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Microwave detection of cosmic rays and multi-messenger analysis of the parameters of ultra-high energy astrophysical sources

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Microwave Detection of Cosmic
Rays and Multi-Messenger
Analysis of the Parameters of
Ultra-High Energy
Astrophysical Sources
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of
Philosophy in the Graduate School of The Ohio State University
By
Nathan E. Griffith, M.S., B.S.
Graduate Program in Physics
The Ohio State University
2015
Dissertation Committee:
Professor James J. Beatty, Advisor
Professor John F. Beacom
Professor Robert J. Perry
Professor Brian Winer
UMI Number: 3710178
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UMI 3710178
Published by ProQuest LLC (2015). Copyright in the Dissertation held by the Author.
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© Copyright by
Nathan E. Griffith
2015
Abstract
The study of ultra-high energy (UHE) cosmic particles is frequently characterized by its
low statistics, and a central problem of the field is to find novel ways to navigate this
challenge. The research presented in this dissertation attempts to address this problem
in two ways: first, by investigating microwave radiation as a new method of UHE cosmic
ray detection, and second, by using a multi-messenger (proton and neutrino) analysis to
determine what current and next generation UHE neutrino detectors may be able to reveal
about UHE astrophysical sources. The cosmic ray detector (called AMBER) is primarily a
joint collaboration between Ohio State and the University of Hawaii. In May/June 2011 the
AMBER experiment was installed at the Pierre Auger Observatory in Malargue, Argentina,
and began taking data in coincidence with the observatory’s surface detector array. This
work presents a description of the experiment, a calibration based on an astrophysical
radio source (the Milky Way galaxy), and an analysis of data. The second half of this
document describes a multi-messenger analysis performed with co-authors Amy Connolly
and Shunsaku Horiuchi on a publication in preparation. Fits to Pierre Auger 2013 data
are used in conjunction with a spectral model and simulations of UHE neutrino detectors
to explore the UHE source parameters of cosmic evolution and source spectrum cutoff.
Constraints provided using the effective areas of the ANITA 3, ARA, and EVA detectors
are considered.
ii
For my wife Zara, my parents Gregory and Mary Lea, and my brother Aaron.
iii
Acknowledgments
The author would like to acknowledge James Beatty for making it all possible, Amy Connolly
for the opportunity to work on a really cool paper, Shunsaku Horiuchi for being an excellent
co-author, John Beacom for many beneficial nudges over the years, Patrick Allison for always
having an answer, Eric Grashorn for writing almost all of the Auger-interfacing code, Jacob
Gordon for helping him when he was stuck, and Michael Sutherland for suggesting useful
edits.
iv
Vita
Spring 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.S. Physics, The Ohio State University
Autumn 2007 to Spring 2009 . . . . . . . . . . . . . . . . . . . Graduate Teaching Assistant, Department of Physics, The Ohio State University
Winter 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.S. Physics, The Ohio State University
Summer 2009 to present . . . . . . . . . . . . . . . . . . . . . . . Graduate Research Assistant, Department of Physics, The Ohio State University
Fields of Study
Major Field: Physics
Studies in Experimental Astrophysics: James J. Beatty
v
Table of Contents
Page
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgments
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Chapters
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
1.2
1
Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.2
The Knee and Ankle . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.3
The GZK Cutoff and Process . . . . . . . . . . . . . . . . . . . . . .
3
1.1.4
GZK Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.1.5
Challenges
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
The AMBER Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2.1
Background: Extensive Air Showers . . . . . . . . . . . . . . . . . .
5
1.2.2
Detection Systems of the Pierre Auger Observatory
. . . . . . . . .
6
1.2.3
Microwave Detection . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.2.4
AMBER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
vi
1.3
Multi-messenger Approach to the Study of Ultra-High Energy Sources . . .
12
1.3.1
Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.3.2
GZK Neutrino Experiments . . . . . . . . . . . . . . . . . . . . . . .
12
2. AMBER: Hardware, Software, and Deployment . . . . . . . . . . . . . . . . . .
2.1
2.2
2.3
2.4
2.5
14
Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.1.2
RF and Analog Components . . . . . . . . . . . . . . . . . . . . . .
14
2.1.3
Digital Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
Software: Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.2.1
Design Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.2.2
Description of Core Acquisition Programs (amberdaq) . . . . . . . .
19
2.2.3
Other Software (neutrino) . . . . . . . . . . . . . . . . . . . . . . . .
20
Software: Observatory Triggers . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.3.2
Trigger Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.3.3
SD Event Processing . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.3.4
Event Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
Software: GUI Event Viewer . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.4.1
Viewing Run Files . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.4.2
Candidate Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
Deployment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.5.1
Site Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.5.2
Timing Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.5.3
Solar Transits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.5.4
Data Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
3. AMBER: Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.2
Hardware Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
vii
3.3
Simulated Transits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
3.4
Calibration Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
4. AMBER: Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
4.2
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
4.3
Results and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
5. Multi-messenger: Spectrum Generation . . . . . . . . . . . . . . . . . . . . . . .
5.1
5.2
45
A Reintroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
5.1.1
Other Approaches to the Problems of UHE Astrophysics . . . . . . .
45
5.1.2
Basic Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
Model for UHE Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
5.2.1
Explanation of Components . . . . . . . . . . . . . . . . . . . . . . .
46
5.2.2
Free Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
5.2.3
Choice of Evolutions . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
6. Multi-messenger: Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
6.1
“Ankle” and “Dip” Models . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
6.2
Weighting Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
6.3
Fit Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
7. Multi-messenger: Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
7.1
Pseudoexperiment Generation . . . . . . . . . . . . . . . . . . . . . . . . . .
54
7.2
Excluding Source Evolutions . . . . . . . . . . . . . . . . . . . . . . . . . .
56
7.2.1
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
7.2.2
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
Excluding Values for the Source Spectrum Cutoff . . . . . . . . . . . . . . .
66
7.3.1
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
7.3.2
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
Estimating the Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
7.3
7.4
viii
7.5
7.4.1
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
7.4.2
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
78
Appendices
A. Expected Power of Microwave Bremsstrahlung Radiation . . . . . . . . . . . . .
79
B. Atmospheric Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
ix
List of Figures
Figure
Page
1.1
The knee and ankle features of the cosmic ray spectrum. From Ref. [1]. . .
1.2
Experimental cosmic ray data and a range of neutrino models. Dated early
2007. From Ref. [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3
2
4
A cartoon visualizing the two major detection system of the Pierre Auger
Observatory: surface detectors (SDs) and fluorescence detectors (FDs). From
Ref. [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4
Detected microwave power plotted versus the power of a man-made shower
during AMBER’s SLAC test. From Ref. [2]. . . . . . . . . . . . . . . . . . .
1.5
9
Distribution of pseudoranges from AMBER data taken during its preliminary
setup on a University of Hawaii rooftop. From Ref. [2]. . . . . . . . . . . . .
2.1
7
11
The configuration of the feed horn array and associated channel numbers as
implemented in the AMBER-01 configuration. Note that this is as seen from
the perspective of the dish—the sky positions of these channels would be
left-right and up-down inverted. . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Data flow through major components of the AMBER acquisition system.
GPS/PPS propagation not shown. Based on Ref. [4]. . . . . . . . . . . . . .
2.3
15
18
A demonstration of some features of the AmViewer application in candidate
mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
26
2.4
AMBER-01 as installed at the Coihueco site of the Pierre Auger Observatory.
From [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5
Data from four AMBER channels during a single tank timing test event. The
black vertical line marks the expected beginning of the pulse. . . . . . . . .
2.6
31
Results of the AMBER calibration to transits through the plane of the Milky
Way galaxy. Channels C1H to C8. . . . . . . . . . . . . . . . . . . . . . . .
3.2
30
The pointing directions of AMBER horns as derived from solar transits.
Channel 9 did not function well and is not shown. From Ref. [6]. . . . . . .
3.1
30
37
Results of the AMBER calibration to transits through the plane of the Milky
Way galaxy. Channels C9 to C16. Fit parameter n is the system noise given
in units of milliwatts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
38
The parameter µavg plotted as a function of Pcut . This analysis restricts to
sub-candidates with showers that have a Gaisser-Hillas function (see Eq 4.7)
value greater than 0.1, with an altitude less than 11 km, and a zenith angle
less than 60 degrees (see Appendix B). Plotted points give µavg values, with
the number of sub-candidates used to calculate each average recorded in a
histogram. A red line marks the completely incoherent case at µavg = 1.
Note that because Pobs is actually excess power (see Eq. 4.8) it is possible
for µavg to fluctuate to a negative value. . . . . . . . . . . . . . . . . . . . .
5.1
44
Plots of the three astrophysical source evolutions considered. Solid lines
represent the nominal shape, while dashed lines indicate a shape used to calculate systematic uncertainties. The stellar formation rate function (SFR) is
derived from information, data, and fits in Refs. [7] [8] [9] [10] [11]. Gamma
ray burst (GRB) evolution and FRII-type AGN evolution are based on modified versions of Ref. [12] and Ref. [13] respectively. Plots are by Amy Connolly
for Ref. [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
49
6.1
Fits to Auger ICRC 2013 data [15, p. 30] in both the ankle and dip models
using all four evolutions. Ankle fits start at the 1018.8 eV data point, while
dip fits start at 1017.6 eV. Created with assistance from Ref. [16]. . . . . . .
7.1
53
Effective areas of the three experiments considered as implemented in this
research. EVA information is from Ref. [17] with numbers from Eugene
Hong [18]. ARA37 (the final 37 station configuration of ARA) values are
from Ref. [19] with numbers again from Eugene Hong [18]. ANITA 3 effective
area figures are estimated by shifting ANITA 2 [20] [21] values 30% lower in
energy [16]. In all cases the last available data point is preserved out to a
log10 (E/eV) of 24.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2
Distribution of events expected for all three experiments in the FRII evolution
using the ankle model fits. For this figure Emax is assumed to be 1021.5 eV.
7.3
55
56
An example of the likelihood distributions −2 ln(Q0 ) and −2 ln(Q1 ) for 15,000
pseudoexperiments each and at various elapsed experiment times. Vertical
lines mark the ±1 σ analogs in the red −2 ln(Q0 ) distribution.
. . . . . . .
59
7.4
Source evolution exclusion for ANITA 3, ankle model, 150 days. . . . . . . .
60
7.5
Source evolution exclusion for ANITA 3, dip model, 150 days. . . . . . . . .
61
7.6
Source evolution exclusion for ARA (37 stations), ankle model, 20 years. . .
62
7.7
Source evolution exclusion for ARA (37 stations), dip model, 20 years. . . .
63
7.8
Source evolution exclusion for EVA, ankle model, 150 days. . . . . . . . . .
64
7.9
Source evolution exclusion for EVA, dip model, 150 days. . . . . . . . . . .
65
7.10 log(Emax /Ev) exclusion plots for ANITA 3. Note log time axis. . . . . . . .
67
7.11 log(Emax /Ev) exclusion plots for ARA (37 stations). Note log time axis. . .
68
7.12 log(Emax /Ev) exclusion plots for EVA. Note log time axis. . . . . . . . . . .
69
7.13 Middle 68% of minimum − ln(L) based log Emax predictions for ANITA 3. .
70
7.14 Middle 68% of minimum − ln(L) based log Emax predictions for ARA37. . .
72
7.15 Middle 68% of minimum − ln(L) based log Emax predictions for EVA. . . .
72
xii
A.1 Intensity versus time in a T471 experiment. This version uses an antenna configuration that cross polarizes against potentially contaminating Cherenkov
emission. From Ref. [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
80
List of Tables
Table
2.1
Page
AMBER horn pointing directions derived from R. Mussa’s solar transit analysis [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.1
Constraints used for fit parameters. . . . . . . . . . . . . . . . . . . . . . . .
36
3.2
AMBER noise temperatures by channel using Equation 3.12 with B = 500
MHz for channels 1-4, and 800 MHz for channels 5-16. . . . . . . . . . . . .
6.1
39
For dip model: Maximum absolute difference induced in α across all evolutions for changes in various parameters. Energy shifts are in units of 14%.
Values calculated by Amy Connolly for Ref. [14]. . . . . . . . . . . . . . . .
6.2
51
For ankle model: Maximum absolute difference induced in α across all evolutions for changes in various parameters. Energy shifts are in units of 14%.
Values calculated by Amy Connolly for Ref. [14]. . . . . . . . . . . . . . . .
52
6.3
Fit values for alpha and norm . . . . . . . . . . . . . . . . . . . . . . . . . .
52
7.1
Expected event numbers for the experiments considered. ANITA 3 and EVA
are calculated for 150 days flight time with a log10 (Emax /eV) of 21.5. ARA37
is calculated for the same Emax with 20 years of experiment time.
xiv
. . . . .
55
Chapter 1
Introduction
1.1 Cosmic Rays
1.1.1 Background
In the most general sense, cosmic rays can be described as charged particles of extraterrestrial origin that have been accelerated by some energetic process [22, p. 77].
The study of cosmic rays is important for a number of reasons. As direct messengers from
the phenomena that create them, they can provide valuable information about astrophysical
processes that are poorly understood [22, p. 294]. They may also yield data relevant to
particle physics, since cosmic ray energetics at the high end of the energy spectrum exceed
the center of mass energies of man-made accelerators by nearly two orders of magnitude [22,
p. 293]. They tell us about the Universe’s most powerful objects, and give us a difficult to
obtain glimpse into the energies of physics outside the Standard Model.
The overall structure of the cosmic ray spectrum is characterized by an inverse power
law, leading to a dramatic reduction in particle flux with increasing energy [23, p. 7].
This means that experiments probing higher energies must use creative methods to garner
acceptable event rates. Just as the quest for higher energy particle accelerators drives
Earthbound particle physics experiments, a similar search for larger and larger apertures
drives the field of high energy particle astrophysics. Indeed, the flux at the highest energies
is so low that relatively tiny numbers of events (compared to particle accelerators) can have
a dramatic effect on the plausibility of theoretical models (for example, the GZK process
1
discussed in Section 1.1.3).
1.1.2 The Knee and Ankle
Energy multiplied log-log plots of the cosmic ray spectrum have two major breaks in the
power law spectrum that appear as a slight bend downward, followed later by a slight
bend back upward. These are respectively referred to as the “knee” and the “ankle” of the
spectrum [22, p. 80].
The knee occurs at an energy near 1015 electron-volts (eV), and the associated increase
in the slope of the spectrum is thought to be associated with the inability of the Milky
Way’s magnetic field to contain particles above this energy [22, pp. 80-81]. These particles
leaving the Galaxy are unavailable for observation, causing a corresponding decrease in the
overall flux. An alternative hypothesis states that the change in spectral index may be due
to the physics of supernovae, and their inability to accelerate cosmic rays in this energy
regime [22, p. 81].
Figure 1.1: The knee and ankle features of the cosmic ray spectrum. From Ref. [1].
2
At 5 × 1018 eV the slope shifts toward a slightly more positive value, creating the
“ankle” of the spectrum [22, p. 80]. This is commonly thought to be where a transition
occurs between the spectrum of cosmic rays created by galactic sources to one that is due to
extra-galactic phenomena [23, p. 9]. However this is not the only model for this feature [24],
and in Section 8.1 I will introduce an alternative hypothesis.
1.1.3 The GZK Cutoff and Process
The cosmic ray spectrum as observed by most experiments sensitive in the ultra-high energy
(UHE) regime shows another feature: a sharp cutoff that begins near 5 × 1019 eV [22, pp.
80, 82-83].
First theorized in 1966, the GZK process (named for Greisen, Zatsepin, and Kuzmin)
[25] [26] posits that cosmic rays at very high energies are above threshold for an interaction
with abundant photons from the cosmic microwave background (CMB). This interaction,
described by,
p + γCM B → p/n + π 0 /π +
(1.1)
causes protons above the threshold to be subject to energy loss, and enforces an effective
horizon for these particles on the scale of tens of mega-parsecs (Mpc) [22, p. 82]. This
cutoff and associated horizon mean that cosmic rays from the end of the spectrum make for
poor probes of both the extremely energetic processes that create them, and the behavior
of the universe outside of our local galactic neighborhood.
1.1.4 GZK Neutrinos
The same process responsible for the cutoff in the ultra high cosmic ray (UHECR) spectrum
is also the source of a spectrum of cosmogenic ultra-high energy neutrinos [23, p. 300].
Consider, for example, the GZK interaction path that results in the production of a charged
pion. The pionic decay yields the following:
π + → µ+ + νµ ,
3
(1.2)
with the resulting muon decaying via
µ+ → e+ + νe + ν µ .
(1.3)
Protons with energies far above the GZK interaction energy may initiate this process many
times, creating additional neutrinos at each collision with a CMB photon [23, p. 300].
Unsurprisingly, the small cross-section of neutrino interactions makes the detection of
this GZK-borne spectrum difficult. Though none have yet been observed, the ultra-high
energy neutrinos resulting from these interactions are expected to have a flux and energy
range comparable to that of cosmic rays in the ultra-high energy regime. However, so little
is known about UHECR sources that much uncertainty remains as to the exact magnitude
and shape of the spectrum [2].
Figure 1.2: Experimental cosmic ray data and a range of neutrino models. Dated early
2007. From Ref. [2].
Searches to date have failed to provide a single unequivocal event [27], but with new
experiments on the horizon the chances of observing GZK neutrinos are better than ever.
4
1.1.5 Challenges
As stated before, the predominant difficulty of ultra-high energy astroparticle physics has
to do with the extremely low event rates associated with the tail end of the cosmic ray
spectrum. In fact above GZK energies (> 1020 eV) the number of expected cosmic ray
events is near 1 km−2 century−1 [2]. The only way around such a difficulty is to compensate
by finding novel (and hopefully inexpensive) ways to instrument vast areas and volumes.
These concerns naturally lead to an examination of related secondary phenomena that
can be observed with sparse instrumentation—a topic of extreme relevance to the research
presented in this dissertation.
However even clever engineering cannot fully make up for the dearth of statistics, and
much about the fundamental nature of cosmic rays is still unknown. While supernovae and
related phenomena are thought to comprise much of the galactic spectrum [23, pp. 9, 62],
the origin of extra-galactic cosmic rays remains a mystery. Black hole-fueled active galactic
nuclei (AGN) have the necessary power for UHECR acceleration, and are often indicated
as candidates [22, p. 168], but as of yet we lack any clear indication from cosmic ray arrival
directions that they are indeed their source [23, pp. 256-257].
The study of the highest energy extra-terrestrial particles demands that we bring both
our best technological and analytical methods to bear. No potential source of data can go
uninvestigated, and existing data must be carefully analyzed for all it is worth.
1.2 The AMBER Experiment
1.2.1 Background: Extensive Air Showers
For cosmic hadrons descending vertically, Earth’s atmosphere has a depth of 11 interaction
lengths, ensuring that collision with ambient gas molecules is a virtual certainty [22, p.
143]. The corresponding burst of daughter particles continue to propagate downward, reinteracting over many generations and spreading out conically until they reach Earth’s
surface. Such an event is termed an extensive air shower (EAS), and their detection serves
as the basis for ground based cosmic ray observatories [22, pp. 156-162].
5
In hadronic collisions, about half of the available energy goes to the creation of new
hadronic particles [22, p. 157]. Pions comprise a majority of these new particles, with
kaon production occurring with a tenth the frequency [22, p. 143]. Pion charges are
produced evenly, with a roughly equal probability for each [23, p. 185]. Neutral pions
decay nearly instantly [23, p. 185] via π 0 → γγ, while charged pions and kaons that do
not re-interact produce muons, which can in turn produce electrons that contribute to
electromagnetic cascades [22, p. 144]. However, the majority of hadrons do re-interact,
and make a contribution to the electromagnetic (EM) component by way of π 0 decay [22,
p. 157]. Since the radiation length for EM particles is only about a third of that of
the interaction length for hadrons, the EM channel of showers is often referred to as the
“soft” component [22, pp. 143-144]. EM particles deposit their energy in the atmosphere
so efficiently that they permit a calorimetric reading for skyward-facing detectors that
correlates well to the energy of the primary cosmic ray [22, pp. 157-159].
By the time an EAS originating from a UHE event reaches sea level the hadronic core
of the shower is nearly spent, and much energy has already been bled off into the soft EM
component [23, p. 203]. While not as accurate, particles that survive to the Earth’s surface
can also be used to gain information about the primary particle’s energy [22, p. 159].
1.2.2 Detection Systems of the Pierre Auger Observatory
The Pierre Auger Observatory (PAO) located in Malargue, Argentina is designed to study
cosmic rays with an energy above 3 × 1017 eV [15, p. 27]. It uses two main methods to
determine the energy and arrival direction of ultra-high energy cosmic rays: fluorescence
telescopes, and surface detectors [15, p. 7]
Auger’s air fluorescence detectors (FDs) work by collecting the ultraviolet light created
by the photon emissions of excited atmospheric nitrogen [15, p. 7]. This magnitude of this
effect is proportional to perturbations caused by the EM component of a shower, which, as
stated earlier, represents a large portion of overall shower energy. Light from shower events
is focused via a spherical mirror onto an array of photomultiplier tubes (PMTs) [15, p. 7].
Using PMT signal and pixel trigger timing information, the energy of the shower and its
6
Figure 1.3: A cartoon visualizing the two major detection system of the Pierre Auger
Observatory: surface detectors (SDs) and fluorescence detectors (FDs). From Ref. [3].
arrival direction can be estimated [15, pp. 7-8].
In addition to the FDs, Auger employs a hexagonal grid of over a thousand surface
detectors (SDs) which instruments an area of 3000 km2 [15, p. 7]. The SDs are tanks of
water equipped with PMTs used to measure the Cherenkov light which is created [15, p.
23] as the surface-surviving components of a shower pass through them. Signal in the tanks
can be used as an estimator of shower energy [23, p. 241], and the timing of tank triggers
can be fit to determine directional information [22, p. 159].
Fluorescence detectors are able to provide a more accurate reading of shower (and thus
primary particle) energetics because they are able to capture a shower’s development at
many points as it traverses the sky, as opposed to the SDs which only receive information
about a single atmospheric depth [22, p. 159]. However, due to the nature of their operation
FDs can only provide data at night, in favorable weather conditions, and in the absence of
interfering moonlight [22, p. 160]. This results in an overall uptime of only 14%, unlike the
SDs which can run day and night without interruption [15, p. 15].
Part of Auger’s design philosophy is to play to the strengths of both these systems. This
7
is done by mapping the signals seen in SD tanks to the energetics observed by the FDs via
a fit to events in which both systems have data [23, p. 241]. These hybrid events form
the backbone of the cosmic ray spectrum as seen by the PAO; currently one of the most
prominent datasets available to those studying ultra-high energy astrophysical phenomena.
1.2.3 Microwave Detection
Another detection technique for cosmic ray showers, first investigated by Gorham et al. [2]
and presented in this and the following section, is that of microwave-band signals created
by a phenomenon called microwave bremsstrahlung radiation (MBR). As an air shower
propagates, it creates a plasma of thermal electrons with a size proportional to the shower
energy. These free electrons pass through the electric fields of neutral atmospheric molecules,
emitting bremsstrahlung photons in the gigahertz range. Like air fluorescence, this radiation
is expected to be unpolarized and distributed isotropically.
The question of how this radio frequency (RF) power scales with the number of electrons,
and thus shower energy, is important in determining how successful MBR based detectors
might be. To this end, a simple model is used to consider the effect of coherence among
plasma electrons. For a number of electrons Ne each with a field ⃗ϵ, the total field can be
expressed as a sum over phase factors
⃗ =
E
Ne
X
⃗ϵ e−iφk .
(1.4)
k=1
⃗ 2 , which in the completely coherent case of φ1 =
Radiated power P is proportional to |E|
φ2 = ... = φNe yields Pcoh = Ne2 Pe , where Pe is the power due to one electron. For the
completely incoherent case of random phases, the result for power is a term that scales with
Ne added with a sum of other terms whose average value is zero, so the incoherent power
is taken as Pinc = Ne Pe (see Ref. [28] for a useful discussion).
To examine a partially coherent case divide the total number of electrons Ne into M
groups, which each contain µe electrons that all share the same phase factor (Ne = M µe ).
The total field can now be written as
8
⃗ =
E
M
X
µe⃗ϵ e−iφk ,
(1.5)
k=1
which, similar to Eq. 1.4, yields a power of Ppar = M µ2e Pe for random phases. Assuming
that an increase in plasma electrons would also increase the size of the coherent groups,
it can be seen that in both the coherent and partially coherent case, radiated power from
MBR would scale quadratically with shower energy. Indeed, this is exactly the behavior
that Gorham, et al. [2] (see Fig. 1.4) saw during a test at the Stanford Linear Accelerator
Center (SLAC) where an electron beam was fired into an anechoic chamber equipped with
a power detector.
Figure 1.4: Detected microwave power plotted versus the power of a man-made shower
during AMBER’s SLAC test. From Ref. [2].
While fluorescence and Cherenkov detection have certainly proved themselves as valuable tools, the development of a microwave radio based system to detect cosmic rays is
attractive in a number of ways. First, RF power in this frequency regime can be detected
using readily available C-band radio antennas instead of PMTs, and second, microwave
9
radiation attenuates at a longer distance than ultra-violet light, and is able to be detected
at all hours and most conditions. An array of them could even be used to achieve shower
development information in the same manner that an FD would. Microwave based detection could combine the best parts of the FD and SD all in one package, making an excellent
complement to existing systems or even a capable standalone detector.
1.2.4 AMBER
A primary focus of this document will be the design, implementation, and data analysis of
the Air-shower Microwave Bremsstrahlung Experimental Radiometer (AMBER), a detector
built in collaboration between the University of Hawaii and The Ohio State University
to test the feasibility of shower detection using GHz band RF. For the initial tests of
AMBER, which took place on the roof of the University of Hawaii’s physics department,
the experiment consisted of four dual-band, dual polarization C and Ku antennas mounted
at the focus of a 1.8 meter parabolic dish. These antennas serve as an analog to the PMT
cluster clusters found in the FDs at Pierre Auger.
For data collected in this initial rooftop environment, AMBER is unable to discern
signals due to astrophysical phenomena from other environmental RF sources. In order to
gain a sense of how many events might be related to air showers a parameter called the
pseudorange, which assumes a perfectly perpendicular crossing through the line of sight at
the speed of light, was calculated for events that appear in two feeds. Given an angular
separation between antennas of θsep , a time between signals ∆t, and the speed of light c,
the pseudorange is given by
Rpseudo =
1
c ∆t
.
2 tan(θsep /2)
(1.6)
Comparison of a distribution of pseudoranges to those generated by simulated cosmic rays
in a Monte Carlo showed general agreement (Fig. 1.5), however for any more rigorous
analyses AMBER would have to be installed in the presence of another cosmic ray detector.
AMBER’s eventual deployment and integration with the surface detector system of the
Pierre Auger Observatory will be discussed in further chapters.
10
Figure 1.5: Distribution of pseudoranges from AMBER data taken during its preliminary
setup on a University of Hawaii rooftop. From Ref. [2].
11
1.3 Multi-messenger Approach to the Study of Ultra-High
Energy Sources
1.3.1 Motivations
Studying the cosmic ray spectrum at the highest energies is difficult work, but the Pierre
Auger collaboration and others have managed to collect an impressive amount of data,
especially when compared to the nascent studies of the ultra-high energy neutrino spectrum,
which have so far only managed to place upper-limit constraints on theoretical spectra [27].
In fact, when used in conjunction with modeling, the current amount of cosmic ray data
is robust enough to provide estimates of the spectral index of ultra-high energy (UHE)
sources—a class of objects that we know very little about. Nevertheless, the cosmic ray
spectrum on its own is limited by the GZK effect in what it can tell us about ultra-high
energy sources.
Because cosmogenic GZK neutrinos are tied directly to the cosmic rays that produce
them, without being subject to the same cutoff or horizon, they provide a perfect complement to cosmic ray data. Their observation would allow us to see past the GZK curtain
and into the realm of UHE sources at very high energies and cosmological distances.
1.3.2 GZK Neutrino Experiments
Research in this document relevant to multi-messenger analysis will focus on the observational potential of three ultra-high energy neutrino experiments in various stages of development: two that are payloads of high-altitude balloons (ANITA and EVA), and a third
ground-based experiment (ARA). All three are RF detectors that operate in Antarctica.
Unlike Auger, these experiments use ice as their interacting medium rather than air or
liquid water. They leverage the Askaryan effect, which is the ability of dielectrics (like ice)
to create coherent radio frequency radiation when they are the medium for the propagation
of relativistic charged particles [29] [30] [31]. The attenuation length of such signals in ice is
quite long (e.g., order 1 km over ANITA 1’s 200–1200 MHz bandwidth) [29], allowing these
experiments to instrument very large volumes and mitigate the tiny cross section and low
12
statistics associated with the object of their search.
ANITA (the Antarctic Impulsive Transient Antenna) is a well established experiment
that has just completed its third run (ANITA 3) [32] [33]. Currently under development, the
EVA (ExaVolt Antenna) experiment is similar to ANITA, but is a more sensitive instrument
owing to the fact it uses balloon surfaces as a part of the antenna apparatus [34]. ARA
(Askaryan Radio Array) is an in-ice detector which is now implemented in a preliminary
version. Eventually ARA will consists of 37 stations, each of which spanning a depth of 200
meters [35].
By constraining a model to Auger data and looking a few years ahead to what these
current and next-generation ultra-high energy neutrino detectors might see, it becomes
possible to use both particle messengers to probe some features of UHE sources. As will be
seen, this approach is powerful for inspecting the redshift evolution of UHE sources, as well
as the spectral cutoff for particle acceleration at the sources.
13
Chapter 2
AMBER: Hardware, Software,
and Deployment
2.1 Hardware
2.1.1 Introduction
The hardware components that comprise the AMBER experiment must solve the problems
of signal acquisition, power detection, analog to digital conversion, data acquisition, triggering, and data storage. The goal of the following sections is to provide an overview of
how these problems are solved, and by which components. Much of the information in this
chapter was gleaned from Refs. [36] [37] [38] [4] and conversations with collaborator Patrick
Allison (currently at The Ohio State University) [39].
Some descriptions in this chapter are only completely accurate for the AMBER-01 configuration, which lasted until late 2013 at which point the camera system underwent a
significant upgrade. A discussion of that iteration of the experiment is left for future work.
The AMBER hardware was designed by University of Hawaii members of the collaboration
under the direction of principal investigator Peter Gorham.
2.1.2 RF and Analog Components
As first deployed at the Coihueco FD site of the Pierre Auger Observatory, AMBER consists
of 16 antennas mounted at the focal point of a 2.4 meter off-axis parabolic dish. The central
four antenna horns (see Fig. 2.1) are arranged in a diamond configuration, and are dual14
band operating in both C and Ku bands with a sensitivity of 3.7–4.2 GHz for C band and
10.95–12.75 GHz for Ku [40] [17]. They are also dual polarization, capable of providing
both the horizontal and vertical components of RF signals. The remaining 12 horns are
arranged in groups of three around the central diamond pattern, and are single polarization
C band with a range of 3.3–4.9 GHz [40] [17].
Figure 2.1: The configuration of the feed horn array and associated channel numbers as
implemented in the AMBER-01 configuration. Note that this is as seen from the perspective
of the dish—the sky positions of these channels would be left-right and up-down inverted.
This antenna cluster can be thought of as AMBER’s version of the light-sensitive pixels
that comprise the detecting surface of more traditional imaging systems, making it analogous to the CCDs of digital cameras or the PMT arrays of Auger’s fluorescence telescopes.
Though they are not as numerous as the PMTs in an FD (440 [23, p. 238] vs AMBER’s
16), they could still be used in a similar fashion to determine some rudimentary information
about shower development and direction.
15
Moving backward down the chain that leads from the antenna horns to data acquisition,
RF signals are first acquired via a low-noise block down-converter (LNB). For C band signals
a Chaparral LNB with an input frequency of 3.4–4.2 GHz is used. These LNBs have an
output frequency of 950–1750 MHz and a typical gain of 65 dB. Ku band frequencies are
handled by a Norsat 4106A LNB. These have an input range of 11.7–12.2 GHz and an output
of 950–1450 MHz with a gain of 64 dB. [41]. Note that for the central feeds AMBER is
limited to the 500 MHz bandwidth imposed by the horns, while the outer feeds are sensitive
along the full range (800 MHz in C band) of the LNBs.
From there the signal propagates to a JEDI (Joining Electronics and Detection Instrument) board, of which there is one per channel. It is the responsibility of these electronics
to convert the received RF signal into a voltage related to power. To do this each JEDI
is equipped with an Analog Devices integrated circuit called an AD8318. The AD8318 is
a power detector with an input range of 1 MHz to 8 GHz, capable of providing logarithmically scaled output in the range of 0–5 V with a 10 ns response time [42]. This power
value (adjusted by the analog front end discussed in the next section) is presented to the
digitization electronics.
2.1.3 Digital Electronics
From the JEDIs, antenna channels are fed into one of the four AMBER data acquisition
boards (DAQ). Each DAQ board accepts eight channels, and utilizes an AD9233 analog
to digital converter (ADC) which digitizes at a resolution of 12 bits and 100 MHz [38]
[43]. The AD9233 accepts voltages in the range 0–2 V via a two-lead differential input
(−1V < (Vin+ − Vin− ) < +1). In order to construct a digitizer-compatible input the signal
is propagated through the analog front end of the DAQ board. In this component the single
voltage that arrives as the output of the AD8318 is transformed into two voltages of the
appropriate separation and level for the AD9233 digitizer via a series of op-amp circuits.
Since AMBER is designed to work with existing observatory detectors, it must have the
capability to retain data for long enough for external triggers to propagate to its systems. To
solve this problem, each DAQ board carries enough on-board RAM to accommodate a ring
16
buffer that holds over 5 seconds of data from all channels. In the event of an observatory
trigger, an indicated time and window (typically ∼ 100 microseconds) are read out from
the device and stored to disk.
The four DAQ boards are attached via a compact PCI crate to a computer (“amberdaq”)
which runs most of the low-level software associated with data acquisition. Also present
in this machine is a piece of hardware called the LTRIG board, which is primarily associated with self-triggering the AMBER system—a feature that is not used in deployment at
Auger. The LTRIG board also accepts the hardware PPS (pulse per second) signals from
an attached USB GPS receiver, and propagates them to the AMBER DAQ boards.
Attached via Ethernet is a second AMBER data acquisition machine called “neutrino,”
which has many terabytes of capacity and serves in part as the on-site data storage computer. One of neutrino’s main computational responsibilities is to process the raw data sent
over from amberdaq into compressed ROOT [44] files. This processed data can then be
pulled down remotely via an Internet connection to university servers. This same network
interface delivers the Auger SD trigger data by way of a UDP connection from Auger’s central data acquisition system (CDAS), the handling of which is neutrino’s second primary
responsibility. A detailed discussion of this trigger follows in section 2.3.
17
Figure 2.2: Data flow through major components of the AMBER acquisition system.
GPS/PPS propagation not shown. Based on Ref. [4].
2.2 Software: Data Acquisition
2.2.1 Design Philosophy
Rather than relying on one single monolithic application to do all the work, the low-level
data acquisition on amberdaq is handled by a suite of interdependent, intercommunicating
background processes. This allows for flexible propagation of data, sidestepping bottlenecks
that a single large all-in-one solution might incur. In all there are six core data acquisition
programs which are linked to together via the D-Bus and TCP/IP protocols. A brief
description of each is provided in this section.
As in the previous section, conversations with Patrick Allison [39] and Ref. [36] were
very helpful in formulating the descriptions in the following passages. AMBER software is
18
largely written by Dr. Allison, with major contributions from myself in the implementation
of the software trigger and the design of the AmViewer application.
2.2.2 Description of Core Acquisition Programs (amberdaq)
The basic set of problems that low-level AMBER data acquisition must solve includes
managing associated hardware, event creation, trigger acceptance, and providing for a user
interface. However these tasks are not quite orthogonal with respect to the amberdaq
processes, and some programs perform more than one duty. What follows is a brief tour of
some of these programs and some of their functions within AMBER.
A piece of software called AmTrigServer can be considered the heart of these tools. It
accepts triggers via TCP/IP connection from three possible sources: self-triggers (unused at
Auger), external triggers from the observatory, and housekeeping triggers associated with
the PPS heartbeat from the GPS. After receiving a trigger, AmTrigServer uses the included
timing information to arrange for the DAQ devices to have the appropriate data queued up
for readout. Next, a process called AmEvGen reads the relevant data from the acquisition
board memory and compiles it into an uncompressed event format.
Non-observatory triggers are generated by two pieces of software: AmTrigClient and
AmTrigPpsClient. Self-triggers are sent by AmTrigClient when they are initiated by the
LTRIG board device, while AmTrigPpsClient sends a message to create a PPS associated
event at 10 second intervals. A program named AmGpsd is tasked with managing and logging
the GPS data, and it informs AmTrigPpsClient via a D-Bus PPS signal.
Aside from sending self-triggers, AmTrigClient has additional responsibilities. One of
these is fetching a housekeeping data sample from the DAQ boards when it receives a
hardware PPS initiated interrupt from the LTRIG board. Ten samples are retrieved once
per second, leading to an overall housekeeping data rate of about 10 Hz. Another function of
AmTrigClient is to provide an interface via D-Bus for real-time monitoring of the AMBER
system.
The final piece, which might be considered the head of these components, is AmCtl
which manages acquisition, the starting and stopping of data runs, and provides for the
19
coordination via D-Bus (an interprocess communication system) of all the other processes.
2.2.3 Other Software (neutrino)
A number of other tools important to the complement of AMBER software are executed on
the second on-site computer, neutrino. One of these is AmMon, which is a GUI application
that accesses the AmCtl process on amberdaq in order to provide a real-time representation
of power reported by housekeeping. This capability is especially useful when performing
tests of the detector in the field.
Also on neutrino is AmRunDaemon, which orchestrates the creation of ROOT files from raw
data sent over from amberdaq. ROOT data classes (and their AMBER-customized descendants) provide for compression and present a well-established software interface (ROOT is
a standard analysis package for particle and astroparticle physicists). As part of its responsibilities AmRunDaemon creates .root files for each four hour run that it finds. Of particular
interest to my research are the .root files associated with the housekeeping data and the
event data from triggers. These files are small enough (∼100 MB for event files, ∼10MB for
housekeeping) that uploads to off-site computers present no difficulties even over the limited bandwidth available to Auger field sites. In the chapters to follow a power calibration
is completed using data from housekeeping files, and an analysis is performed on triggers
spread over many event files.
The neutrino machine also hosts AmTrigAugerClient, which accepts and processes triggers from the SD array of the observatory. This software will be discussed at length in the
next section.
2.3 Software: Observatory Triggers
2.3.1 Introduction
The purpose of AMBER’s deployment at Auger is to record gigahertz RF emissions during
known shower events. To this end, a trigger based on data received from the observatory’s
array of SD tanks was designed. To function optimally, the design of this trigger must
20
account for a major constraint of the AMBER system—namely, the 5 second buffer of
available data. Information from the SD’s must be received promptly, and then processed
promptly.
2.3.2 Trigger Propagation
As detailed by Refs. [45] and [40], triggering of Auger’s surface detectors is accomplished
using three trigger levels. The first trigger, T1, is limited to a single tank and has a rate
of about 100 Hz. For the T2 trigger, the station forwards T1 triggers at a rate near 20 Hz
to Auger’s data acquisition computers. Sets of stations whose trigger timings and locations
are a good match for a shower event are queried by CDAS for their PMT data, and then
the creation of a full T3 trigger can begin.
In order to receive events registered by the SD array in a timely manner, the trigger
information that AMBER receives is based on the list of tanks that CDAS queries when
beginning the construction of a T3 [40]. This SD trigger data is forwarded via UDP to
AmTrigAugerClient on AMBER’s neutrino machine located in the field. The primary
contents of these UDP datagrams are the event time, the associated tank ID’s for the
event (eventually translated into locations via data from an XML file read into memory at
runtime), and microsecond-accurate timing information for each tank. Though not identical
to a completed T3, this data suffices to provide AMBER with some rough information about
shower events in the array.
Once received by AmTrigAugerClient, the datagram is processed for an AMBER trigger
time, relevant data about the trigger is logged, and the trigger is sent by TCP/IP to
AmTrigServer on amberdaq where it induces the creation of an observatory initiated event.
2.3.3 SD Event Processing
To increase the accuracy of the AMBER trigger time, SD data is processed for a shower
arrival direction. This allows for a correction to be made that accounts for the difference
between the time the SD array triggered, and the time that the shower was closest to
AMBER’s LOS (line of site). This fit and associated calculations are performed analytically,
21
insuring a minimal amount of processing time.
The simplest model for an air shower front is that of a plane that moves at the speed of
light c. Taking tank locations ⃗ri = (xi , yi , zi ) and the unit vector in the direction of shower
propagation ⃗s = (sx , sy , sz ), an estimate of shower trigger times in tanks can be given as
1
test,i = (ŝ · ⃗ri ).
c
(2.1)
So given the actual trigger times ti , the following quantity should be minimized (as in
Ref. [46]) to find a best fit value for shower direction:
χ2 =
X
1
(ti − (ŝ · ⃗ri ))2 .
c
(2.2)
Making the approximation that all tanks are at equal height (roughly true), and taking
zi = 0, yields:
χ2 =
The requirements
dχ2
dsx
dχ2
dsy
= 0 and
X
1
1
(ti − sx xi − sy yi )2 .
c
c
(2.3)
= 0 give the following equations:
sx
X
x2i + sy
X
sx
X
x i yi + s y
x2i
P
x i yi = c
X
yi2 = c
X
t i xi
(2.4)
X
ti yi .
(2.5)
Or, in matrix form:

P

P
With
xi yi
  

P
x i yi   s x   c
ti xi 
.
P 2   =  P
yi
sy
c
ti yi

−1 
  
P
P 2 P
ti xi 
xi
xi yi  c
s x  
,
  = P
P 2   P
xi yi
yi
c
ti yi
sy
(2.6)
(2.7)
yielding at last:
sx =
P
P
P
P
c ( yi2
tx −
xy
ty)
P 2 Pi i2
Pi i 2 i i
yi − ( xi yi )
xi
22
(2.8)
sy =
c (−
P
P
P 2P
xy
tx +
x
ty)
P i 2iP 2i i P i 2 i i .
xi
yi − ( x i yi )
(2.9)
This leaves the calculation of sz as a trivial matter.
To find the amount of time to subtract for an AMBER trigger, the vector that describes
the shortest distance between the lines of the AMBER LOS and the propagation of the
shower must be found. This vector has the unique property of being perpendicular to both
the LOS and shower propagation lines. (Refs. [47] and [48] have useful explanations). These
lines are taken to be the vectors
⃗vshower = m ŝ + ⃗rshower
(2.10)
⃗vAM BER = n ˆl + ⃗rAM BER ,
where ⃗rshower
=
(xshower , yshower , zshower ) is the location of the shower core on
the ground (as calculated by a simple average over tank positions), ⃗rAM BER =
(xAM BER , yAM BER , zAM BER ) is given by the coordinates of the AMBER site, and ˆl and ŝ
are the LOS and shower directions. (Note that I take m to be zero when the shower hits
ground level, and negative for earlier times). The shortest-distance vector thus points in
a direction w
⃗ = ŝ × ˆl. Taking wnorm = |ŝ × ˆl|, the closest distance between the LOS and
shower path is given by the absolute value of the parameter
d = (⃗rAM BER − ⃗rshower ) ·
w
⃗
wnorm
.
(2.11)
Now with w
⃗ and d in hand, the following vector equation can be written:
m ŝ + ⃗rshower + d
w
⃗
= n ˆl + ⃗rAM BER .
wnorm
The parameters m and n are easily solved for, and the trigger is adjusted by
(2.12)
m
c
plus time-
of-flight to the detector.
Another step in the processing of Auger triggers is to log information about the received
SD triggers and calculated AMBER trigger. This data is stored in amtrig files, which are
of the *.root filetype and are accessible via ROOT’s standard TTree class. As will be seen
in the next section, these files serve a vital role in the analysis chain.
23
2.3.4 Event Matching
Because the modified Auger SD trigger that AMBER uses operates so quickly, the data it
propagates arrives without an associated official observatory event ID. In fact, the information passed to the amtrig logs also predates the formation of an AMBER trigger ID created
by AmTrigSever. This means that the matching of observatory events with AMBER events
must at first be done using exclusively the timing information of events.
Events are matched in two steps. First, amtrig log information is compared against fits
provided by Auger’s Offline analysis package in conjunction with ROOT data files for the
SD array (credit to Eric Grashorn, formerly of The Ohio State University, for authoring
this Auger-facing analysis code). These fits from the Auger files are much more accurate
than the quick ones performed by AmTrigAugerClient, and are presumed to indicate the
actual arrival directions of showers. An ideal trigger time is arrived at by back-propagating
the shower from the Auger SD reconstruction to the AMBER LOS (see previous section).
If this time is within some window of the actual time that AMBER was triggered (30
microseconds), then I select the event as a candidate for capture.
The second step in the matching process is to connect these trigger times with RF data
collected in the AMBER event files. Since trigger times are stored in AMBER’s ROOTbased event objects, as well as in the amtrig log files that I have already matched against,
this step is relatively trivial. Events with matching triggers are compiled into a .root file
along with information about the observatory reconstruction of the event. These candidate
files are used for the interactive data analysis discussed in the next subsection, as well as
for the power analysis presented in Chapter 4.
2.4 Software: GUI Event Viewer
2.4.1 Viewing Run Files
AMBER’s interactive event viewer, AmViewer, is written in C++ using features from ROOT
and the Qt GUI framework and operates in two modes. The first mode is initiated by
opening a simple AMBER ROOT data file such as those made on neutrino for each data
24
run. In this mode a list of events in the run grouped by trigger type (PPS, self, and Auger)
is provided on the left. The middle provides an interactive diagram of the AMBER horn
array which can be used to select what channel of RF data should be displayed by plots on
the right-hand side. AMBER event ID and trigger time information is also provided above
the horn diagram.
2.4.2 Candidate Mode
The second mode occurs when viewing a file which is a compilation of candidates made
by the second step of the event matching procedure. This mode provides for a number of
windows and dialogs that allow for interactive AMBER data analysis.
In candidate mode, the left-hand menu now gives a list of candidate events grouped by
run number, and the shower reconstruction information given by the observatory is used
extensively to indicate where and when RF signals ought to be visible. The middle horn
icons are colored with a shade of orange if the shower passed through them, with the shade
indicating the relative times between these passages (lighter for horns that would have seen
the shower first, and darker for later ones). Vertical blue lines on the right-hand plots
indicate entry and exit from the FOV, while a vertical red line indicates the point when
the shower crossed nearest that horn’s line of sight. Detailed information about an event is
available in a separate, optional window.
AmViewer’s candidate mode also offers two ways to visualize the reconstructed event.
The first is a 3D interactive plot that shows vectors describing AMBER’s LOS and the
shower’s trajectory, as well as the locations of tanks in the Auger SD array. The second is a
plot with axes of azimuth and elevation, which shows the position and rough angular size of
each horn, as well as the path of the shower through the sky. This is useful for determining
if candidate events passed directly through a candidate horn, or merely grazed them.
Signal processing for candidate events can also be accomplished via AmViewer. By
default, when the “display dBm” box is ticked, an optimal filter using a Gaussian of a
width corresponding to horn-crossing time is enabled. This default filtering be adjusted to
a width of any desired value.
25
26
Figure 2.3: A demonstration of some features of the AmViewer application in candidate mode.
Since a list of candidates can be quite large, AmViewer also provides a way to search for
a subset of events to be displayed on the left-hand list. Search criteria allows for a large
number of simultaneous parameters such as reconstruction angles, number of tanks, and
the atmospheric depth at entry and exit from AMBER field of view. This last parameter
can be very useful for limiting the candidate list to only showers that are of an appropriate
development (e.g. near the point of maximum particle content).
2.5 Deployment
2.5.1 Site Description
In May/June 2011, AMBER was installed at the Coihueco fluorescence detector site of
Pierre Auger by a team of Ohio State and University of Hawaii researchers with support
from the observatory staff and the aid of collaborator Roberto Mussa (INFN Torino). This
deployment consisted of the AMBER dish, horn array, a small shed to house electronics,
and a copper mesh attached to a nearby fence to provide RF shielding. From atop the hill
the Coihueco site rests on, AMBER has a line of sight that looks out over the SD array at
an azimuthal angle of about 70 degrees (East of due North), and an elevation of about 30
degrees. This positioning puts the LOS over a section of the SD called the infill array. This
infill area is instrumented more densely by tanks, which allows for the detection of lower
energy showers in this patch of the observatory [49]—a feature of use to AMBER, whose
detection threshold was untested.
AMBER is located at a latitude of -35.11380 degrees and a longitude of -69.59619
degrees. In the planar UTM coordinates favored by the SD stations this corresponds to a
Northing of 6114174.2 meters and an Easting of 445671.8 meters with an altitude of 1707.34
meters [50].
2.5.2 Timing Tests
In order to test the timing calibration of the AmTrigAugerClient’s triggering, two tests
were carried out using a modification to SD tanks that allowed for them to be triggered by
27
an external RF signal.
I participated on-site in the first of these tests in November 2011, where a modified
tank was triggered via an RF pulse sent from the roof of the Coiheuco site’s FD building.
AmTrigAugerClient was also modified to allow for triggering off of single tank events, with
direction of the RF pulse aligned so that it would be captured by the AMBER horns as it
propagated out toward the target SD tank. The signal observed by AMBER in this test
was seen to be in excellent agreement with expectation.
The second test, conducted on-site in March 2012 with remote support from Ohio State
members, followed a similar procedure but was designed to provide for a more thorough
examination of AmTrigAugerClient’s response to events forwarded from the SD array. In
this version three tanks were modified to generate a trigger from an RF pulse, effectively
emulating a small shower. Unlike the previous trial, this iteration allowed for a test of
the fitting algorithms used in generating observatory-initiated events at AMBER. Again
AmTrigAugerClient behaved as expected, and the RF pulse was visible in the man-made
triggers.
2.5.3 Solar Transits
In May 2012, AMBER collaborator Roberto Mussa used several weeks of solar transits (4
September 2011 to 21 October 2011) to generate an accurate map of the C band channels
that comprise AMBER’s line of sight [6]. This plot shows the total width of the AMBER
field of view (FOV) to be about 16 degrees, with each horn spanning a full width at halfmaximum (FWHM) of 2.7 degrees [51]. The horn pointing directions derived from these
transits (listed in Table 2.1) are used in the AmViewer application, the calibration in Chapter
3, and the data analysis in Chapter 4.
2.5.4 Data Rate
Though its field of view extends over only a small fraction of the SD array, AMBER’s
AmTrigAugerClient accepts and forwards events that originate from tanks all over the
observatory’s detection grid. It also creates a secondary trigger that pulls data from one
28
Table 2.1: AMBER horn pointing directions derived from R. Mussa’s solar transit analysis
[6].
channel
C1H
C1V
C2H
C2V
C3H
C3V
C4H
C4V
C5
C6
azimuth
70.9748
71.0134
74.1202
74.1303
70.7735
70.7839
67.5472
67.6673
73.8496
75.5745
elevation
24.2085
24.1736
26.948
26.9768
29.7183
29.8044
26.8922
26.9224
21.5874
23.0592
channel
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
azimuth
77.2165
77.3872
74.9357
73.5953
67.5521
65.77
64.1074
64.8875
66.7044
68.4769
elevation
24.6442
30.021
31.1156
33.1748
32.5643
30.9526
29.3239
24.371
22.942
21.5122
or both adjacent seconds (possibly useful for some methods of data analysis). The rate of
incoming events from the array is of order one for every 10 seconds. All together this yields
a daily data rate of a few tens of gigabytes, which includes raw event files, processed event
files, housekeeping, and various other log files. The neutrino machine’s onboard capacity is
supplemented by external hard drives to hold the raw data. These drives are swapped out
with new ones when scheduling permits; typically once or twice a year.
29
Figure 2.4: AMBER-01 as installed at the Coihueco site of the Pierre Auger Observatory.
From [5].
Figure 2.5: Data from four AMBER channels during a single tank timing test event. The
black vertical line marks the expected beginning of the pulse.
30
Figure 2.6: The pointing directions of AMBER horns as derived from solar transits. Channel
9 did not function well and is not shown. From Ref. [6].
31
Chapter 3
AMBER: Calibration
3.1 Introduction
Whether or not AMBER succeeds in capturing a clear MBR signature from a passing air
shower, a careful calibration of the system’s internal ADC (analog-to-digital converter) scale
to units of actual power is a crucial component of the experimental analysis.
The calibration procedure described in this chapter is accomplished by parameterizing
sources of uncertainty in AMBER’s RF detection chain. Using this model of the hardware
system, the free parameters can then be fixed using the expected response to a well-known
astrophysical object that emits in the microwave band. There are a number of RF sources
that make good candidates including the Sun, the Moon, the Crab Nebula, and Jupiter.
However, an ideal source would cross the field of view several times without having to
reposition the instrument manually. Thus Solar and Moon transits seem like an ideal
match, but there is a third option. AMBER is sensitive to microwave emission originating
from the galactic plane of the Milky Way (Ref. [52]). Transits through the plane happen
once a day, at the same galactic coordinates, and at a relatively low power, making such
events a perfect match for calibration considerations.
As noted previously AMBER’s central horns are both dual-band and dual-polarization,
but in this and the next chapter I will consider only the C-band and horizontal polarization
of AMBER’s central horns. Except where noted the work presented here is my own, with
helpful input and guidance from Peter Gorham, Patrick Allison, and my advisor James
Beatty.
32
3.2 Hardware Model
In order to perform the fits necessary for an AMBER power calibration, I must have a
hardware model that can take the ADC units of AMBER’s DAQ boards to units of dBm
(decibels per milliwatt). This will be done in several steps, moving from the input of the
AMBER DAQ to the AD8318 power detector (JEDI board), and from there to the LNB as
it accepts an RF signal. Patrick Allison’s Ref. [37] provides the bulk of the information for
this section.
First, the voltage at a DAQ input can be taken to be linear, with
VDAQ = gDAQ K + bDAQ
(3.1)
where gADC is a gain factor, and bADC is an offset. From there, voltage at the power
detector on the JEDI board is
VP D =
VDAQ
aP D
(3.2)
where aP D is due to an attenuator and the normal diminishing of signal due to propagation
through a cable.
The power detector relates output voltage to power as
1
PP D = 10 10
V
( g P D +bP D )
PD
× 1 mW
(3.3)
where gP D is a gain, and bP D is an offset associated with the power detector. Continuing
toward the RF end of the chain, the power at the stage just before the LNB can be given
as
PLN B =
PP D
aJEDI
(3.4)
in order to account for attenuators internal to the JEDI. Finally, the RF power can be
written as
PRF =
PLN B
− nsys
acable gLN B
(3.5)
where acable accounts for signal attenuation due to cable, gLN B is the gain of the LNB, and
nsys is the system noise [39] in units of milliwatts. Here the values of particular interest
33
are the LNB gain, the system noise, and the parameters relevant for to translating ADC
counts to PLN B .
Note that (as discussed in Ref. [37] if the function PLN B (K) were known, and acable
measured, then a simple Y-factor calibration to two known sources (a “hot” higher powered
one and a “cold” lower powered one) could be performed, and the equations
Phot =
PLN B (Khot )
− nsys
acable gLN B
Pcold =
PLN B (Kcold )
− nsys
acable gLN B
would suffice to solve for the parameters of LNB gain and system noise. However since the
figures relevant to translating ADC counts to LNB power are not known, a degeneracy is
introduced that necessitates a more sophisticated solution [37]. In the procedure to follow,
this degeneracy is overcome by considering many values of expected power rather than just
two.
I proceed by combining the hardware model equations to produce the following relation,
1
PRF = 10 10 (KD+X) − nsys
(3.6)
which encapsulates many of the relevant parameters into two less obviously physical ones
called D and X, while preserving nsys . These three variables fully describe AMBER’s
hardware response, and finding their values is the goal of the procedure described in this
chapter.
3.3 Simulated Transits
The other side of this calibration involves a comparison against expected values of power.
To this end, simulated crossings of AMBER’s horns through the Milky Way’s plane are
generated using a radio frequency model of the galaxy which includes data from many
different experiments that operate in the GHz regime [53] [54]. From this model a sky map
is compiled by averaging over samples the in relevant bands. If the function Tsky (f, θ, φ) is
taken to be the temperature of the sky as a function of the frequency f and the galactic
34
coordinates θ and φ, then the sky temperature for the 3.7–4.2 GHz band of the inner horns
can be approximated by
4
Tsky,inner (θ, φ) =
1X
Tsky (3.75 GHz + 0.1 n GHz, θ, φ).
5
(3.7)
n=0
For the outer horns, the band is 3.4–4.2 GHz (as given by the LNB specification), and the
temperature is calculated to be
7
1X
Tsky,outer (θ, φ) =
Tsky (3.45 GHz + 0.1 n GHz, θ, φ).
8
(3.8)
n=0
To complete the information necessary to perform simulated transits, the pointing directions
of the horns located at the focus of AMBER’s dish are taken from fits to data collected
from solar transits through AMBER’s field of view (see Section 2.5.3) [6]. These directions
are used both for this calibration, and for the interactive GUI viewer discussed previously.
The average sky temperature observed by a horn pointing in the direction of (θ0 , φ0 )
(here θ and φ represent the galactic coordinates of latitude and longitude, respectively) is
given by
RR
Tavg,inner/outer =
2
0 ,θ,φ)|
exp(− |ψ(θ0 ,φ
) Tsky,inner/outer (θ, φ) cos(θ) dθ dφ
2σ 2
RR
2
0 ,θ,φ)|
exp(− |ψ(θ0 ,φ
) cos(θ) dθ dφ
2σ 2
(3.9)
where ψ is the angular separation between (θ0 , φ0 ) and (θ, φ), and TM W is the temperature
in Kelvin of the Milky Way at the requested frequency band and coordinates. The parameter
σ is set using a solar transit derived value for the angular width of a horn’s field of view
(FWHM of 2.7 degrees) [51]. These average temperatures are calculated for each horn at
10 second intervals as their lines of sight pass through the galactic equator. The power
readings (in watts) from this simulation can be arrived at via the relation
Psim =
1
kB B Tavg
2
(3.10)
where kB is the Boltzmann constant, and B is the bandwidth of the receiver (taken to be
500 or 800 MHz). The hardware model and the expected power from Milky Way transits
can now be linked.
35
3.4 Calibration Fits
With each component in hand I can now perform a fit over the promised many values of
RF power, and the quantity to minimize for each horn is written
χ2 =
1
X
kB B Tavg (ti )
1
(10 log( 2
) − 10 log(10 10 (K(ti +tshif t )D+X) − nsys ))2 .
0.001
(3.11)
i
Where here the simulated and hardware-modeled powers are expressed in dBm, and the
sum over i denotes a sum over the data points (simulated average temperature Tavg and
observed ADC counts K) at the times ti . The fit parameter tshif t is used to ensure that
the power peaks still align in the case of any timing or geometric discrepancies.
To perform this fit, up to 20 transits captured in AMBER’s housekeeping files are used.
Each night’s data is smoothed using a running boxcar average with a width of 10 seconds,
and then combined and averaged with the data from other available transits. The results
of these fits can be seen in Figures 3.1 and 3.2. In these plots the parameter n is the system
noise given in units of milliwatts, X and D are as in Eq. 3.6, and the value of the fit
function Eq. 3.11 is given by “funcVal.” The noise temperatures for each channel can be
calculated via the relation
Tnoise =
nsys
,
kB B
(3.12)
and these are listed in Table 3.2.
The fit procedure for these plots is as follows: The noise parameter n starts at a value
of zero and is raised in steps of 10−12 until a fit to the remaining parameters fails, at which
point the lowest value of “funcVal” achieved is so far taken to correspond to the best overall
fit. The starting values, constraints, and step sized of parameters are recorded in Table 3.1.
Table 3.1: Constraints used for fit parameters.
parameter
D
X
n
tshif t
start
0.025
-126.9
0.0
0.0
constraint
0.022-0.045
none
first failed fit
-400.0–400.0
36
step size
0.0001
0.01
10−12
1.0
Figure 3.1: Results of the AMBER calibration to transits through the plane of the Milky
Way galaxy. Channels C1H to C8.
37
Figure 3.2: Results of the AMBER calibration to transits through the plane of the Milky
Way galaxy. Channels C9 to C16. Fit parameter n is the system noise given in units of
milliwatts.
38
Fits are performed using ROOT’s [44] interface to the Minuit2 fitter in Migrad mode.
The fit algorithm handles the step size indicated in Table 3.1 for all parameters but n, which
was explained earlier.
Table 3.2: AMBER noise temperatures by channel using Equation 3.12 with B = 500 MHz
for channels 1-4, and 800 MHz for channels 5-16.
channel
C1H
C2H
C3H
C4H
C5
C6
C7
C8
Tnoise
50.3 K
45.6 K
48.1 K
49.5 K
65.4 K
47.7 K
55.8 K
47.8 K
channel
C9
C10
C11
C12
C13
C14
C15
C16
39
Tnoise
54.4 K
57.9 K
50.6 K
53.9 K
62.2 K
66.7 K
53.9 K
47.1 K
Chapter 4
AMBER: Analysis
4.1 Introduction
Since a visual inspection of candidate events using the AmViewer tool yields no obvious
results, a more careful approach must be taken to ascertain if the AMBER system detects
any appreciable signal from MBR emission. To perform this analysis I must link the power
calibration performed in Chapter 3 to the physical mechanism behind MBR, as detailed in
Chapter 1. Again, except where indicated otherwise this work is my own with assistance
from Peter Gorham, Patrick Allison, and my advisor James Beatty.
4.2 Method
As seen in Section 1.2.3 (and Ref. [2]), the expected power from an MBR event consisting
of Ne electrons each with power Pe can be written as
Pinc = Ne Pe
(4.1)
for the completely incoherent case, and as
Ppar = M µ2e Pe
(4.2)
for the partially coherent case, where µe is the size of a coherently emitting group of electrons
and M is the number of those groups. Taking that Ne = M µe and solving for µe , one arrives
at the relation
40
µe =
Ppar
.
Pinc
(4.3)
Now consider a similar parameter called simply µ which is arrived at by the ratio
µ=
Pobs
Pexp
(4.4)
where Pobs is the observed power of a candidate event, and Pexp is the expected power in the
completely incoherent scenario. Values for Pexp are calculated using the method outlined
in Gorham et al. which scales the MBR observed from an accelerator experiment up to the
energies of cosmic ray events (see Appendices A and B).
One problem with the Gorham et al. procedure for calculating expected MBR power
is that it only gives a value for the power at the point of maximum shower development.
This point is represented by the atmospheric depth Xmax , and has the distance-independent
units of grams per centimeter squared. Using a fit from Ref. [55], I can translate the energy
of an event as calculated by Auger to the expectation value of Xmax for showers with that
energy. This, ignoring uncertainties, can be given as
⟨Xmax ⟩ = D10 log10 (E/eV) + 746.8 g/cm2
where
D10 =



86.4 g/cm2
if log10 (E/eV) < 18.27


26.4 g/cm2
if log10 (E/eV) > 18.27
(4.5)
.
(4.6)
From here we can use the Gaisser-Hillas function [56] [55], which describes the scale of a
shower as it passes through the atmosphere:
f (X)/f (Xmax ) =
X − X0
Xmax − X0
Xmax −X0
λ
e
Xmax −X
λ
.
(4.7)
Taking X as the atmospheric depth of the shower at its closest point to the horn’s line of
site (see Appendix B), and using values from Ref. [55] for λ and X0 of 61 g/cm2 and −121
g/cm2 , I can now scale the expected power from an MBR event appropriately.
With this knowledge in hand I calculate µ for each entry in a set of candidate sub-events
(“sub-candidates”), which are defined as instances where a shower passes through the line
41
of sight of one of AMBER’s horns. To do this I also need a figure for the power observed
by AMBER during a candidate MBR event. This is arrived at by first optimally filtering
the data using the amount of time the shower was within that horn’s field of view, and then
selecting the value found at the time that corresponds to the shower’s closest approach to
the line of sight. Calibration parameters arrived at via the fits to galactic plane transits
provide the translation from ADC counts to units of power during this step, and Pobs (before
filtering) can be written as the excess power (in milliwatts),
1
Pobs = 10 10 K D − n,
(4.8)
where again K is ADC counts, D and X are fit parameters relevant to the AMBER hardware, and n is the noise.
From here I consider the parameter µavg as a function of Pcut , which is the average
value of µ calculated using candidate sub-events with Pexp > Pcut . Since µavg is grounded
physically as the size of groups of coherently emitting electrons, it makes an excellent tool
to investigate the power detected by AMBER’s systems.
4.3 Results and Conclusion
In Figure 4.1, µavg is plotted versus increasing values of Pcut , with the number of subcandidates used to calculate each average value shown as a histogram. In creating this plot
I exclude any horns that have a poor calibration fit (C15), or have experienced hardware
difficulty at any time (C3 [57] and C9 [6]). Sub-candidates are considered from data taken
in within a two month interval from 1 July 2011 to 29 August 2011.
The high values that we see for µavg on the left-hand side of the plot are to be expected in
both the detection and non-detection scenarios. These are due to the large µ’s that result
when the ambient RF power background signal is divided by relatively faint predicted
signals. As Pcut grows, these background-inflated µavg values can be expected to fall.
However, as we continue to raise the floor of allowed values of Pexp , there is a divergence in
what can be expected from a plot where MBR is detected versus one where it is not.
42
In the detection scenario, µavg can be expected to either stabilize around a value of
unity (the case of completely incoherent radiation), or possibly to grow as increased shower
energy causes emission to become more strongly coherent. For non-detection, values of µavg
will fall indefinitely, eventually crossing the µavg = 1 threshold, which corresponds to the
non-physical case of less than one electron per coherently emitting group. It is this behavior
that we see in Figure 4.1, and so it can be concluded that AMBER fails to see any power
resulting from MBR phenomena.
While certainly much more AMBER data is available, it should be remembered that
the system is intended as an augmentation to existing detectors, and simple discovery of
MBR is not enough. In order to work as an enhancement, or even competitor to existing
systems a gigahertz detector would need to consistently discern a strong signal. An interval
of two months is certainly enough time to determine whether that is the case. It is worth
noting, however, that the result shown in the figure is not strong enough to be conclusive,
and further research may yet reveal the presence of a weak MBR phenomenon.
Although the analysis presented in this document fails to find any traces of isotropic
air shower induced microwave radiation, gigahertz radio detection remains a promising and
active area of investigation within the field of UHE cosmic ray experiments, especially for
those sensitive to beamed RF signals. Indeed, AMBER shares its home at the Pierre Auger
Observatory with two other microwave detectors (EASIER and MIDAS) [15, pp. 96-99],
and continues to collect data in its current incarnation as AMBER-02.
For now, water Cherenkov tanks and fluorescence telescopes remain the detectors of
choice, but with continued research microwave detectors may yet join their older siblings as
a crucial technology in the quest for more UHE cosmic ray data.
43
Figure 4.1: The parameter µavg plotted as a function of Pcut . This analysis restricts to
sub-candidates with showers that have a Gaisser-Hillas function (see Eq 4.7) value greater
than 0.1, with an altitude less than 11 km, and a zenith angle less than 60 degrees (see
Appendix B). Plotted points give µavg values, with the number of sub-candidates used to
calculate each average recorded in a histogram. A red line marks the completely incoherent
case at µavg = 1. Note that because Pobs is actually excess power (see Eq. 4.8) it is possible
for µavg to fluctuate to a negative value.
44
Chapter 5
Multi-messenger: Spectrum
Generation
5.1 A Reintroduction
5.1.1 Other Approaches to the Problems of UHE Astrophysics
The previous chapters have been a detailed discussion of one experiment’s approach to
increasing the body of information available to the study of UHE astroparticle physics. In
this and later chapters I turn my attention from the attempted observation of events directly
initiated by UHE cosmic rays, to a speculative analysis of events that could be observed by
experiments sensitive to the slightly more indirect but still intimately related phenomenon
of UHE neutrinos. As discussed in Chapter 1, these particles are generated by the same
GZK phenomenon that affects UHE cosmic rays, but are borne across very large distances
with very little energetic loss.
Used in combination with data from the Pierre Auger Observatory, events from current
and next generation neutrino experiments like ANITA 3, EVA and ARA may be able to
provide constraints on where and how these particles are created—one of the most poorly
understood aspects of UHE astroparticle physics.
The research presented in the last chapters of this dissertation is based on a publication
in preparation (Ref. [14]) with co-authors Amy Connolly (Ohio State, CCAPP) and Shunsaku Horiuchi (currently at Virginia Tech). The work presented in this and the following
chapter is based on their efforts.
45
5.1.2 Basic Procedure
This analysis assumes a model for spectrum generation which works for both species of
particle (cosmic rays and neutrinos), and includes several tunable elements relevant to
features of UHE cosmic ray sources. The model is first constrained to available cosmic ray
data (in this case from the Pierre Auger Observatory), and the parameters most well suited
to determination by the cosmic ray spectrum are fixed. With the values of these parameters
in place, I generate neutrino spectra that adhere to the existing PAO data set.
Model generated neutrino spectra are convolved with the acceptance of neutrino experiments (ANITA 3, EVA, and ARA) in a Monte Carlo to create a large number of simulated
experimental runs. A likelihood analysis associated with these “pseudoexperiments” can be
used to place constraints on the remaining neutrino-sensitive parameters.
5.2 Model for UHE Spectra
5.2.1 Explanation of Components
In this work (as in Ref. [58]) the flux at Earth at energy E is given by an integral over
initial energies at the UHE source (Ei ):
d4 Nb
=
Jb (E) =
(d log E) dA dt dΩ
Z
∞
dEi Gϵb (E, Ei )I(Ei ),
(5.1)
0
with Gϵb as the propagation probability for a particle of species b, in cosmic evolution ϵ, at
initial and final energies Ei and E. The factor I(Ei ) gives the injection spectrum for UHE
sources, and is written
I(Ei ) =



1
dN
= I0 Ei−α

dEi

2
2
Ei2 /Emax
exp(1 − Ei2 /Emax
)
Emin < Ei < Emax
,
(5.2)
Emax < Ei
where I0 is a normalization factor, α is the spectral index, and Emax is used as a parameter
for a cutoff in the UHE source spectrum.
The factor Gϵb in Eq. 5.1 is given by the integral
46
Gϵb (E, Ei )
1
=
4π
Z
0
∞
dPb (E, Ei , r) ρϵ (z(r)).
dr d log10 E (5.3)
Here |dPb (Ei , E, r)/d log10 E| gives the probability that a particle of species b created with
energy Ei at distance r arrives at Earth with energy E, and the factor ρϵ (z(r)) describes
how the density of UHE sources evolves with redshift z. Discrete values in steps of 0.1 in
log energy are taken for Ei and E, and the resulting table of Gϵb values is calculated using
the CRPropa 2.0 [59] Monte Carlo software.
5.2.2 Free Parameters
The evolution ρϵ can be thought of as a function that describes the shape of the density of
sources, multiplied by a normalization factor, and can be written as
ρϵ (z(r)) = ρ0,ϵ fϵ (z(r)).
(5.4)
Therefore the overall normalization of the flux Jb (E) is given by the product I0 ρ0,ϵ . This
normalization, along with the source spectral index α are parameters whose values can be
set by fitting to existing cosmic ray spectrum data. The remaining two parameters are the
evolution ϵ and the source spectrum cutoff Emax . These parameters, having to do with
particles created at large distances and very high energies, are not well defined by cosmic
ray data and will be inspected using neutrino pseudoexperiment likelihoods.
5.2.3 Choice of Evolutions
This work limits itself to four UHE source evolutions: three that relate to astrophysical
phenomena, and another corresponding to a case with no source evolution.
The first evolution model follows the stellar formation rate (SFR), and assumes any
UHECR source whose density can be correlated to the birthrate of stars in a region. The
second is based on the occurrence of gamma ray bursts (GRB), which as very high energy
phenomena provide a possible candidate for UHECR acceleration. The final astrophysically
motivated evolution is associated with the density of FRII-type active galactic nuclei—a
popular choice for UHECR production. The last case is that of no cosmic evolution, in
47
which a constant density of UHECR sources is considered for all values of redshift.
An advantage of the selected model is that it has a separable spatial integral: once the
ρ0,ϵ constant is removed, the choice of evolution affects only the G factor of the spectrum.
As would be expected, the generation of these G tables is computationally expensive, but
once calculated the values corresponding to a certain scenario can be used independent
of the other parameters. This results in a notable reduction in the resources needed to
generate a test spectrum compared to other Monte Carlo based methods.
48
Figure 5.1: Plots of the three astrophysical source evolutions considered. Solid lines represent the nominal shape, while dashed lines indicate a shape used to calculate systematic
uncertainties. The stellar formation rate function (SFR) is derived from information, data,
and fits in Refs. [7] [8] [9] [10] [11]. Gamma ray burst (GRB) evolution and FRII-type AGN
evolution are based on modified versions of Ref. [12] and Ref. [13] respectively. Plots are
by Amy Connolly for Ref. [14].
49
Chapter 6
Multi-messenger: Fits
6.1 “Ankle” and “Dip” Models
Fits to Pierre Auger data are performed using two different models which are both intended
to explain the “ankle” feature of the cosmic ray spectrum. In the dip model, the steepening
slope before the ankle feature is explained by energy loss due to electron-positron pair
production from proton interactions with the CMB [24] (a process that is distinct from the
GZK phenomenon). The ankle model, however, posits the ankle feature to be the transition
point between the galactic and extra-galactic cosmic ray spectra [24].
For our purposes, the primary difference between these two models is that they place
the lower bound for cosmic rays of extra-galactic origin at different energies. For the ankle
model we say this shift occurs near the spectral feature of the same name, around an energy
of 1018.8 eV. However for the dip model we place the minimum energy for extra-galactic
cosmic rays much lower, near 1017.6 eV. Since our analysis is only concerned with UHE
cosmic rays of extra-galactic origin (the primary source for the UHE neutrino spectrum),
the overall effect of these two scenarios is to change the minimum data bin to be used in
fits to cosmic ray data.
6.2 Weighting Function
A novel feature of this work is the inclusion of a weighting scheme that de-emphasizes contributions from the Earth’s immediate cosmological neighborhood. The weighting function
50
is written as
Jp (E, rcut ; α, Emax ) (d log10 E)
R rmax
,
ρϵ (z(r)) I(E; α, Emax ) (d log10 E)
Emin 0
we (E; α, Emax ) = R ∞
(6.1)
and gives the flux contribution at Earth from sources more distant than rcut , divided by
the total number of particles created at all energies and distances. We take rcut to be 100
Mpc.
The purpose of this weighting is to correct for the fact that the highest energy particles,
being subject to the 100 Mpc horizon imposed by the GZK effect, may not provide an
accurate picture of processes at work in the universe at large. Without this adjustment,
the fitted functions would in effect oversample the local universe—an undesirable artifact
in an analysis that uses neutrinos to consider UHE sources at all redshifts.
6.3 Fit Results
Fits are performed allowing for a 14% uncertainty in the energy of PAO’s ICRC 2013
data [15, pp. 27-30]. This is treated as a nuisance parameter. We also consider several
other sources of error and their effect on the physical parameter α. These include the
choice of the minimum energy bin that defines the start of the dip and ankle models, the
alternative evolution functions (see Fig. 5.1), the choice of Emax , and the choice of rcut .
The error sources, their perturbations, and their maximum influence on the value of α are
summarized in Tables 6.1 and 6.2. The fits for α and norm achieved using the nominal
values of these parameters is given in Table 8.3.
Table 6.1: For dip model: Maximum absolute difference induced in α across all evolutions
for changes in various parameters. Energy shifts are in units of 14%. Values calculated by
Amy Connolly for Ref. [14].
parameter
energy shift
minimum bin
evolution choice
log10 (Emax /eV)
rcut
nominal value
0.9
17.6
solid line
21.5
100 Mpc
change
±0.5
±0.1
dashed line
±1.0
+100 Mpc / −50 Mpc
51
max. |αnominal − αchange |
0.01
0.16
0.09
0.004
0.004
Table 6.2: For ankle model: Maximum absolute difference induced in α across all evolutions
for changes in various parameters. Energy shifts are in units of 14%. Values calculated by
Amy Connolly for Ref. [14].
parameter
energy shift
minimum bin
evolution choice
log10 (Emax /eV)
rcut
nominal value
0.9
18.8
solid line
21.5
100 Mpc
change
±0.5
±0.1
dashed line
±1.0
+100 Mpc / −50 Mpc
max. |αnominal − αchange |
0.01
0.09
0.1
0.008
0.009
Compared to what is observed by the Pierre Auger Observatory (Fig. 6.1), fits performed
using these methods show what might be considered to be an excess of protons at the highest
energies. This discrepancy can be motivated physically by the presence of the Milky Way in
a region that represents a fluctuation from the average cosmic ray intensity at the highest
energies, or as an indication that the properties of UHE sources are themselves subject to
evolution.
Table 6.3: Fit values for alpha and norm
model
SFR ankle
GRB ankle
FRII ankle
NoEvo ankle
-α
-2.47
-2.45
-2.35
-2.55
norm
14.82
20.64
13.63
14.67
model
SFR dip
GRB dip
FRII dip
NoEvo dip
52
-α
-2.48
-2.57
-2.11
-2.82
norm
15.09
22.77
9.14
19.8
Figure 6.1: Fits to Auger ICRC 2013 data [15, p. 30] in both the ankle and dip models
using all four evolutions. Ankle fits start at the 1018.8 eV data point, while dip fits start at
1017.6 eV. Created with assistance from Ref. [16].
53
Chapter 7
Multi-messenger: Analysis
7.1 Pseudoexperiment Generation
The alpha and normalization values obtained from fits to each model and source evolution
can be trivially combined with Eqs. 5.1 and 5.2 (and the relevant CRPropa-derived probability tables) to calculate the neutrino spectra that correspond to each scenario. When
combined with the effective area of a detector for a certain timespan, I can readily calculate
the expected number of events in an experiment given a total running time. Taking Aef f (E)
to be the relevant effective area, and Jf it (E; Emax ) to be the spectrum that results from
the parameters fixed by the cosmic ray fit, the expected number of neutrino events for an
experimental run of length ∆t can be written as the integral:
Z
µ=
Aef f (E)Jf it (E; Emax ) ∆t dE.
(7.1)
As noted earlier, Emax is not well constrained by cosmic ray data, and can be left as a free
parameter.
In order to gain a sense of the ability of various detectors to discriminate between
source evolutions and different values of Emax , it is necessary to simulate the outcome of
a large number of these experiments. These “pseudoexperiments” are calculated in several
steps. First, a total number events for an experiment is calculated by drawing a random
number from a Poisson distribution that takes µ as the expected number of events. Then,
the energies for each of these events is found by drawing from a distribution that reflects
the shape of the cosmic ray-fitted neutrino spectrum for the scenario as well the relevant
54
Figure 7.1: Effective areas of the three experiments considered as implemented in this
research. EVA information is from Ref. [17] with numbers from Eugene Hong [18]. ARA37
(the final 37 station configuration of ARA) values are from Ref. [19] with numbers again
from Eugene Hong [18]. ANITA 3 effective area figures are estimated by shifting ANITA
2 [20] [21] values 30% lower in energy [16]. In all cases the last available data point is
preserved out to a log10 (E/eV) of 24.0.
detector’s effective area (see Fig. 7.2).
To account for energy resolution, each of the energies assigned to events via this method
is further “smeared” by adding (in log10 (E/eV)-space) a Gaussian function with a standard
deviation of 0.4 (see Ref. [60]). If the result of this places the event outside of the detector’s
sensitivity, the process is repeated starting from the step where an event energy is selected.
Table 7.1: Expected event numbers for the experiments considered. ANITA 3 and EVA are
calculated for 150 days flight time with a log10 (Emax /eV) of 21.5. ARA37 is calculated for
the same Emax with 20 years of experiment time.
model
SFR ankle
GRB ankle
FRII ankle
NoEvo ankle
ANITA 3
1
1
8
0
EVA
4
4
23
1
ARA37
35
34
205
7
55
model
SFR dip
GRB dip
FRII dip
NoEvo dip
ANITA 3
1
1
24
0
EVA
4
2
59
0
ARA37
34
23
433
3
Figure 7.2: Distribution of events expected for all three experiments in the FRII evolution
using the ankle model fits. For this figure Emax is assumed to be 1021.5 eV.
The analyses that follow use many thousands of these pseudoexperiments performed at
a variety of values of ∆t. In this way we can observe how the ability of these detectors to
discriminate between UHE astrophysical parameters improves with time.
For the methods discussed in this chapter I use a Mathematica notebook written by coauthor Shunsaku Horiuchi as a point of departure, with Connolly and Horiuchi providing
guidance as to the design and implementation of the resulting plots.
7.2 Excluding Source Evolutions
7.2.1 Method
Simulated data from each scenario is used in conjunction with the method of likelihood
ratios [61] to determine the degree to which one ultra-high energy source evolution can be
excluded from another. Here the extended unbinned likelihood function [61] is used:
L(E|p) =
N
Y
f (Ei |p)
i=1
e−µ(p) µ(p)N
.
N!
(7.2)
The vector E represents the set of event energies from a pseudoexperiment, N is the total
number of events in the pseudoexperiment, and p is the set of parameters associated with
56
a particular astrophysical scenario. The function f (Ei |p) gives the probability of observing
an event with energy Ei in a scenario defined by parameters p.
Taking two hypotheses corresponding to different evolution scenarios (here called H0
and H1 ), and using the likelihood function L(E|p), I consider the following ratios of
Q
f (Ei,H0 |pH1 )e−µH1 µH1 N /N !
L(EH0 |pH1 )
=Q
Q0 =
L(EH0 |pH0 )
f (Ei,H0 |pH0 )e−µH0 µH0 N /N !
(7.3)
Q
f (Ei,H1 |pH1 )e−µH1 µH1 N /N !
L(EH1 |pH1 )
Q1 =
.
=Q
L(EH1 |pH0 )
f (Ei,H1 |pH0 )e−µH0 µH0 N /N !
(7.4)
and
where pH0 references the parameters relevant to hypothesis H0 , and Ei,H0 denotes the
energies of events from a pseudoexperiment generated using H0 . The variable µH0 is short
for µ(pH0 ) and stands for the number of events that are expected from a scenario with
parameters pH0 . Variables corresponding to the H1 hypothesis are defined similarly.
For each pseudoexperiment I can calculate a corresponding value of −2 ln(Q0 ) or
−2 ln(Q1 ), depending on whether the data is generated using hypothesis H0 or H1 . In
this analysis I generate 15,000 pseudoexperiments for each hypothesis, and consider the
two distributions of −2 ln Q0 and −2 ln Q1 as plotted on a one-dimensional histogram.
At low values of experiment time these two distributions overlap significantly. However
at increasing exposure (later time) they can be seen to drift apart, with the −2 ln(Q0 )
distribution assuming the higher values of the two (see Fig. 7.3).
For Figs. 7.4–7.9, the separation between the distributions of −2 ln(Q0 ) and −2 ln(Q1 )
is used as an indicator of how well data from evolution hypothesis H0 can be differentiated
from data corresponding to hypothesis H1 . To make these plots I first consider two points in
the upper −2 ln(Q0 ) distribution which are arrived at by finding the bounds of the “middle
68.2%” of −2 ln(Q0 ) values (the analog of the ±1 σ points in a Gaussian distribution) (again
see Fig. 7.3). Then, I take the percentage of values from the other −2 ln(Q1 ) distribution
that are less than or equal to “+σ”, and then the percentage of values that are less than “-σ.
These two percentages define the edges of the shaded region in the plots. In general, thinner
shaded bands indicate tighter clustering of −2 ln(Q0 ) values (and thus higher confidence in
57
a single experiment obtaining this result), while higher percentages demonstrate better
exclusion of the alternative H1 hypothesis’s data from data generated using H0 . In this
series of plots the value of log(Emax /eV) is taken to be 21.5.
7.2.2 Discussion
Results of this method are largely intuitive with respect to the expected event numbers
shown in Table 7.1. Namely, the large numbers of events predicted by active galactic
nuclei (FRII) evolution fits provide ample data to differentiate it from other UHE source
distributions. This technique demonstrates that FRII can be easily distinguished from other
scenarios in all considered experiments within a reasonable amount of experiment time.
Since both provide detectors with similar numbers of events, of particular interest is the
question of whether a stellar formation rate (SFR) based evolution of UHE sources can be
distinguished from a gamma ray burst (GRB) one. For the most part this analysis concludes
that events which come from either distribution will look similar, with the notable exception
of the dip model cosmic ray fits, which hint that the two may become dissimilar at very
long experiment times. In particular the lower level of exclusion for ARA37 in the cases of
SFR versus GRB and GRB versus SFR at 20 years reaches greater than eighty percent.
58
Figure 7.3: An example of the likelihood distributions −2 ln(Q0 ) and −2 ln(Q1 ) for 15,000
pseudoexperiments each and at various elapsed experiment times. Vertical lines mark the
±1 σ analogs in the red −2 ln(Q0 ) distribution.
59
60
Figure 7.4: Source evolution exclusion for ANITA 3, ankle model, 150 days.
61
Figure 7.5: Source evolution exclusion for ANITA 3, dip model, 150 days.
62
Figure 7.6: Source evolution exclusion for ARA (37 stations), ankle model, 20 years.
63
Figure 7.7: Source evolution exclusion for ARA (37 stations), dip model, 20 years.
64
Figure 7.8: Source evolution exclusion for EVA, ankle model, 150 days.
65
Figure 7.9: Source evolution exclusion for EVA, dip model, 150 days.
7.3 Excluding Values for the Source Spectrum Cutoff
7.3.1 Method
In this section I employ a similar method to inspect the ability of neutrino experiments
to discriminate between various values of Emax . In this case the H0 hypothesis always
corresponds to a log(Emax /eV) value of 21.5, while the H1 hypothesis is allowed to assume
a range of values for the spectral cutoff of UHE sources. In this way a 2-dimensional plot
indicating the certainty of exclusion at various values for Emax and experiment run time is
generated. We consider each evolution and dip/ankle model scenario separately.
In Figs. 7.10-7.12, light colored red bands indicate that ≥ 90% of pseudoexperiments
have a −2 ln(Q1 ) value less than the +1 σ-analog position in the −2 ln(Q0 ) distribution,
while darker colored bands indicate the same but for the −1 σ-analog position. The beginning of the middle colored band indicates when exclusion surpasses ≥ 90% at the median
level in the −2 ln(Q0 ) distribution.
7.3.2 Discussion
Much as in the previous section large numbers of FRII events allow for very strong exclusion
of a UHE source parameter. However, we see some constraint of Emax across almost every
combination of experiment, evolution, and cosmic ray model. In particular, we see strong
exclusion of lower values for the UHE source spectrum cutoff at 1 year in ARA37. EVA is
less pronounced, but nearly all combinations of evolution and spectrum model show some
constraint within 10 days of flight time. It can also be noted that the improved sensitivity
of the ARA and EVA detectors allows for a dissimilarity with the assumed cutoff at higher
values of Emax in scenarios other than FRII.
This method shows clearly that differences in the spectrum cutoff of UHE sources can
lead to observable changes in the makeup of the events detected in both this and the next
generation of ultra-high energy neutrino detectors.
66
67
Figure 7.10: log(Emax /Ev) exclusion plots for ANITA 3. Note log time axis.
68
Figure 7.11: log(Emax /Ev) exclusion plots for ARA (37 stations). Note log time axis.
69
Figure 7.12: log(Emax /Ev) exclusion plots for EVA. Note log time axis.
7.4 Estimating the Cutoff
7.4.1 Method
Plots that exclude Emax values for FRII-based evolutions exhibit such strong discrimination
from alternative data sets that it may be possible to determine a sense of the Emax cutoff
by doing a simple likelihood analysis of only one experiment’s data set.
In order to investigate how successful such an approach might be, I start with a set of
5000 pseudo-experiments generated using an FRII source evolution scenario with a cutoff
set at a log10 (Emax /eV) of 21.5. Then, for each pseudoexperiment I calculate a best guess
value for Emax by minimizing the negative log-likelihood − ln(L), where L comes from
the simple extended unbinned likelihood given in Eq. 7.2. In the case that each assumed
Emax value returns the same − ln(L), the best guess value for log10 (Emax /eV) is randomly
assigned to be either 20.0 or 24.0. Finally, the middle 68% of these best guesses for the
UHE spectrum cutoff are plotted versus experiment run time.
Figure 7.13: Middle 68% of minimum − ln(L) based log Emax predictions for ANITA 3.
70
7.4.2 Discussion
Though limited by the energy resolution “smearing” discussed in Section 7.1, the method
detailed generally works as expected, delivering an increasingly narrow distribution of best
guesses for the parameter Emax with increasing experiment time. It is worth noting, perhaps, that of the three experiments the two balloon-borne detectors provide the more accurate picture of the true value of the UHE source cutoff assumed. As can be see in Figure
7.2, this may be due to the fact that both ANITA 3 and EVA receive more events in the
higher energy range than the ARA detector, making them more sensitive to the end of the
spectrum.
71
Figure 7.14: Middle 68% of minimum − ln(L) based log Emax predictions for ARA37.
Figure 7.15: Middle 68% of minimum − ln(L) based log Emax predictions for EVA.
72
7.5 Conclusion
If the study of particle astrophysics at the highest energies can be said to have a theme, it is
one of finding ever more innovative ways to capture and analyze the exceedingly rare spaceborne events that comprise the substrate on which the field builds its body of knowledge.
Sometimes, as in AMBER, the way forward leads through the investigation of a potential
new avenue of event detection—in this case, microwave-band radio induced by cosmic ray
air showers.
Other times the field progresses by applying powerful analysis techniques that squeeze
as much information as possible from relatively (compared to other particle physics fields)
tiny collections of events. The research presented in this chapter used this approach as it
looked ahead to the UHE neutrino results of the current and upcoming experiments ANITA
3, EVA, and ARA. Used in conjunction with constraints from Pierre Auger, we saw how,
used appropriately, even mere dozens of events can give us some crucial hints about the
distribution and nature of the energetic powerhouses that, in more than one way, drive
much contemporary astroparticle research.
As the body of information about astrophysical spectra grows in a way that considers
whole new species of particles, it seems likely that multi-messenger approaches will provide
us with an ever more important tool to combat the limitations that are so often a trademark
of the field.
73
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78
Appendix A
Expected Power of Microwave
Bremsstrahlung Radiation
The following is based on the method that Gorham et al. outlined in Ref. [2] to scale the
microwave radiation detected in a laboratory test to that which might be observed from a
cosmic ray air shower at its peak intensity. Here I consider only the completely incoherent
case as per the analysis in Chapter 4.
Performed in 2004, the T471 experiments at the Stanford Linear Accelerator Center
(SLAC) used an electron beam incident on a metallic target in order to simulate an air
shower with an energy of E0 = 6 × 1017 eV. As measured inside an anechoic chamber, the
length of this “reference” shower was L0 = 0.65 m, with a characteristic decay time near
τ = 10−8 s (see Fig. A.1). The reference flux density If 0 (peak intensity over bandwidth)
is taken by Gorham et al. to be 4 × 10−16 W/m2 /Hz, and the distance to the shower axis
is d = 0.5 m.
With these reference shower values in place, I can now turn my attention to the parameters that Ref. [2] uses to describe a C band microwave detector. The dish effective area is
given by
Aef f = η π
D
2
2
(A.1)
where η is the feed efficiency (in this dissertation taken to be 1.0), and D assumes a value of
2.4 m for AMBER as installed at Pierre Auger. The bandwidth of the AMBER-01 detector
∆f is taken as 500 or 800 MHz depending on whether the inner or outer antennae are being
considered.
79
Figure A.1: Intensity versus time in a T471 experiment. This version uses an antenna
configuration that cross polarizes against potentially contaminating Cherenkov emission.
From Ref. [2].
Now the reference shower must be scaled appropriately to a cosmic ray air shower. In
Gorham et al. this is accomplished with two factors. One of these factors, which I will
call ρ(h)/ρ0 , is the ratio of the electron density at a shower altitude h divided by the same
density at sea level (taken to mean the ratio of atmospheric densities). The other scaling
factor comes from considering the ratio Γ = Lτ /L0 , where Lτ = c τ is the characteristic
length of shower development.
Finally, since I am considering only the completely incoherent case, the factor E/E0
suffices to scale from the reference shower energy to any shower of arbitrary energy E.
Taking all these parameters together, I can now write the expected power
Pexp =
E
E0
ρ(h)
Aef f If 0 Γ
ρ0
−2
R
∆f
d
(A.2)
which is a function of the shower energy E, the altitude h, and the distance to the shower
from the detector R.
80
Appendix B
Atmospheric Calculations
In this research (as per Refs. [62] and [63]) I take atmospheric pressure below 11 kilometers
to be described by the equation
P = P0 (1 −
Lh gM
) RL ,
T0
(B.1)
where P0 = 101325 Pa, T0 = 288.15 K, g = 9.80665 m/s2 , L = 0.0065 K/m, M = 0.0289644
kg/mol, and R = 8.31432 J/(mol K). With this in hand I write the density ratio relevant
to calculating expected MBR power (see Appendix A) as
ρ(h)
Lh gM
= (1 −
) RL .
ρ0
T0
(B.2)
Finally, as in Ref. [64], the atmospheric depth X for a shower with a zenith angle θ can
be written as
X=
P 1
,
g cos(θ)
(B.3)
where P is the pressure given in Eq. B.1 and the value of g is given above. Note that this
is an approximation made assuming a flat Earth, and is only valid for showers with zenith
angles less than 60 degrees [65].
81
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