Download Search for Pulsed Very High Energy Gamma Ray

Search for Pulsed Very High
Energy Gamma Ray Emission
from the Millisecond Pulsar
PSR J0437-4715 with H.E.S.S.
Humboldt–Universität zu Berlin
Mathematisch–Naturwissenschaftliche Fakultät I
Institut für 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 gammarays 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 Suche 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 aussichtsreicher Kandidat für die Suche nach gepulster hochenergetischer Gammastrahlung.
Die Beobachtung wurde 2004 mit dem in Namibia stehenden High Energy Stereoscopic
System (H.E.S.S.) durchgeführt. H.E.S.S. wurde im Dezember 2003 in Betrieb genommen
und ist momentan das empfindlichste System der Imaging Atmospheric Cherenkov Telescopes. 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◦ für ein einzelnes Gammastrahlungsphoton.
Im Rahmen dieser Arbeit wurden die Methoden zur Suche nach zeitlichen Periodizitäten in den Ankunftszeiten der Gammastrahlungsphotonen - eine Pulsar-Zeitanalyse
- und eine Prozedur zur Optimierung der Standard-Analyse entworfen und in die bestehende 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. Dafür wurde ein Vergleich mit dem StandardZeitanalyse-Programm für Pulsar-Radioastronomie durchgeführt. Außerdem wurde die
Pulsar-Zeitanalyse mit einer Simulation für gepulste Signale getestet. Mit der PulsarZeitanalyse 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 obere Grenze mit einem Vertrauensbereich von 99% für den gepulsten Fluss integriert ab
100 GeV zu 1.0 · 10−12 cm−2 s−1 berechnet.
Contents
Introduction
1 Pulsars
1.1 Overview . . . . . . . . . . . . . . . . .
1.1.1 Discovery . . . . . . . . . . . . .
1.1.2 Identification with Neutron Stars
1.1.3 Pulsar Population . . . . . . . .
1.2 High Energy Emission Models . . . . . .
1.2.1 Magnetosphere . . . . . . . . . .
1.2.2 Radiation Processes . . . . . . .
1.2.3 Polar Cap Model . . . . . . . . .
1.2.4 Outer Gap Model . . . . . . . .
1.3 PSR J0437-4715 . . . . . . . . . . . . .
1.3.1 Detection in Radio and X-rays .
1.3.2 Theoretical Predictions . . . . .
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3
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2 H.E.S.S. Experiment
2.1 Imaging Atmospheric Cherenkov Technique
2.1.1 Air Showers . . . . . . . . . . . . . .
2.1.2 Cherenkov Radiation . . . . . . . . .
2.1.3 Detection Principle . . . . . . . . . .
2.2 H.E.S.S. Telescope System . . . . . . . . . .
2.2.1 Site and Telescopes . . . . . . . . . .
2.2.2 Optics . . . . . . . . . . . . . . . . .
2.2.3 Camera . . . . . . . . . . . . . . . .
2.2.4 Trigger . . . . . . . . . . . . . . . .
2.2.5 Data Acquisition . . . . . . . . . . .
2.2.6 Monitoring . . . . . . . . . . . . . .
2.3 Monte Carlo Simulations . . . . . . . . . . .
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3 Methods and Algorithms
28
3.1 Standard Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.1 Calibration and Preselection . . . . . . . . . . . . . . . . . . . . . . 28
3.2
3.3
3.4
3.1.2 Geometrical Reconstruction . . . . . . . . . . . . . . . .
3.1.3 Background Suppression . . . . . . . . . . . . . . . . . .
3.1.4 1D Analysis Using 7 Background Regions . . . . . . . .
3.1.5 Energy Estimation . . . . . . . . . . . . . . . . . . . . .
Timing Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Time of Arrival Corrections . . . . . . . . . . . . . . . .
3.2.2 Clock and Frequency Corrections . . . . . . . . . . . . .
3.2.3 Solar System Corrections . . . . . . . . . . . . . . . . .
3.2.4 Relative Motion . . . . . . . . . . . . . . . . . . . . . .
3.2.5 Binary Corrections . . . . . . . . . . . . . . . . . . . . .
3.2.6 Timing Model Parameters . . . . . . . . . . . . . . . . .
3.2.7 Statistical Tests for Periodicity Search . . . . . . . . . .
Cross-Check of Timing Analysis . . . . . . . . . . . . . . . . . .
3.3.1 Optical Crab Data . . . . . . . . . . . . . . . . . . . . .
3.3.2 Comparison with Standard Radio Timing Analysis Tool
3.3.3 Simulation of a Pulsed Signal . . . . . . . . . . . . . . .
Optimisation of Hillas Cuts . . . . . . . . . . . . . . . . . . . .
3.4.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Analysis
4.1 Dataset and Analysis . . . . . . . . . . . .
4.1.1 Quality Checks . . . . . . . . . . .
4.2 Results . . . . . . . . . . . . . . . . . . . .
4.2.1 Low Energy Bin . . . . . . . . . .
4.2.2 Low Zenith Angle Bin . . . . . . .
4.3 Background Dependence on Zenith Angle
4.4 Flux Upper Limits . . . . . . . . . . . . .
4.5 Discussion . . . . . . . . . . . . . . . . . .
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Summary
83
Acknowledgments
91
iii
List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Pulsar illustration . . . . . . . . . . . . . . . . . . . . . . . .
Discovery of the first pulsar . . . . . . . . . . . . . . . . . . .
Pulsar population . . . . . . . . . . . . . . . . . . . . . . . . .
Pulsar Magnetosphere . . . . . . . . . . . . . . . . . . . . . .
Pulsar emission regions . . . . . . . . . . . . . . . . . . . . .
Spectrum of the Vela pulsar with model predictions . . . . .
PSR J0437-4715 phasograms in radio and X-ray wavelengths
PSR J0437-4715 flux prediction from Harding . . . . . . . . .
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3
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2.1
2.2
2.3
2.4
2.5
2.6
View on H.E.S.S. site . . . . . . . . . . . . .
Cherenkov detection principle . . . . . . . .
H.E.S.S. telescope and night sky on site . .
Technical drawing of one H.E.S.S. telescope
Technical drawing of one H.E.S.S. mirror .
H.E.S.S. mirrors and camera . . . . . . . .
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3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
Hillas parameters in camera display . . . . . . . . . . .
Intersecting shower ellipses . . . . . . . . . . . . . . . .
Background estimation . . . . . . . . . . . . . . . . . . .
Leapseconds in UTC . . . . . . . . . . . . . . . . . . . .
Illustration of the Roemer delay in the Solar System. . .
Timing corrections due to the Roemer delay in the SSB
Timing corrections due to the Shapiro delay in the SSB
Timing corrections due to the Einstein delay in the SSB
Timing corrections due to relative motion of the source
Geometry of a Binary System . . . . . . . . . . . . . . .
Timing corrections in a binary system . . . . . . . . . .
Timing phase residuals . . . . . . . . . . . . . . . . . . .
2D random walk in the Rayleigh-Test . . . . . . . . . .
Illustration of the Kuiper-Test . . . . . . . . . . . . . . .
Optical Crab lightcurve . . . . . . . . . . . . . . . . . .
SSB cross-check with TEMPO . . . . . . . . . . . . . .
Binary models cross-check with TEMPO . . . . . . . . .
Simulated phasograms . . . . . . . . . . . . . . . . . . .
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3.19
3.20
3.21
3.22
3.23
3.24
Simulation with S2B 0.0 . . . . . . . . . . . . . . . . . . . . .
Simulation of a pulsed signal with a S2B of 0.1 . . . . . . . .
Simulation of a pulsed signal with a S2B of 0.15 . . . . . . . .
Simulation of a pulsed signal with a S2B of 0.2 . . . . . . . .
Effect of pulsar frequency shift . . . . . . . . . . . . . . . . .
Effective Areas for low energy and H.E.S.S. standard selection
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
Cross-check of the central trigger GPS clock . . . . . . .
Theta squared plots for both cut configurations . . . . .
Phasograms for both cut configurations . . . . . . . . .
Phasograms in the low energy bin . . . . . . . . . . . . .
DC significance dependence on max zenith angle . . . .
Phasograms in the low energy and low zenith angle bin
Zenith angle distribution of the OFF runs relative to the
OFF run event and livetime Distribution . . . . . . . . .
Zenith Rate Distribution . . . . . . . . . . . . . . . . . .
PSR J0437-4715 Pulsed Upper limits . . . . . . . . . . .
v
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configurations
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position
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62
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81
List of Tables
3.1
3.2
3.3
3.4
Timing model parameters . . . . .
Simulation parameters . . . . . . .
Optimisation results for low energy
Standard H.E.S.S. selection cuts .
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events
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4.1
4.2
4.3
4.4
4.5
J0437-4715 observation parameters . . . . . . . . . . . . . . . . . . . . . .
J0437-4715 timing model ephemeris . . . . . . . . . . . . . . . . . . . . .
Phasogram test statistics . . . . . . . . . . . . . . . . . . . . . . . . . . .
Phasogram test statistics and DC significances in the low energy bin . . .
Phasogram test statistics and DC significances in the low energy and low
zenith angle bin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
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70
71
73
75
. 77
Introduction
Pulsars were discovered in 1967, when Hewish and Bell were studying interplanetary scintillations 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 December 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 J04374715 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 Atmospheric 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 Xrays. This chapter about the physics of pulsars provides a physical motivation for the VHE gammaray 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 illuschrotron, 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 working 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. Neutron 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
1.1.2
5
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 independently 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 Ṗ . The power emitted by a rotating
magnetic dipole is given by
8π 4 6 2 2
Ė =
Ω R B0 sin α,
µ0 c3
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
= IΩΩ̇,
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:
19
B0 ' 3.3 · 10
q
(P/s) · Ṗ Gauss.
Here, we assumed an orthogonal rotator (α = 90◦ ) with a radius of R = 10 km. Typical
values of B0 are of order 1012 G, although field strengths up to 1014 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 spindown Ṗ 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
=−
2
ν̇
2Ṗ
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.
τ=
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 Ṗ = 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 Ṗ 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
order of milliseconds) and small spin-downs Ṗ . 10−18 s s−1 . 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
~ ∝ ~v × B).
~ The electric fields are of such a strength that
generating huge electric fields (E
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
gravitational acceleration of about 1011 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
~ ·B
~ 6= 0) are believed to exist above the surface at
a few locations. These spots (where E
~ ·B
~ = 0 where the corotation charge
the polar caps and along the null charge surface, Ω
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
ω=
eB
,
γme
given for an electron with mass me and velocity v in the magnetic field B; γ denotes the
−1/2
usual Lorentz factor γ = 1 − β 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
3γ 2 eB
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
PSR ∝
e4 B 2 β 2 γ 2
.
m2e
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
r=
βcγme
eB
with the radius of curvature rcurvature of magnetic field lines. Hence, we obtain νcrit ∼
γ 3 c/rcurvature . Assuming a dipole field and an electron Lorentz factor of 107 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
to photon energies below ≈ me c2 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
B
pair production γ −→ 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 θEγ )−1 )
lγB ∝
.
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
POLAR GAP
Ω. B = 0
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 & Sutherland (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
photons (γ . 106 ). 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
MeV cm -2 s-1
10 -3
10 -4
EGRET
COMPTEL
OSSE
RXTE
CANGAROO
OPTICAL
ROSAT
PC Model
10 -5
Daugherty & Harding 1996
10 -6
10 -7
Outer Gap Model
Romani 1996
10 -8
Outer Gap Model
Hirotani 2000
10 -9
10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 6
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
~ B
~ = 0), outer gap models, in
between the last open field line and the null charge surface (Ω·
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 photonphoton 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
1.3
Pulsars
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,
a characteristic age of 4.9·109 yr, a magnetic field B0 ∼ 7·108 G, and a rotation energy loss
rate Ė ∼ 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 pulXMM−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
1.3.1
0.1 − 10 keV
Detection in Radio and X-rays
PSR J0437-4715 was discovered as a radio pulsar in 1993 during the Parkes survey of the
0.1 − 2.4 keV
ROSAT
southern sky for millisecond and low-luminosity
pulsars [23]. The radio pulsation consists of one
narrow pulse peak, see Fig. 1.7. PSR J04374715 was the first millisecond pulsar detected
in X-rays [24]. The X-ray satellite ROSAT observed the pulsar and discovered the X-ray pulParkes
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 phasocan, 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
10−6
PSR J0437−4715
GeV/(cm 2 s)
10−7
10
EGRET
CR
−8
GLAST
MAGIC
10−9
10−10
10−11
H.E.S.S.
SR−prim
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 unfortunately 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 Fra̧ckowiak 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 Atmospheric 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
2.1
17
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 gammarays.
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
nitrogen translates into a radiation length of x0 = 36.66 g cm−2 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
− xx
hE (x)i = E0 e
0
where E0 indicates the incident particle energy. Similarly, due to pair production the
intensity of a photon beam is reduced to 1/e after traveling a distance of x = 97 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 Molière 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
Molière radius is RM ' 9 g cm−2 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 undergoes inelastic scattering or collision with air nuclei. Thereby, a significant part of the
primary energy is consumed in nuclear processes, such as excitation, nucleon evaporation, and spallation, resulting in low energetic particles at the MeV scale. Secondly, fast
hadronic particles with large transverse momenta are produced, including protons, neutrons, 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 π 0 ’s.
These pions will decay into two photons, π 0 → 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
cos θc =
cn
c
1
=
=
,
v
nv
βn
(2.1)
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,
m0 c2
Emin = p
1 − 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
y
t
t
ligh
e axis
telescop
camera
v
nko
ere
Ch
lopemen
maximum
nal deve
x
Particle air shower
plane
x
y
~ 10 km
n (x)
rojectio
dinal p
longitu
~ 1 deg
ov li
shower
longitudi
altitude
Gamma−ray
nal
longitudi
renk
Che
lateral
ght
ent
velopem
lateral de
r axis
showe
camera
plane
r
reflecto
ground
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 culminates 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 altitudeazimuth mount to point the telescope at any source
Camera
on the sky, see technical drawing in Fig. 2.4 and telescope image in Fig. 2.3 left side. This tracking system
is a friction drive on rails, with a maximum speed of
about 100◦ min−1 , and an accuracy better than 1000 . 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
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
mirrors are focused to an object distance of 10 km. The
point spread function is over most of the field of view well
contained within a single pixel.
Detailed information about the mirrors, the alignment
and optical characteristics can be found in [43, 44].
2.2.3
23
Glas mirror
Support
frame
Actuators and motors
Camera
Figure 2.5: Technical drawThe 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 photomultiplier 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 gammaray 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
y
φ
len
wi
dth
x
gth
LocalDistance
Figure 3.1: Hillas parameters of the shower
ellipse in the camera online event display.
Each pixel’s color corresponds to its intensity
in PEs.
(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
e2
Tele
scop
Te
le
sc
op
e1
Exploiting the stereoscopic observation mode, i.e. having
the Hillas parameters of the shower ellipse of at least two
y
telescopes, it is possible to reconstruct the shower direction 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 interx
section points are weighted taking into account the sine of
the angle between the two axes, the two image sizes, and
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
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
widthi − hwidthii
1 X
MRSW =
,
Ntels
σiwidth
i=1
(3.1)
Ntels
lengthi − hlengthii
1 X
.
length
Ntels
σ
i
i=1
(3.2)
MRSL =
Here, Ntels is the number of telescopes with data; widthi (lengthi ) is the the Hillas
parameter width (length) of telescope i; hwidthii , hlengthii and σiwidth , σilength 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.
P tels
In contrast to the HEGRA parameters MSW = 1/Ntels N
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
of OFF regions, NBackground = αNOFF . Here, α is the normalization, i.e. #OFF1regions .
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
√ NON
NOFF
1+α
.
S = 2 NON ln
+ NOFF ln (1 + α)
α NON + NOFF
NON + NOFF 3.1.5
(3.3)
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
determine the energy spectrum. In most cases, the differential energy spectrum dN
dE or
is
the
quantity
of
interest.
E 2 dN
dE
The binned differential energy flux in a given energy bin [E, E + ∆E) is given by

dN
1

=
dE
∆tlive ∆E
EiX
∈∆E
i=1...NON
1
−α
Aeff (i)
EiX
∈∆E
i=1...NOFF

1 
Aeff (i)
(3.4)
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. Indeed 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
dN
dE
Methods and Algorithms
multiplied with the effective area Aeff :
dR
dN
(E) =
(E) Aeff (E) .
dE
dE
(3.6)
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
−2.6 , if not specified otherwise.
the Crab spectrum is assumed, i.e. dN
dE ∼ E
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 Gammaray 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
ν (t) = ν0 +
dν
1 d2 ν
(t − t0 ) +
(t − t0 )2 + . . .
dt
2 dt2
(3.7)
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 derivative ν̇ and ν̈ can be approximated
as constant. This is in particular true for millisecond pulsars which are very stable over
time, thus have a very small ν̈.
By serially numbering the pulsar’s rotations with n and taking into account ν = dn/dt
we find
1 dν
1 d2 ν
n = n0 + ν0 (t − t0 ) +
(t − t0 )2 +
(t − t0 )3 + . . .
(3.8)
2 dt
6 dt2
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
f2
+∆tRoemer, + ∆tShapiro, + ∆tEinstein,
t = tTOA + ∆tclock −
+∆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
~r3
SSB
~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
∆tRoemer, = ~robs · ~n.
c
(3.11)
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
∆ tSSB [s]
Sample Roemer delay
200
150
100
50
Figure 3.6: Timing corrections due to the
Roemer delay in the SSB.
0
-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],
g00 = 1 − 2 U + O c−4 ,
g0k = O c−3 ,
gmn = −δmn (1 + 2 γ U ) + O c−4 ,
(3.12)
in which c is the speed of light and U = U (~r) is the gravitational potential (dimensionless),
U (~r) =
X
b
G mb
.
|~r − ~rb | c2
(3.13)
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 ) c2 dt2 − (1 + 2 γ U ) dx2 + dy 2 + dz 2 .
(3.14)
Emitted photons from the pulsar naturally followed the null trajectory, characterized by
ds2 = 0. Integrating over this null trajectory from the space-time point (tpsr , ~rpsr ) of the
40
Methods and Algorithms
∆ tShapiro [µs]
Sample Shapiro delay
0
-20
-40
Figure 3.7: Timing corrections due to the
Shapiro delay in the SSB.
-60
-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
Z
Z tobs
cdt =
tpsr
⇐⇒
~
robs
~
rpsr
r
1 + 2γ U
(dx2 + dy 2 + dz 2 )
1 − 2U
c (tobs − tpsr ) = |~rpsr − ~robs | +
X Gmb ~n · (~robs − ~rb ) + |~robs − ~rb | .
+ (1 + γ)
ln c2
~n · (~rpsr − ~rb ) + |~rpsr − ~rb | (3.15)
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
!
~robs − ~r G M
∆tShapiro, = 2
· ln
· (1 + cos θ) ,
(3.16)
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
∆ tEinstein [ms]
Sample Einstein delay
1.5
1
0.5
Figure 3.8: Timing corrections due to the
Einstein delay in the SSB.
0
-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 2
dt .
(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.
2
A Taylor expansion of the integrand around vc2 ∼ 0 to order O c−2 yields
Z t
1 v2
τ =t−
U+
dt0 .
(3.18)
2 c2
0
dτ 2 = (1 − 2 U (~r)) dt2 − (1 + 2 γ U (~r))
2
Here, the terms U and vc2 can be interpreted as the gravitational redshift and the Doppler
shift. Thus, the Einstein delay amounts to an integral of the expression
d∆tEinstein, X G mb
v2
=
+
− constant
(3.19)
dt
c2 rb
2 c2
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 proper motion correction
∆ tProper motion [ms]
∆ tParallax [µs]
Sample Parallax correction
-7.2
-7.4
-7.6
-7.8
-8
1.5
1
0.5
0
-8.2
-8.4
-0.5
-8.6
-1
-8.8
-1.5
-9
52000
52100
52200
52300
52400
Epoch [MJD]
52000
52500
53000
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,
∆tShlovskii =
vt2 2
t ,
2dc
(3.20)
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 ∆tπ = −
(~robs · ~n)2 − |~robs |2
(3.21)
2cd
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
auxiliary circle
y 00
z 0 , z 00 z
2
P
orbit
pulsar
~r1
y 00
y0
i
1
x, x0
ω
x00
y
center of
ellipse
BB
E
φ
periapsis
x00
~r2
~n
observer
binary companion
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 z 0 -axis with the periastron angle ω leads to the plane of orbit (x00 , y 00 , z 00 ). 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
E − e sin E =
2π
(t − T0 ) ,
Pb
(3.24)
q
2
with e being the eccentricity, i.e. e = 1 − ab 2 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
a 1 − e2
m2
m1
r1 =
r, r2 =
r, r =
= a (1 − e cos E) ,
(3.25)
M
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
U (~r, t) = U1 (~r, t) + U2 (~r, t) =
−G m1
G m2
− 2
.
− ~r1 (t)| c |~r − ~r2 (t)|
c2 |~r
(3.26)
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
dtp
~r˙1
= 1 + U (~r1 , t) − 2 + O c−4 .
dt
2c
(3.27)
By applying
G m22
2
~r˙1 = v12 =
M
2 1
−
r a
,
(3.28)
3.2 Timing Analysis
45
which can be derived using Kepler’s energy relation
E =−
G m1 m2
,
2a
(3.29)
and dropping all constant terms, we find:
dtp
G m2 G m2 1
=1− 2 − 2 2 .
dt
c r
c M r
(3.30)
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 m2 (m2 + M ) Pb
E
c2 a M
2π
G m2 (m2 + M ) Pb
e sin E.
= t−
c2 a M
2π
tp = t −
(3.31)
Next, we discuss the Shapiro delay. Let us begin from the last term of Eqn. 3.15,
1
(3.32)
2 m2 ln 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
∆tShapiro,bin = 2 m2 ln
1 + e cos φ
1 − sin i sin (ω + φ)
.
(3.34)
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
∆tRoemer,bin = − ~n · ~r1
c
a1 1 − e2 sin i sin (ω + φ)
= −
,
c (1 + e cos φ)
(3.35)
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:
1/2
a1
∆tRoemer,bin = − sin i sin ω (cos E − e) + cos ω 1 − e2
sin E . (3.37)
c
Finally, BT incorporate the Einstein delay from Eqn. 3.31. Since tem = tp 1 + O c−2 ,
we can write the complete BT binary correction as
tp = t − α (cos E − e) − (β + γ) sin E,
where
α = χ sin ω,
β = 1 − e2
1/2
χ cos ω,
χ=
a1
sin i,
c
(3.38)
(3.39)
and
G m22 (m2 + M ) Pb e
.
(3.40)
c2 a1 M 2
2π
Yet, Eqn. 3.38 is an implicit equation in that the eccentric anomaly E depends on tp
or equally tem . This, however, can be solved by defining another eccentric anomaly E 0 :
γ=
E 0 − e sin E 0 =
2π
(t + T0 ) .
Pb
(3.41)
Comparing Eqn. 3.41 with the corresponding one for E (tp ), we find
tp − t ≈
Pb
1 − e cos E 0 E − E 0 .
2π
On the other hand, E = E 0 and tp = t to zeroth order. Thus,
tp − t ≈ −α cos E 0 − e − (β + γ) sin E 0 ,
and so
E − E0 = −
2π (cos E 0 − e) + (β + γ) sin E 0
−2
+
O
c
.
Pb (1 − e cos E 0 )
Therewith, Eqn. 3.38 becomes the final BT correction formula
∆tBT,bin = − α cos E 0 − e − (β + γ) sin E 0
−
(α sin E 0 − β cos E 0 ) [2π (cos E 0 − e) + (β + γ) sin E 0 ]
.
Pb (1 − e cos E 0 )
(3.42)
Secular effects can be accommodated by the replacements:
ω → ω + ω̇ · t,
e → e + ė · t,
χ → χ + χ̇ · t,
Pb → Pb + P˙b · t.
Here, ω̇ describes the precession of the longitude of periastron; the change of orbital period
Ṗb can be a damping effect like tidal dissipation or the emission of gravitational waves;
and ė, χ̇ describe changes in the binary orbit. Fig. 3.11 shows typical timing corrections
for the BT and DD model.
47
Sample BT model corrections
4
∆ tDD, bin [s]
∆ tBT, bin [s]
3.2 Timing Analysis
3
2
1
4
3
2
1
0
0
-1
-1
-2
-2
-3
-3
-4
Sample DD model corrections
52000
52005
52010
52015
Epoch [MJD]
-4
52000
52005
52010
52015
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 for ω̇, whereas the DD model contains
both secular and quasi-periodic effects in ω given by
ω = ω0 + k Ae (u)
and ω̇ = 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 + e 1/2
u
Ae (u) = 2 arctan
tan
,
(3.45)
1−e
2
48
Methods and Algorithms
and a generalized Kepler equation is used to relate the eccentric anomaly with the time
"
#
Ṗb t − T0 2
t − T0
−
.
(3.46)
u − e sin u = 2π
Pb
2
Pb
The post Keplerian Roemer delay is given by
∆tRoemer,bin = −χ sin ω [cos u − e (1 + δr )]
h
i1/2
cos ω sin u,
−χ 1 − e2 (1 + δθ )2
(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,
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
cos ω sin u ,
+ 1 − e2
(3.48)
(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 apriori 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 FastFourier-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
2
χ =
N X
φ (ti ) 2
i=1
σi /P
(3.51)
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. Incorrect 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
right ascension
declination
proper motion in α direction
proper motion in δ direction
reference position epoch
solar parallax
dispersion measure
reference epoch
pulsar rotational frequency
time derivative of ν
second time derivative of ν
binary period
epoch of periastron
binary eccentricity
projected semimajor axis
longitude of periastron
gravitational redshift and time dilation
shape of Shapiro delay
distance between binary partners
drift of periastron
damping effect
change of binary orbit
change of binary orbit
FRQ
X
X
X
X
-
Used
SSB
X
X
X
X
X
X
X
-
in
BT
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Type
DD
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
astrometric
α
δ
µα
µδ
t0,pos
PX
D
t0
ν = 1/P
ν̇
ν̈
Pb
T0
e
χ
ω
γ
s = sin i
r
ω̇
Ṗb
ė
χ̇
Short description
pulsar
Parameter
Methods and Algorithms
binary system
50
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
Position displaced
Residuals [ms]
Residuals [µs]
Perfect least-squares fit
30
20
10
350
300
250
200
0
150
-10
100
-20
50
0
-30
-50
-40
49400
49600
49800
50000
50200
Epoch [MJD]
-100
49400
49600
49800
50000
50200
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 H0P
is a flat distribution,
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
y
φN
Figure 3.13: 2D random walk. The angle between the x axis and the i-th step corresponds
to φi .
φ2
d
φ1
x
The test statistic is then given by
b
X
(nj − n0 )2
χ =
n0
2
j=1
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
fk (x) =
x 2 −1
k
2
2 Γ
k
2
− x2
e
,
(3.52)
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 -Test are more effective. Another disadvantage is the dependence on the
tests like the Zm
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

!2
!2 
N
N
X
X
1
2

ZRayleigh
=
sin (φi ) +
cos (φi ) 
(3.53)
N
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 -Test
Zm
A generalization of the Rayleigh-Test proposed by Bucceri et al. (1983) [69] and Buccherie
2 -Test, which copes with higher order sinusoidal
& Sacco (1985) [70] is the so called Zm
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

!2
!2 
m
N
N
X
X
X
2
2

Zm
=
sin (j · φi ) +
cos (j · φi ) 
(3.54)
N
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 -Test, was avoided by the H-Test
The bottleneck of having one free parameter m in the Zm
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
Figure 3.14: Illustration of the KuiperTest. The dashed line is the null hypothesis,
whereas the red solid line is a sample phase
distribution. D+ and D− indicate the maximum and minimum deviations, respectively.
D−
D+
0.0
φ1 φ2 φ3
0.0
...
φN
1.0
φ
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 -Test and χ2 might be more sensitive.
and for very narrow peaks or three peaks the Zm
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) =
(3.56)
φi ≤ φ < φi+1 , 1 ≤ i ≤ N − 1
 N
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 −
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
(3.57)
3.3 Cross-Check of Timing Analysis
55
The probability for obtaining a value z larger or equal to N 1/2 VN assuming H0 is given
by
∞
X
2 2
P N 1/2 VN ≤ z = PN (z) =
2 4k 2 z 2 − 1 e−2k z
k=1
−
8z
√
3 N
X
∞
2 2
k 2 4k 2 z 2 − 3 e−2k z + O N −1 ,
(3.58)
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
Pk (z0 ) = 1 − (1 − P (z ≥ z0 ))k .
(3.59)
−ν2
is a compromise, in that no potential
The choice of the frequency step ∆ν and so k = ν1∆ν
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
,
(3.60)
∆νIFS =
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.
Methods and Algorithms
7320
Frequency Scan
Optical Crab Phasogram
χ2 value
ADC Counts
56
12000
7315
10000
7310
8000
7305
6000
7300
4000
7295
2000
7290
-0.2 -0.1 -0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Phase
0
29.79
29.795
29.8
29.805
29.81
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
3.3.2
57
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
t=−
log (RND)
,
r
(3.61)
58
Methods and Algorithms
SSB Cross-Check, H.E.S.S.-Site
∆ tSSB,TEMPO-∆ tSSB, H.E.S.S. [µs]
∆ tSSB,TEMPO-∆ tSSB, H.E.S.S. [ns]
SSB Cross-Check, Geocenter
10
5
0
-5
-10
2
1.5
1
0.5
0
-0.5
-1
-15
-1.5
52000.5
52001
52001.5
Epoch [MJD]
52000.5
15
10
5
0
-5
52001.5
Epoch [MJD]
SSB Cross-Check, H.E.S.S.-Site
∆ tSSB,TEMPO-∆ tSSB, H.E.S.S. [µs]
∆ tSSB,TEMPO-∆ tSSB, H.E.S.S. [ns]
SSB Cross-Check, Geocenter
52001
2
1
0
-1
-10
-2
-15
52000
52100
52200
52300
Epoch [MJD]
52000
52100
52200
52300
52400
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
0.15
0.1
0.05
0
-0.05
-0.1
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
52000
Binary DD model Cross-Check
∆ tDD,TEMPO-∆ tDD, H.E.S.S. [ns]
∆ tBT,TEMPO-∆ tBT, H.E.S.S. [ns]
Binary BT model Cross-Check
-0.15
52005
52010
Epoch [MJD]
52000
52005
52010
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. Furthermore, 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
Observation time
Signal to Background ratio (S2B)
Total event rate
Signal shape
Ephemeris for
Value
25 h
0.0, 0.1, 0.15, and 0.2
0.02 Hz
gaussian with a width of 1 ms
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 tests that mainly test multiple peak structures, were applied to
of the higher order Zm
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
with a S2B ratio of 0.1, we see a weak indication of the signal in the Z12 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
indication of the included signal visible in the Z12 , 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.
Apparently, the Z12 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
counts/bin
3.3 Cross-Check of Timing Analysis
300
61
phasogram, S2B 0.0
250
200
150
100
50
counts/bin
0
0
300
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8 2
Rotational phase
phasogram, S2B 0.1
250
200
150
100
50
counts/bin
0
0
300
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8 2
Rotational phase
phasogram, S2B 0.15
250
200
150
100
50
counts/bin
0
0
300
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8 2
Rotational phase
phasogram, S2B 0.2
250
200
150
100
50
0
0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8 2
Rotational phase
Figure 3.18: Phasograms showing two phase
cycles for the simulations corresponding from
top to bottom to a S2B ratio of 0.0, 0.1, 0.15,
and 0.2.
Methods and Algorithms
χ2 test
35
χ2 distribution
Entries
χ 2 value
62
2
10
30
25
20
10
15
10
1
5
173.688
173.6885
ν [Hz]
Z21 test
0
10
20
30
χ 2 value
Z1 distribution
Entries
Z21 value
173.6875
14
12
102
10
8
10
6
4
1
2
173.6875
173.688
173.6885
0
ν [Hz]
5
15
Z21 value
H distribution
Entries
H value
H test
10
18
16
102
14
12
10
10
8
6
4
1
2
173.6875
173.688
173.6885
ν [Hz]
0
Kuiper test
10
15
H value
Kuiper distribution
Kuiper value
Entries
0.06
5
102
0.05
0.04
0.03
10
0.02
0.01
0
1
173.6875
173.688
173.6885
ν [Hz]
0
0.02
0.04
0.06
Kuiper value
Figure 3.19: Statistical tests
applied to the simulated data
with a S2B ratio of 0.0.
Left plots: Different statistical tests applied for various frequencies ν. Vertical red line:
Simulated signal frequency.
Right plots:
Corresponding statistical test distributions.
Red bins: Obtained test statistics. Solid black line: Normalized theoretical PFD.
χ2 test
30
63
χ2 distribution
Entries
χ 2 value
3.3 Cross-Check of Timing Analysis
2
10
25
20
10
15
10
1
5
173.688
173.6885
ν [Hz]
Z21 test
0
10
20
30
χ 2 value
Z1 distribution
Entries
Z21 value
173.6875
14
102
12
10
8
10
6
4
1
2
173.6875
173.688
173.6885
0
ν [Hz]
10
15
Z21 value
H distribution
Entries
H value
H test
5
16
102
14
12
10
10
8
6
4
1
2
173.6875
173.688
173.6885
ν [Hz]
0
10
15
H value
Kuiper distribution
Kuiper value
Entries
Kuiper test
5
102
0.05
0.04
10
0.03
0.02
1
0.01
0
173.6875
173.688
173.6885
ν [Hz]
0
0.02
0.04
Kuiper value
Figure 3.20: Statistical tests
applied to the simulated data
with a S2B ratio of 0.1.
Left plots: Different statistical tests applied for various frequencies ν. Vertical red line:
Simulated signal frequency.
Right plots:
Corresponding statistical test distributions.
Red bins: Obtained test statistics. Solid black line: Normalized theoretical PFD.
64
Methods and Algorithms
χ2 distribution
Entries
χ 2 value
χ2 test
40
35
102
30
25
10
20
15
10
1
5
173.688
173.6885
ν [Hz]
Z21 test
0
10
20
30
40
χ 2 value
Z1 distribution
Entries
Z21 value
173.6875
25
102
20
15
10
10
5
1
173.6875
173.688
173.6885
0
ν [Hz]
20
Z21 value
H distribution
Entries
H value
H test
10
25
102
20
15
10
10
5
1
173.6875
173.688
173.6885
ν [Hz]
0
10
20
H value
Kuiper distribution
Kuiper value
Entries
Kuiper test
0.07
102
0.06
0.05
0.04
10
0.03
0.02
1
0.01
0
173.6875
173.688
173.6885
ν [Hz]
0
0.02
0.04
0.06
Kuiper value
Figure 3.21: Statistical tests
applied to the simulated data
with a S2B ratio of 0.15.
Left plots: Different statistical tests applied for various frequencies ν. Vertical red line:
Simulated signal frequency.
Right plots:
Corresponding statistical test distributions.
Red bins: Obtained test statistics. Solid black line: Normalized theoretical PFD.
3.3 Cross-Check of Timing Analysis
χ2 distribution
Entries
χ 2 value
χ2 test
65
50
102
40
30
10
20
10
1
173.688
173.6885
ν [Hz]
Z21 test
0
20
40
χ 2 value
Z1 distribution
Entries
Z21 value
173.6875
40
35
102
30
25
20
10
15
10
5
1
173.6875
173.688
173.6885
0
ν [Hz]
10
30
40
Z21 value
H distribution
Entries
H value
H test
20
40
35
2
10
30
25
20
10
15
10
5
1
173.6875
173.688
173.6885
ν [Hz]
0
20
30
40
H value
Kuiper distribution
Kuiper value
Entries
Kuiper test
10
0.08
0.07
102
0.06
0.05
0.04
10
0.03
0.02
0.01
0
1
173.6875
173.688
173.6885
ν [Hz]
0
0.02
0.04
0.06
0.08
Kuiper value
Figure 3.22: Statistical tests
applied to the simulated data
with a S2B ratio of 0.2.
Left plots: Different statistical tests applied for various frequencies ν. Vertical red line:
Simulated signal frequency.
Right plots:
Corresponding statistical test distributions.
Red bins: Obtained test statistics. Solid black line: Normalized theoretical PFD.
Methods and Algorithms
phasogram
500
counts/bin
counts/bin
66
450
400
500
450
400
350
350
300
300
250
250
200
200
150
150
100
100
50
50
0
0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8 2
Rotational phase
phasogram, ∆ν=10-5 Hz
0
0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8 2
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 (configurations). The results, i.e. the number of events of the two datasets NiMC , NiOFF 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) Aeff (E) ,
dE
dE
where dN
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=
Rassumed .
MC
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
scaled gamma-ray events and background events s · NiMC , NiOFF , 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−Γ with a high index of Γ = 5.0 to represent a spectrum
law gamma-ray spectrum dN
dE ∼ E
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
LOWE (0510)
LOWE (0407)
√
σ/ h
0.43
0.48
min size [PEs]
Excess
72
66
θ2 [deg2 ]
LOWE (0510)
LOWE (0407)
30
40
0.028
0.024
γ efficiency
52%
47%
MRSW [σ]
min max
−2.0 0.75
−2.0 0.7
S2B
tlive [h]
0.019 6.2
0.027 6.2
MRSL [σ]
min
max
−2.0 1.6
−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
(tailcut)
STD (0510)
min size [PEs]
θ2 [deg2 ]
80
0.02
MRSW [σ]
min max
−1.7 0.9
MRSL [σ]
min max
−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
2
Effective Area [m ]
Comparison of Effective Areas
105
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
Livetime
Energy threshold
Zenith angle range
Oct. 2004 with all 4 telescopes
8.2 h
255 GeV (STD), 195 GeV (LOWE)
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
GRO ephemeris
Reference epoch
Pulsar Frequency
Time derivative of barycentric frequency
Binary (orbital) period
Projected semimajor axis of pulsar orbit
Binary eccentricity
Epoch of periastron passage
Longitude of periastron
71
t0
ν
ν̇
Pb
χ
e
T0
ω
= 53145.000000039 MJD
= 173.6879487056748 Hz
= −1.22912 · 10−15 s−2
= 496026.060394 s
= 3.3667067 s
= 0.00001917
= 53146.58539277 MJD
= 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 crosschecked 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
events
Central trigger vs. Telescope time
3500
3000
Figure 4.1: Distribution of the timestamp
deviations between the central trigger timestamps and the different telescope timestamps. The different peaks correspond to the
four telescopes.
2500
2000
1500
1000
500
-0.3
200
180
160
-0.28
-0.26
-0.24
-0.22
-0.2
Central trigger - Telescope [ms]
Theta Squared ON/OFF
823 on, 5568 off, 0.143 norm
0.909 σ
0.316 σ / h
# Events
# Events
0
500
450
400
140
350
120
300
100
250
80
200
60
150
40
100
20
0
0
Theta Squared ON/OFF
2158 on, 14853 off, 0.143 norm
0.732 σ
0.255 σ / h
50
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Theta squared [deg2]
0
0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
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
# Events
# Events
Phasogram ON/OFF
140
120
100
Phasogram ON/OFF
250
200
80
150
60
100
40
50
20
0
0
300
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8 2
Rotational phase
0
0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8 2
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
χ2 / NDF
Z12
Z22
Z32
H
Kuiper
STD
ON
OFF
9.4/9 (0.40) 5.5/9 (0.79)
7.4 (0.03)
1.2 (0.41)
7.5 (0.11)
3.2 (0.11)
9.7 (0.14)
3.4 (0.15)
7.4 (0.05)
1.2 (0.24)
0.06 (0.05)
0.04 (0.82)
LOWE
ON
OFF
12.0/9 (0.22) 4.2/9 (0.90)
4.4 (0.11)
0.3 (0.86)
10.1 (0.04)
3.2 (0.52)
19.1 (0.004)
3.4 (0.76)
11.1 (0.01)
0.3 (0.89)
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.
Analysis
100
Phasogram ON/OFF
Phasogram ON/OFF
# Events
# Events
74
90
80
250
200
70
60
150
50
40
100
30
20
50
10
0
0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8 2
Rotational phase
0
0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8 2
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
Test
χ2 / NDF
Z12
Z22
Z32
H
Kuiper
DC significance
STD
ON
OFF
10.2/9 (0.33) 6.1/9 (0.73)
8.7 (0.01)
1.1 (0.59)
9.8 (0.04)
1.6 (0.81)
11.5 (0.07)
5.4 (0.49)
8.7 (0.03)
1.1 (0.66)
0.07 (0.02)
0.04 (0.88)
0.93 σ
75
LOWE
ON
OFF
11.7/9 (0.23) 5.1/9 (0.83)
4.7 (0.10)
1.5 (0.47)
11.4 (0.02)
6.2 (0.19)
17.8 (0.007)
6.4 (0.39)
9.8 (0.02)
2.2 (0.42)
0.04 (0.04)
0.03 (0.48)
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 = ZAOF F − 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 [σ]
DC significance for E < 500 GeV
2.5
2
1.5
1
0.5
0
24
25
26
27
28
29
30
max zenith angle [deg]
80
Phasogram ON/OFF
Phasogram ON/OFF
# Events
# Events
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.
70
60
140
120
100
50
80
40
30
60
20
40
10
20
0
0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8 2
Rotational phase
0
0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8 2
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
Test
χ2 / NDF
Z12
Z22
Z32
H
Kuiper
DC significance
STD
ON
OFF
15.0/9 (0.07) 6.1/9 (0.73)
10.4 (0.006)
2.9 (0.23)
12.2 (0.02)
4.4 (0.35)
14.7 (0.02)
5.1 (0.53)
10.4 (0.02)
2.9 (0.31)
0.11 (0.002)
0.07 (0.35)
2.20 σ
77
LOWE
ON
OFF
7.8/9 (0.55) 5.9/9 (0.75)
2.8 (0.25)
2.2 (0.34)
5.8 (0.21)
6.6 (0.17)
10.5 (0.11)
6.9 (0.33)
2.8 (0.33)
2.6 (0.35)
0.04 (0.24)
0.04 (0.28)
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.
# Events
Zenith Distribution Off regions
160
140
120
100
Figure 4.7: Zenith angle distribution of the
OFF runs relative to the target position of
PSR J0437-4715.
80
60
40
20
0
-1.5
-1
-0.5
0
0.5
1
1.5
Zenith angle difference: Off - Target [deg]
78
Analysis
Zenith Livetime Distribution
Liveime [s]
# Events
Zenith Event Distribution
1000
800
80000
70000
60000
50000
600
40000
400
30000
20000
200
0
0
10000
10
20
30
40
50
60
Zenith angle [deg]
0
0
10
20
30
40
50
60
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 (E<0.5 TeV)
Rate [#events / s]
Rate [#events / s]
Zenith Rate Distribution
0.022
0.02
0.018
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
0
0.025
0.02
0.015
0.01
0.005
0
0
10
20
30
40
50
60
Zenith angle [deg]
10
20
30
40
50
60
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.
In analogy to Eqn. 3.4, the ONphase flux F in the energy bin [E1 , E2 ) is calculated by
F =
,
tlive
with
=
EiX
∈∆E
i=1...NON
1
Aef f (i)
−α
EiX
∈∆E
i=1...NOF F
1
,
Aef f (i)
where tlive is the observation livetime, NON and NOF F denote the number of events in
the ONphase and OFFphase region lying in the energy bin ∆E = [E1 , E2 ); Aef f (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 σ 2 (), that is
analytically determined by Gaussian error propagation, and the desired confidence level
(CL) via an upper limit function UL (, σ () , CL):
1
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,
Φ (E1 , E2 ) =
F (> E) = Φ (E, ∞) ,
dN
Φ (E, E + ∆E)
(E) =
.
dE
∆E
Upper limits on the energy flux per decade E 2 dN
dE , are calculated in analogy to
with replaced by
0 =
EiX
∈∆E
i=1...NON
Ei2
−α
Aef f (i)
EiX
∈∆E
i=1...NOF F
dN
dE
but
Ei2
.
Aef f (i)
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 Htest 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
10 −6
H.E.S.S. 99% CL
PSR J0437−4715
LOWE conf.
STD conf.
EGRET
GeV/(cm
2
s)
10 −7
CR
10 −8
10 −9
10 −10
10 −11
SR−prim
10 −12
ICS
10 −13
10 −14
10 −5 10 −4 10 −3 10 −2 10 −1 10 0
10 1
10 2
10 3
10 4
10 5
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
[1] W. Baade and F. Zwicky. On Super-novae. Proceedings of the National Academy of
Science, 20:254, 1934.
[2] A. G. Lyne and F. Graham-Smith. Pulsar Astronomy. Pulsar Astronomy, by Andrew G. Lyne and Francis Graham-Smith, pp. . ISBN 0521839548. Cambridge, UK:
Cambridge University Press, July 2005.
[3] A. Hewish, S. J. Bell, J. D. Pilkington, P. F. Scott, and R. A. Collins. Observation
of a Rapidly Pulsating Radio Source. Nature , 217:709, 1968.
[4] F. Pacini. Energy Emission from a Neutron Star. Nature , 216:567–+, 1967.
[5] T. Gold. Rotating Neutron Stars as the Origin of the Pulsating Radio Sources. Nature
, 218:731–+, 1968.
[6] D. J. Thompson. Gamma-ray Pulsars: Observations. In American Institute of Physics
Conference Series, pages 103–+, 2001.
[7] M. S. Longair. High energy astrophysics. Vol.2: Stars, the galaxy and the interstellar
medium. Cambridge: Cambridge University Press, 1994, 2nd ed., 1994.
[8] M. Kramer. Millisecond Pulsars as Tools of Fundamental Physics. LNP Vol. 648:
Astrophysics, Clocks and Fundamental Constants, 648:33–54, 2004.
[9] J. K. Daugherty and A. K. Harding. Electromagnetic cascades in pulsars. Astrophysical Journal , 252:337–347, January 1982.
[10] V. V. Usov and D. B. Melrose. Pulsars with Strong Magnetic Fields - Polar Gaps
Bound Pair Creation and Nonthermal Luminosities. Australian Journal of Physics,
48:571–+, 1995.
[11] K. Hirotani and S. Shibata. One-dimensional electric field structure of an outer gap
accelerator - II. gamma-ray production resulting from inverse Compton scattering.
MNRAS, 308:67–76, September 1999.
[12] A. K. Harding. Gamma-ray Pulsars: Models and Predictions. In American Institute
of Physics Conference Series, pages 115–+, 2001.
[13] Fabian Schmidt. Search for Pulsed TeV Gamma-Ray Emission from Pulsars with
H.E.S.S.. Diplomarbeit, Humboldt–Universität zu Berlin, 2005.
[14] Energetic
Gamma
Ray
Experiment
http://cossc.gsfc.nasa.gov/docs/cgro/cossc/egret/.
Telescope
(EGRET).
[15] P. Goldreich and W. H. Julian. Pulsar Electrodynamics. Astrophysical Journal ,
157:869–+, August 1969.
[16] T. Erber. High-Energy Electromagnetic Conversion Processes in Intense Magnetic
Fields. Reviews of Modern Physics, 38:626–659, October 1966.
[17] J. D. Jackson. Classical electrodynamics. 92/12/31, New York: Wiley, 1975, 2nd ed.,
1975.
[18] G. R. Blumenthal and R. J. Gould. Bremsstrahlung, Synchrotron Radiation, and
Compton Scattering of High-Energy Electrons Traversing Dilute Gases. Reviews of
Modern Physics, 42:237–271, 1970.
[19] P. A. Sturrock. A Model of Pulsars. Astrophysical Journal , 164:529–+, March 1971.
[20] M. A. Ruderman and P. G. Sutherland. Theory of pulsars - Polar caps, sparks, and
coherent microwave radiation. Astrophysical Journal , 196:51–72, February 1975.
[21] K. S. Cheng, C. Ho, and M. Ruderman. Energetic radiation from rapidly spinning
pulsars. I - Outer magnetosphere gaps. II - VELA and Crab. Astrophysical Journal ,
300:500–539, January 1986.
[22] R. W. Romani and I.-A. Yadigaroglu. Gamma-ray pulsars: Emission zones and
viewing geometries. Astrophysical Journal , 438:314–321, January 1995.
[23] S. Johnston, D. R. Lorimer, P. A. Harrison, M. Bailes, A. G. Lyne, J. F. Bell, V. M.
Kaspi, R. N. Manchester, N. D’Amico, and L. Nicastro. Discovery of a very bright,
nearby binary millisecond pulsar. Nature , 361:613–615, February 1993.
[24] W. Becker and J. Trumper. Detection of Pulsed X-Rays from the Binary Millisecond
Pulsar J:0437-4715. Nature, 365:528–+, October 1993.
[25] V. E. Zavlin, G. G. Pavlov, D. Sanwal, R. N. Manchester, J. Trümper, J. P. Halpern,
and W. Becker. X-Radiation from the Millisecond Pulsar J0437-4715. Astrophysical
Journal , 569:894–902, April 2002.
[26] V. E. Zavlin. XMM-Newton observations of four millisecond pulsars. ArXiv Astrophysics e-prints, July 2005.
[27] O. Kargaltsev, G. G. Pavlov, and R. W. Romani. Ultraviolet Emission from the
Millisecond Pulsar J0437-4715. Astrophysical Journal , 602:327–335, February 2004.
[28] J. M. Fierro et al. EGRET High-Energy gamma -Ray Pulsar Studies. II. Individual
Millisecond Pulsars. Astrophysical Journal , 447:807–+, July 1995.
[29] L. Zhang and K. S. Cheng. X-ray and gamma-ray emission from millisecond pulsars.
Astronomy and Astrophysics , 398:639–646, February 2003.
[30] A. K. Harding, V. V. Usov, and A. G. Muslimov. High-Energy Emission From Millisecond Pulsars. AAS/High Energy Astrophysics Division, 8, August 2004.
[31] C. Venter and O. C. de Jager. Gamma-Ray Pulsar Visibility. ArXiv Astrophysics
e-prints, October 2005.
[32] M. Fra̧ckowiak and B. Rudak. Modeling gamma radiation from millisecond pulsars.
Advances in Space Research, 35:1152–1157, 2005.
[33] T. Bulik, B. Rudak, and J. Dyks. Spectral features in gamma-rays expected from
millisecond pulsars. MNRAS, 317:97–104, September 2000.
[34] W. Hofmann. H.E.S.S. highlights. In Proceedings of the International Comsic Ray
Conference Pune, volume 00, pages 101–106, 2005.
[35] C. W. Fabjan and F. Gianotti. Calorimetry for particle physics. Rev. Mod. Phys.,
75:1243–1286, 2003.
[36] J. Jelley. Cherenkov radiation and its application. Pergamon Press, 1958.
[37] A. M. Hillas. Cerenkov light images of EAS produced by primary gamma. In Proceedings of the International Cosmic Ray Conference, pages 445–448, August 1985.
[38] C. A. Wiedner. Site aspects of the HESS project: astronomical and visibility conditions. unpublished, H.E.S.S. internal note, 1998.
[39] O. Bolz. Absolute Energiekalibration der Abbildenden Cherenkov-Teleskope des
H.E.S.S. Experiments und Ergebnisse erster Beobachtungen des Supernova-Überrests
RX J1713-3946. PhD thesis, Universität Heidelberg, 2004.
[40] G. Hermann. private communication.
[41] The H.E.S.S. project an array of Imaging Atmospheric Cherenkov Telescopes.
http://www.mpi-hd.mpg.de/hfm/hess, 2005.
[42] J. A. Hinton. The status of the H.E.S.S. project. New Astronomy Review, 48:331–337,
April 2004.
[43] K. Bernlöhr et al. The optical system of the H.E.S.S. imaging atmospheric Cherenkov
telescopes. Part I: layout and components of the system. Astroparticle Physics,
20:111–128, November 2003.
[44] R. Cornils et al. The optical system of the H.E.S.S. imaging atmospheric Cherenkov
telescopes. Part II: mirror alignment and point spread function. Astroparticle Physics,
20:129–143, November 2003.
[45] A. Kohnle, J. Mattes, G. Hermann, W. Hofmann, and M. Panter. Photodetectors for
HESS. Nuclear Instruments and Methods in Physics Research A, 442:322–326, March
2000.
[46] F. Aharonian et al. Calibration of cameras of the H.E.S.S. detector. Astroparticle
Physics, 22:109–125, November 2004.
[47] S. Funk, G. Hermann, J. Hinton, D. Berge, K. Bernlöhr, W. Hofmann, P. Nayman,
F. Toussenel, and P. Vincent. The trigger system of the H.E.S.S. telescope array.
Astroparticle Physics, 22:285–296, November 2004.
[48] S. Schlenker, U. Schwanke, and C. Stegmann. The Central Data Acquisition System
of the H.E.S.S. Telescope System. In Proceedings of the International Cosmic Ray
Conference, pages 2891–+, July 2003.
[49] D. Heck, G. Schatz, T. Thouw, J. Knapp, and J. N. Capdevielle. CORSIKA: A Monte
Carlo code to simulate extensive air showers. FZKA-6019, 1998.
[50] K. Bernlöhr. CORSIKA and sim hessarray - Simulation of the imaging atmospheric
Cherenkov technique for the H.E.S.S. experiment. unpublished, H.E.S.S. internal
note, 2002.
[51] F. Aharonian et al. Calibration of cameras of the H.E.S.S. detector. Astroparticle
Physics, 22:109–125, November 2004.
[52] A. Daum et al. First results on the performance of the HEGRA IACT array. Astroparticle Physics, 8:1–2, December 1997.
[53] T.-P. Li and Y.-Q. Ma. Analysis methods for results in gamma-ray astronomy. Astrophysical Journal , 272:317–324, September 1983.
[54] TEMPO. http://www.atnf.csiro.au/research/pulsar/tempo/.
[55] International Earth Rotation Service. http://hpiers.obspm.fr.
[56] E. M. Standish. Orientation of the JPL Ephemerides, DE 200/LE 200, to the dynamical equinox of J 2000. Astronomy and Astrophysics , 114:297–302, October 1982.
[57] International
Earth
Rotation
http://maia.usno.navy.mil/.
and
Reference
Systems
Service.
[58] I. I. Shapiro. Fourth Test of General Relativity. Physical Review Letters, 13:789–791,
December 1964.
[59] D. C. Backer and R. W. Hellings. Pulsar timing and general relativity. Ann. Rev.
Astron. Astrophys., 24:537–575, 1986.
[60] C. M. Will and K. J. Nordtvedt. Conservation Laws and Preferred Frames in Relativistic Gravity. I. Preferred-Frame Theories and an Extended PPN Formalism. Astrophysical Journal , 177:757–+, November 1972.
[61] L. Fairhead et al. The Time Transformation Tdb-Tdt an Analytical Formula and Related Problem of Convention. In IAU Symp. 128: The Earth’s Rotation and Reference
Frames for Geodesy and Geodynamics, pages 419–+, 1988.
[62] R. Blandford and S. A. Teukolsky. Arrival-time analysis for a pulsar in a binary
system. Astrophysical Journal , 205:580–591, April 1976.
[63] T. Damour and N. Deruelle. Ann. Inst. H. Poincaré (Phys. Théor.), 44:263, 1986.
[64] T. Damour and N. Deruelle. Ann. Inst. H. Poincaré (Phys. Théor.), 43:107, 1985.
[65] J. H. Taylor and J. M. Weisberg. Further experimental tests of relativistic gravity
using the binary pulsar PSR 1913 + 16. Astrophysical Journal , 345:434–450, October
1989.
[66] The Australian Pulsar Group. http://www.atnf.csiro.au/people/pulsar/.
[67] W. T. Eadie, D. Drijard, and F. E. James. Statistical methods in experimental physics.
Amsterdam: North-Holland, 1971, 1971.
[68] D. A. Leahy, R. F. Elsner, and M. C. Weisskopf. On searches for periodic pulsed
emission - The Rayleigh test compared to epoch folding. Astrophysical Journal ,
272:256–258, September 1983.
[69] R. Bucceri et al. Search for pulsed gamma-ray emission from radio pulsars in the
COS-B data. Astronomy and Astrophysics , 128:245–251, November 1983.
[70] R. Buccheri and B. Sacco. Time Analysis in Astronomy: Tools for Periodicity
Searches. In Data Analysis in Astronomy, 1985.
[71] R. J. Protheroe. Periodic analysis of gamma-ray data. Proceedings of the Astronomical
Society of Australia, 7:167–172, 1987.
[72] O. C. de Jager. Ph.D. Thesis, 1987.
[73] O. C. de Jager, B. C. Raubenheimer, and J. W. H. Swanepoel. A poweful test for
weak periodic signals with unknown light curve shape in sparse data. Astronomy and
Astrophysics , 221:180–190, August 1989.
[74] O. C. de Jager. On periodicity tests and flux limit calculations for gamma-ray pulsars.
Astrophysical Journal , 436:239–248, November 1994.
[75] L. Jetsu and J. Pelt. Searching for periodicity in weighted time point series. Astronomy and Astrophysics Supplement Series , 118:587–594, September 1996.
[76] C. P. Masterson and H. E. S. S. Collaboration. Optical Observations of the Crab Pulsar Using the First H.E.S.S. Cherenkov Telescope. In Proceedings of the International
Cosmic Ray Conference, pages 2987–+, July 2003.
[77] Andreas Franzen. Diplomarbeit, Ruprecht-Karls-Universität Heidelberg, 2003.
[78] S. Schlenker and T. Eifert. Optimisation of Hillas selection cuts. in preparation,
H.E.S.S. internal note.
[79] F. Aharonian et al. Discovery of VHE gamma rays from PKS 2005-489. Astronomy
and Astrophysics , 436:L17–L20, June 2005.
[80] G. J. Feldman and R. D. Cousins. Unified approach to the classical statistical analysis
of small signals. Physical Review D, 57:3873–3889, April 1998.
[81] O. C. de Jager. private communication.
Acknowledgments
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 impressive 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!
Erklärung
Hiermit bestätige ich, daß ich die vorliegende Arbeit ohne unerlaubte fremde Hilfe angefertigt habe. Mit der Auslage meiner Diplomarbeit in der Bibliothek der Humboldt–
Universität zu Berlin bin ich einverstanden.
Berlin, den 9. Dezember 2005
Till Eifert