γ as in 3.40 in the BT model. Next, the Shapiro delay ∆tShapiro,bin = 2r log 1 − e cos u − s sin ω (cos u − e) 1/2 + 1 − e2 cos ω sin u , (3.49) introduces two additional measurable parameters s = sin i and r which characterize the shape and range of the Shapiro delay. Last, the aberration delay is determined by

∆tAb,bin = −A [sin (ω + Ae (u)) + e sin ω]

−B [cos (ω + Ae (u)) + e cos ω] , (3.50) where A and B are new parameters. Indeed, the parameters A, B, δr, and δθ are very small quantities and nearly degenerated with other parameters. Thus, they are not measurable over a reasonable time scale, as DD point out.

3.2.6 Timing Model Parameters The timing correction models contain several free parameters, which are not known a- priori or only with limited accuracy at the time the pulsar is ﬁrst analysed. According to their occurrence in the models, these parameters can be categorized into three groups.

• Astrometric parameters: Including the pulsar position, its proper motion, and the solar parallax. Clearly, the position is known from the observation, though its precision can be highly improved within the parameter determination, which is explained below. Proper motion and the parallax are only observable after long time-spans. 3.2 Timing Analysis 49

• Pulsar / Spin parameters: That is, the rotational pulsar frequency and its time derivatives. While the frequency itself, obviously, is essential for timing analysis, its time derivatives become significant only for longer observations. The first detection of a pulsar requires quite some computational effort to find the proper frequency. In most cases, this search is performed in the frequency space by applying a Fast- Fourier-Transformation to the TOAs.

• Binary parameters: For pulsars belonging to a binary system, these parameters characterize the orbital motion. Moreover, by introducing post Keplerian parameters, it is possible to test diﬀerent theories of gravity.

A complete list of all parameters is given in Table 3.1. To determine the model parameters, a least-squares ﬁt is performed

N 2 X φ (ti) χ2 = (3.51) σ /P i=1 i using N rotational phases φ, calculated from the corrected TOAs ti with the loose parameters and the discussed timing models. Here, σi is the estimated uncertainty of ti and P is the pulsar period. Along with the partial derivatives ∂φ/∂pj for the fitted parameters pj, a standard minimization of the test statistic of Eqn. 3.51 yields the desired values of the parameters. Admittedly, the whole process of fitting the TOAs to find the timing model parameters is elaborate and not entirely straight forward. For instance, with a minimal set of starting parameters, one typically begins with a small set of TOAs close in time and a fit for only a few parameters. A more detailed description can be found in [65], in which the very common software package TEMPO [54] is utilized to perform the timing corrections and the least-squares fit. Once the model fit is successful, the postfit phase residuals should have the character of random white noise, see Fig. 3.12, left side. Incor- rect or incomplete timing models cause systematic structures in the phase residuals which can be used to identify the incorrect adjusted or missing parameter, compare Fig. 3.12 right side. So far, only radio observations allow a determination of the model parameters. In contrast to high energy measurements, radio telescopes uncover a pulse profile already after a few minutes of observation, if the correct rotational frequency is known. Thus, radio observations comprise a dramatically higher statistics and furthermore each TOA almost certainly represents a phase time, not some arbitrary background arrival time. Therefore, in high energy pulsar searches, all model parameters are taken from previous model fits which were carried out with radio data. The set of fitted model parameters is typically called ephemeris and can be obtained from pulsar databases. Such a database can be found, for example, at the Australian Telescope National Facility (ATNF) Pulsar Group [66]. The ephemerides provided by ATNF are produced with TEMPO and are typically available for several timing models, such as BT and DD. For high energy observations, the ATNF provides a reduced ephemeris, called Gamma Ray Observations (GRO) ephemeris. These GRO ephemerides include the most important astrometric and pulsar parameters 50 Methods and Algorithms

Parameter Short description Used in Type FRQ SSB BT DD α right ascension - XXX δ declination - XXX µα proper motion in α direction - XXX µδ proper motion in δ direction - XXX

t0,pos reference position epoch - XXX astrometric PX solar parallax - X -- D dispersion measure - X -- t0 reference epoch X --- ν = 1/P pulsar rotational frequency X --- ν˙ time derivative of ν --- X pulsar ν¨ second time derivative of ν X --- Pb binary period - - XX T0 epoch of periastron - - XX e binary eccentricity - - XX χ projected semimajor axis - - XX ω longitude of periastron - - XX γ gravitational redshift and time dilation - - XX s = sin i shape of Shapiro delay - - - X r distance between binary partners - - - X ω˙ drift of periastron - - XX binary system ˙ Pb damping eﬀect - - XX e˙ change of binary orbit - - XX χ˙ change of binary orbit - - XX Table 3.1: Parameters of the timing models grouped by their type. The ’Used in’ columns indicate in which part of the correction models each parameter is used: FRQ stands for pulsar frequency corrections, embodied in Eqn. 3.8; SSB refers to all corrections within our Solar System; BT and DD denote binary correction terms in the two diﬀerent models, respectively. 3.2 Timing Analysis 51

Perfect least-squares fit Position displaced s] µ 30 350 300 20 250 10 Residuals [ Residuals [ms] 200 0 150 -10 100

-20 50 0 -30 -50 -40 -100 49400 49600 49800 50000 50200 49400 49600 49800 50000 50200 Epoch [MJD] Epoch [MJD]

Figure 3.12: Timing phase residuals from PSR J0437-4715 radio observations, provided by and analysed with TEMPO [54]. Left: postfit phase residuals for a perfect fit. Right: phase residuals for the same data and model parameters but with the position parameter displaced by 0.5 arc-seconds. Note the different scales on the y-axes. and optionally the main BT parameters. Since most parameters describing time derivatives are missing, GRO ephemerides are valid for a short time only. The developed H.E.S.S. timing software is able to read in original TEMPO parameter files for the BT and DD model as well as GRO files.

3.2.7 Statistical Tests for Periodicity Search

Once all the arrival times have been corrected, it is possible to search for deviations from a flat distribution in the phasogram. A quantitative approach is to use statistical tests for the examination of all event phases. In the following, a list of statistical tests for periodicity searches is given. Each test is briefly explained and advantages and disadvantages with regard to pulsar profiles are discussed.

Pearson χ2-Test

In the simple and well established χ2-Test, see [67] for a complete reference, the N continuous phases are arranged into a certain number b of bins. Since this test depends on the number of bins, it is advisable to ﬁx b taking into account statistical and experimental considerations. In order to retain the χ2 distribution the expected value for each bin is required to be above ∼ 5. The experimental aspect is to consider the measurement errors in the phase and choose a comparable or wider bin size. Within the scope of this thesis all phasograms have 10 bins, meaning each bin corresponds to roughly 0.6 ms for the rotational period of PSR J0437-4715. Obviously, the null hypothesis H0 is a ﬂat distribution, P i.e. nj = n0. Here nj denotes the entries in the j-th bin and n0 = nj/b is the average number of entries per bin. 52 Methods and Algorithms

φN Figure 3.13: 2D random walk. The angle be- φ2 tween the x axis and the i-th step corresponds d to φi.

φ1 x

The test statistic is then given by

b 2 X (nj − n0) χ2 = n j=1 0 and is a measurement for how much the data diﬀers from the null hypothesis. The probability density function (PDF) for a given test statistic x is

k −1 x 2 x − 2 fk (x) = k e , (3.52) 2 k 2 Γ 2 where k is the degree of freedom, here b−1. The probability is then obtained by integrating over the PDF. In particular, the incomplete Gamma function k x Γ , 2 2 represents the probability to obtain a χ2-value which is smaller than the observed value x. Thus, this is the probability that the null hypothesis is wrong. The χ2-Test is a valuable method to search for narrow and high S2B peaks in the phasogram, like it is seen in most radio observations. On the other hand, it is not that suﬃcient if the expected signal is wide and relatively small. In those cases, other statistical 2 tests like the Zm-Test are more eﬀective. Another disadvantage is the dependence on the arbitrary number of bins.

Rayleigh-Test In contrast to the χ2-Test, the Rayleigh-Test probes for fundamental sinus and cosinus harmonics. Therefore, this test is the most powerful one for sinusoidal pulse profiles, typically observed in the case of X-ray pulsars. However, it is not equally sensitive for most other more complex profiles, e.g. double peak structures or more narrow profiles. For more details and a comparison with the χ2-Test consult [68]. In this test, each event phase φi is taken as a rotational phase, i.e. a number between 0 and 2π. The test statistic is then calculated by

 N !2 N !2 1 X X Z2 = sin (φ ) + cos (φ ) (3.53) Rayleigh N  i i  i=1 i=1 3.2 Timing Analysis 53 where N is the total number of events. This test can also be regarded as a 2-dimensional random walk having a unit step for each event and the rotational phase φi as the direction angle, compare Fig. 3.13. For a large number of events and in the absence of periodic pulsation, the test statistic, suitably normalized, is approximately χ2 distributed with 2 degrees of freedom,

− x f (x) = e 2 .

2 Integrating the PDF from 0 to ZRayleigh yields the probability that the phasogram is indeed described by a sinusoidal proﬁle.

2 Zm-Test A generalization of the Rayleigh-Test proposed by Bucceri et al. (1983) [69] and Buccherie 2 & Sacco (1985) [70] is the so called Zm-Test, which copes with higher order sinusoidal proﬁles. Thus, unlike the Rayleigh-Test, it is not bound to broad and one peak proﬁles. The test statistic is calculated for a given index m, denoting the harmonics of the period, by the formula

m  N !2 N !2 2 X X X Z2 = sin (j · φ ) + cos (j · φ ) (3.54) m N  i i  j=1 i=1 i=1 which is the summation of the Fourier power in the first m harmonics. For m = 1, this recovers the Rayleigh-Test except for a marginal factor. As in the Rayleigh-Test, for a uniform phase distribution the test statistic has a PDF asymptotically as χ2 with 2m degrees of freedom, see Eqn. 3.52. Power studies of this test [71, 72, 73] show that more harmonics are needed when the pulse profile gets more narrow. In fact, each fixed index m is powerful against a specific range of pulse profiles and rather poor against the rest. This, however, introduces the problem of choosing the parameter m when no knowledge about the expected pulse shape is available a-priori.

H-Test

2 The bottleneck of having one free parameter m in the Zm-Test, was avoided by the H-Test proposed by de Jager, Swanepoel, Raubenheimer in 1989 [73] and further speciﬁed in [74]. This extended test performs an automatic scan for the optimal harmonics number m, as can be seen from the construction of the test statistic

2 H = max Zm − 4m + 4 . (3.55) 1≤m≤20

The maximum harmonics of 20 was chosen for practical reasons, whereas the number 4 was chosen with respect to power studies. In absence of a signal, the PDF was found from simulations to be approximately exponential with a mean of 0.4. Hence for a large number of events N > 100 the probability of obtaining a value larger than H is 54 Methods and Algorithms

F (φ)

1.0

D− Figure 3.14: Illustration of the Kuiper- Test. The dashed line is the null hypothesis, D+ whereas the red solid line is a sample phase distribution. D+ and D− indicate the maximum and minimum deviations, respectively.

0.0 φ1 φ2 φ3 ... φN φ 0.0 1.0

P (H) ∼ e−0.4H .

Detailed power studies [73] demonstrate the good sensitivity of the H-Test, without having arbitrary parameters. Nevertheless, for sinusoid shapes the Rayleigh-Test is better 2 2 and for very narrow peaks or three peaks the Zm-Test and χ might be more sensitive.

Kuiper-Test

A completely different approach is carried out by the so-called Kuiper-Test. This test searches for the maximum deviation with regard to the uniform phase distribution. It is a modification of the Kolmogorov-Smirnov statistics. For a detailed description see [75]. H0, the null hypothesis, is defined such that all phases φi are uniformly distributed between 0 and 1. Hence, the cumulative distribution is F (φ) = φ. First, all N phases φi are arranged corresponding to their phase value. Then, a phase distribution  0 φ < φ1  i FN (φ) = N φi ≤ φ < φi+1, 1 ≤ i ≤ N − 1 (3.56)  1 φ ≥ φN is calculated and compared with the uniform distribution, see Fig. 3.14. The Kuiper test statistic is then given by + − VN = D + D (3.57) where D+ and D− denote the maximum and minimum deviation, respectively.

+ D = max FN (φ) − F (φ) φ − D = max F (φ) − FN (φ) φ 3.3 Cross-Check of Timing Analysis 55

1/2 The probability for obtaining a value z larger or equal to N VN assuming H0 is given by ∞ 1/2 X 2 2 −2k2z2 P N VN ≤ z = PN (z) = 2 4k z − 1 e k=1 ∞ 8z X 2 2 − √ k2 4k2z2 − 3 e−2k z + ON −1 , (3.58) 3 N k=1 where the residual is negligible for N & 20. A power study of the Kuiper-Test using a simulated pulsed signal is brieﬂy described in section 3.3.3 on page 57.

Periodicity Scan In most cases, the pulsar frequency is known from radio ephemerides with high accuracy and therefore the statistical tests are applied to the calculated phases for exactly this frequency. Nonetheless, for some reasons it might be more favorable to assume a frequency interval which is then probed in small frequency steps. This is reasonable, when the given pulsar frequency is not known with suﬃcient accuracy or when there are reasons to assume a frequency shift between the pulsed emission in the radio and high energy domains. The disadvantage, however, is that with an increasing number of independent frequencies tested, the probability for a randomly high test statistic is increasing. Thus, the signiﬁcance of the applied test is decreasing. If k independent frequencies are probed for periodicity, the overall probability of a test statistic of z0 is given by k Pk (z0) = 1 − (1 − P (z ≥ z0)) . (3.59)

ν1−ν2 The choice of the frequency step ∆ν and so k = ∆ν is a compromise, in that no potential frequency should be left out, on the other hand when considering too many frequencies, Eqn. 3.59 no longer holds. The latter eﬀect is called oversampling and occurs when ∆ν is smaller than the so-called independent Fourier Spacing 1 ∆νIFS = , (3.60) Tobservation where Tobservation is the total observation time.

3.3 Cross-Check of Timing Analysis

All steps of the timing analysis were tested. This check was conducted in three different ways. First, timestamps from the measurement of optical Crab data were corrected. The resulting phasogram was then compared with the known radio phasogram. Second, a detailed comparison of the different timing correction terms with the standard radio pulsar analysis tool TEMPO was performed to obtain absolute timing errors with respect to TEMPO. Last, pulsed signals were simulated and then analysed. This allowed to compare different timing models, check the response of small variations in timing model parameters, and test the sensitivity of the statistical tests. 56 Methods and Algorithms

Optical Crab Phasogram Frequency Scan 7320 12000 value 7315 2 χ 10000

ADC Counts 7310 8000 7305 6000 7300 4000 7295

7290 2000

0 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 29.79 29.795 29.8 29.805 29.81 Phase ν [Hz]

Figure 3.15: Timing analysis results of the optical Crab data. Left: Phasogram showing one phase cycle. Right: χ2-Test applied for various frequencies ν. red vertical line: known radio pulsar frequency. Note: a high χ2 value corresponds to a low probability of the phasogram being compatible with a ﬂat distribution.

3.3.1 Optical Crab Data

In 2003, one H.E.S.S. telescope was modiﬁed to measure pulsed optical emission making use of the large mirror area. A detailed description of the experimental methods and measurements can be found in [76, 77]. The target of the optical observation was the Crab pulsar, which is known as a pulsed emitter from radio wavelengths up to energies of ∼ 10 GeV. This experiment demonstrated the functionality of the timing analysis including hardware and software by reconstructing the optical Crab phasogram. The long term time stability was estimated by determining the main peak jitter over time, resulting in a time stability of the order of 30 µs [77].

TOAs of a data sample comprising ∼ 2 min were corrected using the H.E.S.S. software and a Crab ephemeris from ATNF [66]. Fig. 3.15 shows on the left side the obtained optical Crab phasogram. The two-peak structure resembles that of radio phasograms. Also the peak phase positions are the same. On the right side of Fig. 3.15, a χ2-Test is applied for various rotational frequencies ν. Using a frequency step ∆ν = 0.0001 Hz, the maximum χ2 is found at 29.8004 Hz in excellent agreement with the known radio frequency (red vertical line) at 29.80039496 ... Hz.

However, the Crab pulsar neither has an observable proper motion nor a parallax, and it is not part of a binary system. Thus, all correction terms corresponding to these eﬀects were not tested. Further, the Crab pulsar has a comparably large period of about 30 ms which makes small timing errors of the order of ∼ 1 ms marginal. This is obviously not the case for millisecond pulsars. 3.3 Cross-Check of Timing Analysis 57

3.3.2 Comparison with Standard Radio Timing Analysis Tool

To allow for the effects not included in the optical Crab test, a detailed comparison between TEMPO and the developed H.E.S.S. timing corrections was performed. TEMPO is the most widely used program to perform pulsar timing analysis and almost all pulsar ephemerides are produced with TEMPO. In this comparison, the different timing correction terms were calculated with both H.E.S.S. and TEMPO. Therefore, TOAs and an observatory position are needed. For the TOAs, a list of times covering one full year was generated and used (we are only interested in the correction terms, thus the TOAs are arbitrary). For the observatory position, the H.E.S.S. site and the Earth Geocenter were used. The following figures show the deviations of the timing corrections between the two programs. First, Fig. 3.16 illustrates the deviations for the SSB timing corrections including all discussed correction terms with typical parameter values. The upper two plots are for a time period of one day, whereas the lower two plots are for about one year. On the left side, the observatory position is the Earth’s Geocenter, thus no transformation from the Earth’s surface to the SSB frame is required. On the right side, plots use the H.E.S.S. site position. The timing corrections are in very good agreement, i.e. the difference is . 2 µs. When using the Geocenter, the agreement is even better, i.e. . 15 ns. The coordinate transformation from the Earth’s frame to the SSB’s frame, which also takes into account such subtle corrections as the polar motion, causes a systematic but negligibly small timing difference. Next, we check the binary correction terms. Fig 3.17 represents the time deviation of the corresponding correction terms, left for the BT model and right for the DD model. In both cases, the parameter values were taken from an ephemeris for PSR J0437-4715. Since both models were implemented in analogy to the ones in TEMPO, it is not very surprising that the deviations are well below 1 ns. Concluding, it is safe to say the developed H.E.S.S. timing corrections have a maximal over-all deviation of order 2 µs from TEMPO. Therefore, the developed H.E.S.S. timing corrections, implemented within this thesis, satisfy the needs for millisecond pulsar analysis.

3.3.3 Simulation of a Pulsed Signal

Finally, a simulated pulsed signal was used to test the whole timing analysis chain, compare the sensitivity of diﬀerent statistical tests with respect to the signal to noise ratio, and also check the response of the phasogram to little variations in timing model parameters. The simulation generates a certain number of event timestamps, some belonging to the assumed signal and the others to randomly distributed noise. The ratio of these two numbers is ﬁxed by the Signal to Background (S2B) parameter. The time between two events is exponentially distributed and hence calculated by

log (RND) t = − , (3.61) r 58 Methods and Algorithms

SSB Cross-Check, Geocenter SSB Cross-Check, H.E.S.S.-Site s] µ [ [ns] 10 2 1.5 5

SSB, H.E.S.S. SSB, H.E.S.S. 1 t t ∆ ∆ - 0 - 0.5

-5 0 SSB,TEMPO SSB,TEMPO t t

∆ ∆ -0.5 -10 -1 -15 -1.5 52000.5 52001 52001.5 52000.5 52001 52001.5 Epoch [MJD] Epoch [MJD] SSB Cross-Check, Geocenter SSB Cross-Check, H.E.S.S.-Site s] µ [ [ns] 15 2 10 1 SSB, H.E.S.S. SSB, H.E.S.S.

t 5 t ∆ ∆ - - 0 0

SSB,TEMPO -5 SSB,TEMPO t t -1 ∆ ∆ -10 -2 -15 52000 52100 52200 52300 52000 52100 52200 52300 52400 Epoch [MJD] Epoch [MJD]

Figure 3.16: SSB cross-check with TEMPO. Upper Left: One day comparison with the observatory in the Geocenter. Upper Right: One day comparison with the H.E.S.S. site as the observatory position. Lower Left: One year comparison with the observatory in the Geocenter. Lower Right: One year comparison with the H.E.S.S. site as the observatory position. 3.3 Cross-Check of Timing Analysis 59

Binary BT model Cross-Check Binary DD model Cross-Check

[ns] 0.15 [ns] 0.15

0.1 0.1 BT, H.E.S.S. DD, H.E.S.S. t t ∆

0.05 ∆

- 0.05 -

0 0 BT,TEMPO DD,TEMPO t t

∆ -0.05

-0.1 -0.1

-0.15 -0.15 52000 52005 52010 52000 52005 52010 Epoch [MJD] Epoch [MJD]

Figure 3.17: Binary cross-check with TEMPO over a period of 10 days. Left: For the BT model Right: Using the DD model where r is the given event rate and RND denotes a random number between 0 and 1 according to a flat distribution. To correctly simulate signal events, the obtained random arrival times have to be slightly shifted in time until they match a signal pulse. Further- more, the signal events are smeared by a given function in order to produce a certain pulse profile. To better reflect realistic H.E.S.S. observation conditions, the events are grouped into runs of roughly 30 min length. The start time of each run can be specified and thus allows to spread the simulated runs over any given time span. In the simulations performed here, 25 h observation time and a gaussian signal shape with a width of 1 ms were assumed, see Table 3.2.

Parameter Value Observation time 25 h Signal to Background ratio (S2B) 0.0, 0.1, 0.15, and 0.2 Total event rate 0.02 Hz Signal shape gaussian with a width of 1 ms Ephemeris for PSR J0437-4715

Table 3.2: Parameters of the pulsed timing simulation.

The choice of a signal having a gaussian profile is reasonable, considering the large spread of the X-ray pulse profiles in comparison to the radio profile. However, we do not know the possible peak shape for VHE gamma-rays. The assumed total event rate of 0.02 Hz corresponds to roughly 35 events per run after selection cuts. This is a reasonable average number of events passing all Hillas selection cuts including the θ2 cut for typical observation conditions. The simulation was carried out with S2B ratios of 0, 0.1, 0.15, and 0.2. Then, the 60 Methods and Algorithms generated event timestamps were analysed using the H.E.S.S. timing analysis. In the case of no signal events, the phasogram should be flat and all statistical tests should follow their PDF. With an increasing S2B, on the other hand, the gaussian signal shape should become visible in both the phasogram and the statistical tests (indicated by the probability). Therefore, these simulations allow us to compare the power of the different statistical tests, assuming a gaussian signal. Fig. 3.18 shows two rotational phases of all four phasograms corresponding from top to bottom to a S2B ratio of 0.0, 0.1, 0.15, and 0.2. In the top figure, that is for no signal events at all, we find the phasogram to fluctuate a little. This, however, is statistically in agreement with the assumed exponentially distributed time between any two events. Below, the three figures increasingly reveil the simulated gaussian signal. A more quantitative approach to estimate the pulse significance is given by the statistical tests, see Figs. 3.19, 3.20, 3.21, and 3.22. All statistical tests, with the exception 2 of the higher order Zm tests that mainly test multiple peak structures, were applied to each S2B phasogram. In each case, a frequency scan around the simulated frequency was performed to make sure that the correct frequency is found and also to gain some statistics for the test distributions. The frequency step ∆ν was chosen to be 10−6 Hz, which is one order of magnitude below the limit to flatten the whole phasogram, and a total of 1000 frequency steps was used. On the left, for each statistical test, the test statistic as a function of the frequency is shown. A vertical red line indicates the frequency used for the simulation of the signal. Plots on the right side show the corresponding test distributions with the obtained test statistics shown as red bins . The normalized theoretical test PDFs are drawn as solid black lines. All statistical tests nicely follow their PDFs for a S2B ratio of 0.0, compare Fig. 3.19. This assures the test statistics are distributed as predicted under the null hypothesis. Next, when looking into Fig. 3.20 which depicts the statistical tests for the simulation 2 with a S2B ratio of 0.1, we see a weak indication of the signal in the Z1 test and more vaguely in the Kuiper test. The χ2 and H tests also give quite high test statistics with good probabilities but are not powerful enough to distinguish the correct frequency among the 1000 scanned frequencies. In the next higher S2B simulated dataset, having a ratio of 0.15, we find a clear 2 indication of the included signal visible in the Z1 , H, and Kuiper tests, see Fig. 3.21. Yet, the χ2 test statistic has not changed much compared to the S2B ratio of 0.1. Therefore, it is still not sensitive enough to determine the correct frequency. Finally, for a S2B ratio of 0.2 all statistical tests evidently detect the simulated signal among all scanned frequencies, compare Fig. 3.22. 2 Apparently, the Z1 test followed by the Kuiper test seem to be the most powerful tests for such a gaussian signal shape in that they detect the signal for the lowest S2B ratios and further fairly well reject wrong signal frequencies. In addition to the power studies of the statistical tests, the simulation was used to check the required precision of the timing model parameters. Not very surprisingly, it turned out that the most sensitive parameter with respect to tiny variations is the pulsar frequency itself. A simple demonstration is displayed in Fig. 3.23. A simulated dataset, comprising 50 h observation time and a very high S2B ratio of 10.0, all other parameters 3.3 Cross-Check of Timing Analysis 61

phasogram, S2B 0.0 300

250 counts/bin 200

150

100

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Rotational phase phasogram, S2B 0.1 300

250 counts/bin 200

150

100

0 Figure 3.18: Phasograms showing two phase 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Rotational phase cycles for the simulations corresponding from phasogram, S2B 0.15 top to bottom to a S2B ratio of 0.0, 0.1, 0.15, 300 and 0.2.

250 counts/bin 200

150

100

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Rotational phase phasogram, S2B 0.2 300

250 counts/bin 200

150

100

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Rotational phase 62 Methods and Algorithms

χ2 test χ2 distribution 35

2 value

Entries 10 2 30 χ

20 10 15

5 1

173.6875 173.688 173.6885 0 10 20 30 ν [Hz] χ2 value

2 Z1 test Z1 distribution 14 value Entries 2 1

Z 12 102 10

8 Figure 3.19: Statistical tests 10 6 applied to the simulated data

4 with a S2B ratio of 0.0. 1 2 Left plots: Diﬀerent statistical tests applied for various fre- 173.6875 173.688 173.6885 0 5 10 15 ν [Hz] Z2 value 1 quencies ν. Vertical red line: Simulated signal frequency. H test H distribution 18 Right plots: Correspond- Entries

H value 16 102 ing statistical test distributions. 14 Red bins: Obtained test statis- 12 10 tics. Solid black line: Normal- 10 8 ized theoretical PFD. 6

4 1 2

173.6875 173.688 173.6885 0 5 10 15 ν [Hz] H value

Kuiper test Kuiper distribution 0.06

2 0.05 Entries 10

Kuiper value 0.04

0.03 10

0.02

0.01 1 0 173.6875 173.688 173.6885 0 0.02 0.04 0.06 ν [Hz] Kuiper value 3.3 Cross-Check of Timing Analysis 63

χ2 test χ2 distribution 30 102 value Entries 2

χ 25

20 10 15

10 1 5

173.6875 173.688 173.6885 0 10 20 30 ν [Hz] χ2 value

2 Z1 test Z1 distribution

14 value Entries 2 1 Z 12 102

10 Figure 3.20: Statistical tests 8 10 6 applied to the simulated data

4 with a S2B ratio of 0.1. 1 2 Left plots: Diﬀerent statistical tests applied for various fre- 173.6875 173.688 173.6885 0 5 10 15 ν [Hz] Z2 value 1 quencies ν. Vertical red line: Simulated signal frequency. H test H distribution Right plots: Correspond- 16 Entries

H value 2 14 10 ing statistical test distributions. 12 Red bins: Obtained test statis- 10 10 tics. Solid black line: Normal- 8 ized theoretical PFD. 6

4 1 2

173.6875 173.688 173.6885 0 5 10 15 ν [Hz] H value

Kuiper test Kuiper distribution

102

0.05 Entries

Kuiper value 0.04

10 0.03

0.02 1 0.01

0 173.6875 173.688 173.6885 0 0.02 0.04 ν [Hz] Kuiper value 64 Methods and Algorithms

χ2 test χ2 distribution 40

value 2 Entries 2 35 10 χ 30

20 10

10 1 5

173.6875 173.688 173.6885 0 10 20 30 40 ν [Hz] χ2 value

2 Z1 test Z1 distribution value

25 Entries 2 1 Z 102 20 Figure 3.21: Statistical tests 15 10 applied to the simulated data 10 with a S2B ratio of 0.15. 5 Left plots: Diﬀerent statisti- 1 cal tests applied for various fre- 173.6875 173.688 173.6885 0 10 20 ν [Hz] Z2 value 1 quencies ν. Vertical red line: Simulated signal frequency. H test H distribution Right plots: Correspond-

25 Entries H value ing statistical test distributions. 102 20 Red bins: Obtained test statis-

15 tics. Solid black line: Normal- 10 ized theoretical PFD. 10

5 1

173.6875 173.688 173.6885 0 10 20 ν [Hz] H value

Kuiper test Kuiper distribution

0.07 2 Entries 10 0.06

Kuiper value 0.05

0.04 10 0.03

0.02 1 0.01

0 173.6875 173.688 173.6885 0 0.02 0.04 0.06 ν [Hz] Kuiper value 3.3 Cross-Check of Timing Analysis 65

χ2 test χ2 distribution

50 value Entries 2 2 χ 10 40

30 10 20

10 1

173.6875 173.688 173.6885 0 20 40 ν [Hz] χ2 value

2 Z1 test Z1 distribution

40 value Entries 2 1

Z 35 2 30 10 25 Figure 3.22: Statistical tests 20 10 applied to the simulated data 15 with a S2B ratio of 0.2. 10 Left plots: Diﬀerent statisti- 5 1 cal tests applied for various fre- 173.6875 173.688 173.6885 0 10 20 30 40 ν [Hz] Z2 value 1 quencies ν. Vertical red line: Simulated signal frequency. H test H distribution Right plots: Correspond- 40 Entries H value 35 ing statistical test distributions. 2 30 10 Red bins: Obtained test statis- 25 tics. Solid black line: Normal- 20 10 ized theoretical PFD. 15

5 1

173.6875 173.688 173.6885 0 10 20 30 40 ν [Hz] H value

Kuiper test Kuiper distribution

0.08 Entries 0.07 102

Kuiper value 0.06

0.05

0.04 10 0.03 0.02

0.01 1 0 173.6875 173.688 173.6885 0 0.02 0.04 0.06 0.08 ν [Hz] Kuiper value 66 Methods and Algorithms

phasogram phasogram, ∆ν=10-5 Hz 500 500 450 450 400 400 counts/bin counts/bin 350 350 300 300 250 250 200 200 150 150 100 100 50 50 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Rotational phase Rotational phase

Figure 3.23: Phasograms showing two phase cycles for a simulated pulsed signal with 50 h observation time and a S2B ratio of 10. Left: Phases reconstructed using the correct simulated frequency ν. Right: Phases reconstructed using a frequency shifted by ∆ν = 10−5 Hz. as shown before in Table 3.2, was used for the reconstruction of the phasogram. Fig. 3.23 shows the phasogram for the correct pulsar frequency on the left side. On the right side of Fig. 3.23, the phasogram is ﬂat. This phasogram was obtained by using a slightly shifted pulsar frequency, i.e. ∆ν = 10−5 Hz. This frequency shift corresponds to a period shift of only ∆P = 0.3 ns, illustrating the sensitivity of the analysis of data to changes in ν. For millisecond pulsar analysis, high precision ephemerides are obviously required.

3.4 Optimisation of Hillas Cuts

Mainly focusing on a good overall performance for Crab like sources, the standard H.E.S.S. selection cuts, that are applied on the Hillas parameters to reduce the hadronic background, are not very eﬃcient at low energies. Pulsars, on the other hand, are predicted to possess an exponential energy cutoﬀ in the range of some GeV, see section 1.2 on page 8. Given these predictions, as well as the fact that no pulsed gamma-ray emission has been observed in the VHE range, a procedure to optimise the selection cuts, with special emphasis on low energy events, was developed. The outcome was a general software package that can be used to optimise any selection cuts for any assumed source spectrum. A complete documentation of this package can be found in [78].

3.4.1 Method The optimisation procedure runs on two distinct datasets. Gamma-ray Monte Carlo simulations (MC) are used for the assumed signal, whereas observational OFF runs, i.e. without gamma-ray contamination, provide the background. The observational and simulated zenith angles have to be the same. By weighting the Monte Carlo gamma-ray events as a 3.4 Optimisation of Hillas Cuts 67 function of their energy, any possible gamma-ray spectrum can be generated. Next, the standard analysis chain is applied to the two datasets. In this context, all selection cuts considered for the optimisation are varied simultaneously within a specified range. In other words, the analysis is performed for a large set of selection cuts (configu- MC OFF rations). The results, i.e. the number of events of the two datasets Ni ,Ni passing the configuration i and the total livetime tlive of the OFF runs, are stored in a database for a subsequent evaluation. In contrast to the OFF runs, the MC runs do not possess an intrinsic livetime. Thus, MC the numbers of gamma-ray events Ni passing the analysis chain have to be scaled to match the OFF livetime and an assumed signal rate. This scaling is done with respect to the well-known Crab gamma-ray rate RCrab which is calculated by integrating over the differential Crab rate, dR dN (E) = (E) A (E) , dE dE eff dN where dE is the known differential Crab flux and Aeff is the H.E.S.S. effective area, as described in section 3.1.5 on page 32. Note that the obtained rate itself depends on the applied selection cuts. As an example, for 20◦ zenith angle and relaxed selection cuts, the expected rate of gamma-rays from the Crab is ∼ 45 min−1. The scaling factor is calculated by tlive · RCrab s = MC Rassumed. Nrelaxed

Here, tlive · RCrab represents the number of expected gamma-ray events from the Crab, MC Nrelaxed is the number of MC events passing the same relaxed cuts that were used to calculate RCrab, and Rassumed is the assumed signal rate in units of the Crab rate. With the MC OFF scaled gamma-ray events and background events s · Ni ,Ni , the signal significance is calculated using Eqn. 3.3 for each configuration i. In the last step, the significance of all configurations is evaluated. Among 1% of the configurations with the highest significances, the one with the highest gamma-ray excess is chosen as the final configuration. This is supposed to prevent configurations with high significance but small gamma-ray efficiency.

3.4.2 Results The optimised configuration for low energy events (LOWE) was derived assuming a power- dN −Γ law gamma-ray spectrum dE ∼ E with a high index of Γ = 5.0 to represent a spectrum with an exponential energy cutoff. Further, a weak gamma-ray rate of 1% of the Crab was assumed. Corresponding to the observation conditions, the optimisation datasets were chosen with 20◦ zenith angle and 0.5◦ camera offset. In the process of optimisation the following parameters were considered: both tailcuts (0407, 0510), θ2 cut, mean reduced scaled parameters (MRSW, MRSL), and the minimum size (min size). These parameters are explained in section 3.1 on page 28. The parameter variation ranges and steps were chosen from reasonable standard parameter values. In the process of optimisation, the ranges were expanded until the final cut configuration did not reach any parameter limit. 68 Methods and Algorithms

√ Configuration σ/ h Excess γ efficiency S2B tlive [h] LOWE (0510) 0.43 72 52% 0.019 6.2 LOWE (0407) 0.48 66 47% 0.027 6.2 min size [PEs] θ2 [deg2] MRSW [σ] MRSL [σ] min max min max LOWE (0510) 30 0.028 −2.0 0.75 −2.0 1.6 LOWE (0407) 40 0.024 −2.0 0.7 −2.0 1.2 √ Table 3.3: Optimised configurations for low energy events, for both tailcuts. σ/ h is the significance per square root hour; Excess is the gamma-ray excess; γ efficiency is the percentage of gamma-ray events passing all selection cuts; S2B is the signal to background ratio after selection cuts; min size, θ2, MRSW, and MRSL are explained in section 3.1.

Conﬁguration min size [PEs] θ2 [deg2] MRSW [σ] MRSL [σ] (tailcut) min max min max STD (0510) 80 0.02 −1.7 0.9 −2.0 1.3

Table 3.4: Standard H.E.S.S. selection cuts

The final configuration is listed in Tab. 3.3 with the obtained parameter values, significance, gamma-ray excess, gamma-ray efficiency, and S2B ratio. The results are presented for the two tailcuts separately. For comparison, the H.E.S.S. standard selection cut configuration (STD) is given in Tab. 3.4. In Fig. 3.24, the effective areas using the LOWE (0407) and STD configurations are plotted in blue and red, respectively. As can be seen, the LOWE configuration yields a larger effective area for energies below ≈ 1 TeV. Furthermore, the LOWE (0407) configuration was tested with observational data. PKS 2005-489 is the VHE gamma-ray source with the highest photon index of Γ = 4.0 that was detected by H.E.S.S. [79]. The dataset comprises ∼ 24 h livetime from 2003 and 2004. This dataset was analysed with the STD and LOWE configurations for events with energies below 500 GeV only. The obtained significances were 6.3 σ, 6.9 σ and the gamma-ray excesses 220, 575 for the STD and LOWE configurations, respectively. Although the S2B ratio of the LOWE configuration is low (compare Tab. 3.3), that does not necessarily mean the final S2B ratio of an analysed source is that low since we do not know its gamma-ray rate. In fact, by increasing the assumed gamma-ray rate to 10% of the Crab gamma-ray rate the S2B ratio rises to 0.26, while the selection cut values remain unaltered. 3.4 Optimisation of Hillas Cuts 69

Comparison of Effective Areas ] 2

105 Effective Area [m

104

103 10−1 1 10 102 Energy [TeV]

Figure 3.24: Eﬀective Areas using the LOWE (0407) and H.E.S.S. standard selection conﬁgurations in blue and red, respectively. Chapter 4

Analysis

In this chapter the results of the PSR J0437-4715 analysis are presented. The analysis consists of the standard H.E.S.S. analysis (DC) and the timing analysis (AC) which were both introduced in the last chapter. The full analysis was carried out with two Hillas selection cut configurations, the standard H.E.S.S. (STD) and a configuration optimised for low energy events (LOWE). Statistical tests were applied on the set of event phases to determine the agreement with a flat distribution. For a better signal to background ratio, some further selection cuts, that will be motivated, were applied. Additionally, the background rate as a function of the zenith angle was estimated to exclude a systematic background overestimation. Finally, flux upper limits for pulsed emission are given in a model independent way.

4.1 Dataset and Analysis

PSR J0437-4715 was observed by H.E.S.S. in October 2004 by all four telescopes for about 10 h. Most of the time, these observations were conducted with the lowest possible zenith angle of 23.9◦. The key observation parameters are listed in Tab. 4.1. Two sets of cut conﬁgurations were used in the analysis. The STD selection cuts (see Tab. 3.4), which were the H.E.S.S. standard at the time of analysis, and the LOWE

PSR J0437-4715 Data from Oct. 2004 with all 4 telescopes Livetime 8.2 h Energy threshold 255 GeV (STD), 195 GeV (LOWE) Zenith angle range 23.9◦ - 30◦

Table 4.1: J0437-4715 observation parameters. Energy thresholds are calculated from the eﬀective areas after the corresponding selection cuts and assuming a Crab spectrum and a zenith angle of 24◦. More details about the STD and LOWE conﬁgurations are given in the text below. 4.1 Dataset and Analysis 71

GRO ephemeris Reference epoch t0 = 53145.000000039 MJD Pulsar Frequency ν = 173.6879487056748 Hz Time derivative of barycentric frequencyν ˙ = −1.22912 · 10−15 s−2 Binary (orbital) period Pb = 496026.060394 s Projected semimajor axis of pulsar orbit χ = 3.3667067 s Binary eccentricity e = 0.00001917 Epoch of periastron passage T0 = 53146.58539277 MJD Longitude of periastron ω = 1.643376 deg

Table 4.2: J0437-4715 Timing model parameters obtained from an ATNF GRO ephemeris [66] valid in the observation time span.

configuration (see Tab. 3.3), which was optimised for low energy events. The LOWE configuration was obtained by the Monte Carlo optimisation process described in the previous chapter assuming a very soft (Γ = 5.0) spectrum and a very weak gamma-ray rate (0.01% of Crab). While the STD configuration applies 0510 tailcuts, the LOWE configuration uses 0407 tailcuts which are less conservative as less events are rejected. The full timing analysis was performed within the H.E.S.S. software framework, i.e. it was based on the new timing corrections. Timing model parameters were obtained from an ATNF GRO ephemeris [66] that was valid in the observation time span, see Tab. 4.2.

4.1.1 Quality Checks

Observation runs were required to pass several selection criteria before they were included in the run list for analysis. Critical observation parameters that were considered include: System rate, which should stay constant over the run duration; Ceilometer and Radiometer data which indicate clouds and dust; Camera monitoring showing any hardware malfunctions. Timestamps of the events generated by the central trigger GPS clock were cross- checked with the local telescope timestamps generated by the telescopes’ GPS clocks. This test allows to identify problems with the central trigger GPS clock. Telescopes generate their GPS timestamps when digitizing the recorded camera image after the central trigger signal and thus after the central trigger timestamps are produced. Consequently, we expect a time delay between telescope and central trigger event timestamps. Fig. 4.1 shows the distribution of this time deviation for one observation run with all Hillas but no θ2 selection cut applied, leaving about 5000 events. As expected, a time delay of the order of 0.25 ms can be seen between the diﬀerent peaks, corresponding to diﬀerent telescopes. Thus, no obvious problems of the central trigger event timestamps have been found. 72 Analysis

Central trigger vs. Telescope time

3500 events 3000 Figure 4.1: Distribution of the timestamp 2500 deviations between the central trigger times- 2000 tamps and the diﬀerent telescope times- 1500 tamps. The diﬀerent peaks correspond to the

1000 four telescopes.

500

0 -0.3 -0.28 -0.26 -0.24 -0.22 -0.2 Central trigger - Telescope [ms]

Theta Squared ON/OFF Theta Squared ON/OFF 200 500 180 823 on, 5568 off, 0.143 norm 450 2158 on, 14853 off, 0.143 norm 0.909 σ 0.732 σ

# Events 160 0.316 σ / h # Events 400 0.255 σ / h 140 350 120 300 100 250 80 200 60 150 40 100 20 50 0 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Theta squared [deg2] Theta squared [deg2]

Figure 4.2: θ2 distribution of the ON region (black points) and OFF regions normalized (red solid). left panel: STD and Right panel: LOWE selection cut conﬁgurations.

4.2 Results

Fig. 4.2 shows the DC results for the STD (left) and the LOWE (right) configurations. The plots display the number of events as a function of θ2, i.e. the squared angular distance between the reconstructed event position and the test position. For the signal (ON) region, the test position is the source position, whereas for the seven background (OFF) positions it is the center of the background region. The ON regions events are displayed as black points, while the normalized background is shown as red solid bars. The black vertical lines represent the θ2 selection cuts. The DC significance calculation is based on the number of ON and OFF events passing the θ2 cut, as described in section 3.1.4 on page 31. DC significances are found to be 0.9 σ and 0.7 σ for the STD and LOWE configurations, respectively. Turning to the AC results, Fig. 4.3 shows the phasograms containing two phase cycles for the STD (left) and LOWE (right) configurations. The ON region events are displayed as black crosses representing their statistical errors, the normalized background is shown in 4.2 Results 73

Phasogram ON/OFF Phasogram ON/OFF 300 140 250

# Events 120 # Events

100 200

80 150 60 100 40 50 20

0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Rotational phase Rotational phase

Figure 4.3: Phasograms showing two rotational phases cycles of the ON region (black points) and the OFF regions normalized (red solid). Left panel: STD and Right panel: LOWE selection cut conﬁgurations.

Test STD LOWE ON OFF ON OFF χ2 / NDF 9.4/9 (0.40) 5.5/9 (0.79) 12.0/9 (0.22) 4.2/9 (0.90) 2 Z1 7.4 (0.03) 1.2 (0.41) 4.4 (0.11) 0.3 (0.86) 2 Z2 7.5 (0.11) 3.2 (0.11) 10.1 (0.04) 3.2 (0.52) 2 Z3 9.7 (0.14) 3.4 (0.15) 19.1 (0.004) 3.4 (0.76) H 7.4 (0.05) 1.2 (0.24) 11.1 (0.01) 0.3 (0.89) Kuiper 0.06 (0.05) 0.04 (0.82) 0.04 (0.04) 0.02 (0.87)

Table 4.3: ON and OFF phasogram test statistics for both selection cut conﬁgurations. The numbers in brackets represent the probabilities for the phasograms being compatible with a ﬂat distribution.

red solid bars again. Tab. 4.3 lists the results of the statistical tests applied on the event phases of the ON and OFF regions for both selection cut configurations. Numbers in brackets represent the probabilities for the phasogram (one phase cycle) being compatible with a flat distribution. Inspecting the background, we find the test statistics compatible with a flat distribution. The ON region’s statistical tests indicate a lower compatibility with a flat distribution. In particular, the H and Kuiper tests , which were shown to be the most sensitive tests for X-ray like pulse profiles (see section 3.2), yield probabilities between 1% and 5% for the ON region to be compatible with a flat distribution. Compared to the X-ray and radio phasograms (Fig. 1.7), we find the ON fluctuation at the same phase position around 1. 74 Analysis

Phasogram ON/OFF Phasogram ON/OFF 100 90 250

# Events 80 # Events 70 200 60 150 50 40 100 30

20 50 10 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Rotational phase Rotational phase

Figure 4.4: Phasograms showing two rotational phases cycles of the ON region (black points) and the OFF regions normalized (red solid) for events with energies below 500 GeV. Left panel: STD and Right panel: LOWE selection cut conﬁgurations.

4.2.1 Low Energy Bin Based on the pulsar VHE emission models (section 1.2), we expect pulsed emission to be close to the energy threshold of the H.E.S.S. experiment, if there is any. Therefore, a low energy bin, i.e. considering events with a reconstructed energy below 500 GeV, was analysed. Fig. 4.4 shows the the low energy bin phasograms for both configurations, STD (left) and LOWE (right). The total number of ON events decreased from 823 to 681 and from 2158 to 1966, for the STD and LOWE configurations, respectively. Thus, most of the events are left in this low energy bin below 500 GeV. Tab. 4.4 lists all AC test statistics results as well as the DC significances. The background is still in good agreement with a flat phase distribution, whereas the ON region’s test statistics has remained at a low compatibility with a flat distribution. Compared to the test statistics for all energies (Tab. 4.3), slightly higher test statistics, meaning smaller likelihood to be flat, are found in the low energy bin for the STD configuration. In contrast, the LOWE test statistics have slightly decreased.

4.2.2 Low Zenith Angle Bin As aforementioned in section 3.1.5 on page 32, the rate of low energetic gamma-rays decreases with the zenith angle. This is due to the increasing distance between an air shower’s maximum and the telescopes which leads to less Cherenkov photons collected by the cameras. Low energetic air showers, generating only little Cherenkov light, thus become less frequently observed. To increase the signal to background ratio for low energies, a low zenith angle bin was introduced to the analysis in addition to the low energy bin. Fig. 4.5 illustrates the DC signiﬁcance (STD) obtained in the low energy bin for all zenith angles smaller than a maximum zenith angle (ZAmax). The DC signiﬁcance increases with smaller ZAmax. The 4.3 Background Dependence on Zenith Angle 75

Test STD LOWE ON OFF ON OFF χ2 / NDF 10.2/9 (0.33) 6.1/9 (0.73) 11.7/9 (0.23) 5.1/9 (0.83) 2 Z1 8.7 (0.01) 1.1 (0.59) 4.7 (0.10) 1.5 (0.47) 2 Z2 9.8 (0.04) 1.6 (0.81) 11.4 (0.02) 6.2 (0.19) 2 Z3 11.5 (0.07) 5.4 (0.49) 17.8 (0.007) 6.4 (0.39) H 8.7 (0.03) 1.1 (0.66) 9.8 (0.02) 2.2 (0.42) Kuiper 0.07 (0.02) 0.04 (0.88) 0.04 (0.04) 0.03 (0.48) DC signiﬁcance 0.93 σ 0.55 σ

Table 4.4: ON and OFF phasogram test statistics and DC significances for both selection cut configurations and the event energies below 500 GeV. The numbers in brackets represent the probabilities for the phasograms being compatible with a flat distribution.

red vertical line represents the zenith angle cut of 25◦ which was chosen for the further timing analysis. Applying this 25◦ maximum zenith angle selection cut on the data, leaves us with 406 and 1087 events for the STD and LOWE configurations, respectively. Thus, the number of events decreased by a factor of about 2 and correspondingly the livetime diminished to about 5 hours. The resulting phasograms and test statistics are presented in Fig. 4.6 and Tab. 4.5. Once again, the background is in good agreement with a flat phase distribution. The test statistics for the STD configuration increased further, yielding probabilities that the phasogram is incompatible with a flat distribution exceeding 90%. Inspecting the LOWE configuration results, we find decreased test statistics. Also the DC significance decreased to 0.2 σ for the LOWE configuration. A gamma-ray signal is above the mean background level at all phase positions. Of course, this is also valid for a pulsed signal. This, however, seems not to be the case in the phasograms shown here.

4.3 Background Dependence on Zenith Angle

Investigating in the puzzle of the ON events minimum being below the background level in the phasograms, in this section studies of a zenith angle effect are described. The system rate depends on the zenith angle, hence a zenith angle difference between ON and OFF regions leads to a systematic effect in the background estimation. Fig. 4.7 displays the zenith angle distribution of all OFF events relative to the source position. We see a fairly symmetric distribution. The mean value is slightly shifted to a lower zenith angle (ZA) h∆ZAi = −0.06◦, where ∆ZA = ZAOFF − ZAON . This means that the background regions were taken with too small zenith angles on average. Thus, the background was overestimated. 76 Analysis

DC significance for E < 500 GeV ] σ 2.5

DC significance [ 1.5

0.5

0 24 25 26 27 28 29 30 max zenith angle [deg]

Figure 4.5: DC signiﬁcance in the low energy bin (below 500 GeV) calculated with the STD conﬁguration, considering only events below a maximum zenith angle. The red vertical line represents the maximum zenith angle which was chosen for the low zenith bin analysis.

Phasogram ON/OFF Phasogram ON/OFF 80 140 70

# Events # Events 120 60 100 50 80 40

30 60

20 40

10 20

0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Rotational phase Rotational phase

Figure 4.6: Phasograms showing two rotational phases cycles of the ON region (black points) and the OFF regions normalized (red solid) for events with energies below 500 GeV and zenith angles < 25◦. Left panel: STD and Right panel: LOWE selection cut conﬁgurations. 4.3 Background Dependence on Zenith Angle 77

Test STD LOWE ON OFF ON OFF χ2 / NDF 15.0/9 (0.07) 6.1/9 (0.73) 7.8/9 (0.55) 5.9/9 (0.75) 2 Z1 10.4 (0.006) 2.9 (0.23) 2.8 (0.25) 2.2 (0.34) 2 Z2 12.2 (0.02) 4.4 (0.35) 5.8 (0.21) 6.6 (0.17) 2 Z3 14.7 (0.02) 5.1 (0.53) 10.5 (0.11) 6.9 (0.33) H 10.4 (0.02) 2.9 (0.31) 2.8 (0.33) 2.6 (0.35) Kuiper 0.11 (0.002) 0.07 (0.35) 0.04 (0.24) 0.04 (0.28) DC signiﬁcance 2.20 σ 0.25 σ

Table 4.5: ON and OFF phasogram test statistics and DC significances for both selection cut configurations and the event energies below 500 GeV and zenith angles < 25◦. The numbers in brackets represent the probabilities for the phasograms being compatible with a flat distribution.

Zenith Distribution Off regions

160

# Events 140 120 100 Figure 4.7: Zenith angle distribution of the 80 OFF runs relative to the target position of 60 PSR J0437-4715. 40

20 0 -1.5 -1 -0.5 0 0.5 1 1.5 Zenith angle difference: Off - Target [deg] 78 Analysis

Zenith Event Distribution Zenith Livetime Distribution

1000 80000

# Events 70000

800 Liveime [s] 60000 50000 600 40000 400 30000 20000 200 10000 0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Zenith angle [deg] Zenith angle [deg]

Figure 4.8: Event (Left) and livetime (Right) distribution over the zenith angle of 156 OFF runs.

A quantitative estimation of this eﬀect is presented here. For this purpose, the event rate after selection cuts (STD) for 156 OFF runs (from many diﬀerent observations) covering a large zenith angle range was calculated. Fig. 4.8 shows in the left panel the event distribution and in the right panel the amount of corresponding livetime spent in each zenith angle bin.

The two histograms were calculated by taking the mean zenith angle and the elapsed livetime between two events in the same OFF region. Therefore, the θ2 selection cut was lowered from 0.02 deg2 to 0.01 deg2 in order to decrease the zenith angle range in each OFF region. Dividing the number of events by the corresponding livetime, yields the event rate per zenith angle bin, see Fig. 4.9. On the left side all energies are taken whereas on the right side only events with an energy below 500 GeV were used. In agreement with expectations, the low energy rate is falling off more quickly caused by the rapid fall off of the effective area. For events above 500 GeV the rate is even increasing with the zenith angle, as the effective area does.

In the range of 20 deg to 50 deg the rate distribution can be described by a linear function. A ﬁt yields a slope of m = −0.000462 Hz/deg for the low energy bin rate distribution. Together with the observed zenith angle diﬀerence, we obtain a rate deviation of ∆Rate = m · h∆ZAi = 2.8 · 10−5 Hz. Correspondingly, the background rate was overestimated by about 0.2% which is negligible (it corresponds not even to one full event in the STD low energy, low zenith angle bin phasograms).

Thus, a systematic zenith angle effect can not describe the high background level with respect to the minimum of the ON events in the phasograms. Assuming a signal of pulsed emission, the background level remains an open issue, though the effect is within statistical fluctuations of the background. 4.4 Flux Upper Limits 79

Zenith Rate Distribution Zenith Rate Distribution (E<0.5 TeV) 0.022 0.025 0.02 0.018 0.02 0.016 0.014 Rate [#events / s] Rate [#events / s] 0.015 0.012 0.01 0.01 0.008 0.006 0.005 0.004 0.002 0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Zenith angle [deg] Zenith angle [deg]

Figure 4.9: Zenith rate distribution of 156 OFF runs. Left for all energies, Right for energies below 500 GeV.

4.4 Flux Upper Limits

Pulsed flux upper limits were calculated using a model independent approach. A detailed description can be found in [13]. The method used for the analysis here, however, was slightly modified as will be explained below. The pulsed flux upper limit determination is also briefly presented in the following. The rotational phase is divided into a signal (ONphase) and a background (OFFphase) region. The ONphase region is defined by the pulse signal position in the X-rays and/or radio waves. In the case of PSR J0437-4715, the phasograms of radio and X-rays peak at the same phase position. Therefore, the phase region [0, 0.1] ∪ [0.8, 1] was chosen as the ONphase region. phase In analogy to Eqn. 3.4, the ON flux F in the energy bin [E1,E2) is calculated by F = , tlive with E ∈∆E E ∈∆E iX 1 iX 1 = − α , Aeff (i) Aeff (i) i=1...NON i=1...NOFF where tlive is the observation livetime, NON and NOFF denote the number of events in phase phase the ON and OFF region lying in the energy bin ∆E = [E1,E2); Aeff (i) is the effective area as a function of the energy, zenith angle, and offset of event i; and α is the normalization factor, i.e. the ratio of the ONphase and OFFphase phase areas. In addition to the use of the target region’s OFFphase region for the background estimation, an improved determination and thus a smaller statistical error can be obtained by adding the OFFphase region in the seven OFF background regions. Then, of course, α has to be modified to additionally take the number of background regions into account. 80 Analysis

The integral flux upper limit is therewith calculated from the , its error σ2 (), that is analytically determined by Gaussian error propagation, and the desired confidence level (CL) via an upper limit function UL (, σ () , CL): 1 Φ(E1,E2) = UL (, σ () , CL) . tlive Here, tlive denotes the livetime of the observation. The upper limit function UL is calculated in the unified approach of Feldman and Cousins [80]. Integrated and differential flux upper limits are calculated by, dN Φ(E,E + ∆E) F (> E) = Φ (E, ∞) , (E) = . dE ∆E 2 dN dN Upper limits on the energy flux per decade E dE , are calculated in analogy to dE but with replaced by

E ∈∆E E ∈∆E iX E2 iX E2 0 = i − α i . Aeff (i) Aeff (i) i=1...NON i=1...NOFF Using the full PSR J0437-4715 dataset, i.e. no energy or zenith angle constraints, and a phase ON region between [0, 0.1] ∪ [0.8, 1], the H.E.S.S. upper limits on the energy flux per decade with a confidence level of 99% are given in Fig. 4.10. In addition, EGRET upper limits [28] and polar cap predictions from Harding [30] are shown. The H.E.S.S. upper limits are drawn as points with downward arrows. The different components of the polar cap model prediction are represented by black lines (see section 1.3). The integrated H.E.S.S. upper limit above 200 GeV for pulsed emission in the phase region [0, 0.1] ∪ [0.8, 1] with a confidence level of 99% is 1.0 · 10−12 cm−2 s−1.

4.5 Discussion

The VHE gamma-ray phasograms of PSR J0437-4715 yield a probability above 90% of being incompatible with a flat distribution, i.e. a pulsed signal. This probability rises further to more than 98% for the STD configuration, when the data is limited to low energies (E < 500 GeV) and low zenith angles (ZA < 25◦). In addition to the phasograms’ high probabilities of being not flat, the phasograms’ maximum is at the same phase position as the radio and X-ray pulses. The statistical tests, however, do not assume any particular phase position. Exploiting the phasograms’ agreement in pulse phase, the degrees of freedom of the statistical tests can√ be decreased. The AC significance due to the H- test is then simply given by σH = H according to de Jager [81]. This results in a significance of more than 3.2 σ corresponding to a probability of more than 99.7% (which was already given by the Kuiper-Test without any phase position information) for the STD configuration in the low energy and low zenith angle bin. The background is in all cases in good agreement with a flat distribution. On the other hand, there are some points that are not easily explained in the picture of a pulsed signal. First, as already discussed, the phasogram background level is higher than 4.5 Discussion 81

10−5 PSR J0437−4715 H.E.S.S. 99% CL 10−6 LOWE conf. STD conf. −7 EGRET 10 CR 10−8 s) 2 10−9

10−10 GeV/(cm SR−prim 10−11

10−12 ICS

10−13

10−14 10−5 10−4 10−3 10−2 10−1 100 101 102 103 104 105 Energy (GeV)

Figure 4.10: Energy flux per decade upper limits and model predictions. Lines represent the different components of polar cap model predictions from Harding [30]. The two black upper limits are from EGRET [28] and the blue and light blue points are the H.E.S.S. upper limits. While the dark blue (most left) point was determined with LOWE, the other three upper limits were calculated with the STD configuration. 82 Analysis the ON profile minimum. Assuming a gamma-ray excess, we would always find the ON phase profile on top of the flat background level. Studies of the background dependence on the zenith angle could not explain this deficit. However, a background fluctuation could explain this deviation within statistics. Second, the LOWE configuration (optimised for low energy events), does not yield test statistics with higher probabilities for the phases to be incompatible with a flat distribution. Moreover, these probabilities do not increase when limiting the data to the low energy bin and low zenith angle bin. Indeed, the opposite is true. As can be seen from the simulation of a pulsed signal (section 3.3.3) and the LOWE configuration (Tab. 3.3), about 10% of the Crab gamma-ray rate is required for a sufficient S2B ratio with the LOWE selection cuts. Below 10% Crab gamma-ray rate, the LOWE configuration is not efficient anymore. As a consequence, pulsed flux upper limits were calculated. They represent the first upper limits of PSR J0437-4715 in the VHE range. Polar cap predictions from Harding et al. [30] could, however, not be constrained. To compare with the predicted integrated flux by Bulik et al. [33], the integrated H.E.S.S. flux upper limit has to be scaled from 200 GeV to 100 GeV. Therefore, a photon index of Γ = 2 and no energy cutoff between 100 GeV and 200 GeV is assumed. This is reasonable with regard to the predicted ICS peak around 1 TeV by Bulik et al.. The derived integrated H.E.S.S. upper limit for pulsed emission above 100 GeV with a confidence level of 99% is then 2 · 10−12 cm−2 s−1. This is below the predicted value of 8 · 10−12 cm−2 s−1 to 200 · 10−12 cm−2 s−1. Summary

This work represents a study of the millisecond pulsar PSR J0437-4715 dedicated to the search for pulsed very high energy (VHE) gamma-ray emission within the observational data taken with the High Energy Stereoscopic System (H.E.S.S.). H.E.S.S. is an array of four large Imaging Atmospheric Cherenkov Telescopes situated in Namibia. The H.E.S.S. telescopes detect VHE cosmic gamma-rays ranging from 100 GeV up to 100 TeV with an energy resolution better than 15% and an angular resolution below 0.1◦ for a single gamma-ray photon. Pulsars are generally accepted to be fast spinning, highly magnetized neutron stars. The 1600 known pulsars can be classified into normal and millisecond pulsars with respect to their rotational period. Pulsed emission was detected in a wide range of wavelengths, ranging from radio waves, optical waves, and X-rays, up to gamma-rays. Yet, no detection in the VHE gamma-ray domain above 20 GeV has been made. Two regions are considered for the origin of VHE gamma-ray emission, giving rise to two classes of emission models: Polar cap models suggest radiation near the pulsar surface at the magnetic poles, whereas outer gap models assume the radiation to take place in vacuum gaps in outer regions of the pulsar magnetosphere. A detection of pulsed VHE gamma-rays may hold the key to distinguish between the two models. PSR J0437-4715 is the closest and brightest millisecond pulsar known at radio and X-ray wavelengths. Its proximity to the Earth, the relatively low magnetic field, and the high spin-down flux make PSR J0437-4715 an excellent candidate for the search of pulsed VHE gamma-ray emission. The H.E.S.S. standard analysis reconstructs the energy and direction of the observed VHE gamma-rays and rejects the hadronic background (mainly cosmic protons) which is dominating by a factor of about 104. To search for time periodicities in the observed arrival times - pulsar timing analysis - a set of new methods was added to the H.E.S.S. analysis software framework. The key components of the pulsar timing analysis are the timing corrections. Timing corrections have to be applied to the arrival times since the observation frame is not inertial with respect to the pulsar. For pulsars in a binary system - such as PSR J0437-4715, binary timing corrections need to be applied to account for further acceleration in the binary system. After applying all necessary timing corrections to the arrival times, a set of statistical tests calculates the significance of a possible pulsed signal. All steps of the pulsar timing analysis were extensively tested. A cross-check with the standard timing analysis tool for pulsed radio astronomy - TEMPO - yielded relative errors for all timing corrections. It was found that the developed timing corrections in the 84 Analysis

H.E.S.S. software have a maximum over-all deviation of . 2.0 µs with respect to TEMPO. This satisfies the needs for the pulsar timing analysis. Furthermore, a simulation of a pulsed signal was performed to test the complete pulsar timing analysis. In particular, the sensitivity of the different statistical tests were compared with respect to the S2B ratio. Since most theoretical models for the VHE gamma-ray emission of pulsars predict an energy cutoff in the range of some GeV, the H.E.S.S. standard selection cuts were optimised for gamma-rays with low energies. This lead to a reduction of the energy threshold from 255 GeV to 195 GeV. The 8.2 h of data, passing the quality selection criteria, taken on PSR J0437-4715 with the H.E.S.S. telescopes were analysed using both the standard and low energy selection cuts and the developed pulsar timing analysis. No significant unpulsed signal was found from the direction of PSR J0437-4715. The search for periodicity in the data yielded no significant signal, although the probabilities for a pulsed signal were above 90% for both the standard and low energy selection cuts. These probabilities increased further to more than 98% when limiting the data to a low energy and low zenith angle range. The radio and X-ray phase pulse positions were found to be in agreement with the maximum phase position of the VHE gamma-ray phasogram. An upper limit on the integrated pulsed gamma-ray flux above 100 GeV was calculated to be 1.0 · 10−12 cm−2 s−1 at 99% confidence level, which constrains model predictions. Subsequent H.E.S.S. observations of PSR J0437-4715 were proposed to follow the hint of pulsed emission. Bibliography

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This thesis would not have been possible without all the support, discussions, and motivation from many people. In particular, I thank

• Christian, for convincing me to do experimental (astro)particle physics, for bringing me to H.E.S.S. and the pulsar topic, for all his support and motivation. Simply: for being the best adviser I can think of!

• Thomas, for making all this possible, all the valuable physical discussions, his im- pressive stories about physics as well as many other topics (e.g. mountain climbing), and the fantastic time in Namibia!

• Schlenk - Mr DAQ: The solution to any problem, for helping me whenever I needed help, for all the fun in Berlin and Paris!

• Ulli, Nukri, Martin, and Veronika for the priceless support and the gorgeous time here!

• Fabian for all his initial advice about pulsars!

• Sebastian and Matthias for all the perfect moments in the Diplomandenraum!

• Felix, der andere Felix, Magdalena, Tilo, Regina, and Jan for making all the physics studies so awesome and for all the wonderful kayak trips

• Aileen for helping and motivating me, for the perfect time! Erkl¨arung

Hiermit best¨atige ich, daß ich die vorliegende Arbeit ohne unerlaubte fremde Hilfe an- gefertigt habe. Mit der Auslage meiner Diplomarbeit in der Bibliothek der Humboldt– Universit¨at zu Berlin bin ich einverstanden.

Berlin, den 9. Dezember 2005

Till Eifert