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Measurement of the cosmic microwave background temperature and Galactic emission at 8.0 and 8.3 GHz with the ARCADE 2 experiment

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UNIVERSITY OF CALIFORNIA
Santa Barbara
Measurement of the Cosmic Microwave Background
Temperature and Galactic Emission at 8.0 and 8.3 GHZ with the
ARCADE 2 Experiment
A Dissertation submitted in partial satisfaction of the
requirements for the degree Doctor of Philosophy
in Physics
by
Jack Edward Singal
Committee in charge:
Professor Philip Lubin, Chair
Professor James Hartle
Professor Joseph Incandela
December, 2006
UMI Number: 3245904
UMI Microform 3245904
Copyright 2007 by ProQuest Information and Learning Company.
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
ProQuest Information and Learning Company
300 North Zeeb Road
P.O. Box 1346
Ann Arbor, MI 48106-1346
The Dissertation of Jack Singal is approved.
_________________________________________
James Hartle
_________________________________________
Joseph Incandela
_________________________________________
Philip Lubin, Committee Chair
December, 2006
Measurement of the Cosmic Microwave Background Temperature and
Galactic Emission at 8.0 and 8.3 GHz with the ARCADE 2 Experiment
Copyright © 2006
by
Jack Edward Singal
iii
“You can choose a ready guide
in some celestial voice…”
-Rush, Freewill
This work is dedicated to my mom and dad.
ACKNOWLEDGEMENTS
My graduate education having been uniquely split between a first two years at
Santa Barbara and a final two years at NASA Goddard, I have been extremely
fortunate to bask in the mentorship of great scientists. The ARCADE project has
presented me this opportunity to work with and study under the field’s finest, and I
hope to convey my deep appreciation. From my advisor, Philip Lubin, from Al
Kogut, the PI of ARCADE, and from Dale Fixsen, Ed Wollack, Michele Limon, and
Paul Mirel I have learned everything that I know how to do and I thank them for their
tutelage and patience. Special thanks to Ed Wollack for his feedback on this
document. Also, without the work of Adam Bushmaker, Jane Cornet, Paul Cursey,
Keith Feggins, Sarah Fixsen, Steve Levin, Luke Lowe, Alexander Rischard, and
Michael Seiffert, ARCADE would not have gotten off the ground.
For showing a complete novice how to operate in a lab and think about the
cosmos, I am indebted to Peter Meinhold, Alan Levy, and Brian Williams. For
receiving the greatest physics education possible, I have to thank the outstanding
iv
professors I’ve had at Santa Barbara, including the members of my committee,
Professors Hartle and Incandela.
I must also say that my time in graduate school would not have amounted to
anything without the friendship, collaboration, and conversation, academic and not so
academic, of Feraz aka person X, Bangstrom, Johannes, Puneeth, Hadrian, Jiminez,
Lawrence, Mendozaaaaa, “H.R.” Patterson, Pillsbury, and the Stromberg Constant.
Who would have guessed it was possible to fit so much wry observing, trash talking,
geo-political problem solving, naked beach running, and rockin’ music making into
two years?
Then there are those that I am so lucky to call my best friends of many years:
Ethan, Ray, Tom, Dave, Uncle Jeed, Lytwyn, Brandon, Sub, Keith, John, Frank, and
Balls. Thank you all for being part of my life. Thanks to my grandma and grandpa
for always being on my side, and I also appreciate Jess and Roberta for their support
over the course of my time as a graduate student. Finally, saving the best for last,
thanks to my love Danielle - here’s to our wonderful future together!
The material in this dissertation is based on work supported by the National
Aeronautics and Space Administration under the Science Mission Directorate.
Additional funding was obtained through the CalSpace fellowship.
v
Jack Singal
Curriculum Vitae, December 2006
Education:
B.S. in Physics, 2000, New York University (attended 1996-2000)
Magna Cum Laude with minor in History
M.A. in Physics, 2004, Ph.D. in Physics, 2006
University of California, Santa Barbara (attended 2002-2006)
Teaching Appointment:
Graduate Teaching Assistant, UCSB Physics Department (2002-2004)
Research Appointments:
UCSB Experimental Cosmology Group (2003-2004)
ARCADE lab, NASA Goddard Space Flight Center (2004-2006)
Experience in mechanical design, CAD design, microwave antenna design,
microwave component fabrication, cryogenics, thermal design and analysis, thermal
calibration, integration and testing, field launch campaigns, and analysis of CMB data
Publications:
“The Cosmic Microwave Background Temperature and Galactic Emission at 8.0 and
8.3 GHz.” J. Singal, D. Fixsen, A. Kogut, S. Levin, M. Limon, P. Lubin, P. Mirel, M.
Seiffert, and E. Wollack, 2006, Astrophysical Journal, 653, 835
“Design and Performance of Sliced-Aperture Corrugated Feed Horn Antennas.”
J. Singal, E. Wollack, A. Kogut, P. Lubin, M. Limon, P. Mirel, and M. Seiffert, 2005,
Review of Scientific Instruments, 76, 124703
“A Compact Microwave Calibrator.” D. Fixsen, E. Wollack, A. Kogut, M. Limon,
P..Mirel, J. Singal, and S. Fixsen, 2006, Review of Scientific Instruments, 77, 064905
“Radiometric Waveguide Calibrators.” E. Wollack, D. Fixsen, A. Kogut, M. Limon,
P. Mirel, and J. Singal, IEEE Trans. Instrum. Meas., accepted
Honors and awards:
- Elected to Sigma Pi Sigma physics honor society.
- Received NYU physics department prize for highest undergraduate GPA in major.
- Tied for second place in New York City bicycle messenger race.
vi
ABSTRACT
Measurement of the Cosmic Microwave Background Temperature and Galactic
Emission at 8.0 and 8.3 GHz with the ARCADE 2 Experiment
by
Jack Edward Singal
This work presents a measurement of the radiometric temperature of the
Cosmic Microwave Background (CMB) and of the intensity of Galactic emission at
8.1 and 8.3 GHz. These are the science results of the first flight of the ARCADE 2
instrument, on which the author’s design, fabrication, and data analysis work forms
the basis of this dissertation.
ARCADE 2 (Absolute Radiometer for Cosmology, Astrophysics, and Diffuse
Emission) is a balloon-borne instrument designed to perform measurements of the
radiometric temperatures of the sky at six microwave frequency bands, from 3 to 90
GHz, to milliKelvin precision. ARCADE 2 features a novel cryogenic design and
sophisticated radiometry as described herein. During the first flight of the
instrument, a mechanical failure allowed for the accumulation of scientifically
meaningful data in only one frequency band, and those results are not as well
constrained as that from future flights will be. However, the measurement
presented here of the radiometric temperature of the CMB is in fact the one of most
well constrained below 10 GHz, and the measurement of Galactic free-free and
synchrotron emission presented here is a potentially significant confirmation of
existing results.
vii
The temperature of the CMB at 8.0 and 8.3 GHz is found to be 2.90 ± .12 K
and 2.77 ± .16 K respectively. The level of Galactic synchrotron emission at these
frequencies is found to be that which would be expected by naively interpolating
the previously available data at other frequencies, and the level of Galactic free-free
emission is found to be two-thirds as high, providing an independent confirmation
of changes recently announced in the three year Galactic foreground results release
from the WMAP satellite.
The first section of this work is a comprehensive review of important topics in
cosmology, the CMB, and deviations from a blackbody spectrum therein, as well as
Galactic microwave emission. The second section describes the ARCADE 2
instrument and instrumental considerations, with some emphasis on design and
fabrication contributions by the author. The third section presents the data obtained
from the first flight of the instrument, the data analysis as carried out by the author,
and the science results.
viii
TABLE OF CONTENTS
Preface………………………………………………………………………………
1
Section I – Theory
Chapter 1. Cosmology and Cosmography…………………………………….…....
3
I. The Big Bang and the expanding universe………………………....…….
3
II. Redshift and cosmography……………….…………………….….….…
5
III. Our universe of dark matter and dark energy…………………....….…. 12
Chapter 2. The CMB and CMB Cosmology……………………………………...... 16
I. The Cosmic Microwave Background Radiation……...…………..…...… 16
II. Structure in the CMB…………………………...………….…….…..…. 19
III. Inflation……………………………………………………..……..…… 24
IV. Universe timeline and other backgrounds…………..……….……….... 26
Chapter 3. Deviations from Blackbody in the Microwave Background…………… 29
I. The CMB as blackbody and existing constraints…………………..……. 29
II. Free-free emission and reionization………………………………….…. 32
III. Energy injection into the primordial plasma…………………….……... 36
Chapter 4. Galactic Microwave Emission……………………………...………….. 42
I. Galactic microwave foregrounds………………………………………… 42
II. Galactic Free-free and Synchrotron radiation………………..…………. 44
Section 2 – Experiment
Chapter 5. Experimental Challenge and Instrument Overview……………………. 50
I. The experimental challenge……………………………………………… 50
II. ARCADE instrument concepts…………………………………………. 52
III. The ARCADE 2 instrument overview…………………………………. 56
IV. ARCADE 2 thermometry, control, and read out electronics…………... 60
V. Doctoral timeline………………………………………………………... 63
ix
Chapter 6. ARCADE 2 Radiometers………………………………………………. 65
I. Switching radiometers…………………………………………………… 65
II. ARCADE 2 radiometers………………………………………………… 69
III. Steelcast as emitter…………………………………………………….. 79
IV. ARCADE 2 internal loads……………………………………………... 83
V. 2005 flight 8 GHz radiometer performance…………………………….. 87
Chapter 7. ARCADE 2 Feed Horn Antennas…………………………………….... 92
I. Corrugated feed horns…………………………...………………………. 92
II. ARCADE 2 feed horns………………………………………………….. 97
III. ARCADE 2 low band horns………………………………...…………. 101
IV. ARCADE 2 high band horns…………………………………………... 110
V. ARCADE 2 waveguide transitions……………………………………... 114
Section 3 – Results
Chapter 8. Data and data analysis………………………………………………….. 117
I. The ARCADE 2 2005 flight……………………………………………... 117
II. The data and data reduction…………………………………...………... 120
III. Data analysis strategy………………………...………………………... 126
IV. Pointing solution……………………………………………………….. 136
Chapter 9. Results………………………………………………………………….. 140
I. Results from iterative fit…………………………………………………. 140
II. Galactic emission levels………………………………………………… 143
III. Estimation of TCMB………………………………………...…………... 149
IV. Uncertainty estimates…………………………………………………... 153
V. Discussion………………………………………………………………. 157
Appendix A: Mechanical and Thermal Design Considerations…………………… 159
Appendix B: The ARCADE 2 Target……...………………………………………. 172
Appendix C: 2006 flight and future prospects……………………………………... 183
References………………………………………………………………………….. 187
x
LIST OF FIGURES
2-1. The CMB angular power spectrum……………………………………………. 23
3-1. A blackbody distribution for two different temperatures……………………... 30
3-2. CMB radiometric temperature as measured by FIRAS……………………….. 31
3-3. Maximum allowable distortions to the CMB spectrum in the ARCADE 2
frequency range given existing constraints………………………………………… 40
4-1. Base-level of Galactic foreground signals away from the Galactic center as a
function of frequency……………………………………………………………….
4-2. The HASLAM all sky survey showing Galactic emission at 408 MHz……….
4-3. WMAP 2003 Galactic free-free emission map at 22 GHz…………………….
4-4. WMAP 2003 Galactic synchrotron emission map at 22 GHz…………………
5-1. Plot of CMB, atmospheric, and Galactic emission levels in the frequency
range from 10 to 500 GHz………………………………………………………….
5-2. Schematic of the ARCADE 2 instrument……………………………………...
5-3. Photograph of the ARCADE 2 instrument as flown in 2005 being lowered
into the dewar………………………………………………………………………
5-4. Photograph of the ARCADE 2 horn apertures………………………………...
5-5. Photograph of the lazy susan being placed on top of the aperture plane………
6-1. Schematic of ARCADE 2 radiometer chain…………………………………...
6-2. Photograph of the cold stage of the 8 GHz radiometer………………………..
6-3. Photograph of the warm radio stage of the 8 GHz radiometer………………...
6-4. Real and imaginary components of the dielectric constant of Steelcast……….
6-5. Photograph of radiometric side of the 8 GHz buffy load……………………...
6-6. Measured reflectivity of 8 GHz buffy load at room and cryogenic
temperatures, along with predicted reflectivity…………………………….……….
6-7. Photograph of underside of 10 GHz buffy load, showing thermometer wires...
6-8. Photograph of mold for 8 GHz buffy load……………………………….…….
6-9. Cross section schematic of 30 GHz split block load…………………………..
6-10. Amplitude spectrum of the 8 GHz low channel AC lockin output for the high
gain data……………………………………………………………………………..
6-11. Amplitude spectrum of the 8 GHz high channel AC lockin output for the
high gain data……………………………………………………………………….
6-12. Amplitude spectrum of the 8 GHz low channel AC lockin output for the null
data………………………………………………………………………………….
6-13. Amplitude spectrum of the 8 GHz high channel AC lockin output for the
null data……………………………………………………………………………..
xi
43
46
47
47
53
56
57
59
59
69
71
72
82
84
85
85
86
87
89
89
90
90
7-1. Profile of ARCADE 2 30 GHz horn………………………………………...… 94
7-2. Schematic of waveguide/transition/horn system……………………………… 99
7-3. Profile of the 10 GHz horn as designed……………………………………….. 102
7-4a. Predicted and measured E-plane beam pattern for the 10 GHz horn at 10.11
GHz…………………………………………………………………………………. 105
7-4b. Predicted and measured H-plane beam pattern for the 10 GHz horn at 10.11
GHz…………………………………………………………………………………. 106
7-4c. Predicted and measured cross polar response for the 10 GHz horn at 10.11
GHz………………………………………………………………………………… 106
7-5. Predicted return for the 10 GHz horn and measured return loss for the
rectangular-to-circular waveguide transition and 10 GHz horn combination……… 107
7-6a. Predicted E- and H-plane beam patterns for the 8 GHz horn………………… 107
7-6b. Predicted E- and H-plane beam patterns for the 8 GHz horn, close up of
main lobe…………………………………………………………………………… 108
7-7. Predicted return loss for the 8 GHz horn……………………………………… 108
7-8. Photograph of 10.2 GHz horn…………………………………………………. 109
7-9. Photograph of 3 GHz horn…………………………………………………….. 109
7-10. Predicted E-plane beam patterns for the 30, 30# and 90 GHz horns………… 112
7-11. Photograph of 30 GHz horn………………………………………………….. 112
7-12. Photograph of 30# (narrow beam) horn……………………………………… 113
7-13. Photograph of 90 GHz horn………………………………………………….. 113
7-14. Photograph of 3 GHz (WR 284) waveguide transition………………………. 115
8-1. Photograph of July 2005 launch of ARCADE 2…………………………………….. 118
8-2. Photograph of ARCADE 2 at 2005 landing site………………………………. 120
8-3. Time ordered 8 GHz low channel AC lockin output following the opening of
the instrument lid…………………………………………………………………… 124
8-4. Selected instrument temperatures during and shortly after the null period…… 125
8-5. Portion of sky viewed during data records used for analysis, in Galactic
coordinates, on top of the WMAP K band Galactic emission map………………… 139
9-1. Time ordered 8 GHz high channel AC lockin output for the null period
(upper), and residuals of the least squares fit for null data period with falling
switch temperature (lower)………………………………………………………… 142
9-2. Time ordered 8 GHz low channel AC lockin output for the null period
(upper), and residuals of the least squares fit for null data period with falling
switch temperature (lower)…………………………………………………………. 143
9-3. Time ordered 8 GHz high channel AC lockin output for records 11825-12375,
a representative sample of the high gain period (upper), and residuals of the least
squares fit for those records (lower)……………………………………….
147
9-4. Time ordered 8 GHz low channel AC lockin output for records 11825-12375,
a representative sample of the high gain period (upper), and residuals of the least
squares fit for those records (lower)……………………………………………….. 147
xii
9-5. Plot showing time ordered 8 GHz high channel AC lockin output for records
12450-12750 (solid), along with the best fit model from the least squares fit
(dashed) and a model with αff = αsynch = 1 (dotted)………………………………... 148
9-6. Plot showing time ordered 8 GHz low channel AC lockin output for records
12450-12750 (solid), along with the best fit model from the least squares fit
(dashed) and a model with αff = αsynch = 1 (dotted)………………………………... 148
9-7. Plot of high channel AC lockin output with horn, galactic contribution, and
back end offset taken out vs. internal load temperature, for null records 7095-7397 151
9-8. Plot of low channel AC lockin output with horn, galactic contribution, and
back end offset taken out vs. internal load temperature, for null records 7095-7397 151
9-9. Plot of high channel residuals vs. switch temperature for records 6770-6970... 154
9-10. Plot of low channel residuals vs. switch temperature for records 6770-6970.. 154
A-1. The major elements of the instrument core as designed……………………… 160
A-2. Photograph of the major elements of the instrument core as built……………. 160
A-3. The 3 GHz horn collar as designed…………………………………………… 162
A-4. Aperture plane as designed…………………………………………………… 163
A-5. Shop drawing and photograph of completed ‘Anti-Knuckle’ part……………. 168
A-6. Shop drawing of 3 Collar part………………………………………………… 169
B-1 Photograph of the ARCADE 2 target, radiometric side………………………. 173
B-2. Photograph of the ARCADE 2 target, back side, in place on the lazy susan…. 173
B-3. Photograph of the ARCADE 2 target, radiometric side, in place on the lazy
susan…………………………………………………………………………………174
B-4. Photograph of a target cone…………………………………………………… 176
B-5. Photograph of a batch of cone cores ready for casting………………………... 177
178
B-6. Photograph of mold for casting Steelcast over cone cores to form target
cones, along with metal positives around which the mold was formed…………….
B-7. Photograph of Instrumented target cone core with Ruthenium-Oxide
resistance thermometer at tip of wire……………………………………………….. 179
B-8. Locations of thermometers in instrumented cones……………………………. 180
B-9. Schematic of target as assembled for the 2005 flight…………………………. 180
B-10. Temperatures in the target during selected records from the 2005 flight……. 182
C-1. Section of 3 GHz low channel radiometer output from the 2006 flight……… 184
xiii
LIST OF TABLES
3-1. Previous limiting low frequency CMB measurements and their uncertainties. 32
5-1. The ARCADE science team………………………………………………….. 55
6-1. Specifications for off the shelf components of the ARCADE 2 8 GHz
radiometer………………………………………………………………………….. 77
6-2. Specifications for ARCADE 2 8 GHz radiometer back end…………………. 78
6-3. 8 GHz radiometer flight performance properties…………………………….. 91
7-1: Parameters for ARCADE 2 low band horns…………………………………... 104
7-2: Specifications for ARCADE 2 low band horns……………………………….. 104
7-3: Parameters for ARCADE 2 high band horns…………………………………..111
7-4: Specifications for ARCADE 2 high band horns………………………………. 111
8-1. Telemetry data frame from ARCADE 2 2002 flight, words 0-8……………... 121
8-2. Telemetry data frame from ARCADE 2 2002 flight, words A-D……………. 122
8-3. Summary of data used for analysis…………………………………………… 125
9-1. Coefficients from the least-squares fit, low channel…………………………. 141
9-2. Coefficients from the least-squares fit, high channel………………………… 142
9-3. Uncertainty summary for TCMB………………………………………………. 156
B-1. Measured attenuation of reflections of ARCADE 2 target when viewed with
ARCADE 2 band horns……………………………………………………………. 175
xiv
PREFACE
I feel it necessary to briefly explain the organization, emphasis, and length of
this work. The ARCADE 2 project seeks to measure the radiometric temperature of
the CMB at six frequency channels to the milliKelvin level, and as a secondary
purpose, to gain new data on Galactic emission foregrounds. This dissertation
presents the results from the summer 2005 first flight of the instrument. As described
herein, mechanical and other failures during that flight yielded meaningful science
data in only one frequency channel, that at 8 GHz, and the CMB temperature results
presented for that channel, although new, do not constrain any cosmology and are
potentially less scientifically interesting than the Galactic results presented here.
However, because the primary purpose of ARCADE 2 is CMB measurements,
and future measurements with the instrument may constrain cosmological parameters,
I view this still as fundamentally a CMB project. There is then, in the first three
chapters, a thorough review of cosmology and the CMB. The fourth chapter is a
review of Galactic microwave emission.
The second section, Chapters 5, 6, and 7, describe the instrument. Again, I
have striven for thoroughness, but the overall content of these chapters is entirely
relevant to the data and results presented in the third section. For completeness, I
wanted to include chapters on the instrument mechanical design work and the
calibrator target fabrication that I did. Although these are important for the
instrument and its future results, they are not of direct relevance to the measurements
1
presented in this dissertation, so in the interest of cohesion, I have relegated them to
appendices. A third appendix reports the successful 2006 fight of the instrument and
discusses the future prospects for improvement.
The third section, Chapters 8 and 9, as mentioned, presents the science data,
data analysis, and results from the 2005 flight. I have attempted to maintain a
cohesive narrative starting with a review of cosmology, the CMB and Galactic
microwave emission, through an overview of the instrument and a component by
component description, to the 2005 data and results, although this cohesion was
naturally in tension with my desire to be thorough. I hope I have succeeded in
walking that line.
2
CHAPTER 1: COSMOLOGY AND COSMOGRAPHY
I. The expanding universe and the big bang
A conception of the origin, fate, and nature of the universe on the largest
scales is one of the central goals of physics. While mankind’s scientific
understanding of the nature of the cosmos and notion of our place in it have been
evolving for millennia, the canonical cosmology for 70 years has been that of the ‘hot
big bang’, in which the universe was previously in a very dense state and has
expanded since. The big bang idea was primarily motivated by the discovery,
through spectroscopic observations in the 1920s of the redshifts of galaxies, that other
galaxies are moving away from our own and that those that are farther are moving
away faster, suggesting an expanding universe; and by the discovery that General
Relativity allows a spatially homogeneous and isotropic universe in which the
expansion is an initial condition.
The Einstein Equation of General Relativity. R µν −
1
8π G
Rg µν = 4 Tµν , relates
2
c
the curvature of spacetime, expressed by the metric tensor g µν and its derivatives
which form R µν and R, to the stress-energy tensor Tµν within the spacetime, thus
explaining the gravitational interaction as the effect of a spacetime that is curved by
the energy within. To describe our universe it is convenient to use so-called
3
Chapter 1. Cosmology and Cosmography
comoving coordinates, in which the spatial locations of objects that are not subject to
local motion on the largest scales are constant. In these coordinates, a universe that is
spatially homogeneous and isotropic must by definition have a metric with isometries
in the special dimensions of the spacetime, and there must exist a timelike coordinate
that is independent of all spacetime coordinates. Thus the metric for our universe on
the largest scales must have the form ds 2 = −c 2 dt 2 + R ( t ) 2 d £ 2 where d £ 2 is a
homogeneous spatial 3-D metric and R(t) is a scale factor which may change with the
time coordinate. The possibilities for a homogeneous and isotropic 3-space in 4
dimensions are the surface of a 3-sphere, a flat Euclidean space, or the surface of a 3hyperboloid, giving, as expressed by Robertson and Walker in 1924,


 sin 2 χ 
 2 
 2
2
2
2
2
2
2
2 
ds = −c dt + R ( t ) dχ +  χ (dθ + sin θ dφ ) ,
sinh 2 χ 






or, with the substitution dχ =
dr
1 − kr 2
with k=1 for the 3-sphere case, k=0 for the
Euclidean case, and k=-1 for the 3-hyperboloid case,
 dr 2

ds 2 = −c 2 dt 2 + a 2 ( t ) 
+ r 2 (dθ2 + sin 2 θ dφ2 ) ,
2
1 − κr

where κ =
k
R(t)
, and a ( t ) =
is a dimensionless scale factor.
2
R0
R0
We see that in light of General Relativity, a universe with a scale factor that is
changing with time is a natural consequence. In our universe, the scale factor is
4
Chapter 1. Cosmology and Cosmography
increasing with time, and this expansion, as far as General Relativity is concerned, is
an initial condition. At some point in the past then, the universe had a singularity
where a=0, and this is the big bang, the point from which the universe began
expanding. It is worth mentioning that although physics is the same in every frame,
the frame where the universe is described in comoving coordinates with the time
coordinate independent of all spatial coordinates is especially convenient for treating
cosmology. To that end, whenever cosmological notions of time are referred to in
this treatment, it is understood to mean the time as measured by comoving observers
in these coordinates.
II. Redshift and cosmography
II–A. Redshift
In General Relativity, massless particles including photons travel on null
geodesics; in terms of the 4-momentum, pµ pµ = 0 . In the Robertson-Walker metric,
2
1 r2
 hν 
and with E = hν for photons, this becomes −   + 2
p = 0 . The presence of
 c  a (t)
killing vectors in all special directions means that the component of the 4-momentum
r
in all special directions is conserved, so solving for p 2 at the time of a photon
2
r
r2
 h ⋅ ν emitted ⋅ a emitted 
emission, p 2 = 
 , and plugging this conserved value for p into
c


5
Chapter 1. Cosmology and Cosmography
the above equation at the time of photon observation gives
ν observed
a
= emitted , which
ν emitted
a observed
shows that light from objects is redshifted from the time of emission to that of
observation by the ratio of the scale factors at those times. The redshift z of an object
is defined as the fractional change in wavelength z ≡
λ obs − λ em ν em
=
− 1 , with the
λ em
ν obs
redshift of the present being 0. With this definition, given observation taking place
today where a=1, then a em =
1
. The more distant an object, the higher the
1+ z
redshift of the object, as the scale factor at the time in which the object emitted light
that reaches us at the present will be smaller than that at the present for an expanding
universe.
In the potentially curved space of an expanding universe the instantaneous
physical distance between objects is not a particularly useful concept. However, for
close-by objects, the curvature can be neglected and the instantaneous
distance d p = a 0 r can be differentiated to give an instantaneous velocity
v 0 = a& 0 r =
a& 0
d p = H 0d p . H 0 is the Hubble constant, the present-day constant of
a0
proportionality between the distance of a nearby object and its velocity. For a nearby
object, the redshift will be a Doppler shift proportional to the velocity, and this is the
distance redshift correlation that was observed by Hubble in the 1920s.
6
Chapter 1. Cosmology and Cosmography
II–B. The scale factor and the content of the universe
Being a theory of stress-energy induced geometry, General Relativity allows
us to relate the geometry of the universe to its content. Our universe is assumed and
observed to contain contributions to the energy density in the form of matter,
radiation, and vacuum energy. The 0-component of the stress-energy continuity
equation ∇ µ Tµν = 0 in the Robertson-Walker metric yields
 da 3 
d 3
a ρ = − p
 , which
dt
 dt 
[ ]
has a nice interpretation in light of the first law of thermodynamics, in that the left
side is, up to a constant factor, the change with time of a total energy and the right
side is, up to the same constant factor, the change with time of the pressure times a
total volume. Matter is pressureless, so a 3ρ is constant and ρ m ( t ) =
ρm (t 0 )
, or,
a(t)3
3
equivalently ρ m (z) = ρ m 0 (1 + z ) , which is just a statement of the familiar notion that
the energy density in matter is just inverse linear in the volume. The equation of state
ρ (t )
1
for radiation, from statistical mechanics, is p r = ρ r , and so ρ r ( t ) = r 4 0 ,
3
a
4
ρ r ( z ) = ρ r 0 (1 + z ) , which is a statement that the energy density in radiation is
inversely proportional to the volume, plus another factor of the scale factor to account
for the redshift of the photons. For vacuum energy, p Λ = −ρ Λ , and so ρ r ( t ) = ρ r ( t 0 ) .
7
Chapter 1. Cosmology and Cosmography
It is important to note that temperatures have a linear evolution with redshift;
that is T ( z ) = T0 (1 + z ) . This can be seen by considering the result from statistical
mechanics that the energy density of a photon emitting object is proportional to the
fourth power of the object’s temperature: ρ rad ∝ T 4 , and that as above radiation
evolves as a quartic function of redshift, hence the linear relation for temperature.
Solving the Einstein Equation with the Robertson-Walker metric yields the
Friedmann equation,
H2 =
8π G
κc 2
ρ− 2 ,
3
a
with H =
a&
being the Hubble parameter. It can be seen that the value of the energy
a
density that gives a spatially flat κ=0 universe, the critical density, is ρ c =
3H 2
, and
8π G
the density parameter Ω i is the ratio of the present density in a phase to the critical
density, Ω i =
ρi ( t 0 )
. Evaluated at the present time, then, the Friedmann equation
ρc
gives 1 − Ω tot
− κc 2
=
. At a given redshift, the energy density of a given component
H 02
n
n
will be ρ i ( z ) = ρ i 0 (1 + z ) i = Ω iρ C 0 (1 + z ) i , where ni is the factor by which the
component scales with redshift, derived above. The total density will be the sum of
8
Chapter 1. Cosmology and Cosmography
the components, and so the Friedman equation can be expressed as

n
2
H 2 = H 02 ∑ Ω i (1 + z ) i + (1 − Ω tot )(1 + z )  ,
 i

or in full,
3
2
2
H ( z ) = H 0 Ω m (1 + z ) + Ω r (1 + z ) + Ω Λ + (1 − Ω m − Ω r − Ω Λ )(1 + z ) .
The quantity 1 − Ω tot can be interpreted as the density parameter that derives from
curvature, and the energy density due to curvature then evidently evolves as
2
ρ k ( z ) = ρ k 0 (1 + z ) .
In general, determining the evolution of the scale factor in the presence of
arbitrary contributions to the total energy density from matter, radiation, and vacuum
energy is a nontrivial calculation involving integrating the Friedmann equation.
However, since the energy densities from the three different sources evolve at
different rates, there are long epochs in the history the universe where one will
dominate. At the earliest times in our universe, then, radiation was the dominant
contribution to the total energy density, later it was matter, and will in the future be
vacuum energy. If one source dominates and the curvature can be neglected, the
Friedmann equation reduces to a& ≈
8πG
ρ i 0 * a 1−n 2 , where n is the power by which
3
the density component evolves with redshift. This can be immediately integrated to
obtain a ∝ t 2 n , so for a radiation dominated universe, a ∝ t 1 2 , for a matter dominated
universe a ∝ t 3 2 , and for a vacuum energy dominated universe a ∝ e t .
9
Chapter 1. Cosmology and Cosmography
II-C. Distance measures
The redshift is the most directly observable property of a distant object, but in
order to take stock of the universe we would like to relate the redshift to notions of
spatial distance. An important quantity is the comoving coordinate distance DM from
an observer to an object observed at a given redshift. From ds 2 = 0 for photon
trajectories, and of course light is how we observe the object, and dθ = dφ = 0 for a
DM
radial path that terminates at the observer, we have
∫
0
1
a&
H
with
= (1 + z ) , dz = − 2 dt = dt , and then
a
a
a(t)
t obs
t obs
dr
=
1 − κr 2
c
∫ a( t) dt .
First,
t em
z
em
c
c
∫t a ( t ) dt = z∫ H (z ) dz ≡ D C .
em
obs
This is an important integral.
Next, evaluating the dr integral, D M =
(
)
f DC κ
, where f(x)=sin(x) for the open case
κ
where κ<1, f(x)=x for the flat κ=1 case, and f(x)=sinh(x) for the closed κ>1 case.
With 1 − Ω tot =
DM =
− κc 2
, then
H 02
c
H 0 1 − Ω tot
 H0
*f


1 − Ω tot
c

DC  .


At any given redshift, an object’s proper size d is simply its size in comoving
coordinates D times the scale factor, so d =
D
. The angular diameter distance DA
1+ z
10
Chapter 1. Cosmology and Cosmography
of an object at a constant distance from the observer and therefore constant redshift is
defined such that the angle it subtends δα on the observer’s sky times the angular
diameter distance is the object’s proper size, δα * D A = d =
D
, the way an arc
1+ z
length is related to an angle and the radius. The actual size of an object at a constant
radius in comoving coordinates is D = D M dα , so D A =
DM
. The luminosity
1+ z
distance DL is the distance we infer given the total flux received from an object F and
its absolute luminosity L, 4 π D 2L ≡
the observer,
L
. In terms of the area A of a sphere centered on
F
L
= A , and the area is A = 4 π D 2A . However, the luminosity of an
F
object in an expanding universe is diminished by one power of the redshift as the
photons lose energy, and another power of the redshift as the time between surface
2
crossing events increases, so D L = (1 + z ) D A . Another important quantity is the
horizon size, the maximum scale of causal influence, as influence can only propagate
as far as the speed of light multiplied by the time since the big bang. This is
obviously c ∗ D M , with zem evaluated at ∞. Lastly, the elapsed time between now and
when the light from an object of redshift z was emitted is the lookback time,
t0
t L = t 0 − t em
1
da
= ∫ dt = ∫
=
a H (a )
t em
a em
z em
dz
∫ (1 + z ) H (z ) .
0
11
Chapter 1. Cosmology and Cosmography
III. Our universe of dark matter and dark energy
It is clear that from the above discussion that knowing the spatial shape and
fate of the universe involves knowing Ω m , Ω r , and Ω Λ . If Ω tot < 0 , then the universe
will be spatially open, while if Ω tot = 0 the universe will be spatially flat, and if
Ω tot > 0 the universe will be spatially closed. Additionally, given that the vacuum
energy density is constant with time while the energy density in matter goes as a −3 ,
and that the scale factor of a universe dominated by vacuum energy evolves as a ∝ e t ,
almost any universe with any nonzero true vacuum energy at the present will continue
to expand forever.
In fact, astronomy and astrophysics have progressed to the point where the
questions of astrophysics on the largest scales, from the shape and fate of the universe
to galactic evolution, can be answered by knowing just a small number of parameters.
As this introduction is concerned primarily with cosmology, I will be concerned here
with the Hubble Constant, H0, which is conceptually easy to determine by knowing
the velocities and distances of many nearby galaxies, and the three density parameters
Ω m , Ω r , and Ω Λ . Ω r is the smallest, consisting almost entirely of the cosmic
microwave background, which is discussed in the next chapter, and any possible
relativistic neutrino background, explained there as well. Although it may seem
counterintuitive as the night sky seems to be full of the light from stars, Ω r ~ 10 −4 .
12
Chapter 1. Cosmology and Cosmography
Ω m is more interesting. Adding up all of the matter that we see in stars, the
interstellar medium, stellar nebulae, that which is thought to exist in brown dwarf and
extinct starts, and the intergalactic medium and multiplying it by the sizes and
densities of galaxies, this ‘ordinary,’ or so-called baryonic matter has Ω baryon ≈ 0.04
(Carroll). Additionally, depending on the existence and abundance of massive
neutrinos, there may be an Ω neutrino on the same order.
However, there is overwhelming evidence for far more matter in the universe.
As early as the 1970s, measurements of the rotation speeds of galaxies, specifically
the rotation speed as a function of distance from the center, which should be only a
function of the gravitational potential and therefore the total mass within, were
inconsistent with the amount of observed matter in these galaxies. It was inferred
then that galaxies contained much more matter than was visible, the so called ‘dark
matter.’ This matter must be dark, in that is that it does not interact
electromagnetically, or else we would see it directly. In fact, every galaxy is believed
to be sitting within a much bigger ‘halo’ of dark matter.
Additional overwhelming evidence for dark matter comes from observations
of gravitational lensing by galaxies, the process by which intervening massive objects
bend the light of objects behind them. The lensing properties are a function of the
mass, and again it is clear that there must be much more mass in galaxies than we see.
In fact, both observations argue for there being more than 5 times as much dark
matter as baryonic matter. It is worth mentioning that the very fact that dark matter
13
Chapter 1. Cosmology and Cosmography
does not interact electromagnetically means that it cannot be as clumped as baryonic
matter. Baryonic matter can cool by emission of photons to the environment, losing
gravitational potential energy by radiating electromagnetic energy, and hence can
coalesce into solar systems, stars, and planets. Dark matter cannot cool by radiating
photons, and so is less able to clump and must exist as the ‘halos’ previously
described.
Every known particle in the standard model of particle physics has been ruled
out as a candidate for the dark matter. Given that the dark matter is by far the
dominant matter component in the universe, it must have been the dominant
component in gravitational structure formation in the evolving universe, such as that
of galaxies, clusters, and superclusters. This leads to the conclusion that the dark
matter must have been essentially stationary in the early universe and therefore it is
referred to as ‘cold’ dark matter. One possible result of the measurements undertaken
n the ARCADE 2 project is an upper limit on the mass, lifetime, and abundance of
dark matter particles that may have existed in the early universe.
An observational handle on Ω Λ has been achieved only recently, in the past
ten years. One method is by using type I-A supernova studies. These supernova are
the most distant standard candles known, meaning that their intrinsic optical
properties are largely independent of redshift of origin. With an inferred absolute
luminosity and measured flux, their luminosity distance can then be known. Given a
14
Chapter 1. Cosmology and Cosmography
measured redshift and the previously derived luminosity distance formula, these can
be used to solve for Ω Λ and Ω m .
The result of type I-A supernova studies, taken together with anisotropies in
the cosmic microwave background, which are discussed in the next chapter, have
achieved a remarkable convergence on Ω Λ ≅ 0.7 and Ω m ≅ 0.3 . And so there is the
amazing result that everything that we know about, everything that is made of the
elements - ourselves, stars, planets, all the light that we see, and the enormous amount
of hydrogen in the intergalactic medium – is only 4% of the energy in the universe.
Another 26% is the dark matter, surrounding us but whose effects we have only seen
gravitationally, and fully 70% is the so called ‘dark energy,’ the unknown vacuum
energy that turns out to dominate our universe.
With the values of Ω Λ and Ω m that we now have, the age of the universe can
be determined. Computing the lookback time to the big bang, z = ∞ , gives the age of
the universe as around 13.7 billion years (Bennett, 2003).
This current view of the universe is called the ΛCDM model, for the vacuum
energy and the cold dark matter that overwhelmingly comprise the universe. The
realization, primarily in the past 10 years, that we live in a ΛCDM universe where
baryonic matter is a small component will likely be viewed years from now as a
revolution as significant as Copernicus’ sun-centered solar system.
15
CHAPTER 2: THE CMB AND CMB COSMOLOGY
I. The Cosmic Microwave Background Radiation
That the universe having previously existed in a hotter, denser state implies
the existence of a background of electromagnetic radiation was first realized as early
as the 1940s by George Gamow. The early universe was sufficiently hot that that the
photons and charged baryons formed a plasma of free charged particles and photons.
Following the very early nucleosynthesis era the charged particles were mostly
hydrogen nuclei and electrons. As the universe expanded and cooled, a low enough
temperature was reached where it was statistically favorable for electrons and protons
to adopt a lower energy state and combine to form neutral hydrogen, a process
misleadingly called recombination. At this point, the density of free charged particles
was greatly reduced and the photons, lacking charged scatterers, could stream away.
These are the photons that we detect as the Cosmic Microwave Background Radiation
(CMB). Recombination was sufficiently abrupt and rapid that the CMB photons are
said to originate from a relatively thin ‘surface of last scattering’ which surrounds us
at a great distance.
Quantitatively, the progress of recombination can be approximated by the
Saha equation, which has a nice intuitive derivation. The ratio of the probability of
16
Chapter 2. The CMB and CMB Cosmology
two states is the ratio of their partition functions. Let χ e =
ne
be the ratio of the free
nb
electron number density to the total baryon number density, so n e = χ e n b = χ e η n γ ,
where η =
z bound = 2e
z free
nb
Pr ob( bound ) z bound 1 − χ e
is the baryon to photon ratio. So
=
=
, with
nγ
Pr ob(free )
z free
χe
−I
kT
2V
=
(2πh )3
where I=13.6 eV is the ionization energy of hydrogen, and
∫
0
3
4 πp 2 dp
∞
− p2
e
2 mec 2 kT
−1
 m c 2 kT  2
2V
 ξ(3) 3 integrating over the possible
=  e
h
 2π 
position and momentum states of a free electron where V =
1
1
=
is the
n e χeη n γ
volume available to the electron. Putting it together,
3
−I
1 − χe
(2π h )
= ξ(3)η n γ
e kT .
3
2
χe
(2π m ec 2 kT ) 2
This equation implies a dramatic falloff for χ e with increasing temperature,
corresponding to a relatively narrow recombination era. A more careful analysis
must take into account that a photon emitted from a direct free to ground state
electron transition will simply reionize another atom, as will several photons emitted
from less energetic combination events. Thus during the recombination era, atoms
continually combine and ionize. However, two-photon emission from the 2S to 1S
transition breaks the chain because one of these photons individually is too weak to
excite a transition and the probability of simultaneous double photon absorption is
17
Chapter 2. The CMB and CMB Cosmology
vanishingly small. This explains why there is not a line in the observed photon
background corresponding to the ionization energy of hydrogen or any of its atomic
transitions. The two photons emitted in the 2S to 1S transition can share the energy
in any way, and given that η ~ 10 9 , they have a vanishing effect on the background.
In the full analysis of recombination, the redshift at which CMB photons were last
scattered is well fitted by a Gaussian of mean 1065 and standard deviation 80
(Peacock).
Experimentally, researchers were attempting to detect the CMB by the early
1960s, but the first detection, by A. Penzias and R. Wilson using a Bell Laboratories
microwave telescope, was, famously, unintentional, as the CMB manifest itself as an
irreducible background in a telescope intended to observe nearer field objects.
Subsequent observations showed the CMB to be highly isotropic with a spectrum
close to that of a blackbody. Thus the CMB stands as an impressive confirmation of
hot big bang cosmology, as a spatially isotropic blackbody background could only be
produced by a last scattering surface of constant redshift. If the CMB were produced,
for example, by dust emission, its spectrum would be at best be a superposition of
blackbodies of different temperatures.
18
Chapter 2. The CMB and CMB Cosmology
II. Structure in the CMB
Interesting structure is not entirely absent from the CMB, and the
characterization of that structure has provided, and will continue to provide,
enormously important information about the universe. The structure present in the
CMB is 1) temperature anisotropies - variations in temperature from place to place on
the sky, 2) polarization and polarization anisotropies, and 3) deviations in the
spectrum from that of a blackbody. The latter is the primary concern of the project
described in this dissertation, but the first two are of the utmost importance in CMB
cosmology, so they will be outlined briefly here.
II-A. Temperature anisotropy
Temperature anisotropies can be caused by conditions at the last scattering
surface, so called primary anisotropies, or conditions since, the so-called secondary
anisotropies, and there is also a dipole anisotropy taken to be caused by the particular
motion of the Earth relative to the background and the resulting dipole doppler shift.
Primary anisotropies result from the presence of overdense and underdense regions at
the last scattering surface, the so-called Sachs-Wolfe effect, and by doppler shift due
to peculiar velocities of the photons at decoupling. The early universe contained
density fluctuations on all length scales, which are presumed to be the seeds of later
19
Chapter 2. The CMB and CMB Cosmology
structure. Baryons in the early universe undergo oscillation as they fall into the
overdense regions, compressing and eventually reaching a density where the radiation
pressure is greater than the gravitational attraction, at which point they move outward
until the gravitational attraction again takes over.
At the last scattering surface, photons from regions that are overdense will be
gravitationally redshifted while those from underdense regions will be gravitationally
blueshifted, and since frequency shift is linear in gravitational potential in the weak
field limit,
δT
δφ
( grav ) = 2 . However, there is a competing effect in that in that
T
c
overdense regions are hotter and therefore undergo recombination slightly later and
so the CMB photons will be less redshifted relative to regions that undergo
recombination slightly earlier. To work out the net effect, following White and Hu
(White and Hu, 1997), I note that in an overdense region in the weak field limit time
as measured by an outside observer runs slower as dt ' =
(1 − Φ ) dt , so in doing a
transformation from t to t’ in moving to a denser region
δt
Φ
= − 2 . In a matter
t
c
c2
dominated universe, as it was at the last scattering era, the scale factor a ∝ t
δa =
2
3
so
2
2Φ
δt = − 2 . Finally, with the scale factor and temperature related by
3t
3c
aT = const , then
δT − δa
δT
2Φ
=
, and so
(full ) = 2 . The doppler shifts at
T
a
T
3c
decoupling will in general be out of phase with the Sachs-Wolfe blueshift, as regions
20
Chapter 2. The CMB and CMB Cosmology
that are fully compressed or rarefracted will have no net velocity while those that are
at the midpoint of an oscillation will have maximal velocity. However, the doppler
term is subdominant.
For characterizing measured anisotropies on the 4π sterradian sky, the sky is
naturally expanded in a spherical harmonic basis, where the quantity being observed,
X(n) is expanded as X ( n ) = ∑
l
∑a
lm
Ylm ( n ) , with a lm = ∫ dΩ n ∆T ( n )Yl*m ( n ) . An
l >0 m = − l
anisotropy is characterized by its angular power. The angular power observed on the
2
sky at a given multipole moment l is the ensemble average of power at that l, (Csky
l ) ,
where C sky
=
l
1
2
a lm . Large angular power at a given l means that there is a
∑
2l + 1 m
lot of structure of angular size corresponding to the angular scale of that l. In the
case of temperature, the temperature anisotropy at each multiple moment is
∆Tl =
C sky
l l(l + 1)
.
2π
One would expect to see angular power in the CMB on scales corresponding
to the size at which a region could undergo one compression by the time of last
scattering, resulting in an overdense region and the temperature change described
above, and at the size for one compression and one rarefraction, resulting in an
underdense region, and so on for additional harmonics. The size d of a region that
would compress once by the time of last scattering is given by the speed of the
baryons undergoing compression, which in a relativistic fluid is one third the speed of
21
Chapter 2. The CMB and CMB Cosmology
light, times the time from the beginning of the universe to last scattering, quantities
which are known. The measured angular size of such a region today is this proper
size divided by the angular diameter distance, δα =
d
, as in chapter 1, and the
DA
angular diameter distance is given by
DA =
sin
c
(1 + z )H 0
1 − Ω tot
 H 0 1 − Ω tot
* 1× 

c
sinh 

c
dz
∫0 H(z )  as derived there, so the

z lss
angular size of the peak corresponding to this one compression in the measured
angular power spectrum of the CMB is a powerful measurement of Ωtot. Angular
sizes larger than this are causally disconnected at last scattering, and so we would not
expect peaks in angular power there. The angular size of region that undergoes one
compression and one rarefraction by last scattering will also feature a peak. The
magnitude of this peak should be diminished relative to the first because the potential
well is stronger once the baryons are compressed so compressions are enhanced over
rarefractions. The magnitude of the second peak relative to the first, then, is
dependent on Ωbaryon.
The definitive measurement of CMB temperature anisotropies to date has
been the results of the WMAP satellite experiment (Bennett et al., 2003a). The
temperature-temperature angular power spectrum from WMAP’s first year results are
shown in Fig, 2-1. The results clearly point to a flat Ω tot = 1.02 ± 0.02 universe with
22
Chapter 2. The CMB and CMB Cosmology
Ω baryon = 0.044 ± 0.004 , and WMAP’s results are one of the most important pillars of
modern ΛCDM cosmology.
Fig. 2-1. The CMB angular power spectrum. From the Lambda CMB data
archive. Originally in Bennett et al., 2003.
II-B. Polarization
Linear polarization of the CMB results from Thomson scattering of CMB
photons by electrons in which the photons incident on the electrons are anisotropic in
temperature. As the Thomson scattering cross section is proportional to the dot
product of the incident and outgoing polarization directions, a quadrupole incident
temperature anisotropy will generate polarized light upon scattering. As multiple
scattering events would tend to suppress polarization by damping the temperature
anisotropy incident on a given scattering electron, the scattering events giving rise to
an observable polarization signal would have occurred during two different cosmic
23
Chapter 2. The CMB and CMB Cosmology
epochs, one at the last scattering surface when the photons would in large part not
scatter again, and one much later when the universe again contained free charged
particles but is mostly optically thin to CMB photons. As the horizon size was very
different in these two eras, the characteristic angular scales of the polarization signal
are different.
An m=0 quadrupole anisotropy from so-called ‘scalar’ temperature
perturbations gives rise a polarization pattern different from that of the quadrupole
anisotropy with an m = ±1 pattern which is from so-called ‘vector’ perturbations that
arise from vorticial doppler motions of the incident photons, and a third distinct
polarization pattern is from so-called ‘tensor’ perturbations which are red and blue
shifts in the incident photons arising from perturbations in the underlying spacetime
metric. WMAP has presented a temperature-polarization angular power spectrum
(Kogut et al., 2003) which is sensitive to polarization at the level where the effect of
scalar temperature perturbations can be detected. Some implications of this are
discussed in Chapter 3, header II.
III. Inflation
The high degree of isotropy of the CMB is a major motivation for the idea of
inflation, that the universe expanded at a very rapid rate at very early times due to the
24
Chapter 2. The CMB and CMB Cosmology
presence of an early short-lived vacuum energy dominated era. Different regions of
the surface of last scattering that we see today were causally disconnected at the time
of last scattering. This can be seen by considering the calculated horizon size at the
time of last scattering, using the formula derived in Chapter 1, which is about
2c
,
H0
which is considerably less than the comoving distance to the CMB, at redshift ~1100,
from the Earth, about
6 × 10 −2 c
. So when we look at widely separated parts of the
H0
last scattering surface, we are seeing regions that were causally disconnected at last
scattering, and yet they somehow ‘know’ to be at very close to the same temperature,
a situation called the horizon problem. However, if the universe underwent an initial
period of very rapid expansion very early on, where the scale factor increased
exponentially, then the horizon at last scattering would be very much larger, owing to
the very large H(z) for very early times.
Inflation is also useful in solving the flatness problem, which arises from
considering that in the Friedmann equation the curvature term is proportional to a −2
while the energy density due to matter is proportional to a −3 . Thus a presently flat
universe that has a contribution to the total energy density from matter, such as our
own, would have had to be exactly flat at early times, and any slight perturbation
from exactly flat would grow to provide for a non-flat universe at the present.
However, with an inflationary era where the scale factor increases exponentially due
to vacuum energy, a ∝ e Ht , the curvature contribution can be driven arbitrarily small.
25
Chapter 2. The CMB and CMB Cosmology
This can be seen on a hiruistic visual level as well. If there is curvature present but
the scale of the universe increases tremendously in an inflationary era, then the local
effects that of curvature can be washed away. Theories of inflation involve the
presence of an energetic quantum field, and fluctuations in this field can be the origin
of density perturbations. These relic perturbations are thought to be the seed of all of
the later structure in the universe. Conceivably, measurements of polarization in the
CMB due to tensor perturbations would contain information about perturbations to
the metric which would be the result of the inflationary era.
IV. Universe timeline and other backgrounds
In particle physics, the rates of interactions are determined by densities, cross
sections, and energies. In the course of the evolution of universe, then, the expansion
and cooling lead to dramatic changes in reaction rates. As the photon component
largely decoupled from the rest of the universe at the time of last scattering, so have
others. A timeline of the universe is as follows.
At the very earliest times, Grand Unified Theory (GUT) era, the energies were
high enough that the standard model of particle physics is not valid, although the
physics of the GUT era is thought to have given rise to inflation and a matter antimatter asymmetry. Following the GUT era the physics can be understood. By
26
Chapter 2. The CMB and CMB Cosmology
around 10-6 seconds the universe underwent quark confinement in which the
previously free quarks and gluons combined into hadrons. Because of the slight
matter/anti-matter asymmetry, particle/anti-particle annihilation leads to a matter
dominated universe with a photon to baryon ratio of ~109. The numbers of neutrons
and neutrinos ‘freeze out’ at ~14 seconds, as energies and densities become too low
for the weak interaction to maintain equilibrium between protons and neutrons, and
neutrinos and leptons. Free neutrons are unstable, and so all neutrons that are not
confined to nuclei by ~900 seconds will decay, and this gives a calculable relic
abundance of deuterium, helium, and the other light elements, and these abundances
from so-called big bang neucleosynthesis can be compared with those observed
today, and this serves as a major pillar of big bang cosmology.
Neutrinos, on the other hand, are stable and thus we can expect a thermal
cosmic background of neutrinos analogous to that of photons. Electron-positron
annihilation after neutrino decoupling adds energy to the photon baryon plasma, so
the neutrino background will be at a lower temperature than the photon background.
The CMB photons themselves, as discussed in this chapter, decouple at
recombination, when neutral atoms become dominant. The universe then enters the
so called ‘cosmic dark ages’ when few new photons are produced. The density
perturbations, however, give rise to structure, the first luminous objects are formed,
and galaxies evolve, and stars and supernovae synthesize the heavier elements,
gradually bringing about the universe we know today. Light from luminous objects is
27
Chapter 2. The CMB and CMB Cosmology
absorbed and reemitted by the dust that dominates galaxies and the interstellar
medium, giving rise, along with redshifted light from the era of galaxy formation, to a
non-thermal Cosmic Infrared Background (CIRB). There is also a non-thermal X-ray
and γ-ray background primarily of photons emitted from Active Galactic Nuclei
(AGNs), in which radiation is emitted from accretion into super massive black holes
at the center of these galaxies. Supernovae also eject huge quantities of neutrinos,
giving rise to a second thermal neutrino background at a higher temperature than that
arising from the early universe.
Photons from the first stars reionize the neutral atoms in galaxies and the
intergalactic medium, and the time of this reionization is an unsolved question in
cosmology. It is a primary goal of the experiment described in this dissertation to
address the reionization question, as discussed in the next chapter. Another open
question is how thoroughly and how early the dark matter component was decoupled
from the photon-baryon component, and constraining these is also a goal of such
measurements.
28
CHAPTER 3: DEVIATIONS FROM BLACKBODY
IN THE MICROWAVE BACKGROUND
I. The CMB as blackbody and existing constraints
A blackbody spectrum is that spectrum of photons which results from those
photons being in complete thermal and chemical equilibrium with a substance at a
temperature T, meaning practically that there are sufficient scattering events for the
photons and charged particles to be in thermal energetic equilibrium, and sufficient
photon creation events for the number of photons not to be constrained by the number
of charged particles. From statistical mechanics, the occupation of any given energy
state n i = − kT
∂ ln Z
− Ei
− εi n i
where Z is the partition function Z = ∑ e kT = ∑ e kT , and
∂ε i
i
i
ε i is the energy of the ith state. As photons in equilibrium have been readily absorbed
and emitted as necessary, there is no restriction on photon number and the partition
function is simply
Z=
∑e
n1 ,n 2 ,L
− n1ε1 kT
e −n 2ε2
kT
 ∞
L =  ∑ e − n1ε1
 n1 = 0
and with ε i = hν i , n ( ν i ) =
kT
  ∞ − n 2ε 2
 *  ∑e
 n 2=0
 
1
e
hν1 kT
−1
kT
1


 *L = 
− ε1
1
−
e


kT
1
 
*
−ε 2
1
−
e
 
kT

 *L

. To get the energy density in an infinitesimal
frequency range, one must integrate over a spherical shell in momentum space,
29
Chapter 3. Deviations from Blackbody in the Microwave Background
2
3
1
dν
E
 E  8π hν
ε( ν )dν = 2 ∫ 4 π 2  hν kT d   =
. This used the standard
3
hν kT
∫
−1  c 
c
e
−1
c  e
relation for massless particles from Chapter 1, p =
a(t) E
, where a(t) is the scale
c
factor, normalized to 1 at the present. It is important to note that a blackbody
spectrum retains its shape with redshift, as both frequency and temperature are
proportional to (1+ z ) , as discussed in Chapter 1 under header II-B. A blackbody
distribution for two different temperatures is plotted in figure 3-1.
Fig. 3-1. Blackbody distributions for two different temperatures. The flux of
light energy across a surface is reported, which is the volume energy density
multiplied by the speed of light.
The spectral shape of a blackbody features a low frequency portion where the
relationship between frequency and energy density is roughly linear, the so-called
30
Chapter 3. Deviations from Blackbody in the Microwave Background
Raleigh-Jeans region, the peak, and beyond the peak the Wien tail, where energy
density falls off with frequency.
It is customary and convenient to quantify deviations from a blackbody
spectrum at a given frequency in terms of the radiometric temperature at that
frequency. This is the temperature the radiation would be if it were an exact
blackbody with the intensity observed at the frequency in question.
The FIRAS instrument on the COBE satellite firmly established the CMB as a
blackbody of temperature T = 2.728 ± .004 K at frequencies above 60 GHz (Fixsen et
al., 1996), as shown in Fig. 3-2, corresponding to a blackbody spectrum with a peak
at around 160 GHz.
Fig. 3-2. CMB radiometric temperature as measured by FIRAS. The 1σ
uncertainties are a small fraction of the line thickness. On the vertical axis,
1Jy=10-26 J/m2. On the horizontal axis, multiply by c=3x1010 cm/s to express
frequency in Hz. From Fixsen et al., 1996.
31
Chapter 3. Deviations from Blackbody in the Microwave Background
Below 60 GHz, however, deviations are much less constrained. Table 2-1 shows the
existing limiting low frequency CMB measurements, which are the current
experimental constraints on deviations from blackbody. The primary purpose of the
ARCADE project is to determine the CMB spectrum below 60 GHz, to as low as 3
GHz, which can constrain questions of cosmology as outlined below.
Table 3-1. Previous limiting low frequency CMB measurements and their uncertainties, from Fixsen
et al., 2004.
Frequency
Radiometric
Uncertainty (mK)
Source
(GHz)
Temperature (K)
30
2.694
32
ARCADE I (Fixsen et al., 2004)
25
2.783
25
Johnson & Wilkinson, 1997
10.7
2.730
14
Staggs et al., 1996b
10
2.721
10
ARCADE I (Fixsen et al., 2004)
7.5
2.64
60
Levin et al., 1992
2
2.55
140
Bersanelli et al., 1994
1.47
2.26
200
Bensadoun et al., 1993
1.4
2.66
320
Staggs et al., 1996a
1.28
3.45
780
Raghunathan & Subrahmanyan, 2000
I shall now examine how the observed CMB spectrum could be distorted from
that of a blackbody at low frequencies. There are three distinct known processes, 1)
Free-Free emission and 2) Energy injection into the primordial plasma, and 3)
Sunyaev-Zelldovich effect.
II. Free-Free emission and reionization
A distortion due to free-free emission is expected. Free electrons present in
galaxies, galactic halos, and the intergalactic medium (IGM) will scatter off of each
32
Chapter 3. Deviations from Blackbody in the Microwave Background
other emitting bremsstrahlung photons, some of which will stream towards us as
additions to the background since last scattering.
The distortion due to free-free emission as derived by Bartlett and Stebbins
(Bartlett and Stebbins, 1991) is
T ( ν) − Tγ 0
Tγ 0
1
= 2
x
t0
∫ κ (1 − n
γ
( ν) x e )dt ≈
t min
Yff
, where
x2
x=
hν
hν
, xe =
, n γ ( ν) is the photon occupation number, and
kT( ν )
kTe
κ=
8πe 6 h 2 n 2e g
with ne being the free electron number density, Te being
3
3m e (kTγ 0 ) 6π m e kTe
the electron temperature, Tγ0 being the current temperature of the CMB, and g being
the Gaunt factor, which has a weak frequency and temperature dependence. Yff is the
t0
‘optical depth’ to free-free emission, Yff ≡
∫
t max
Te − Tγ
Te
κdt , t0 is the present, and tmin is
the time corresponding to the first presence of free electrons. The approximation is
valid when the photon temperature distortion is small, and in this approximation Tγ in
Yff is the blackbody temperature of the photon gas.
The distortion of the photon temperature due to free-free goes as
1
∝ λ2 , and
2
ν
so is a rapidly increasing function of wavelength, hence the distortion below 10 GHz
could be quite significant. The factor ne2 is the hallmark of a scattering probability in
a two body process.
33
Chapter 3. Deviations from Blackbody in the Microwave Background
As the spectral distortion due to free-free emission is a function of the square
of the free electron number density integrated over times dt, a measurement of such
spectral distortion is a measurement of the integrated amount of ionization looking
back, and therefore such a measurement could shed light on the amount of ionized
gas and when it first appeared. This would put a powerful constraint on the time of
formation of the first stars, as the reionization of the ISM and IGM is due to UV
radiation from stars and supernova events, which is sufficiently energetic to ionize
hydrogen atoms.
The reionization history of the universe is still an open question in
astrophysics. During the recombination era of z~1000, as discussed previously, the
universe underwent a phase transition from overwhelmingly ionized, with free
protons and electrons, to overwhelmingly neutral hydrogen. Today the universe is
again highly ionized. There are two major pillars of evidence for a highly ionized
IGM. The first is the absence of a Gunn-Peterson trough in the spectra of distant
quasars. These spectra show distinct wavelengths where the intensity of certain
frequencies is severely diminished. This is due to the light from the distant quasar
being absorbed by intervening neutral hydrogen, then reemitted isotropically, the
result being that the observed intensity of photons with frequency corresponding to
the n=1 to n=2 atomic transitions of hydrogen, the Ly-α line, is diminished. The light
is redshifted between the quasar and the intervening absorbers, creating multiple
frequencies to the blue of the Ly-α line where the intensity is diminished,
34
Chapter 3. Deviations from Blackbody in the Microwave Background
corresponding to multiple absorbers at different redshifts, and therefore to discrete
clouds of neutral hydrogen, dubbed the Lyman-α forest. However, a neutral IGM
would cause the spectra to have a continuous trough in a range of in the frequencies
to the blue of the Ly-α line, corresponding to the absorption of Ly-α photons at a
continuous range of redshifts, and the absence of such a Gunn-Peterson trough
implies an ionized IGM. Spectra of quasars to near redshift z~6 have shown no such
trough, indicating that the universe was highly ionized beginning around then (Fan et
al., 2002).
The other major evidence for a highly ionized IGM is from CMB polarization
results. As discussed in Chapter 2 under header II-B, polarization signal in the CMB
on angular scales larger than those corresponding to the horizon size at last scattering
tracks the properties of free electron scatterers in the IGM. Results from the
temperature-polarization power spectrum of WMAP point to a universe that is
reionized beginning at redshift 11 < z r < 30 (Kogut et al., 2003). This is in
agreement with the distant quasar measurements in that the universe is presently and
has been highly ionized for quite some time. However, there is significant
disagreement as to the time of reionization, leaving this as an open question that
measurements of the spectral distortion to the CMB from free-free emission could
potentially help to answer.
Being a two body ne2 process, The spectral distortion due to free-free emission
is highly sensitive to the amount of clumping of baryonic matter in galaxies and
35
Chapter 3. Deviations from Blackbody in the Microwave Background
galactic halos. It has been argued by P. Oh (Oh, 1999) that the sensitivity of the freefree spectral distortion to clumping will wash out too much information about the
integrated amount of ionization looking back to place a meaningful constraint on the
era of first star formation. In this case, a measurement of the magnitude of spectral
distortion due to free-free emission could provide powerful information on the
clumping in galaxies and halos (Cooray and Furlanetto, 2004).
The current constraint on Yff is from a ground-based measurement at 2 GHz,
and is Yff < 1.9 × 10 −5 (Bersanelli et al., 1994). The maximum allowable distortion
to the CMB spectrum in the ARCADE 2 frequency range due to free-free emission
given that constraint is shown in Fig. 3-3.
III. Energy Injection to the Primordial Plasma
Equilibrium between photons and baryons is maintained through scattering
events of photons and charged particles. As the timescales for these scattering events
are not instantaneous, equilibrium requires a finite amount of time to be reached.
Compton scattering exchanges energy between photons and charged particles, leading
to thermal equilibrium between the two components and thus the energy sharing
described by equilibrium statistical mechanics. However, as Compton scattering
cannot add or remove photons, other processes, bremsstrahlung emission and double
Compton scattering, are needed to achieve chemical equilibrium such that the photon
number is not constrained and the photons can then obey Planck statistics.
36
Chapter 3. Deviations from Blackbody in the Microwave Background
A distortion away from blackbody in the CMB, then, contains information
about the energetic history of the universe. An injection of energy into the baryon
component or the photon component would be preserved in a distortion of the photon
spectrum, provided that the spectrum had insufficient time to relax back to Planckian
through scattering processes.
`
Comptonization proceeds much faster than chemical equilibration. The
timescale for Compton scattering to achieve thermal equilibrium between photons
and electrons is t C =
1 m ec 2
(Burigana et al., 1991a), where σT is the Thomson
n e σ T c kTe
scattering cross section. Following Burigana et al., a dimensionless Comptonization
z
dz t exp
, where texp is the expansion timescale
z tC
0
parameter can be defined y e = ∫
t exp ( z ) =
a
1
(z ) =
. For energy injected sufficiently late when y e > ~ 1 , Compton
a&
H (z)
scattering will not have time to thermalize the photons, and the photon spectrum will
be represented by a superposition of blackbody spectra. As shown by Zeldovich and
Sunyaev (Zeldovich and Sunyaev, 1969), in the full scattering and statistical
mechanical analysis, in the limit of the low frequencies in the Raleigh-Jeans region of
the spectrum,
∆TRJ
∆E
= −2 y e , and the fractional energy injected is
= E 0 e 4 ye − 1 .
TCMB
E0
(
)
Burigana et al. (Burigana et al., 1991a) have done numerical studies
determining the redshift ztherm corresponding to y e = 1 for a variety of cosmological
37
Chapter 3. Deviations from Blackbody in the Microwave Background
scenarios. For the presently widely accepted cosmological parameters, z therm ≈ 10 4 .
For energy injections after this redshift the photon spectrum will be as above.
For energy injections before this ztherm, the photon spectrum will thermalize
with the charged particles. However, as the timescale for chemical equilibrium is
much longer, the photon number may be constrained and the spectrum will be that of
an equilibrium Bose-Einstein distribution with a nonzero chemical potential µ:
n( ν i ) =
1
e
hν1 kT −µ ( νi )
−1
.
The redshift ztherm is before last scattering, and so a CMB spectrum distorted
in such a manner would be the signature of energy injected into the primordial
photon-baryon plasma from a component previously decoupled from it. Possible
candidates include decays or annihilations of heavy supersymmetric dark matter
particles, which, though weak interactions, could release baryons or photons. A
measurement of such a resulting distortion in the CMB spectrum would constrain a
combination of the mass, abundance, and lifetime of such particles, which would be
an exciting result.
As bremsstrahlung emission and double Compton scattering are more efficient
at longer wavelengths, while there is a tendency for Compton scattering to upscatter
photons from the Raleigh-Jeans region to higher frequencies, the chemical potential
of the CMB spectrum as observed, if it is present, will in general be frequency
dependent µ = µ(ν) . There exists a frequency νc where the Compton scattering of
38
Chapter 3. Deviations from Blackbody in the Microwave Background
photons is balanced by creation of new photons. For ν << ν c the photon creation
processes dominate and the chemical potential vanishes, while for ν >> ν c Compton
scattering dominates over the photon creation processes and the chemical potential
νc
ν
approaches a constant value µ0, and µ(ν ) ≈ µ 0e (Burigana et al., 1991a). The
resulting spectrum features a drop in radiometric temperature at frequencies in the
few GHz range. For small chemical potential distortions, in the full scattering and
statistical mechanical analysis, if the energy injection events are not accompanied by
the production of significant numbers of photons, µ 0 =
1 ∆E
.
4 E0
The current constrains on µ and ye are again from FIRAS and are µ < 9 × 10 −5
and y e < 1.4 × 10 −5 (Fixsen et al., 2004). The maximum allowable distortion due to a
chemical potential in the ARCADE 2 frequency range given that constraint is shown
in Fig. 3-3.
39
Chapter 3. Deviations from Blackbody in the Microwave Background
Fig. 3-3. Maximum allowable distortions to the CMB spectrum given
existing constraints at the 95% confidence level (Haiman and Loeb, 1997,
McDonald et al., 2001). The FIRAS data points are shown (Fixsen et al.,
2004), and the ARCADE 2 frequency range is from 3 to 90 GHz.
For energy injections after Ztherm, the photon spectrum is a superposition of
blackbodies as above. Energy is certainly added to the baryon component by the
heating of the baryons by luminous objects once galaxies and stars form and the
universe reionizes. Like free-free emission, this resulting Sunyaev-Zelldovich effect is
another potential distortion to the microwave background arising from processes
since last scattering. In this case, background photons Compton scatter off of the
warmer free electrons in the IGM or in galaxies and halos, warming the photons and
depressing the Raleigh-Jeans portion of the photon spectrum in favor of the higher
frequencies. Given the current constraint on ye from FIRAS, the maximum possible
distortion to the spectrum at low frequencies is, from the formula above,
40
Chapter 3. Deviations from Blackbody in the Microwave Background
∆TRJ
= −2.8 × 10 −5 . This is at a level two orders of magnitude below the capabilities
TCMB
of ARCADE 2 or any such absolute temperature experiment to detect. However, this
is not below the capabilities of anisotropy measurements, and indeed point source
Sunyaev-Zelldovich measurements can be used to measure properties of galaxies
through which the CMB photons pass.
41
CHAPTER 4: GALACTIC MICROWAVE EMISSION
I. Galactic Microwave Foregrounds
Radiation from four emission processes dominate the microwave radiation
flux from our Milky Way galaxy: 1) Galactic free-free emission, 2) synchrotron
radiation, 3) thermal dust emission, and 4) spinning dust emission. Of these, only the
first two are relevant at the frequencies at which measurements are reported in this
dissertation. Each foreground has a characteristic dependence of the radiometric
temperature of the emission with frequency, and this is the so-called spectral index.
All Galactic foregrounds, resulting as they do from the presence of matter of one
form or another, are dramatically brighter in a narrow band toward the center of the
Galaxy than elsewhere. The radiometric temperatures of the dominant foregrounds
are shown in Fig. 4-1.
42
Chapter 4. Galactic Microwave Emission
Fig. 4-1. Base-level of Galactic foreground signals away from
the Galactic center as a function of frequency. A model of the
residual atmospheric emission at an altitude of 30 km is shown
as well, as is the sensitivity of a possible future satellite CMB
low frequency absolute temperature experiment. Plot courtesy
of Alan Kogut.
Thermal dust emission is approximately thermal radiation from dust in the
Galaxy. It has a positive spectral index, and becomes important at frequencies greater
than 100 GHz. At the high end of the ARCADE 2 frequency range at 90 GHz, the
radiometric temperature of thermal dust emission it is below 1 mK even toward the
Galactic center, and is significantly less off of the Galactic center (Bennett et al.,
2003b). It is utterly irrelevant to the ARCADE 2 measurements at lower frequencies.
43
Chapter 4. Galactic Microwave Emission
Spinning dust emission is electric dipole emission from spinning dust grains.
It is not well characterized, but is not thought to be relevant except in the frequency
range from 20 to 40 GHz (Bennett et al., 2003b). At the high end of this range there
is a sharp cutoff due to the limited rotation rate at which a dust grain can spin, while
at the low end, Galactic emission is completely dominated by free-free and
synchrotron radiation (Draine and Lazarian, 1998). It is possible that the integrated
spinning dust contribution from many galaxies could show up as a small increase in
the radiometric temperature of the CMB at ~30 GHz.
II. Galactic Free-free and Synchrotron radiation
The ARCADE 2 Galactic foregrounds are the free-free and synchrotron
signals. The Galactic free-free signal is from the same free-free emission process
described under Chapter 3, header II, however, because it is from our own Galaxy
which is highly concentrated in certain directions, it is anything but an isotropic
background, unlike the collected free-free from the rest of the universe. As shown in
Chapter 3, the radiometric temperature of free-free emission should go roughly as ν-2,
although in the Galactic emission case one is not integrating over a distance looking
back and the emission is highly concentrated in the Galactic plane. For Galactic freefree, which does not dominate the Galactic signal at any frequency, it is known that
the emission pattern should track the intensity of Hα emission lines (hydrogen n=3 to
44
Chapter 4. Galactic Microwave Emission
2), as the same regions with ionized gas giving rise to free-free emission will have
neutral hydrogen in excited states. Maps of Hα emission, such as the Finkbeiner
composite Hα map (Finkbeiner 2003) can be used as an additional input to free-free
maps. However, it is important to note that a correction must be made for the
extinction of Hα emission due to absorption and scattering off of intervening dust.
Galactic synchrotron radiation is the emission from cosmic ray electrons
spiraling in magnetic fields. The diffuse magnetic field in the Galaxy varies from 1 to
5 Gauss, while the fields in the remnants of type Ib and Type II supernovae are
typically ~75G (Bennett et al., 2003b), so in principle the Galactic synchrotron signal
could contain two components, a diffuse one arising from the Galactic magnetic field,
and a concentrated one arising from supernova remnants. Its frequency dependence
may be approximated as a power law T ~ νβ, with spectral index β = -2.8 between
408 MHz and 20 GHz and steepening to -3.3 above 40 GHz (Platania et al., 1998,
Bennett et al., 2003). The synchrotron spectral index varies across the sky, and is
steeper at the poles than in the Galactic plane (Reich and Reich, 1988, Bennett et al.,
2003).
The Galactic free-free and synchrotron signals are not exceptionally well
known in the ARCADE 2 frequency range. A given Galactic emission signal cannot
easily be resolved into free-free and synchrotron components, and can only be done
so based on the frequency dependence observed from measurements at different
frequencies. Even then, however, the spectral indices are not necessarily known a
45
Chapter 4. Galactic Microwave Emission
priori, and for synchrotron radiation the spectral index has a spatial variation as
mentioned above.
A map of Galactic emission at 408 MHz, the Haslam all sky survey (Haslam
et al., 1982), ground based microwave telescope measurements undertaken in the
middle and late 1970s, is shown in Fig. 4-2. Other important full-sky data comes
from the WMAP CMB anisotropy experiment, which produced Galactic free-free and
synchrotron emission maps at all of its observing frequencies. The lowest WMAP
frequency Galactic emission maps from the WMAP first year data release, at 23 GHz,
are shown in Figs. 4-3 and 4-4. It is seen that both free-free and synchrotron are
overwhelmingly concentrated in the direction of the Galactic center, with an
additional bright region, more dramatic in the free-free emission map, in the direction
of the Cygnus spiral arm to the right of the center that is viewed face on.
Fig. 4-2. The Haslam all sky survey showing the radiometric temperature of
Galactic emission at 408 MHz, log scale, in Galactic coordinates. There are
0.2º per pixel. Data from Lambda CMB data archive.
46
Chapter 4. Galactic Microwave Emission
Fig. 4-3. WMAP 2003 Galactic free-free emission map at 23 GHz, log scale,
in Galactic coordinates, 0.2º per pixel. Data from Lambda CMB data archive.
Fig. 4-4. WMAP 2003 Galactic synchrotron emission map at 23 GHz, log
scale, in Galactic coordinates, 0.2º per pixel. Data from Lambda CMB data
archive.
The WMAP free-free and synchrotron emission maps were obtained primarily
with only WMAP data at its five frequencies and Hα maps. However, being a
47
Chapter 4. Galactic Microwave Emission
differencing and not an absolute measurement, WMAP can only create a relative map
from its own data. To assign absolute temperatures to the synchrotron map, a zero
level must be extrapolated from Haslam results, while a zero level for free-free
emission must be assigned using Hα results. The conversion from Hα emission
intensity to free-free emission temperature is not known a priori. It is estimated as
11.4 µK/Rayleigh from the WMAP first year data set (Bennett et al., 2003) and 8
µK/Rayleigh from WMAP three year data (Hinshaw et al. 2006), and this change has
been independently confirmed by the results reported in Chapter 9 of this dissertation.
With only differing spectral indices to separate out the two signals and that of
thermal dust emission, and with the synchrotron spectral index thought to have spatial
variation, these WMAP emission maps, and our knowledge of Galactic free-free and
synchrotron emission, must be treated as quite incomplete.
There is a gap in full sky measurements of Galactic emission between the
Haslam 408 MHz survey and WMAP K band data at 23 GHz. Surveys covering at
least half of the sky in the intervening frequencies have been carried out at 1.42 GHz
(Reich and Reich, 1982, 1988), and 2.326 GHz (Jonas et al., 1998). Partial sky
surveys have been carried out at 1.4 GHz (Altenhoff et al., 1970), 2.6 GHz (Altenhoff
et al., 1970), 5 GHz (Haynes et al., 1978; Altenhoff et al., 1970), 8.35 and 14.35 GHz
(Langstrom et al., 2000), and 19.2 GHz (Cottingham et al., 1988). In this frequency
range, WMAP K band, Haslam, and Reich 1.42 GHz survey data are available as
48
Chapter 4. Galactic Microwave Emission
input models in the HEALPIX scheme, which is the standard employed here for sky
analysis.
The task of developing models of Galactic free-free and synchrotron emission
at 8 GHz is discussed at length in Chapter 9 of this dissertation. These Galactic
signals are both an input to, and a result from, ARCADE 2 experimental results.
49
CHAPTER 5: EXPERIMENTAL CHALLENGE AND
INSTRUMENT OVERVIEW
I. The experimental challenge
The main obstacle to an accurate measurement of the radiometric temperature
of a ~3 K sky signal is, as can be expected, that it is potentially cold compared to
things which are ambient, and potentially weak compared to emission and noise from
components of the instrument. The number of photons at a given frequency from a
thermal emitter, after all, is a rapidly increasing function of the temperature. Such an
absolute measurement of a cold signal is in contrast to an anisotropy measurement,
where the goal is to measure the relative intensity among different places on the sky,
where the contamination from ambient sources and the instrument cancels to first
order. Both of these types of radiometry, which are employed in CMB
measurements, are themselves in marked contrast to measurements of point sources,
such as with radio telescopes, where the challenge is to collect a flux of photons
which may be highly non-thermal from a very small angular area. An absolute
measurement, therefore, must be undertaken with care to reduce the systematic
effects of warmer ambient emitters and noise from components of the instrument.
A radiometer is an instrument that measures the power of electromagnetic
radiation in a given frequency band by converting it to a voltage. The antenna is the
part of the radiometer that couples electromagnetic radiation to a radiometer, and will
50
Chapter 5. Experimental Challenge and Instrument Overview
have a characteristic beam pattern, which is how the received power varies as a
function of angle. The antenna temperature of a source at a given frequency is equal
to the radiometric temperature of that source if the source fills the beam of the
antenna, and is a function of both the radiometric temperature of the source and the
beam pattern of the antenna if the source does not fill the beam, as discussed in
chapter 7. The antenna temperature as measured by a radiometer looking at the sky,
therefore, is TA = TA ,SKY + TA ,IB + TA ,GD , where TA,SKY is the antenna temperature of
the sky, TA,IB is the antenna temperature of parts of the instrument in view of the
antenna including the antenna itself, and TA,GD is the contribution from any parts of
the ground that might be in view. TA,SKY itself is composed of contributions from the
CMB, Galactic foregrounds, and emission from the atmosphere:
TA ,SKY = TCMB + TA ,GAL + TATM , assuming that the beam is sufficiently wide that
contributions from point sources are negligible.
However, there are sources of contamination in the temperature as measured
by a radiometer. The most basic element of a total power radiometer at these
frequencies is the detector, which is what physically converts an electromagnetic field
strength input to a voltage output. However, a cold stage of amplification is
necessary before a warm detector if viewing a cold signal, or else the radiometer
output will be dominated to an unacceptable extent by emission and instability from
the warm components before the detector. To this end, cold amplifiers are employed
to amplify the weak signal before the warm components, so that proportionally
51
Chapter 5. Experimental Challenge and Instrument Overview
emission and instability from the warm components in the total signal are greatly
reduced. This necessitates an entire cold stage, where the cold amplifier and
everything in the path of the radiation before it are kept at cryogenic temperatures.
However, cold amplification comes with noise and drifts in the gain of the amplifier,
that is noise and drifts in the ratio of the output signal power to the input signal
power. As discussed in Chapter 6, the noise manifests itself in statistical uncertainty
in the measured TA through a nonzero system noise temperature, Tsys, while drifts,
which have the characteristic pattern of so-called 1/f noise, would render accurate
radiometry impossible.
A good celestial absolute radiometer experiment, then, must strive to reduce
Tsys, TATM, TA,IB, and TA,GD, must contain a cryogenic temperature stage to
accommodate all components of the radiometer up to a cold amplifier, and must
effectively deal with 1/f noise.
II. ARCADE instrument concepts
The problem of 1/f noise was reduced in the 1940s by microwave pioneer
R.H. Dicke by using a switching radiometer, in which the sky signal is constantly
switched and differenced with another signal, and this has been employed by
radiometers ever since, including those on the ARCADE instruments. The concepts
of a switching radiometer are discussed in chapter 6.
52
Chapter 5. Experimental Challenge and Instrument Overview
ARCADE is also not unique in how it deals with TATM. Fig. 5-1 shows a plot
of the atmospheric emission, Galactic foregrounds, and the CMB in the ARCADE
frequency range. It is obvious that a satellite mission would avoid the atmospheric
effects altogether, but a much less expensive option is to fly the instrument on a high
altitude balloon. Along with weather, atmosphere, and cosmic ray experiments, CMB
measurements are one of the major uses for such scientific ballooning. Balloons and
launch and recovery facilities are provided by the Columbia Scientific Balloon
Facility, a NASA operation based in Palestine, TX.
Fig. 5-1. Plot of CMB, and Galactic emission levels in the frequency range from
10 to 500 GHz, along with a common model of atmospheric emission. The
quantum limit of HEMT radiometers is discussed in Chapter 6. Plot courtesy of
Brian Williams.
ARCADE is unique in its lack of mirrors, windows or apertures, with a novel
thermal design that allows the maintenance of all cold radiometric components at less
53
Chapter 5. Experimental Challenge and Instrument Overview
than 3K, and this is to dramatically reduce the systematic effects of the instrument
and TA,IB. The entire instrument is contained within a giant open bucket dewar, with
liquid helium pumps carrying helium from the helium bath to all needed places for
cooling. Fig. 5-2 shows a schematic of the ARCADE system. At the altitude of
balloon flights, around 120,000 ft, the boiling point of helium is around 1.4K, so the
helium bath is well below the superfluid temperature of 2.1768 K. It thus does not
respond to mechanical pumping, and must be pumped with superfluid pumps. These
pumps have a body with a ceramic plug which only molecules in the superfluid phase
can penetrate. The body is heated converting the superfluid inside to regular liquid,
which then creates a gradient of superfluid across the plug, causing more superfluid to
rush in and push liquid up the pump line.
Another ARCADE innovation is that the antennas point 30º off of zenith to
reduce the contribution to TA,IB from the flight train and the balloon, but in order to
maintain the antenna apertures on a flat horizontal aperture plane, the antennas, which
are conical corrugated feed horns, are sliced at 30º. This is discussed further in
Chapter 7.
ARCADE also employs and external calibrator, or ‘target’, to dramatically
reduce the systematic effects of the instrument. The antennas alternate between
viewing the sky and viewing the target. The target is a very good blackbody emitter
at microwave frequencies and its temperature is known and adjustable. The
radiometer output when looking at the sky and when looking at the target can then be
54
Chapter 5. Experimental Challenge and Instrument Overview
compared in order to cancel systematic effects of the radiometer to first order. In
theory, the temperature of the target can be adjusted until the radiometer output in a
frequency band is equal when looking at the target to when looking at the sky, then
the radiometric temperature of the sky is just the physical temperature of the target, as
the target is a blackbody emitter. During the flight where the data presented in this
thesis was collected, a mechanical failure prevented the use of the target, but in future
flights the target will improve the error bars on ARCADE 2 results. The ARCADE 2
target is discussed further in Appendix B.
ARCADE is a collaboration between NASA Goddard Space Flight Center,
NASA Jet Propulsion laboratory, and UCSB. Table 4-1 lists the ARCADE science
team.
Table 5-1. The ARCADE science team
NASA Goddard
NASA JPL
Al Kogut – PI
Steve Levin
Dale Fixsen
Michael Seiffert
Michele Limon
Paul Mirel
UCSB
Edward Wollack
Philip Lubin
Jack Singal
55
Chapter 5. Experimental Challenge and Instrument Overview
Fig. 5-2. Schematic of the ARCADE 2 instrument.
III. The ARCADE 2 instrument overview
The first phase of the ARCADE project was in building an instrument,
ARCADE 1, to test the radiometric and thermal design innovations. ARCADE 1 had
radiometers at two frequencies, 10 and 30 GHz, and flew for data taking in the
summer of 2003. This is when I joined the project, and within a few weeks of joining
attended the launch. The concept having been deemed a success, work began on
ARCADE 2, a much larger instrument with radiometers centered at 3.3, 5.6, 8.2, 10.2,
30.3, and 90 GHZ, with an additional radiometer at 30 GHz, designated ‘30#’
featuring a significantly narrower antenna beam.
56
Chapter 5. Experimental Challenge and Instrument Overview
ARCADE 2 has a core within the dewar consisting of the horns and their
attached radiometers hanging from a flat aperture plane, with a rotating ‘lazy susan’
structure on top allowing the target to swing to cover various of the horns while
others view the sky. Fig. 5-3 is a photograph of the entire ARCADE 2 instrument as
flown in 2005 hanging over the dewar.
Fig. 5-3. Photograph of the ARCADE 2 instrument core as flown in
2005 being lowered into the dewar.
57
Chapter 5. Experimental Challenge and Instrument Overview
Fig. 5-4 is a view of the horn apertures on the aperture plane, on top of which turns
the lazy susan, containing both the target and a hole for horns to view the sky, which
is shown in figure 5-5. On the top of the lazy susan, the hole for sky viewing is
surrounded by cold stainless steel flares which curve out and block the far edge of the
antenna beam from things on the lazy susan, and direct the boiloff helium gas away
from the horn apertures. The cold components for each of the seven radiometers sit
in pans of liquid helium fed by superfluid helium pumps from the helium bath with
levels maintained as in a cascading duck pond, earning the nickname ‘the hanging
gardens of Arcadia,’ which are partially visible in Fig. 5-3. The central axis of the
lazy susan is a rod extending into the dewar through which helium pump lines and
thermometer and heater wires are sent.
Thermometry signals are carried by twisted pair copper and brass cryo-wire
within the dewar. All thermometry and heater signals are carried to the outside world
through military connectors on a ‘mate collar’ around the rim of the dewar.
Radiometer signals are propagated in coaxial cable or waveguide from the cold stage
out of the dewar to the warm stage. The warm stage of the radiometers and the
electronics are housed in an electronics housing, which is affixed to a simple external
frame which holds the dewar and allows for attachment of the flight train with cables.
58
Chapter 5. Experimental Challenge and Instrument Overview
Fig. 5-4. Photograph of the ARCADE 2 horn apertures
Fig. 5-5. Photograph of the lazy susan, containing the target and a hole for sky
viewing, being placed on top of the aperture plane.
59
Chapter 5. Experimental Challenge and Instrument Overview
The ARCADE 2 radiometers and horns are treated in detail in Chapters 6 and
7 respectively.
IV. ARCADE 2 thermometry, control, and read out electronics
IV-A. ARCADE 2 Thermometry
120 cryogenic temperatures are measured within the ARCADE 2 dewar for
flight data taking. These include 26 thermometers in the target, 24 on or above the
lazy susan, 22 on components of the radiometers such as internal loads, switches,
amplifiers, duck pond pans, and antennas, 24 so called ‘housekeeping thermometers’
at locations within the dewar, and 24 thermometers on a discrete liquid helium level
sensor. The cryogenic thermometers are Ruthenium-Oxide resistors, the resistance of
which is a sharp and steady function of temperature at cryogenic temperatures. The
resistances are read with a 4-wire measurement where a known square wave current
is driven through the resistor and the resulting voltage across the resistor is read
through a high impedance circuit, thus drawing a miniscule amount of current. In this
way, the resistance of the rest of the circuit, which may be an unknown function of
temperature, cancels.
The cryogenic thermometers require calibration, as their resistancetemperature curves are not identical. To this end, three were sent to the National
Institutes of Standards and Technology for professional calibration. The rest were
calibrated based on the NIST calibrated ones, by taking to ~1.4 K a test dewar
60
Chapter 5. Experimental Challenge and Instrument Overview
containing the thermometers needing calibration and the NIST calibrated
thermometer, built up with sufficient wiring to carry around 50 four-wire signals. An
additional handful of non-cryogenic temperatures outside of the dewar are read with
commercially available temperature transducers, which output a current proportional
to temperature for a wide range of temperatures.
Precision temperature control of cold components is maintained through SPID
control. This is where a set point temperature (TS), proportional term (P), integral
term (I), and differential term (D) are set and a software or firmware program outputs
a voltage through a heater circuit where V = P ⋅ (TS − T ) + I ⋅ ∑ (TS − T ) + D ⋅ , where T
time
is the current temperature. Setting the P, I, and D parameters roughly correctly
results in a steady temperature being maintained.
IV-B. ARCADE 2 readout and control electronics
The electronics for readout and control are contained on numerous electronics
boards housed in a box along with the warm radiometer stages. The data readout
system is organized around a data frame that contains 16 ‘words’ in each of which a
given board can report two bytes of information to the data stream, in 32 ‘sentences’
allowing 32 data channels per board. The centerpiece of the electronics is a board
known as ‘Digital’ which contains the master clock along with counters to form the
frame interval, which are both carried on two separate pins of a parallel bus linking
all of the electronics boards. Another pin of the parallel bus carries the data output,
61
Chapter 5. Experimental Challenge and Instrument Overview
which is converted on the Digital board to the RS232 standard for interface with
standard DB9 computer connectors and the CIP data transmission system used by the
Columbia Scientific Balloon Facility which carries out the balloon flights of
instruments such as ACRADE 2. The Digital board also writes the frame number and
input digital logic levels to the data stream in some its channels, accepts two byte
input commands to the parallel bus, and writes those commands to the data stream in
four of its channels.
All boards that report to the data stream contain a firmware chip known as
‘English’ which instructs the board to report its data to the data pin of the parallel bus
at the appropriate word in each sentence, based on the counting of the clock. The
boards contain a switch so that the reporting word of the board can be altered. The
word and sentence assignments for reporting in the ARCADE 2 2005 flight are
discussed in Chapter 8. The output data stream contains 1) digital logic values as
described above, 2) temperatures, which are reported by thermometer boards, 3) the
radiometer voltage outputs, reported by a lockin board, as discussed in Chapter 6, 4)
various housekeeping voltage levels, such as those from the room temperature
thermometers and various voltage and current sensors, which are reported by analog
input boards, and 5) SPID levels reported by a SPID board, which also carries out
SPID control.
In addition to SPID temperature control, the electronics also provide control
for output analog voltage levels to heaters and motors on two ‘analog out’ boards,
62
Chapter 5. Experimental Challenge and Instrument Overview
bias currents to the amplifiers from a ‘bias’ board, and the current pulse signal for the
switches in the radiometers from a ‘switch driver’ board. Commands for control are
handled in a similar fashion, where the first byte of a two byte command contains the
word address and those boards that respond to and execute commands ‘know’ their
commanding address. In the 2005 flight, two distinct varieties of boards, known as
‘low boost’ and ‘high boost,’ provided amplification for output voltages.
V. Doctoral timeline
After attending the final the flight of ARCADE I, a two channel prototype
instrument, in the late summer of 2003, I spent three weeks at Goddard designing the
ARCADE 2 antennas, which are described Chapter 7. In the following few months, I
worked on producing CAD and machinist drawings for the horns, a task that was
complicated by the nontrivial geometry. For example, the 3 GHz horn required 60
pages to specify. Meanwhile the electronics for control and readout were engineered
by the ARCADE team in my absence.
Upon arriving at Goddard permanently in the summer of 2004, my first task
was to mechanically design the aperture plane and core support structure. This work
is described in Appendix A. Next I fabricated the internal loads for the radiometers
and the 299 cone elements of the target, as described in Chapter 6 and Appendix B
respectively. I worked on calibrating the most of the cryogenic target cone and
63
Chapter 5. Experimental Challenge and Instrument Overview
housekeeping thermometers as described above, including building the testing setup,
carried out and analyzed cold-tests of some of the radiometers, and built up the target.
The two months before the first flight of ARCADE 2 in summer 2005 were spent
physically building up the instrument core from scratch and fabricating the cryowiring harnesses to instrument the thermometry and heating within the dewar.
The maiden flight of ARCADE 2 occurred on July 28, 2005. The flight, and
various mishaps before and during, are described in Chapter 8, the end result being
that we obtained meaningful science data with only the 8 GHz radiometer. Upon
returning to Goddard, I carried out an engineering analysis of problems with the
target from the flight, and then the data analysis in the 8 GHz channel, as discussed in
Chapters 8 and 9. By June of 2006 I was helping to prepare the instrument for the
second flight, which occurred on July 21, 2006. In that flight I managed the 5 and 8
GHz systems. Since then, I have worked on contributing to the data analysis from
that flight, and building a new cryogenic test setup at Goddard with the goal of
recalibrating the thermometers in the target cones in an evacuated pressure vessel.
64
CHAPTER 6: ARCADE 2 RADIOMETERS
I. Switching radiometers
As discussed in the pervious chapter, a CMB radiometer must contain a cold
amplifier before any warm components. Amplifiers are characterized by a gain,
which is the ratio of the output signal power to the input signal power. Over the
decades, special amplifiers were developed for microwave frequencies, and these are
HEMTs (High Electron Mobility Transistors). The unique features of HEMTs which
are desirable are low noise and rapid response, which is necessary when dealing with
fields that are oscillating in the GHz range. HEMT transistors achieve these features
because, in contrast to other field-effect transistors, the semi-conductor layer is
effectively 2-dimensional, meaning the electrons have far fewer collisions with
impurities and can respond faster with fewer diversions. However, as with any
amplifier, HEMT based amplifiers are subject to some amount of white noise and 1/f
gain fluctuations.
White noise is random uncorrelated fluctuations in the gain of the amplifier.
This can be due to any number of causes, but there is a fundamental limit to how low
noise an amplifier can be. This quantum limit arises because the magnitude of the
electric field strength in a passing wave is a function of time and time and energy are
complimentary variables subject to quantum uncertainty. The energy carried by the
65
Chapter 6. ARCADE 2 Radiometers
incoming field to the amplifier in an infinitesimal period of time, then, is uncertain by
∆E ~
h
= h ν , and the output energy will vary within this range. At frequencies
∆T
much above 100 GHz, this limit allows too much noise, and cold amplifier based
detectors are not useful.
The white noise of a HEMT amplifier places a lower limit on the statistical
uncertainty of a radiometer measurement. As the reading of the voltage level across
the detector cannot be continuous, there is some characteristic integration time in
which the signal is effectively averaged. The integrating time allows the white noise
fluctuations to be subject to statistics, so that the uncertainty in the measured antenna
temperature due to the HEMT amplifier will be inversely proportional to the square
root of the integrating time. In fact, this uncertainty is given by the so-called
radiometer equation
∆TA ( whitenoise) =
2 Tsys
Bτ
,
where Tsys is the characteristic system noise temperature of the radiometer, the
temperature that would be measured if the radiometer were viewing internal and
external sources at absolute zero, τ is the integration time, B is the bandwidth, and the
square root of 2 is a factor due to switching, as described below. In the equation
above, the units of ∆TA(white noise) are temperature, while the white noise itself has
units of temperature per reading, or temperature per root frequency of reading. For
clarity in this work, the temperature magnitude associated with white noise
66
Chapter 6. ARCADE 2 Radiometers
fluctuations will be referred to as the ‘white noise floor’ to distinguish it from the
purely statistical white noise. The system noise temperature can also be directly
measured by extrapolating the total power output of the radiometer vs. total
temperatures of viewed sources curve to zero total viewed temperature. In this
context, it is often called the Y factor temperature.
Amplifiers have the additional issue of 1/f noise. This is where the gain
fluctuations are correlated on timescales, so the longer one waits, the more the gain
will have drifted. For a bit of context, 1/f noise turns out to be surprisingly common
in physical and biological systems, and shows up, for example, in the number of
individuals of a species inhabiting a certain area, and the fidelity with which human
subjects can duplicate a given temporal interval. In radiometers, 1/f noise would
cripple the ability to assign a reliable input power to an output voltage over a period
of time long compared to the period in which the 1/f noise begins to dominate over
the white noise.
As mentioned in the previous chapter, switching is employed in order to get
around the 1/f noise problem inherent in cold HEMT amplifier radiometers. In a
switching radiometer, the radiometer switches between observing the signal from the
source being studied and a signal from an internal reference load. A lockin amplifier
after the detector multiplies the voltage signal by +1 and –1 respectively, in phase
with the switching. In this way, the signal after the lockin has any gain fluctuations
of the amplifier that are on timescales longer than the switching period subtracted out.
67
Chapter 6. ARCADE 2 Radiometers
The signal is then integrated for a chosen period of time, such that the output voltage
is equal to some constant of proportionality times the difference in emitted power
between the source and the internal load, averaged over the integration time. If the
load is a reasonable blackbody emitter, which it should be designed to be, then the
emitted power over the frequency band of the radiometer is proportional to its
temperature, and Vout = C ⋅ TA ,SOURCE − TLOAD . The constant C is the then gain or
‘responsivity’ of the radiometer. In principle, if the load temperature is known, and
the radiometer is calibrated, then the antenna temperature of the source averaged over
the integration time is a direct measured quantity, up to systematic offsets caused by
any different emissivity of components on the load and source ends of the radiometer.
It is important to note that complete cancellation of gain fluctuations on
timescales longer than the switching period is only true in the limit where the source
antenna temperature and the load temperature are very close. If these temperatures
are farther apart, then the cancellation will not be complete and some 1/f fluctuations
can be seen, as with the off null data from the 2005 flight, as discussed in under
header V below. Also 1/fluctuations on timescales shorter than the switching will not
cancel. The 1/f properties of a radiometer are often expressed by the ‘1/f knee’,
which is the frequency at which the amplitude of fluctuations in output is 2 above
that of the white noise.
68
Chapter 6. ARCADE 2 Radiometers
II. ARCADE 2 radiometers
Fig. 6-1. Schematic of ARCADE 2 radiometer chain.
Fig. 6-1 shows a schematic of the ARCADE 2 radiometers as flown in the
2005 flight. From the antenna the incident radiation propagates through a circular-torectangular waveguide transition. The antennas and transitions are discussed in
Chapter 7. The radiation propagates in rectangular waveguide to the switch, where it
is switched with radiation from the internal load. The radiation is then transferred to
coaxial cable with a launcher, and is amplified by the cold HEMT amplifier, which
contains three HEMT transistors for three stages of amplification. In each case, the
input potential is across the gate and the source, while the output potential is across
the drain and the source. Fig. 6-2 shows a photograph of the cold stage of the
ARCADE 2 8 GHz radiometer. The fields then propagate in either coaxial cable or
69
Chapter 6. ARCADE 2 Radiometers
waveguide out of the dewar and into the warm stage. Waveguide is used for the two
30 GHz and the 90 GHz radiometer because coax is too lossy at these frequencies,
owing to the imaginary component of the dielectric constant of the dielectric between
the conductors, and the fact that the ground plane is not an ideal conductor. It is
important to note here that at GHz frequencies, wires cannot be used to transmit
signal power any appreciable distance, because due to the short wavelengths,
macroscopic twisted wire pairs do not provide phase cancellation and the signal
power would be radiated away. Hence the use of radiative transmission line and
components in all microwave frequency parts of the radiometer system.
Once at the warm stage, the radiation is then amplified again by a warm stage
HEMT amplifier, and then a band pass filter selects the desired frequency band. To
this point additional frequencies were allowed to propagate, as the amplifiers provide
some amplification to frequencies outside of the band, but as individual frequencies
propagate independently, those for which the radiometer was not designed are of little
consequence. There is then a second warm HEMT amplifier for further
amplification, and a power divider which splits the radiation and allows for a lower
and a higher frequency channel in each radiometer. The individual high and low
channels then each have a band pass filter, and then the radiation is incident on the
detector across which a voltage is generated that is proportional to the power of the
radiation incident on it. Fig. 6-3 shows a photograph of the radio components of the
warm stage of the ARCADE 2 8 GHz radiometer. Since the gain of the HEMT
70
Chapter 6. ARCADE 2 Radiometers
amplifiers are not a perfectly flat function of frequency, an amplifier tilt factor
expresses the difference in the total gain in the high and low channels once they are
split.
Fig. 6-2. Photograph of the cold stage of the 8 GHz
radiometer.
71
Chapter 6. ARCADE 2 Radiometers
Fig. 6-3. Photograph of the warm radio stage of the 8 GHz radiometer.
With the exception of the internal loads, which are described in detail under
header III, and the horns and waveguide transitions which are discussed in chapter 7,
the remaining radio frequency components are ‘off the shelf,’ meaning that they are
purchased from outside vendors rather than designed and fabricated by the ARCADE
team. The specifications for these off the shelf components of the ARCADE 2 8 GHz
radiometer as flown in 2005 are presented in Table 6-1, and, although for the
purposes of the ARCADE project they can be considered to be black boxes that work
as advertised, their workings are briefly discussed below.
The switches used in the 2005 flight of ARCADE 2 were ferrite waveguide
switches. Along with the circulators, they are microwave devices that use ferrite
phase shifting techniques. These rely on the properties of waveguide loaded with
72
Chapter 6. ARCADE 2 Radiometers
slabs of ferrite media, media which have permanent magnetic dipoles which precess.
Left handed and right handed circularly polarized radiation have different propagation
constants in ferrite materials, owing to the interaction of the circularly polarized
radiation and the precessing magnetic dipoles. Linearly polarized radiation, which
can be represented as a superposition of left-handed and right-handed circular
polarizations, then has unique propagation properties. A waveguide with a slab of
ferrite material can cause a phase shift in the radiation propagating in the guide, and
with this in mind a guide section with three ports and a slab of ferrite in the middle
can be constructed so that the phase shifts caused by the ferrite cause interference
patterns such that radiation propagates from any given port only to one other. The
direction of the dipole precession in the ferrite, and therefore the interference patterns
and the allowed ports, can be changed by applying a strong bias voltage to the ferrite,
resulting in a functional waveguide switch.
Coax launchers consist of simply a piece of coax extending into a capped
waveguide section, with the location of the coax finely tuned such that the electric
field of the TE10 propagation mode of the waveguide maximally excites the coax.
The launchers used in the ARCADE 2 radiometers reflect less than 1/1000 of the
incident power.
The microwave band pass filters used are of two types that are common in
microwave devices. Both contain several propagating sections with impedance
mismatches, with the length of the sections and frequency dependent impedances
73
Chapter 6. ARCADE 2 Radiometers
being tuned such that in sum along the length of the structure there are large
reflections for wavelengths outside of the band of interest and transmission for
wavelengths inside. Reflections from impedance mismatching are discussed in
Chapter 7 under header IV. In the type of band pass filter used for the high
frequencies, the incoming radiation propagates into a waveguide structure with metal
fins creating bottlenecks in the waveguide with different impedances than the nonbottlenecked waveguide sections. In the type used for the low frequency bands, the
incoming radiation from coax is transferred to a microstrip structure, that being a
radiation transmission line similar to coax but with a rectangular cross section and a
dielectric layer on one side between the center conducting strip and the ground plane
which concentrates the fields in the dielectric region. With different shapes of the
strip conductor along the length of the filter having different impedances, the desired
impedance mismatches and lengths can be achieved.
The power divider can be as simple as a split in the coax, however some care
must be taken to reduce reflections of incident radiation and allow reflections of
radiation coming back from the legs. In a so-called Wilkinson style power divider, the
two legs have an impedance half that of the incident coax which then gradually
increases to the standard coax impedance. A resistor is placed between the two
conducting lines of the legs so that fields from radiation that is out of phase between
the two legs conduct across the resistor and are dissipated, severely attenuating
backwards transmission from the legs.
74
Chapter 6. ARCADE 2 Radiometers
Before and after the cold the amplifier a circulator severely attenuates
reflected radiation propagating backward. A circulator, although it propagates
radiation in microstrip, works in the same way as the ferrite switches, but with the
allowed ports permanently fixed such that forward propagating radiation goes from
an input port to an output port, but backward propagating radiation from the output
port is sent to the third port which leads to a dielectric absorber. Before each band
pass filter is an attenuator which ensures that reflections from the filters are
attenuated at least twice, once coming in and once on the reflection, which render
them inconsequential.
The detector consists of diode between the center conductor of the coax and
ground, converting the sinusoidally varying potential between the center conductor
and ground plane into a half-rectified wave. The inherent capacitance in the diode
combined with a resistor create a low pass filter so that the output is a more or less
‘DC’ level compared to the GHz at which the fields are oscillating in the coax.
Because the voltage across the diode is in the quadratic regime of the diode’s
response, the DC level voltage is proportional to the power across the diodes,
averaged over some period as determined by the characteristics of the filter. Beyond
the detector, a so-called video frequency amplifier or ‘preamp’, has another low pass
filter, and then inverts the voltage level so that the DC level is now referenced to its
own inversion, rather than the ground of the coax. The signal, which is now a level
75
Chapter 6. ARCADE 2 Radiometers
proportional to the power of radiation across the diode, is then carried to the readout
electronics.
The radiometer lockin and integration circuitry is on a specially designed
lockin board, of which there are two. The lockin board performs the differencing in
phase with the switching and the integration, digitizes the final voltage level, and
places it into the data stream in the appropriate sentence of the data frame. In
addition to this demodulated or “AC” signal for both the high and low band of each
radiometer, there is also a total power or “DC” signal reported to the data stream for
each. This is the same, but without the multiplication by 1 and –1 in phase with the
switch. Some specifications for the detector diode, preamp, and lockin board are
presented in Table 6-2.
76
Chapter 6. ARCADE 2 Radiometers
Table 6-2. Specifications for ARCADE 2 8 GHz radiometer back end. Courtesy of Michele
Limon.
Specification
Detector
Diode Resposivity
Diode Voltage
Demodulation Factor
Radiometer White Noise Limit @ Diode
Units
WARM OP.
Low
High
Band
Band
COLD OP.
Low
High
Band
Band
[mV/mW] 750
750
[mV]
10.4
14
[-]
2
2
[mK/rtHz] 11.2
11.2
[nV/rtHz] 3519.5 4747.7
750
1
2
0.4
346.2
750
1.4
2
0.4
467
Preamp
Diode Video Resistor
Preamp Video Bandwidth
Video Resistor Physical Temperature
Video Resistor Johnson Noise
Preamp Op-Amp Input Noise
Preamp Gain
Total Noise @ Preamp Output
Preamp Output Voltage
Variance with Preamp Video Bandwidth
[Ohms]
[Hz]
[K]
[nV/rtHz]
[nV/rtHz]
[-]
[uV/rtHz]
[V]
[mV-rms]
7500
5300
300
11.1
4
100
35
0.1
2.5
7500
5300
300
11.1
4
100
47
0.1
3.4
Lock-in
Lock-in Input Resistor
Lock-in Integration Period
Lock-in Bandwidth
Lock-in Input Resistor Physical Temperature
Lock-in Input Johnson Noise
Lock-in Op-Amp Input Noise
[Ohms]
100000 100000 100000
[sec]
0.533 0.533 0.533
[Hz]
0.25
0.25
0.25
[K]
300
300
300
[nV/rtHz]
41
41
41
[nV/rtHz]
4
4
4
Notes:
78
7500
5300
300
11.1
4
100
352
1
25.6
7500
5300
300
11.1
4
100
475
1.4
34.6
100000
0.533
0.25
300
41
4
Chapter 6. ARCADE 2 Radiometers
III. Steelcast as emitter
III-A. Emitters, reflection, and blackbodies
The most important property of a load is that it be a good blackbody emitter at
the frequencies in question. This is so that the voltage output by the radiometer is
proportional to the time averaged difference over an integration period of the antenna
temperature of the source and the physical temperature of the load, as above. Being a
blackbody at microwave frequencies is synonymous with having a low reflectivity,
meaning that incident radiation at a given frequency is not reflected back but rather
absorbed, incrementally raising the temperature of the load and causing it to emit a
spectrum corresponding to an incrementally increased temperature. Otherwise, if
incident radiation were reflected, the observed spectrum of the load would not be a
backbody but would have some large peak at the incident radiation. Reflectivity is 1
minus the emissivity. A perfect blackbody would have an emissivity of one, so loads
should be highly emissive. The term ‘black’ is often used to refer to the reflectivity
of something that is not very reflective. Reflections are quoted in decibels, where the
( )
decibels of a ratio 1 is dB = 10log10 1 .
x
x
There exist waveguide loads which are commercially available that have a
reflectivity of less than –40 dB, and were used for ARCADE 1. These loads have a
slab of dielectric plastic near the end of a capped waveguide section. However, the
temperature of the dielectric slab is difficult to read, and subject to large
79
Chapter 6. ARCADE 2 Radiometers
nonuniformities due to the generally low thermal conductance of the dielectric plastic.
The ARCADE 2 internal loads are custom built to be very emissive and to be more
isothermal. The emissivity of a load is determined by the dielectric properties of the
material and by the geometry. In general, a higher imaginary component of the
dielectric permittivity of a material results in more attenuation of radiation. As
expressed in Jackson, the wave number k in a dielectric has real and imaginary parts
and can be expressed as k = β + i
α
. The intensity of the wave being the square of
2
the amplitude, it then falls off as e − αx where x is the distance into the material, and α
and β are related to the dielectric permittivity by β 2 −
βα =
α 2 ω2  ε 
= 2 Re
 and
4
c
 ε0 
ω2  ε 
Im 
 . The geometry is important because flat surfaces normal to the
c2
 ε0 
incident radiation are reflective, and geometries that trap radiation with many bounces
are not.
III-B. Steelcast
A relatively thermally conductive emissive substance was needed for the
ARCADE 2 loads and the target. The most widely used microwave emitters are the
commercially available Eccosorb products, which are generally substrates loaded
with iron filings, and feature castable, machineable, and foam forms. However,
Eccosorb does not stand up well to cryogenic conditions, has very different thermal
80
Chapter 6. ARCADE 2 Radiometers
contraction properties than the metals, and causes allergic reactions in many people.
With this in mind, the ARCADE team developed ‘Steelcast,’ an emitter which can be
cast to almost any shape, adheres to metal very well, has thermal expansion behavior
more similar to metals, and is more thermally conductive. Steelcast consists of a
commercially available thermally conductive black epoxy, Emerson & Cummings
Stycast 2850 FT, mixed with 20 micron powderized stainless steel either 25% or 30%
by volume, with more stainless steel powder in that range providing for slightly more
emissivity.
Fig. 6-4 shows the real and imaginary parts of the dielectric permittivity of
Steelcast as a function of frequency. In the notation here, the permittivity ε is related
to the dielectric constant εr and its real er’ and imaginary er’’ parts by
ε = ε*r ε0 = (e r '+ie r '') ⋅ ε0 , where ε0 is the permittiviyt of free space. These were
measured by preparing thin waveguide samples filled with Steelcast and measuring
the transmittance and reflectance on a vector network analyzer. The predicted values
are determined with Garnet effective media theory where in the limit of spherical
randomly distributed inclusions in a host medium, the permittivity of the composite
εeff is related to that of the host εh and the inclusion εinc by
(Wollack
ε eff − ε h
ε − εh
= f ⋅ inc
ε eff + 2ε h
ε inc + 2ε h
et al, 2006a), where f is the volume filling fraction of the inclusion medium.
For comparison, the dielectric constant of Eccosorb is roughly (4 + 0.5i ) ⋅ ε 0 . The
thermal conductivity of Steelcast was measured to be 75*T nW/mK at cryogenic
81
Chapter 6. ARCADE 2 Radiometers
temperatures. Thermal conductivity as a general concept is discussed in Appendix A.
The radiometric and thermal properties of Steelcast are discussed in greater detail in
Wollack et al, 2006a.
Permittivity [ - ]
20
er' - Measurement
er" - Measurement
er' - Theory
er'' - Theory
15
10
5
0
1
10
100
1000
Frequency [GHz]
Fig. 6-4. Real ( ε r ' ) and imaginary ( ε r ' ' ) components of the dielectric
constant of Steelcast, with 30% stainless steel powder loading. From
Wollack et al., 2006a.
The process of casting Steelcast is as follows. A mold is prepared with
Silicone RTV in the shape of the desired product. Dow Corning Silicone RTV is a
commercially available mold making material that transforms from a viscous liquid to
a rubbery solid in 24 hours when mixed with a catalyst. The liquid can be poured into
a shape, so in general a metal positive of the shape desired must be machined and the
mold made around it. Hardening Silicone RTV will not stick to metal if the metal is
brushed with a mixture of 90% mineral spirits and 10% petroleum jelly.
Stycast epoxy cures from a very thick viscous liquid to a hard solid when a
catalyst is apllied. The Steelcast is prepared by first mixing the Stycast epoxy with its
82
Chapter 6. ARCADE 2 Radiometers
CAT 24 catalyst, which takes 24 hours to cure and is thinner than the catalysts that
cure the Stycast in less time. Then the stainless steel powder is mixed in to the
required volume fraction, and the mixture is placed in a vacuum oven and evacuated
to remove the considerable amount of air bubbling that has accumulated during
mixing. The Steelcast is then poured or scooped into the mold. In general, the
Steelcast shape will be wanted as a layer on an aluminum substrate, so the metal can
be pressed into the Steelcast in the mold, and the mold will have been prepared with
this in mind. In order to ensure that the Steelcast doesn’t stick to the mold, the mold
is brushed with a commercially available mold release compound.
The powder and Stycast mixture is quite thick, and does not readily flow into
small spaces to make sharp points. To this end, if the geometry of the mold allows it,
the mixture can be cut with ~10% acetone to make it thinner. During the curing
process, which with the addition of the acetone takes one week, the acetone will outgas. The target cones, discussed in Appendix B were cast in this way. The Stycast
can also be heated prior to mixing, which reduces the thickness somewhat.
IV. ARCADE 2 internal loads
The ARCADE 2 internal loads come in three designs, all using Steelcast as
the emitter. The so called ‘buffy load’ design, used by the 8 and 10 GHz radiometers,
is the most novel. The Steelcast is cast as a layer on top of an aluminum spike that
83
Chapter 6. ARCADE 2 Radiometers
gradually fills the waveguide from front to back. The angle of the front surface and
the thickness of the Steelcast layer were chosen with microwave modeling software to
make for an adequately black load, and the aluminum provides for more
isothermality. Fig. 6-5 shows a picture. Fig. 6-6 shows the reflectivity of the 8 GHz
buffy load across a large frequency range. It is at least 99.95% black across the entire
radiometer band. Prior to casting, I epoxied Ruthenium-Oxide resistance
thermometers to the aluminum surface, to read the emitting temperature of the load,
and Fig. 6-7 shows the ‘underside’ of a load core which is channeled to allow the
thermometer wires to traverse the load. Fig. 6-8 shows the mold for casting the 8
GHz buffy load. The buffy loads are discussed in more detail in Wollack et al.,
2006b.
Fig. 6-5. Photograph of radiometric side of the 8 GHz buffy load.
84
Chapter 6. ARCADE 2 Radiometers
0
Compound Taper: Tamb=296K
Reflection [dB]
Compound Taper: Tamb=77K
HFSS: er=10+2i, L=4ao, t~0.7bo
-20
HFSS: er=10+2i, L=4ao, t~0.5bo
-40
-60
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
Frequency [f/f c ]
Fig. 6-6. Measured reflectivity of 8 GHz buffy load at room and cryogenic
temperatures, along with predicted reflectivity from simulations with HFSS, a
microwave simulation program. The frequency is shown as divided by the
cutoff frequency, which for WR112 waveguide is 5.263 GHz. From Wollack
et al., 2006b.
Fig. 6-7. Photograph of underside of 10 GHz buffy load, showing thermometer
wires.
85
Chapter 6. ARCADE 2 Radiometers
Fig. 6-8. Photograph of mold for 8 GHz buffy load. The aluminum core
screws to the plate, the Steelcast is spooned into the mold, and the core is
pushed against the Steelcast.
The loads for the two 30 and the 90 GHz radiometers are of a ‘split block’
design, where wedges of Steelcast run parallel to the broadwall envelope of the
waveguide, as in Fig. 6-9. The loads for 30 GHz are is at least –40 dB (99.99%)
black across the entire band. Here as well there are thermometers epoxied an
aluminum surface over which the Steelcast was cast. There were two loads made for
each radiometer, and I fabricated all of the buffy and split block loads.
I fabricated buffy loads for 3 and 5 GHz as well, but upon measurement they
turned out to be too reflective. This puzzling result most likely indicates a problem
with the modeling. Instead coaxial loads are used, in which a coax termination was
painted with Steelcast with a thermometer embedded. These have the disadvantage
that they will be less thermally stable. The buffy loads in their waveguide, the split
block loads surrounding theirs, and the coax loads were all encased in sheaths of
86
Chapter 6. ARCADE 2 Radiometers
insulating foam, surrounded by a sheet metal casing with resistance heaters on it. The
waveguide loads have stainless steel waveguide sections adjacent to the load sections.
In this way, the temperatures of the loads could be controlled and heating would not
set up large thermal gradients.
Fig. 6-9. Cross section schematic of 30 GHz split block load. From Wollack
et al., 2006b.
V. 2005 flight 8 GHz radiometer performance
As discussed in Chapter 8, the useful 2005 flight 8 GHz data is in two distinct
chunks of records. The first, consisting of 625 records at the lowest AC lockin gain
setting of 01 with the instrument near null, and the second consisting of 6372 records
at a higher lockin gain of 16 with the sky over 1 K warmer than the internal load. For
shorthand, these are referred to as the ‘null’ and ‘high gain’ sets. The null set,
87
Chapter 6. ARCADE 2 Radiometers
therefore, has a lower signal to noise but a much better 1/f knee than the high gain set.
The differing usefulness of the two chunks for data analysis is discussed in Chapter 8,
while the flight radiometer performance is specified here.
Fig. 6-10 shows the ‘amplitude spectrum’ - meaning the data in the frequency
domain - of the 8 GHz low channel AC lockin output for the high gain data, while
Fig. 6-11 shows the same for the 8 GHz high channel. These figures clearly show the
1/f structure of the data with the Galactic crossings and harmonics of it overlayed.
Figs. 6-12 and 6-13 show the amplitude spectrum for the null data for the 8 GHz low
channel and high channel respectively. The performance properties obtained from
these spectra are summarized in Table 6-3. The Tsys reported is calculated from the
observed white noise floor with the radiometer equation described under header I
above. Given the measured Y factor temperature and the system noise temperatures
inferred from the white noise, it is clear that for the null data, noise from the lockin
amplifier dominates the total noise, while for the high gain data, it is noise from the
cold HEMT amplifier that dominates.
88
Chapter 6. ARCADE 2 Radiometers
Fig. 6-10. Amplitude spectrum of the 8 GHz low channel AC lockin output
for the high gain data. The Galactic crossings and harmonics of it are
overlayed on the 1/f structure.
Fig. 6-11. Amplitude spectrum of the 8 GHz high channel AC lockin output
for the high gain data. The Galactic crossings and harmonics of it are
overlayed on the 1/f structure.
89
Chapter 6. ARCADE 2 Radiometers
Fig. 6-12. Amplitude spectrum of the 8 GHz low channel AC lockin output
for the null data.
Fig. 6-13. Amplitude spectrum of the 8 GHz high channel AC lockin output for
the null data.
90
Chapter 6. ARCADE 2 Radiometers
Table 6-3. 8 GHz radiometer 2005 flight performance properties
8 GHz low channel
8 GHz high channel
Band
7.8 GHz - 8.15 GHz
8.15 GHz – 8.5 GHz
Bandwidth (MHz)
350
350
Integration time (sec)
0.533
0.533
Lockin offset (counts)
-77.5
-102.1
DC lockin offset
-78.8
-107.6
Null data
High Gain data Null data
High Gain data
System gain (counts / K)
93.4
1487.3
74.7
1189.3
White noise floor (mK)
12.4
2.9
12.7
2.9
White noise (mK / rt Hz)
9.1
2.1
9.3
2.1
calculated Tsys (K)
86.9
20.3
89.8
20.3
1/f knee (mHz)
22
350
18
400
DC gain (counts / K)
32.7
not measurable
34.2
not measurable
measured Y factor temp (K)
22
not measurable
21
not measurable
Notes: The system gain is determined from the coupling to the internal load, as discussed in
Chapters 8 and 9. The lockin offset is reported as determined in Chapter 9. The DC gain is also
determined from the coupling to the internal load. The DC lockin offset is determined from raw
data early in the flight that featured the cold amplifier off. The DC gain and Y factor
temperature are not measurable in the high gain data because the internal load temperature was
too constant. The white noise is determined from the amplitude spectra of the data, while the
white noise floor reported is the root of the Gaussian variance of the data in temperature units.
91
CHAPTER 7: ARCADE 2 FEED HORN ANTENNAS
I. Corrugated feed horn antennas
Corrugated feed horns are often useful for precision applications at microwave
frequencies because, as opposed to simple dipole or non-corrugated horn antennas,
they feature symmetrical beam patterns, where the received intensity from a source is
a function only of the angle from the axis of the horn of the source, and low sidelobe
response, meaning that the received power from angles outside of the ‘main beam’
where the response is locally high is significantly suppressed. This is ideally suited to
astronomical radiometer applications such as the ARCADE 2 experiment, where it is
important for the horns to have a low sidelobe response in order to minimize
radiometric pick up from undesired portions of the sky, the instrument, and the Earth.
An important concept in antenna design is antenna reciprocity. In all
antennas, the received electric field magnitude and phase distribution in space is
exactly the same as the broadcasted electric field magnitude and phase. Therefore, in
designing an antenna, one can treat the problem as either a broadcast or reception
problem and the beam pattern is identical. A way to characterize the beam pattern is
v
v
v
v
v
Ik
v . I k is the intensity
by the gain G k in any given direction k , where G k =
I iso k
()
()
() ()
()
v
v
of the antenna in the direction k , while I iso k is the intensity of a hypothetical
()
92
Chapter 7. ARCADE 2 Feed Horn Antennas
antenna having equal radiation intensity in all directions. The intensity is the power
v P
received or radiated in an infinitesimal solid angle element, so I iso k = total . The
4π
()
v
v
v
relative gain in any direction k is the ratio of G k to the peak G k
()
v
gain G k
()
max
()
max
. The peak
is often used as a figure of merit for horn antennas, and is called the
directivity in this case.
Horn antennas, as the name implies, are those that are shaped such that
broadcast radiation propagates within conducting walls from a narrow throat to a
wider aperture. It is plainly intuitive that horns with cylindrical symmetry will
provide the most azimuthally symmetric beam patterns, which are very desirable for a
radiometer application in which the antenna may be rotating. In a cylindrically
symmetric horn, a boresight axis can be defined, which is the axis of cylindrical
symmetry. The beam pattern will be a function of angle from boresight. If the horn
is conical, then a flare angle can be defined, as the angle formed by the wall and the
boresight axis.
Corrugated horns, often known as ‘feed horns,’ contain a regular pattern of
grooves and teeth on the interior surface, as shown in Fig. 7-1. Generally, at the
boundary of a conductor, the normal component of the electric field is zero. The
grooves and teeth serve to cancel the tangential component of the electric field as
well, leading to an electric field strength that is zero at the wall and maximum along
the axis of the horn. In fact, the electric field strength distribution at the aperture is
93
Chapter 7. ARCADE 2 Feed Horn Antennas
roughly a gaussian function of angle from the center. This gives a far-field electric
field strength distribution that is also close to a gaussian function of the angle from
boresight, as the far field distribution is essentially a fourier transformation of the
aperture distribution, as given in classical electrodynamics,
r
E( x ) ∝
∫e
rr
i k⋅ r
aperture
r
n̂ ⋅ R ' r
E( x ' ) da ' , where n̂ is the unit normal to the horn aperture and
R'2
r
R ' is the vector from the point of integration to the observation point. The power is
proportional to the magnitude squared of the electric field intensity, and so the farfield beam pattern is also roughly a gaussian.
Fig. 7-1. Profile of ARCADE 2 30 GHz horn.
A typical feed horn will have a beam pattern where the received power is
overwhelmingly dominated by angles near boresight, known as the main lobe, with
small regions at farther angles where the received power is locally high, known as the
side lobes. The side lobes in feed horns result from diffraction effects of the electric
field around the aperture. As stated previously, one of the desirable properties of feed
94
Chapter 7. ARCADE 2 Feed Horn Antennas
horns is low sidelobe response. The design challenge is then in achieving the desired
beam pattern in the main lobe, and the relative terms ‘wide’ and ‘narrow’ to describe
the beam refer to the main lobe. The standard measure of the width of a beam is the
full width at half power, which is twice the angle from boresight where the received
power is half that of boresight.
In this diffraction limited system, for a given finite length horn, a wider
aperture will result in a narrower beam. This stems from the fact that the aperture
will not be a surface of exactly constant phase in the electric field distribution, as the
propagating radiation will try to assume a more spherical wavefront. So, in the
integral above to transform to the far field electric field distribution, this phase error
among nearby points in the plane of the aperture will serve to wash out to some extent
the peaked nature of the field strength distribution at the aperture about boresight. A
wider aperture minimizes the phase error among nearby points, and thus results in a
narrower far-field beam.
Heuristically, corrugated feeds cancel the tangential component of the electric
field at the wall by setting up surface currents in the wall which make the impedance
at the wall infinite at a wavelength λx in the HE11 propagation mode. λx is called the
hybrid mode wavelength, and can be tuned by varying the depths of the grooves along
the length of the horn. The HE11 mode is the waveguide mode in which the radiation
of the frequency band the horn was designed for propagates in the horn. The Hybrid
Electric (HE) and Hybrid Magnetic (HM) modes are linear superpositions of TE and
95
Chapter 7. ARCADE 2 Feed Horn Antennas
TM modes which comprise a complete basis for waveguide propagation that is
convenient for describing the propagation in structures such as feeds.
In describing a beam pattern, one specifies how the received intensity varies
as a function of angle in two orthogonal angular directions. If the antenna in
broadcast mode is ultimately fed by radiation from a rectangular waveguide, as in
ARCADE 2, the two angular directions customarily used are the E-plane and the H-
plane. The E-plane beam pattern is with the angle measuring rotation from boresight
in the direction of the short dimension of the rectangular waveguide, while the Hplane beam pattern is with the angle measuring rotation from boresight in the
direction of the long dimension of the rectangular waveguide. The names arise from
the directions of the electric and magnetic field vectors of the propagating radiation in
the TE10 waveguide mode.
With a known beam pattern, the antenna temperature of objects that do not fill
the entire beam can be calculated by convolving the temperature of the object with
the beam pattern. In practice, this usually involves a finite element breakup.
Besides beam pattern, another important radiometric design consideration is
return loss, the extent to which reflections of power broadcast into the antenna are
attenuated. A severe attenuation of reflections can be achieved by transitioning from
a groove depth of roughly a half wavelength at the throat to a groove depth of roughly
a quarter wavelength over the first few tooth and groove cycles. This so-called mode
converter converts the TE11 circular mode of propagation in that enters the throat to
96
Chapter 7. ARCADE 2 Feed Horn Antennas
the HE11 mode in which the radiation propagates in the feed. Mode converters are
analyzed in James, 1981. Beyond the mode converter is the flare section, where the
groove depth is roughly a quarter wavelength as described above. For ease of
manufacturing, all of the grooves in a given ARCADE 2 horn have the same width.
If required, the technique of Zhang (Zhang, 1993) can be used to achieve a low return
loss across a wider band by varying the groove width in the first few sections.
II. ARCADE 2 feed horns
As discussed in chapter 5, ARCADE 2 imposed constraints on the horn
design, resulting in the necessity of having sliced horns, and horns with a curved
profile and varying groove depth in the flare section at the low frequencies. The
horns point 30º from zenith to minimize pick-up of emission from the balloon, and to
view the target at an oblique angle, which further reduces reflections from it.
However, it is necessary to have a flat horizontal aperture plane so the antennas must
be sliced at a 30º angle across the antenna aperture. The sliced aperture horns were
sufficiently novel as to merit a publication in Review of Scientific Instruments
(Singal et al., 2005).
In addition to low sidelobe response, the experiment requires beams in which
the main lobe is sufficiently narrow, in order to resolve pointing on the sky and get a
handle on emission from the Galaxy. For ease of analysis and general simplicity, we
97
Chapter 7. ARCADE 2 Feed Horn Antennas
wanted the seven horns to have very identical beam patterns. The maximum length
and aperture size for the largest horn, at 3.3 GHz, is set by the geometry of the
instrument, as the aperture must fit onto an ellipse that fits onto a third of the aperture
plane, and the length must not allow the horn to protrude beyond the radius of the
dewar wall or lower than the dewar floor. We also needed the horns to feature a
return loss of at least –30 dB across the entire band.
As the ARCADE 2 radiometers use standard rectangular waveguide sizes
appropriate to the frequency band in question, the horns propagate radiation from a
rectangular guide to free space as follows (see Fig. 7-2): The radiation transitions in a
rectangular-to-circular waveguide transition from the TE10 mode in rectangular
waveguide to the TE11 mode in a circular waveguide with cutoff frequency equal to
that of the rectangular guide. This allows for the use of a compact stepped
rectangular-to-circular waveguide transition with homogeneous wave propagation in
each section, as discussed under heading V. The radius of the throat of the horn is
thus fixed at the radius of this circular waveguide.
98
Chapter 7. ARCADE 2 Feed Horn Antennas
Fig. 7-2. Schematic of waveguide/transition/horn system.
The design strategy was as follows, then. I designed a 3 GHz horn to achieve
the narrowest beam and highest return loss given the physical size constraints, then
designed horns at the other frequency bands to have close to identical beam patterns
to the 3 GHz horn. The exception is the 30# horn, which was supposed to have much
narrower beam. I found it necessary to use a profiled rather than strictly conical horn
shape at 3 GHz, and to vary the groove depth along the length of the horn, and
continued this shape through the 10 GHz horn. However, for simplicity of
manufacture I switched to a strictly conical profile for the three high frequency horns,
so the ARCADE 2 horns are of two basic designs, profiled for the four low frequency
band horns, and conical for the three high frequency band horns.
The adjustable parameters for each horn, then, were the width of one tooth
and groove section (the ‘cycle width’ U), the with of a groove W, the number of tooth
and groove sections n, the number of transition grooves in the throat as described
99
Chapter 7. ARCADE 2 Feed Horn Antennas
above m, and the value for the hybrid mode wavelength λx. The profiled horns had as
adjustable parameters the profile factor A, and a ‘transition factor’ p which appears in
the formula for the groove depth in the first few sections as discussed below, and the
conical horns had an adjustable parameter in the flare angle α.
I simulated the radiometric performance of the horns under design by
modeling unsliced horns using mode matching software, CCHORN, based on the
work of G.L. James (James, 1982). The code treats each tooth or groove section as a
separate piece of circular waveguide, solves for the propagation modes available, and
matches the boundary conditions with the previous section to solve for the power in
each mode. Upon reaching the aperture, the modes are summed to give a total
electric field strength and phase distribution, which is then Fourier transformed to
give the far field electric strength and phase distribution, from which a power
distribution is obtained, as described above. The mode matching software also
calculated an overall return loss.
The major effect of the 30˚ slice on the beam is to reduce the effective
aperture radius and overall length to that corresponding to an unsliced horn with an
aperture at roughly the middle of the slice. Thus one can predict the behavior of the
sliced horns under design by modeling them as shorter, unsliced horns.
The dominant observable effect of the slice on the beam pattern is the
asymmetry in the H-plane response (see Fig. 7-4b), in which the first sidelobe on the
side with the slice is suppressed and the first sidelobe level on the opposite side is
100
Chapter 7. ARCADE 2 Feed Horn Antennas
increased, both by a few dB. This asymmetry arises from diffraction effects. In an
unsliced horn, or in the E-plane of the horns in question, the horn aperture is
symmetric about the boresight axis, thus in a corrugated feed horn with a symmetrical
beam diffraction around the aperture and the effects of this diffraction in the far field
are symmetric about this axis. However, in the H-plane of the sliced horns in
question, the aperture of the horn is not symmetric about the boresight axis, and
diffraction around the aperture and the far field effects of this diffraction are
correspondingly asymmetric about this axis.
The effect on the main lobe is shown to be negligible, and there are no
peculiarities in the beam at or near 30º from boresight, which corresponds to the
zenith when the antennas are mounted in the instrument. Thus, for analysis, the
ARCADE 2 horns can be treated as having cylindrically symmetric beam patterns.
III. ARCADE 2 low band horns
In order to achieve an acceptably narrow beam given the aperture size and
horn length constraints on the 3.3 GHz horn, I used a tapered rather than linear
profile. The profile of the horn is given by
z


r = rt + (rapt − rt ) (1 − A ) + A sin 2  π z 
2
L


L

with r being the radius corresponding to a distance z along the axis from the throat, L
being the total length, rt being the radius of the throat, rapt being the radius of the
101
Chapter 7. ARCADE 2 Feed Horn Antennas
aperture, and A being a profile factor. The aperture radius rapt of the 3 GHz horn was
set by the maximum allowable size, and for the other profiled horns was scaled
directly from the 3 GHz by the ratio of wavelengths. Fig. 7-3 shows the profile of the
10 GHz horn as designed.
Fig. 7-3. Profile of the 10 GHz horn as designed. The extent of the modeled
unsliced horn is shown, as are directions of the E- and H- plane cuts.
I also set the groove depth along the length of the horn in the flare section to
vary according to
λx
λ
D r = x ⋅ e 5π r .
4
The groove depth is varied in order to allow the HE11 mode to adiabatically propagate
along the length of the horn. These techniques are suggested in Olver et al. (Olver,
1994)
In the mode converter region, the groove depth of the ith section is given by
λ
 m − i + 1 
D i = D r +  x − D r 

 2
 m 
p
102
Chapter 7. ARCADE 2 Feed Horn Antennas
where m is the number of groove-tooth sections in the transition and p is a ‘transition
factor.’
I determined through the use of a Powell optimization routine that at 3.3 GHz,
given the constraints imposed on the overall length and aperture size of the horn by
the instrument geometry, the best values for A in order to achieve the narrowest full
width half power, and for p and m, in order to achieve maximal return loss, were
0.839, 1.843, and 5 respectively. For simplicity I applied this to all four horns. The
remaining parameters for the other horns were chosen by progressive honing so that
the beam patterns were as close as possible to that of the 3 GHz horn. Table 7-1 lists
the values for the adjustable parameters for the four low frequency horns. The horns
have a 12º full width half power.
The exterior profiles of the horns were chosen to minimize the overall mass.
They are machined in sections out of aluminum, with the 3 GHz horn containing
three stainless steel sections to reduce thermal conductivity. Table 7-2 reports the
overall scale of the low band horns. The total length reported is from the throat to the
center of the aperture slice. The slice renders the aperture shape itself as
approximately an ellipse, and the major and minor axes of that ellipse are reported.
103
Chapter 7. ARCADE 2 Feed Horn Antennas
Table 7-1: Parameters for ARCADE 2 low band horns.
Channel
Standard
Hybrid Mode
Cycle Width
Center
Waveguide
‘U’
Wavelength ‘λx’
Frequency
Used
(cm)
(cm)
3.3 GHz
WR284
9.3112
2.665
5.6 GHz
WR187
5.3920
1.5432
7.8 GHz
WR112
3.6724
1.0510
10.2 GHz
WR90
3.0392
0.8698
Profile factor (A)=0.839
Transition factor (p)=1.843
Number of Transition grooves (m)=5
Groove
Width ‘W’
(cm)
2.3575
1.2357
0.9298
0.7694
Number of
Sections
56
48
59
59
Table 7-2: Specifications for ARCADE 2 low band horns.
Channel Center
Frequency
3.3 GHz
5.6 GHz
7.8 GHz
10.2 GHz
Standard
Waveguide
Used
WR284
WR187
WR112
WR90
Total length
(cm)
Total mass
(kg)
149
73
62
51
41
11
4
2.5
Aperture
major axis
(cm)
67.6
40.5
27.3
22.6
Aperture
minor axis
(cm)
58.7
35.6
23.7
19.6
The far field beam pattern of the 10 GHz horn was measured at the GEMAC
(Goddard ElectroMagnetic Anechoic Chamber) compact range. Figs. 7-4a and 7-4b
present the predicted and measured E and H plane beam patterns for the 10 GHz horn.
In the measurement setup used, the E and H plane cuts are aligned as shown in Fig. 73, with the plane of the aperture slice making a 30º angle with the E plane. The effect
of the sliced aperture is visible in the H plane cut. Fig. 7-4c shows the cross polar
response, and Fig. 7-5 shows the combined return loss of the horn and circular-torectangular waveguide transition combination. The measured data for the 10 GHz
horn indicates that the modeling during the design phase is correct. The other low
band horns should thus have actual beam patterns that are nearly identical to that of
the 10.2 GHz horn, and nearly identical to their own simulated beam patterns, as well
104
Chapter 7. ARCADE 2 Feed Horn Antennas
as the reflection loss as predicted. The predicted E and H plane beam patters for the 8
GHz horn are shown in Fig. 7-6a and 7-6b, and the predicted return loss is shown in
Fig. 7-7.
Fig. 7-4a. Predicted and measured E-plane beam pattern for the 10 GHz horn at
10.11 GHz.
105
Chapter 7. ARCADE 2 Feed Horn Antennas
Fig. 7-4b. Predicted and measured H-plane beam pattern for the 10 GHz horn
at 10.11 GHz.
Fig. 7-4c. Predicted and measured cross polar response for the 10 GHz horn at
10.11 GHz. The measured response is near or below the noise floor of the
measurement at most angles.
106
Chapter 7. ARCADE 2 Feed Horn Antennas
Fig. 7-5. Predicted return loss for the 10 GHz horn and measured return
loss for the rectangular-to-circular waveguide transition and 10 GHz
horn combination.
Fig. 7-6a. Predicted E- and H-plane beam patterns for the 8 GHz horn. Shown
are the beam patterns for the lower radiometer band edge (7.8 GHz), the
middle of the band (8.15 GHz), and the high band edge (8.5 GHz). The
predicted beam patterns are symmetric about boresight and for resolution only
angles less than 50° are shown. The full width at half power is 11.6°.
107
Chapter 7. ARCADE 2 Feed Horn Antennas
Fig. 7-6b. Predicted E- and H-plane beam patterns for the 8 GHz horn, close up
of main lobe.
Fig. 7-7. Predicted return loss for the 8 GHz horn.
Figs. 7-8 and 7-9 are photographs of the 10 GHz and 3 GHz horns, respectively.
108
Chapter 7. ARCADE 2 Feed Horn Antennas
Fig. 7-8. Photograph of 10.2 GHz horn.
Fig. 7-9. Photograph of 3 GHz horn.
109
Chapter 7. ARCADE 2 Feed Horn Antennas
IV. ARCADE 2 high band horns
As stated previously, for simplicity and cost reduction in manufacturing, I
chose to make the three high frequency horns conical, and without a varying groove
depth in the flare section. The shape of the horn is given by
r = rt + z ⋅ tan(α) , where r is the radius at a given length z down the axis of the horn, rt
is the radius of the throat, and α is the flare angle. The groove depth in the flare
section is simply D r =
λx
, and the groove depth in the transition section in the throat
4
  m − i + 1 
is D i = D r 1 + 
 .
  m 
The parameters for the two non-narrow horns were chosen by progressive
honing so that the beam patterns were as close as possible to that of the 3 GHz horn,
while the parameters for the 30# horn were chosen to give a very narrow 4º full width
half power beam. Table 7-3 lists the values for the adjustable parameters for the three
high frequency horns.
The two 30 GHz horns are machined in sections out of aluminum, with the
30# horn containing a stainless steel section to reduce thermal conductivity, while the
90 GHz horn, in which the groove dimensions are too small for machine tools to
provide squared corners, was electroformed in copper. Table 7-4 reports the overall
110
Chapter 7. ARCADE 2 Feed Horn Antennas
scale of the high band horns, with the measures described as in the low band case
above.
Table 7-3: Parameters for ARCADE 2 high band horns.
Channel
Standard
Hybrid
Cycle
Center
Waveguide
Mode
Width ‘U’
Frequency
Used
Wavelength
(cm)
‘λx’ (cm)
30 GHz
WR28
.2617
.2755
30# GHz
WR28
.2617
.2755
90.1 GHz
WR10
.08635
.0909
Number of Transition grooves (m)=5
Groove
Width ‘W’
(cm)
Flare
angle
‘α’
Number of
sections
.1378
.1378
.0454
7
5.4
9
67
409
67
Table 7-4: Specifications for ARCADE 2 high band horns.
Channel
Center
Frequency
30 GHz
30# GHz
(narrow)
90 GHz
Standard
Waveguide
Used
WR28
WR28
WR10
Total
length
(cm)
18
113
Total mass
(kg)
.23
3.5
Aperture
major axis
(cm)
8.2
26
Aperture
minor axis
(cm)
7.2
23
6
.1
2.7
2.2
Fig. 7-10 show the modeled far-field beam patterns for the high band horns. Figs. 711, 7-12, and 7-13 are photographs of the 30 GHz, 30#, and 90 GHz horns,
respectively.
111
Chapter 7. ARCADE 2 Feed Horn Antennas
Fig. 7-10. Predicted E-plane beam patterns for the 30, 30# and 90 GHz horns.
Shown are the beam patterns for the center band frequencies.
Fig. 7-11. Photograph of 30 GHz horn.
112
Chapter 7. ARCADE 2 Feed Horn Antennas
Fig. 7-12. Photograph of 30# (narrow beam) horn.
Fig. 7-13. Photograph of 90 GHz horn.
113
Chapter 7. ARCADE 2 Feed Horn Antennas
IV. ARCADE 2 waveguide transitions
I did not electromagnetically design the circular-to-rectangular transitions, but
I did specify them for machinists to make. The purpose of the transitions, as stated
above, is to convert radiation from propagating in waveguide of circular cross section,
as it does at the throat of the horn, to waveguide of rectangular cross section, as is
used in the cold stage of the radiometers. The goal is to create a transition with a high
return loss, so that reflections of radiation propagating into the rectangular end are
severely attenuated. The ARCADE 2 transitions use a step design, where there are
four sections of different cross sectional shape that progress from a circle to a
rectangle. Fig 7-14 shows a photograph of the ARCADE 2 3 GHz transition, where,
because of the size, the steps are most readily visible in a photograph.
114
Chapter 7. ARCADE 2 Feed Horn Antennas
Fig. 7-14. Photograph of 3 GHz (WR 284) waveguide transition.
The strategy for designing these transitions was as follows. At the boundary
of each of the steps where the cross section changes, there will be a transmitted and
reflected component of incident radiation. The reflection coefficient is
Γi =
Z i −1 − Z i
, where Zi is the impedance of the ith section. The impedance and
Z i −1 + Z i
length of each step section can be chosen so that the amplitude and phase respectively
of the combined reflections cancel as much as possible at the boundary between the
end rectangular and adjacent section, resulting in a very low total reflection at the
rectangular end. The impedance of a section is a function of the shape, as the
impedance is Z =
µ b
where λg is the guide wavelength as below, and b is the
ε λg
short cross-sectional dimension of the waveguide. The shape of the sections in
115
Chapter 7. ARCADE 2 Feed Horn Antennas
stepping from a circular cross section to a rectangular one were then chosen within
the impedance constraints so that the guide wavelength for the center band frequency
was the same in each section. The guide wavelength λg is related to the cutoff
wavelength λc by
λg
λ0
=
1
λ
1 −  0 
 λc 
2
, where the cutoff wavelength is a function of
the long cross-sectional dimension of the waveguide. Thus, in principle, shapes and
lengths can be determined. Keeping the same guide wavelength and therefore cutoff
wavelength in each section simplifies the problem and allows for a compact
transition. The transitions are electroformed out of copper. Fig. 7-5 above shows the
measured return loss of the 10 GHz transition and horn combination. It should be
noted that circular-to-rectangular transitions are polarization selecting, that they only
transmit one linear polarization from the circular to rectangular guide, and reject the
other.
116
CHAPTER 8: DATA AND DATA ANALYSIS
I. The ARCADE 2005 flight
After a very rushed preparation in the spring of 2005, first at NASA Goddard
and then at the National Scientific Balloon Facility (NSBF) in Palestine, TX, since
renamed the Columbia Scientific Balloon Facility (CSBF), the ARCADE team
declared the ARCADE 2 payload flight ready in mid-July. On July 22, the payload
was picked up for launch by the NSBF launch vehicle, however a wiring failure in the
NSBF supplied rigging harness caused the firing of the termination package and
separated the balloon from the payload upon launch. After this failure and an internal
NASA investigation, the payload was successfully launched one week later at 1:42
UT on July 29, 2005 (8:42 local time on July 28). Fig 8-1 shows a photograph of this
launch.
The instrument lid was opened for observation at 5:30 UT (0:30 LT) at a float
altitude of 36 km. At this point, a motor gear failure prevented any movement of the
lazy susan structure, freezing the instrument configuration in a position such that the
3 GHz horn was largely viewing the external calibrator target, the 8 GHz horn was
viewing the sky, the 10 GHz, 30 GHz, and 90 GHz horns were completely obscured,
and the 5 GHz horn was partially obscured. Thus, meaningful sky data was only
117
Chapter 8. Data and Data Analysis
available in the 8 GHz channel, and this dissertation presents only results at that
frequency.
Fig. 8-1. Photograph of July 2005 launch of ARCADE 2. The payload is
suspended from the launch vehicle while the balloon rises.
After almost a nine hour flight time, the payload was cut down by NSBF and
landed upright at 10:21 UT (5:21 LT) in West Central Texas, as seen in Fig. 8-2..
118
Chapter 8. Data and Data Analysis
The only major damage upon impact was to some of the external frame members, and
to the 3 GHz waveguide switch.
The failure mode for the lazy susan motor in the 2005 flight was
straightforward to reconstruct. It seems that nitrogen ice accumulated between the
lazy susan and aperture plane, causing some amount of resistance to turning the lazy
susan, at which point the torque output by the motor that turns the lazy susan
exceeded the torque that its gearbox could handle and stripped the gearbox. In
retrospect, we should have specified a gearbox that could handle a torque beyond that
of the maximum stall torque of the motor. This was done for the 2006 flight and the
lazy susan was moved successfully throughout that flight.
Additonally, in the 2005 flight we were unable to cool the target to below 4 K,
indicating that there was an unacceptable a heat leak to the target. A brief summary
of the analysis of this failure that I carried out is presented in Appendix B. Additional
problems included an unacceptably high failure rate of cryogenic thermometers,
which was due to the hurried nature of the flight preparation, and unacceptably high
loss in the 3 and 5 GHz waveguide switches. For the 2006 flight, the 3 and 5 GHz
radiometers were rebuilt to have coax loads and switches.
119
Chapter 8. Data and Data Analysis
II. The data and data reduction
The ARCADE 2 payload data records were transmitted via NSBF telemetry to
the ground and recorded. An on-board data recorder was present but failed. The
ARCADE 2 data frame consists of 36 channel number ‘sentences’ each containing 16
‘words’ reported in the data stream in sequence., as described in Chapter 5 under
header IV-B. Each populated word is the report from one of the data boards. Tables
8-1 and 8-2 summarize the data frame from the 2005 flight. Longitude, latitude, and
altitude were determined throughout the flight with global positioning systems
provided by CSBF and later given to the ARCADE team for data analysis.
Fig. 8-2. Photograph of ARCADE 2 at 2005 landing site.
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Chapter 8. Data and Data Analysis
Table 8-1. Telemetry data frame from ARCADE 2 2005 flight, words 0-8. Words 4, 6, and 9
had no boards reporting. Channels 1-12 of the word 3 are inputs from the AD590 temperature
transducers, which are the thermometers for the warm parts of the instrument.
Channel
(sentence)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Word 0
Digital
LU=1
Lockin board 1
LU=2
Lockin board 2
cmd echo
3L-AC
3L-DC
3H-AC
3H-DC
5L-AC
5L-DC
5H-AC
5H-DC
8L-AC
8L-DC
8H-AC
8H-DC
10L-AC
10L-DC
10H-AC
10H-DC
3L-AC
3L-DC
3H-AC
3H-DC
5L-AC
5L-DC
5H-AC
5H-DC
8L-AC
8L-DC
8H-AC
8H-DC
10L-AC
10L-DC
10H-AC
10H-DC
30L-AC
30L-DC
30H-AC
30H-DC
30#L-AC
30#L-DC
30#H-AC
30#H-DC
90L-AC
90L-DC
90H-AC
90H-DC
Frame #
cmd echo
cmd echo
cmd echo
LU=3
Analog input
LU=7
Voltage out
(Heaters)
Susan_Mtr.
Lid_Mtr.
Camera
Magnetometers
Boost-I
Boost-x
Video_Tx1
Video_Tx2
Lithium_Bats
Lead-Acid_Bat
Electronx_Box
Rotor_Mtr
X Clinometer
Y Clinometer
Camera
Magnetometers
Rotor Mtr.
Susan 2
Lid Mtr.
Susan 1
Boost-x
Electronx Box
Video 1
Video 2
Lithium Bats
Lead-Acid Bats
Susan Angle
Mag-X (Hon)
Mag-Y (Hon)
Mag-Z (Hon)
Mag-X (APS)
Mag-Y (APS)
Mag-Z (APS)
LU=8
SPID
settings
Flares
Horn 3A
Pump3
Load3
Pump5
Load5
Pump8
Load8
PumpAp1
Pump10
Load10
Horn10
Pump30
Load30
Horn30
Pump#
PumpAp2
PumpTgt1
Target 1
Horn90
Horn#
Horn5
Horn8
Load#
Boiloff
PumpTgt2
Target 2
Horn3B
Pump90
Load90
Pump Fl.
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Chapter 8. Data and Data Analysis
In total, there are nearly four hours of flight observations with the instrument
lid open, corresponding to 13770 records. The first half of this time was occupied
with unsuccessful attempts to move the lazy susan, including a heating of the aperture
plane to near 100 K and a subsequent slow cooling. With cryogenic temperatures
then restored, there are 550 records with the lockin amplifier gain set to the lowest
setting and the internal load temperature controlled so that the radiometer is near null.
A null condition is when the radiometer output is zero counts, corresponding to no
temperature difference between the internal load and the object being viewed.
Following that, the 8 GHz lockin gain was set to 16 times higher, and we
simultaneously lost the ability to control the internal load temperature, with it
dropping to the liquid helium bath temperature of ~1.5 K. There then follow more
than 6000 records of this high gain data with the radiometer far from null. After this,
the lid was closed and the flight was terminated at 10:10 UT. Fig. 8-3 shows the 8
GHz low channel radiometer output for records in which the lid was open.
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Chapter 8. Data and Data Analysis
Fig. 8-3. Time ordered 8 GHz low channel AC lockin output following the
opening of the instrument lid. The spatial structure of the Galaxy is visible in
the high gain data, while the structure in the low gain data results from
changing temperatures within the system, primarily that of the internal load.
The null data, from records 6771-7396, is useful because it is in the regime
where the radiometer is insensitive to 1/f drifts in amplifier gain. However, because
this data is at low lockin gain, noise contributions from the lockin amplifier dominate
the total noise resulting in a relatively poor signal-to-noise ratio. Conversely, the
high gain data, from records 7397-13769 is useful in that the spatial variation on the
sky is very prominent, however, since the radiometer is far off of null, it shows large
1/f gain drifts. A summary of the data used for analysis is presented in Table 8-3.
Figs. 8-4 shows some of the important and potentially important instrument
temperatures during the null data period.
124
Chapter 8. Data and Data Analysis
Table 8-3. Summary of data used for analysis
Null data
High gain data
System gain, low channel (counts / K)
93.4
1487.3
System gain, high channel (counts /K)
74.7
1189.3
Internal load temperature (K)
2.473 to 2.718
1.446 to 1.460
White noise, low channel (mK / rt Hz)
9.0
2.1
White noise, high channel (mK / rt Hz)
9.3
2.1
1/f knee, low channel (mHz)
22
350
1/f knee, high channel (mHz)
18
400
Total records
625
6372
Number of excised records
29
700
Notes: The system gains are determined from the least squares fitting procedure
discussed in this and the following chapter.
Fig. 8-4. Selected instrument temperatures during and shortly after the null
period. Shown are the 8 GHz internal load (blue), the 8 GHz horn throat (red),
the 8 GHz switch / amplifier (dashed), the aperture plane (solid), and the
bottom of the flares (jagged). The change to the high gain state is at record
7397 for the low channel and 7400 for the high channel, and this is also where
temperature control of the internal load failed. The temperature at the top of
the flares oscillated between roughly 30 and 50 K during this period.
125
Chapter 8. Data and Data Analysis
In all, I excised 728 data records, or 10%, of the 6997 total for the combined
null and high gain period from analysis. 39 data records were removed during the
time when the internal load temperature was rapidly cooling to the bath temperature.
96 records during an anomalous dip in the low channel radiometer were removed, as
were three sections of less than 20 records each corresponding to anomalous spikes in
radiometer output. There were additional spikes in radiometer output that repeated
over several rotations, consistent with a very warm object such as the moon or
Galactic center passing through the sidelobe of the antenna beam as reflected off of
the reflective shield. These records were excised along with corresponding records at
the same rotational phase for several periods before and after the spikes are visible in
the time ordered data, for a total of 513 records. An additional 18 records were
removed due to being single point outliers, possibly as a result of transmission errors
from the payload to the ground.
III. Data analysis strategy
III-A. A switching differencing radiometer – the semi-ideal case
As designed, ARCADE 2 was to operate as a doubly-nulled instrument where
the temperature of the sky would be compared to that of the calibrator target. In that
case, many of the radiometer effects considered in this analysis would be irrelevant.
However, because the target could not be used in the 2005 flight, the measurement
126
Chapter 8. Data and Data Analysis
presented in this dissertation is a singly-nulled differencing radiometer measurement
where the signal from the sky is switched with that from an internal load and then
demodulated and integrated, with the load used as an absolute temperature standard.
In general, the output of the radiometer in counts can be modeled linearly as


D = G (Tamp ) ⋅ TA ⋅ (1 − a ) − b ⋅ Tload + a ⋅ Thorn + c ⋅ Tswitch + ∑ d i ⋅ Ti  + E ,
i


where G is the system gain, the conversion factor between counts and temperature,
which maybe a function of the cold HEMT amplifier temperature Tamp, TA is the total
antenna temperature of sources external to the horn aperture, Tload is the internal load
temperature, Thorn is the temperature of the feed horn, a is the feed horn emissivity,
Tswitch is the temperature of the ferrite waveguide switch, di and Ti are the couplings
and temperatures respectively of any other objects within the system, and E is a
constant offset signal generated by the lockin amplifier. The nonzero emissivity of
the horn results in it having a small emission contribution and a small effect in
absorbing some power from the incoming signal, hence the factors of a and 1-a.
The factors b and c are related to the imperfect nature of the switch,
specifically the different attenuation in each of the two arms and different amounts of
leakage. The behavior of the switch can be characterized by a 2x2 switch matrix with
elements α, β, δ, and ε such that the temperature ‘seen’ at the output of the switch
when it is in the horn viewing state is Tout = α ⋅ Text + β ⋅ Tload + (1 − α − β ) ⋅ Tswitch ,
while that of the load viewing state is Tout = δ ⋅ Text + ε ⋅ Tload + (1 − δ − ε ) ⋅ Tswitch , where
127
Chapter 8. Data and Data Analysis
Text is the effective temperature of all components feeding radiation to the switch, as
in Text = TA ⋅ (1 − a ) + a ⋅ Thorn . In a perfect switch α=ε=1 and β=δ=0. In the imperfect
case, however, when the radiometer differences the load signal from the sky signal in
phase with the switch, the temperature seen will be
Tdiff = (α − δ) ⋅ (1 − a ) ⋅ TA − (ε − β ) ⋅ Tload + (δ + ε − α − β ) ⋅ Tswitch . To express this effect
on the radiometer output in counts, we can divide through by a common factor α-δ,
and define b ≡
ε −β
δ + ε − α −β
and c ≡
to achieve the radiometer output equation
α−δ
α−δ
in the previous paragraph.
The antenna temperature TA is itself a combination of the background CMB
temperature TCMB, the small contribution from the instrument train and balloon Tit,
the antenna temperature of Galactic free-free and synchrotron emission as well as that
of the CMB dipole, convolved with the antenna beam, and emission from the
instrument train. We can express it as
v
v
v
TA = TCMB + Tff (x ) + Tsynch (x ) + Tdipole (x ) + Tit
where Tff and Tsynch are the Galactic free-free and synchrotron emission signal, Tdipole
is the temperature of the CMB dipole at the point on the sky in question, all three of
which are spatially varying, and a spatially isotropic ‘monopole’ temperature which
includes both TCMB and Tit, the antenna temperature of emission from the instrument
train. In this analysis, I neglect the contribution to the antenna temperature from
integrated extragalactic synchrotron emission, as this will be less than half a
128
Chapter 8. Data and Data Analysis
milliKelvin at the frequency band in question, according to the formula given by
Reich et al. (2004) in which the contribution from extragalactic
sources TEX


ν

= 30 K ⋅ 
 179MHZ 
−2.9
,
In theory, with Tload, Thorn, Tswitch, Tamp, the Ti’s, and b and c known, a leastsquares fit as described under header D below could be performed to achieve TA in
every frame, as well as a and the di’s for the system and a monopole level.
III-B. ARCADE 2 2005 Flight – the far from ideal case
In the case of the data from the ARCADE 2 2005 flight, however, such a
straightforward analysis is not possible, as I discovered after many attempts to pursue
it. First, the switch matrix in flight is not known. Given that in a doubly-nulled
measurement using the external target would have rendered switch considerations
irrelevant, it was simply not a priority to characterize the switch performance in the
flight configuration before launch. After the flight, upon impact or transport back to
the balloon base, the switch accumulated additional dirt and rust in one of the arms,
changing its properties. The switch matrix elements were measured in ground testing
once before and twice after flight, and the values vary considerably in these
measurements. The switch matrix values are a function, among other things, of the
drive current level and current pulse width used to drive the switch, neither of which
were characterized in the chaos before launch.
129
Chapter 8. Data and Data Analysis
Furthermore, the ARCADE 2 thermometry is such that we don’t have a direct
measurement of the temperature of the cold HEMT amplifier, and it can only be
inferred from the temperature of the adjacent switch. Therefore the switch coupling
and temperature dependent portion of the gain are highly degenerate, and are also
degenerate with other components having a similar temperature profile in time.
Again, this is an artifact of the intended doubly-nulled measurement, in which the
switch and amplifier temperatures would be unimportant.
A second reason why I could not ultimately do the straightforward analysis
under header B above is the 1/f drifts in the high gain data. Such an analysis would
result in the drifts showing up as large, unreal differences in TA over the course of
many frames, making a monopole determination and a sky map entirely unreliable.
III-C. Analysis as carried out
Rather than attempting to achieve a TA in each frame therefore, I model the
sky as a superposition of monopole, dipole, and Galactic emission and fit to model
parameters to get the best-fit Galactic and monopole components. I model
v
v
v
mod el v
(x ) ,
Tff (x ) = α ff ⋅ Tffmod el (x ) and Tsynch (x ) = αsynch ⋅ Tsynch
mod el
where Tffmod el and Tsynch
are template beam-smoothed maps of Galactic free-free and
synchrotron emission. The templates used are described in Chapter 9 under header II.
In this way, the flight data can be used to test predicted models of free-free and
synchrotron emission at 8 GHz based on scaling emission maps from other
130
Chapter 8. Data and Data Analysis
frequencies. As the portion of the sky viewed was largely in the direction of the nadir
of the CMB dipole, I use existing knowledge of the dipole as an input to the analysis.
In summary, I am performing a least-squares fit of the radiometer output to
v
mod el v
(x ) such that
temperatures within the system and Tffmod el (x ) and Tsynch


D = G ⋅ TA ⋅ (1 − a ) − Tload + a ⋅ Thorn + ∑ d i ⋅ Ti  + E ,
i


with
v
v
synch v
TA = TCMB + α ff Tffmod el (x ) + αsynch Tmod
el (x ) + Tdipole (x ) + Tit ,
in to obtain couplings to the instrumental temperatures and αff and αsynch.
I then determine the monopole temperature, which includes both TCMB and Tit,
using a period of the null data where the switch temperature was more steady, by
performing a linear fit of the load temperature to the radiometer output with the back
end offset E and the couplings to the instrument and αff, and αsynch subtracted out, to
obtain the load temperature at which the radiometer output was zero counts. This is
then the monopole temperature, which follows from the discussion of switching
radiometers in Chapter 6 under header I. The relative magnitude of the swing in the
switch temperature during its fall and the residuals from those records can then be
used to quantify the uncertainty due to switch and amplifier temperature effects.
It was also necessary, given the 1/f drifts in the high gain data, to
simultaneously fit the high gain data for power in fourier modes corresponding to
periods ranging from 18 times to 36 the instrument rotational period. I chose modes
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Chapter 8. Data and Data Analysis
with periods representing multiples of the instrument rotational period because the
power in such modes would be almost orthogonal to the Galactic signal, with thirty
six times this period being the minimum frequency allowed by the Nyquist theorem
before sampling would effect the zero level. I chose 18 times this period as the
highest frequency mode as a compromise between fitting for maximum 1/f power and
risking absorbing Galactic signal into the fit to these fourier modes. To verify this
technique, I generated simulated data sets with an input Galactic emission model and
random realizations of 1/f noise, and performed the simultaneous fit to the Galactic
model and fourier modes as described above. I was able, with two different input
Galactic emission models and 1000 random realizations of 1/f noise each, to recover
an average of 1.00 times the input Galactic signal with a standard deviation of .02.
I weighted all data points with the inverse of the Gaussian variance. This
results in the high gain data having a factor of four larger weight. Given that the null
low gain data was the only time the load and horn temperatures were moving,
therefore being the useful data for fitting for the system gain G and other couplings to
instrument temperatures, while the high gain data was more useful for fitting for the
Galactic structure, I pursued an iterative strategy where I used the first 200 records of
the null low gain data, where the switch temperature is falling, to perform a
simultaneous least-squares fit for the instrument temperature couplings. Then, with
the determined values of the instrumental couplings as fixed inputs, both the null low
gain and off-null high gain data were used to simultaneously fit for αff, and αsynch, as
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Chapter 8. Data and Data Analysis
well as a parameter accounting for the offset difference in the system between the null
and non-null conditions (s). The values for αff and αsynch , were then fed back and the
200 records of null low gain data were again used to determine the best fit instrument
temperature couplings.
I then attempted to quantify the extent of the unknown effects of the switch
and amplifier temperature. I conservatively bound this effect by comparing the
residuals of the least-squares fit during the 200 records of the null data where the
switch temperature was falling to the change in the switch temperature, dividing the
standard deviation of the residuals of these records by the total swing in the switch
temperature during the period to obtain an estimate for the factor by which the switch
temperature is to be multiplied to cause a 1σ swing in the monopole level. Being
unable to determine the factor b, I assumed the value to be unity with a ± 0.1%
possible swing, based on looking at previously measured switch matrices. The
uncertainty in the monopole temperature and Galactic couplings induced by this is
insignificant when compared to other sources.
The results of this analysis are presented in Chapter 9. I carried out the data
analysis in the IDL programming environment.
III-D. Least squares fitting – general
The task is to achieve a best fit solution relating an output in the data D i , to
various known or unknown input parameters H iα . H iα forms a matrix of basis
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Chapter 8. Data and Data Analysis
functions that contains the value of each parameter at the time of each output. In the
case of the ARCADE 2 2005 analysis described here, the D i are the radiometer lockin
outputs at the data records i, and the α are temperatures and template Galactic
emission levels. We want fits A α to form a model M i = A α H iα in the linear
approximation, where the residuals R i are R i ≡ D i − M i ≅ D i − A α H iα . A perfect
model would have exactly Gaussian distributed residuals.
The data points can have various weights w ij corresponding to their relative
usefulness, and any irreducible correlations between them. The data analyzed in this
dissertation is assumed to be uncorrelated record to record such that w ij is diagonal.
Squaring both sides of the residuals definition to achieve a weighted square of the
residuals, we have (D i − H iα A α ) w ij (D j − H βj A β ) = R i w ij R j = R i R i ≡ χ 2 . The best fit
model is the one that minimizes χ 2 . We must find the fits A α where χ 2 is a
minimum, thus where
∂χ 2
= −2 H iα w ij D j + 2 H iα w ij H βj A β = 0 .
∂A α
−1
Then Aβ = (H iα w ij H βj ) (H iα w ij D j ) .
The data should be weighted by the standard deviation at each data point. Of
course, the standard deviation of a given point is not usually known a priori, so as the
data contains all of the structure resulting from dependence on all of the parameters
α, the best estimate for the Gaussian standard deviation of the data is the standard
134
Chapter 8. Data and Data Analysis
∑R R
i
deviation of the residuals, σ i ≡
i
i
, where n is the number of data points.
n
With properly weighted data and a model that successfully accounts for all structure
in the data, then χ 2 should be equal to the number of degrees of freedom, that is the
number of data points minus the number of model parameters. The deviation of χ 2
from this value is a marker for the goodness of fit. In practice, in analysis situations
such as the one presented in this dissertation, where the gaussian standard deviation
of the data is not known going in, then the model fits are determined as above with
the residuals and their standard deviation then calculated and the data weighted
accordingly. A properly normalized χ 2 can then be calculated, however this χ 2 is
not necessarily then a true measure of the goodness of fit.
The variance of the data is V ij = w ij−1 . The variance of the fits V αβ can be
found from V αβ =
−1
−1
−1
∂A α ij ∂A β
V
, then V αβ = (H βj w ij H iα ) H βj w ij V ij w ij H iα (H iα w ij H βj )
i
j
∂D
∂D
−1
−1
= (H βj w ij H iα ) H βj w ij H iα (H iα w ij H βj ) , resulting in V αβ = (H βj w ij H iα ) . The
correlations Cαβ between the fits can be found by normalizing the variances,
C αβ =
V αβ
. If the model is highly non-linear in one or more of the
V αα ⋅ V ββ
(
n >1
parameters, as in M i = A α H iα + O (A α )
), then the true variance must be expressed
135
Chapter 8. Data and Data Analysis
as V
αβ
j
β
= (N w ij N
i −1
α
)
∂M i
σ , where N =
| α
, and this reduces to the form
∂A α A =solution
2
i
α
above for the linear case.
IV. Pointing solution
Given that I am fitting the data to template free-free and synchrotron emission
maps, it is necessary to determine the pointing of the 8 GHz antenna beam on the sky
for each record. The ARCADE 2 instrument was equipped with a three-axis
magnetometer to determine rotation about the vertical axis of the payload by giving
the orientation of the instrument relative to the Earth’s magnetic field. Additionally,
two clinometers were on board to measure the pitch and roll of the instrument. The
coordinate system convention for the payload is that the z axis is vertical, the 8 GHz
horn points 30º from vertical in the negative y direction, and the x axis is oriented
such that the coordinate system is right handed. Positive ‘pitch’ and ‘roll’ are defined
as right handed rotations about the positive y axis and positive x axis respectively.
In order to reconstruct the pointing of the 8 GHz antenna beam on the sky in
each frame, I first use a coordinate rotation to transform the antenna beam from the
payload coordinate system in which it is
{− sin(roll)x̂,− sin(60° + pitch )ŷ,
}
cos 2 (60° + pitch ) − sin 2 (roll )ẑ to an earth-centered
coordinate system in which the x axis is East, the y axis is North, and the z axis is up.
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Chapter 8. Data and Data Analysis
To determine this coordinate rotation, I used the latitude, longitude, and altitude data,
along with the time, to calculate the direction of the Earth’s magnetic field at the
position of the payload in any given data frame via an available IDL routine utilizing
International Geomagnetic Reference Field (IGRF) data. The direction of this field as
read by the magnetometer provides the orientation of the magnetometer and therefore
the instrument with it in Earth fixed coordinates. I form a unit vector summing the
output on the x, y, and z axes of the magnetometer, while that output by the IGRF
routine is already unitized. Since the instrument is rotating about the vertical axis
only, the coordinate transformation from payload coordinates to earth-centered
coordinates simply a rotation about the z-axis, where an exercise in geometry reveals
magfield
 x magfield 
−1  x measured 
−
tan
the rotation angle to be φ = tan −1  IGRF
 y magfield  .
magfield 
 y IGRF 
 measured 
It is then easy to transform to a standard earth-centered azimuth-elevation


x
z
 and az = tan −1   . I then use
coordinate system where el = sin −1 
 x 2 + y2 + z2 
y


available IDL routines to transform first from azimuth-elevation to RA-DEC, using
the time and latitude and longitude as inputs, and then from RA-DEC to Galactic
coordinates, using the time as an input.
Unfortunately, the clinometers were not zeroed to a level aperture plane prior
to launch, and so are not a reliable determination of absolute pitch and roll, although
they can be used to confirm that the pitch and roll did not vary during the lid open
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Chapter 8. Data and Data Analysis
phase of the flight. To determine the pitch and roll, I optimized a simultaneous fit to
template beam smoothed maps of Galactic emission as outlined above, while varying
the pitch and roll of the instrument. I determined the pitch of the instrument to be a
constant -2.2º, and the roll to be negligible.
I recorded the pointing of the 8 GHz antenna beam in Galactic coordinates
into a blank channel in the flight data file. The pointing was recorded in the nested
HEALPIX pixelization scheme, a commonly used freeware scheme for sky analysis
in IDL which divides a spherical shell in which points are specified by an azimuthal
and latitudinal coordinate, such as in Galactic coordinates, into a number of equally
sized number indexed pixels given by n pix = 12 ⋅ n 2 , where n is a power of 2. I chose
n=64 for 49152 pixels, giving each pixel a resolution of 0.9°.
Fig. 8-5 shows the portion of the sky viewed by the centroid of the antenna
beam during the records used for data analysis, in Galactic coordinates, on top of a
WMAP 2003 K band combined Galactic emission map. For clarity, the centroid of
every tenth data record is highlighted.
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Chapter 8. Data and Data Analysis
Fig. 8-5. Portion of sky viewed during data records used for analysis, in
Galactic coordinates, on top of the WMAP K band Galactic emission map.
The position of the centroid of the antenna beam is sweeping out the circle
shown clockwise, and drifting downward over time.
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CHAPTER 9: RESULTS
I. Results from iterative fit
As discussed in the previous chapter, I performed an iterative least-squares fit
of the radiometer output to temperatures within the system, fourier modes, and
v
mod el v
(x ) such that the radiometer output in counts is modeled as
Tffmod el (x ) and Tsynch


D = G ⋅ TA ⋅ (1 − a ) − Tload + a ⋅ Thorn + ∑ d i ⋅ Ti  + E ,
i


v
synch v
with TA = Tmono + α ff ⋅ Tffmod el (x ) + αsynch ⋅ Tmod
el (x ) ,
to obtain couplings to the instrumental temperatures, a monopole level, and αff and
αsynch.
As discussed previously, in the equations above, G is the system gain, the
conversion factor between counts and temperature, Tload is the internal load
temperature, Thorn is the temperature of the feed horn, a is the feed horn emissivity, di
and Ti are the couplings and temperatures respectively of any other objects within the
system, E is a constant offset signal generated by the lockin amplifier, Tffmod el and
mod el
Tsynch
are template beam-smoothed maps of Galactic free-free and synchrotron
emission respectively, αff and αsynch are the respective couplings to those maps, and
Tmono is a monopole level that includes the CMB monopole and contributions from
the flight train and balloon.
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Chapter 9. Results
The results of the iterative least-squares fit are summarized in Tables 9-1 and
9-2. All values converged to within their statistical uncertainties by the second
iteration. In addition to the parameters listed there, I determined that the radiometer
had vanishingly small sub-milliKelvin couplings to the payload altitude and its first
derivative, many temperatures within the dewar and lazy susan and various
derivatives of temperatures, as expected, and a statistically insignificant coupling to
both the aperture plane temperature and the temperature of the flares. I also modeled
the emission from the flares by convolving the antenna beam with a realistic model of
the flares featuring a temperature of 36 K at the top and 4 K at the bottom, and
confirmed a contribution to the total antenna temperature of less than 1 mK.
The only significant couplings therefore, were to the load and horn
temperature (G and a), to the template Galactic emission maps (αff and αsynch), and to
an offset in counts for the non-null condition relative to the null one. The residuals of
the null data period for the high and low channels are shown in Figs. 8-1 and 8-2
respectively.
Table 9-1. Coefficients from the least-squares fit, low channel
Fit
Value from
Statistical
from
fit
uncertainty in
fit value
System gain (G) (counts/K)
N
93.4
.7
Antenna emissivity (a)
N
.006
.001
System offset for non-null (s)
N&H -5.1
.2
N&H .69
.02
Multiple of model free-free map (αff)
N&H 1.21
.04
Multiple of model synchrotron map (αsynch)
Notes: N=null data, H=high gain data.
Parameter
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Chapter 9. Results
Table 9-2. Coefficients from the least-squares fit, high channel
Fit
Value from
Statistical
from
fit
uncertainty in
fit value
System gain (G) (counts/K)
N
74.7
.7
Antenna emissivity (a)
N
.007
.001
System offset for non-null (s)
N&H -4.6
.2
N&H .68
.02
Multiple of model free-free map (αff)
N&H .93
.05
Multiple of model synchrotron map (αsynch)
Notes: N=null data, H=high gain data.
Parameter
Fig. 9-1. Time ordered 8 GHz high channel AC lockin output for the null
period (upper), and residuals of the least squares fit for null data period with
falling switch temperature (lower). The dominant structure in the time ordered
data is caused by the internal load temperature changing.
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Chapter 9. Results
Fig. 9-2. Time ordered 8 GHz low channel AC lockin output for the null
period (upper), and residuals of the least squares fit for null data period with
falling switch temperature (lower). The dominant structure in the time ordered
data is caused by the internal load temperature changing.
II. Galactic emission levels
II-A. The template Galactic emission maps
The model Galactic emission maps used in the least-squares fit were
generated by extrapolating or interpolating WMAP K band and Haslam all-sky
survey data. The template free-free emission map is the result of isotropically scaling
the first year 2003 WMAP K band free-free emission map with a –2.1 spectral index
and smoothing with an 11.6° Gaussian. The template synchrotron emission map was
generated by removing from the Haslam map an extrapolated free-free contribution
from the 2003 WMAP K band free-free emission map, determining an average
143
Chapter 9. Results
spectral index of –2.78 by comparing the first year WMAP K band synchrotron
emission map to the free-free removed Haslam map, and scaling to 8.0 and 8.3 GHz
the free-free removed Haslam map by this index. The scaled maps were then
smoothed with an 11.6° Gaussian. The ARCADE 2 beams are sufficiently wide at
11.6° that the correction for the differing beam widths of WMAP K band and Haslam
data is negligible. As WMAP is a differencing experiment, the absolute level of the
input WMAP K band free-free emission map is informed by Hα emission maps,
corrected for dust extinction, as discussed in Chapter 4 (Bennett et al., 2003). The
zero level of the free-free signal at 23 GHz was estimated by multiplying the zero
level of the Finkbeiner Hα emission map by 11.4 µK/Rayleigh, the WMAP first year
data estimate for the conversion between Hα intensity and free-free emission
temperature (Bennett et al. 2003; Finkbeiner 2003). Using a different value will only
scale the zero level of the resulting free-free and synchrotron interpolations and will
not affect the spatial structure. Using first year rather than three year WMAP data as
an input to the template maps allows this analysis to serve as an independent
confirmation of changes to the conversion between Hα intensity and free-free
emission temperature as reported by the WMAP team in the three year results
(Hinshaw et al., 2006).
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Chapter 9. Results
II-B. Galactic couplings
As seen in Tables 8-1 and 8-2, I obtained αff. = 68 ± .02 and .69 ± .02 for the
high and low channels respectively, and αsynch = .93 ± .05 and 1.21 ± .04. This means
that ARCADE 2 observed 69% as much free-free and approximately as much
synchrotron signal as expected from the naive scaling described above. This
corresponds, at an effective frequency of 8.15 GHz over the portion of the sky
covered, to an average observed 34 mK beam smoothed peak signal for free-free
emission in the Cygnus spiral arm, and an observed 19 mK beam smoothed peak
signal for synchrotron on the outer reaches of the Galactic bulge. For comparison, the
naive extrapolation and interpolation would give an average of 50 mK beam
smoothed peak signal for free-free emission and a 17 mK signal for synchrotron in
these directions. The data are consistent with the simple synchrotron model but prefer
a modestly lower free-free signal in the spiral arm viewed.
An examination of the data and residuals, as in Figs. 8-3 and 8-4, and a plot of
the data and the best fit Galactic model, as in Figs. 8-5 and 8-6, shows that these
amounts of Galactic coupling provide for the obvious spatial structure in the data. I
also performed the fitting process with a different template synchrotron emission map
generated by scaling the free-free removed Haslam map by the different spectral
index determined by comparing this map and the WMAP K band synchrotron
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Chapter 9. Results
emission map pixel by pixel. The coupling to this map was lower, at .43 for the high
channel and.42 for the low channel, with a considerably lower significance.
I repeated the Galactic fitting with yet another template synchrotron emission
map, one generated by removing a free-free signal from and isotropically scaling as
above the 1420 MHz northern sky survey carried out by Reich and Reich (1982,
1988). The coupling to this map, at 89 ± .01 for the high channel and 1.17 ± .01 for
the low channel is roughly similar to that of the free-free removed isotropically scaled
Haslam map, indicating both the consistency of the radio surveys and the
repeatability of this analysis. In summary, I achieve a best fit synchrotron spectral
index from 408 MHz to 8.15 GHz of -2.6 ± 0.2 and a best fit index from 1.42 GHz to
8.15 GHz of -2.7 ± 0.2.
Finally, I repeated the Galactic analysis with cuts of the data, including cutting
various sections of the data records, and cutting records from Galactic latitudes
greater or less than 15° from the Galactic equator. Removing sections of data
records and removing Galactic latitudes more than 15° from the Galactic equator
yielded results consistent with the above analysis. Removing Galactic latitudes
within 15° of the Galactic equator leads to greatly inflated statistical error bars such
as to render the free-free fit insignificant. This is as expected, as the spatial structure
lies almost entirely within the plane of the Galaxy.
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Chapter 9. Results
Fig. 9-3. Time ordered 8 GHz high channel AC lockin output for records
11825-12375, a representative sample of the high gain period (upper), and
residuals of the least squares fit for those records (lower). The raw data are
converted from counts to temperature by dividing by the system gain.
Fig. 9-4. Time ordered 8 GHz low channel AC lockin output for records
11825-12375, a representative sample of the high gain period (upper), and
residuals of the least squares fit for those records (lower). The Galactic signal
and 1/f drift are clearly visible in the time ordered data.
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Chapter 9. Results
Fig. 9-5. Plot showing time ordered 8 GHz high channel AC lockin output for
records 12450-12750 (solid), along with the best fit model from the least
squares fit (dashed) and a model with αff = αsynch = 1 (dotted).
Fig. 9-6. Plot showing time ordered 8 GHz low channel AC lockin output for
records 12450-12750 (solid), along with the best fit model from the least
squares fit (dashed) and a model with αff = αsynch = 1 (dotted).
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Chapter 9. Results
III. Estimation of TCMB
The CMB temperature (TCMB) and the antenna temperature of emission from
the instrument train and balloon (Tit) in combination comprise the spatially isotropic
‘monopole’ portion of the sky signal. As discussed in the previous chapter, I
determined the monopole temperature, using a period of the null data where the
switch temperature was more steady, by performing a linear fit of the load
temperature to the radiometer output with the back end offset D and the couplings to
the horn temperature a, and the Galactic emission maps, which include the Galactic
zero level, subtracted out, to obtain the load temperature at which the radiometer
output was zero counts. This is then the monopole temperature. The relative
magnitude of the swing in the switch temperature during its fall and the residuals
from those records can then be used to quantify the uncertainty due to switch and
amplifier temperature effects.
The constant offset E in the radiometer generated by the lockin amplifier was
measured in ground testing many times and in the flight configuration prior to launch,
and was shown to vary between –110 and –100 counts in the high channel and
between –75 and –80 counts in the 8 GHz low channel. I estimated the magnitude of
this offset during the null data period of the flight by comparing the radiometer output
just prior to, and just after, the gain change. I modeled the radiometer output for
several data records surrounding the gain change as a baseline signal multiplied by
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Chapter 9. Results
the gain plus a constant offset, and, having measured the ratio of the gains in ground
testing to be 15.921 in the high channel and 15.924 in the low channel, performed a
least-squares fit to a fifth order polynomial for the signal and offset for these records.
I performed this calculation multiple times allowing the number of records used for
the fit to vary from 3 to 30 before the gain change and from 3 to 20 after. Averaging
all results with a standard deviation below 0.6 counts, I determined that the offset in
flight was –102.1 ± 0.3 counts in the high channel and –77.5 ± 0.6 counts in the low
channel.
Fig. 8-7 shows a plot of the high channel AC lockin output with the horn,
Galactic, and back end offset contributions taken out vs. the internal load temperature
during the null records where the switch temperature was steadiest, and Fig. 8-8
shows the same for the low channel. The monopole temperature is the load
temperature where this reduced radiometer output is zero counts, divided by one
minus the antenna emissivity. Performing a linear fit, this give a Tmono of 2.776 in
the high channel and 2.907 in the low channel.
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Chapter 9. Results
Fig. 9-7. Plot of high channel AC lockin output with horn, galactic
contribution, and back end offset taken out vs. internal load temperature, for
null records 7095-7397, during which the switch temperature was steadiest.
The monopole temperature is load temperature where the radiometer is exactly
nulled, ie where this reduced radiometer output is zero counts.
Fig. 9-8. Plot of low channel AC lockin output with horn, galactic
contribution, and back end offset taken out vs. internal load temperature, for
null records 7095-7397, during which the switch temperature was steadiest.
The monopole temperature is load temperature where the radiometer is exactly
nulled, ie where this reduced radiometer output is zero counts.
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Chapter 9. Results
To achieve TCMB from the monopole temperature, the emission and reflection
contributions from the flares, the reflective shield, and parts of the flight train visible
at the edges of the antenna beams, including a piece of the balloon, a video camera,
lights, a bar at the bottom of the shield from which the dewar hangs, and the cables
connecting the dewar to this bar, must be modeled and subtracted. To model the
antenna temperature contribution from each component, I assumed that the object is
both emitting and reflecting the warm ground. I assigned a reflectivity and an
emissivity, and assumed the temperature of the object to be 250 K and that of the
ground to be 300 K, then convolved the object with the antenna beam using existing
IDL code. The largest contribution is from the shield, which, as a smooth dirty
aluminum surface, was modeled with an emissivity of .005 and a glint to ground
power reflection of .001. The reflected ground component is small, and tip scans
with the previous generation ARCADE instrument (Fixsen et al. 2004) verified this
emisssivity to be accurate within 30%. I repeated the calculation with the shield
suspension angle varying ± 2.5º and the emissivity to varying up to 30% to give a
shield antenna temperature contribution of 7.9 ± 3.5 mK. Contribution from the other
components totals 2.2 mK, to which I can assign an uncertainty of 30% or 0.7 mK.
With the instrument emission subtracted from Tmono I determined a value for TCMB of
2.766 K at 8.3 GHz and 2.897 at 8.0 GHz.
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Chapter 9. Results
IV. Uncertainty estimates
The raw statistical uncertainty in the values for the TCMB and the Galactic map
multipliers were determined from the least-squares fit. Additionally, the value of
TCMB is subject to significant sources of systematic uncertainty, described below. I
add the statistical and systematic sources of uncertainly in quadratures to determine a
total uncertainty of ± 116 mK in the low channel and ± 160 mK in the high channel.
An uncertainty summary for TCMB is presented in Table 9-3.
The dominant source of uncertainty in the values for TCMB arises from the
unknown coupling of the amplifier and switch temperatures to the radiometer output.
In order to quantify the extent of this uncertainty, I conservatively bound this effect
by comparing the residuals of the least-squares fit during the 200 records of the null
data where the switch temperature was falling to the change in the switch
temperature, as in Figs. 8-9 and 8-10. I divide the standard deviation of the residuals
of these records by the total swing in the switch temperature during the period to
obtain an estimate for the factor by which the switch temperature is to be multiplied
to cause a 1σ swing in the monopole level. This factor is .08 for the low and .11 for
the high band. The steady switch temperature of 1.455 K during the records where
the monopole temperature was determined is then multiplied by this factor to obtain
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Chapter 9. Results
an upper limit uncertainty due to switch and amplifier temperature effects of ± 116
mK in the low band and ± 160 mK in the high band.
Fig. 9-9. Plot of high channel residuals vs. switch temperature for records
6770-6970, a period where the switch temperature was falling.
Fig. 9-10. Plot of low channel residuals vs. switch temperature for records
6770-6970, a period where the switch temperature was falling.
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Chapter 9. Results
Other contributions to the error budget are small. Uncertainty in the antenna
emissivity will contribute to uncertainty in the monopole temperature. To estimate
this uncertainty, I calculate the swing in the monopole temperature at the 1σ extremes
of the antenna emissivity from the least squares fit, leading to a value of ±6 mK for
the monopole level in both channels. The corresponding changes in the values for
the Galactic map multipliers are an order of magnitude below the level of their
statistical uncertainty, and are consequently ignored.
Another potential source of systematic errors is thermometry. By observing
the λ superfluid transition in thermometer calibration and flight data, I can bound
thermometry errors in flight to less than ± 3 mK. To first order this is only relevant in
the monopole value. Higher order effects, which could arise because of the nonlinearity of the thermometer temperature resistance curves, are negligible.
There are three sources of systematic uncertainty in the measured value of
TCMB which are irrelevant in the determination of the Galactic map multipliers. These
are uncertainties in level of constant offset in the radiometer generated by the lockin
amplifier, in the determination of flight train emsission, and in the zero level of
Galactic emission. The first two effects were discussed and their uncertainties
quantified under header III. The uncertainty in the zero level of Galactic emission at
8.0 and 8.3 GHz springs from a combination of the uncertainty in the zero level of the
input foreground maps which were then interpolated, and the change in zero level
between the different model synchrotron maps considered. The uncertainty in the
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Chapter 9. Results
zero level of synchrotron emission due to the uncertainty in αsynch is less than 1 mK.
I observe a 1 mK difference in the zero level of synchrotron emission between the
model synchrotron emission maps considered. The uncertainty in the zero level of
the Haslam map at 408 MHz is reported as ± 3 K (Haslam et al. 1982), which, scaled
by a –2.8 spectral index to 8.0 and 8.3 GHz, corresponds to ± 1 mK. We thus report a
± 1 mK uncertainty due to the Galactic zero level.
The level of uncertainty in the Galactic map multipliers due to all of the
systematic effects considered is well below their raw statistical uncertainty, as is the
effect on their value of the system gain uncertainty. Thus I report an uncertainty of
.02 for αff and .04 for αsynch at 8.0 GHz and .03 for αff and .05 for αsynch at 8.3 GHz.
Table 9-3. Uncertainty summary for TCMB
Low channel
High channel
Source
Size of
Uncertainty
Size of
Uncertainty
effect (mK)
Contribution
effect (mK)
Contribution
(mK)
(mK)
Statistical uncertainty
N/A
3
N/A
1
Switch temperature effects
N/A
116
N/A
160
Radiometer back end offset
812
6
1369
3
Antenna emissivity
19
6
13
6
Shield emission
7.9
3.5
7.9
3.5
Other flight train components 2.2
0.7
2.2
0.7
Thermometry
N/A
3
N/A
3
Galactic zero level
4
1
4
1
Total uncertainty
116
160
Notes - Uncertainty estimates are discussed in §5. Uncertainties are added in quadrature.
Employment of the external calibrator in future flights will eliminate the uncertainty contributions
from the switch temperature effects and the antenna emissivity.
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Chapter 9. Results
V. Discussion
I report the ARCADE 2 2005 determination of TCMB to be 2.892 ± .116 K at
8.0 GHz and 2.766 ± .160 K at 8.3 GHz. These measurements are consistent with
FIRAS results at higher frequencies and ground-based measurements at lower
frequencies, and are among the most well constrained to date below 10 GHz.
However, the uncertainties are much too large to place any further constrains on
deviations from blackbody in the CMB spectrum, and in turn, any additional
constraints on reionization or relic decay. This will have to await future ARCADE 2
results.
Fitting the spatial structure in the time ordered data to template free-free and
synchrotron emission maps generated by scaling 2003 WMAP K band and Haslam
408 MHz sky survey data, I recover a mean best fit synchrotron amplitude of 1.07 ±
0.05 times the template map, corresponding at an effective frequency of 8.15 GHz to
a physical peak signal height on the outer reaches of the Galactic bulge of 19 mK
after the signal is convolved with the 11.6° antenna beam. This implies a best fit
synchrotron spectral index from 408 MHz to 8.15 GHz of -2.6 ± 0.2.
I recover a best fit free-free emission amplitude of .69 ± .02 times the template
map, based on scaling WMAP first year results, corresponding at an effective
frequency of 8.15 GHz to a peak signal height in the Cygnus spiral arm viewed of 34
mK when the signal is convolved with the antenna beam. Recent WMAP three year
157
Chapter 9. Results
results state a lower conversion of 8 µK/Rayleigh between Hα intensity and free-free
emission temperature (Hinshaw et al. 2006), leading to a free-free emission intensity
that is 70% as high as the WMAP first year result . The free-free coupling result
presented here can therefore be seen as an independent confirmation of the new lower
conversion between Hα intensity and free-free emission temperature.
There is repeatability in the Galactic analysis demonstrated by substituting
Reich 1420 MHz northern sky data for Haslam data. Given that the difference in
αsynch between the low and high channels is present with both Reich and Haslam data
as inputs to the template map points to this being related to the more diffuse nature of
the synchrotron signal relative to the free-free signal in the region of the Galaxy
observed. With the synchrotron signal less sharply peaked, it is much more
susceptible to confusion with residual 1/f noise in the data.
158
APPENDIX A: MECHANICAL AND THERMAL
DESIGN CONSIDERATIONS
Given that some of my early work on the ARCADE 2 experiment was in
designing the physical layout of the cryostat, I will present here an overview of the
techniques used in that design process. The task was to suspend the seven horns and
their attached radiometers from a flat aperture plane that would also support the
weight of the ‘lazy susan’ in such a way that 1) the horns could be arrayed to point in
two opposite directions, 2) that their entire apertures would fit within the cylindrical
envelope of the dewar and within three ellipses the size of the aperture of the largest
horn centered on the center of the aperture plane and formed by rotating that aperture
120° and 240°, 3) that the entirety of the horns and attached radiometers would fit
within the cylindrical envelope of the dewar, 4) that the very center of the cylinder be
clear for a pipe to be allowed to stretch from bottom to top to carry wires and helium
lines up to the lazy susan, 5) that the whole load would be borne by the bottom of the
dewar in such a way that it could withstand 10 gs of acceleration and 6) withstand the
thermal contractions brought about by cooling from room temperature to cryogenic
temperatures, and 7) would not transmit an unacceptable heat load in flight from the
warmer top of the dewar to the colder bath. The physical layout and component parts
were drawn and specified with the SolidWorks CAD design software. Figure A-1
shows an overview of the mechanical core as designed, and Fig. A-2 shows the core
as built.
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Appendix A. Mechanical and Thermal Design Considerations
Fig. A-1. The major elements of the instrument core as designed.
Fig. A-2. Photograph of the major elements of the
instrument core as built.
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Appendix A. Mechanical and Thermal Design Considerations
The first task tackled was the layout of the horn apertures on the aperture
plane. This was an exercise in trial and error, arraying CAD parts of the horns on a
flat plane until a configuration was achieved in which the 3 GHz horn occupied one
aperture region, the 5 and 8 GHz horns occupied another, and the high frequency
horns occupied the third, with all horns and their attached waveguide transitions
fitting within the envelope of the dewar (at the time named the ‘curtain of fate’) and
the apertures fitting within the three rotated ellipses described above, and with a rod
of 1.125” diameter just being able to fit down the middle. This configuration featured
the boresight of the 3 GHz horn pointing in one direction with all of the others
pointing in the opposite direction, adhering to the pointing requirement.
The next task was to design the aperture plane to support the hanging horns
and the load of the lazy susan while allowing the susan to rotate. The horns being
aluminum except for the smallest one, the aperture plane components would have to
be aluminum as well to prevent differential thermal contractions which could warp
the horns or aperture plane. Given that the middle of the cryostat was already to be
full of horns, the only possible layout was for the aperture plane to be supported at the
edge. And so, I formed the circumference of the aperture plane out of a sturdy ‘mule
ring’ 1/2” thick and 3” deep, so named because it carried the load of the aperture
plane. Next, I designed ‘collars’ for the horns to bolt into from below, individual
ones for the 3, 5, and 8 GHz horns, and one ‘ring complex’ part for the four higher
frequency horns. These turned out to be very intricate, as they contained elliptical
161
Appendix A. Mechanical and Thermal Design Considerations
holes, elliptical notches, elliptical bolt patterns, and various protrusions and cuts to fit
around each other and other pieces of the aperture plane, as can be seen in figure A-4.
Fig. A-3 shows the CAD part of the 3 GHz horn collar.
Fig. A-3. The 3 GHz horn collar as designed.
The middle of the aperture plane needed a flat surface where a plastic disk
bearing could support the lazy susan and allow it to turn. In my design this surface
was the top of a rod, named the ‘evil axis’ because of its many cuts and protrusions,
which was hollow to allow for the lazy susan servicing pipe to pass through. The
weights of the horns hanging from the collars and the lazy susan on the evil axis are
transmitted through each other and through fins running below to the mule ring as can
be seen in Fig. A-4. The fins, and the collars in several places, are bolted to the mule
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Appendix A. Mechanical and Thermal Design Considerations
ring with the use of arc-shaped joints. To complete the aperture plane, once it was
assembled thin sheet metal was cut to shapes to fill in the gaps between the horn
collars, and screwed down into tapped holes on a lip which I had designed into the
edge of each of the collars.
Fig. A-4. Aperture plane as designed.
Next I designed a truss structure to bear the weight of the mule ring. The
truss, as can be seen in Figs. A-1 and A-2, is composed of load bearing members
which form triangles. The members are hollow stainless steel tubes, and at each end
of a member is a knuckle, a part which allows the member to be bolted to a
corresponding anti-knuckle while allowing the knuckle/anti-knuckle joint to pivot.
The anti-knuckles then bolt into ‘knuckle heads’ on the top and ‘knuckle feet’ on the
163
Appendix A. Mechanical and Thermal Design Considerations
bottom. The knuckle heads support the mule ring and bolt to it from the inside, and
the knuckle feet rest on a protruding bolt circle at the bottom of the dewar. Both the
knuckle heads and knuckle feet must have arc-shaped tangential surfaces to bolt to
their round mating surfaces. A structure with six fold symmetry, meaning six
knuckle heads, six knuckle feet, and twelve members was chosen, with fewer being
less rigid and more running up against the impossibility of fitting additional knuckle
heads at regular intervals onto the circumference of an already crowded mule ring.
The specification of the truss member tubes illustrates some principles in
mechanical design. The length of the members was determined by the desired height
of the aperture plane above the bottom of the dewar. However, a material, an inner
wall diameter, and an outer wall diameter had to be chosen to provide all of the
following: 1) adequate strength, 2) acceptably low heat leak from the top of the dewar
to the bath, 3) acceptably low differential thermal contraction between the members
and the stainless steel dewar wall, 4) acceptably low contribution to the overall
weight, and 5) reasonable cost.
When subject to compression or tension, materials undergo a stress, defined
as the load per unit area, σ =
F
, where A0 is the original cross sectional area and F
A0
is the force of the applied load. An applied stress causes a strain defined as the
change in length per unit length ε =
l − l0
, and different materials have different
l0
stress/strain curves. There is generally a linear region at low stresses, where the ratio
164
Appendix A. Mechanical and Thermal Design Considerations
of the stress to strain is the Young’s modulus or modulus of elasticity, E =
σ
. For
ε
higher stresses, there are two general classes of materials. A ductile material is
plastic, meaning that it will permanently deform when subject to stress beyond the
linear stress/strain region. The stress at which plasticity sets in is the elastic limit. At
stresses slightly beyond the elastic limit is the yield strength, SY, where the material
begins to deform more readily. Beyond the yield point is the material’s ultimate
tensile strength, SUT, considered the highest stress it can take before breaking. A
brittle material subject to stress beyond the linear stress/strain region will not
permanently deform but will fracture, with a yield strength and a fracture point
slightly beyond the linear region.
Columns of a support structure of made of a ductile material will generally
fail by elastic instability if under static loading, where the load causes a bending of
the column which eventually rips it with the stress concentrated on a fraction of the
cross section. This is because in a real world the load will not be exactly transverse in
the column, and the column will not be perfectly uniform, and so bending results. If
the column is supporting varying loads, then it may fail by fatigue under a smaller
load than in the static load case. Under static loading, for a column of constant cross
section with both ends pinned, the maximum load the column can take without
buckling is Fmax =
π 2 EI A
, where l is the length and IA is the moment of inertia of a
l2
cross section about the central axis (Roark). In the case of a tube of circular cross
165
Appendix A. Mechanical and Thermal Design Considerations
π(ro4 − ri4 )
section, I A =
. The maximum load the truss can withstand then is 12*Fmax,
4
because there are 12 members. The load being designing for was the weight of the
core and the lazy susan, multiplied by a factor of 10 as payloads in ballooning can be
subject to 10 gs, multiplied by a safety factor of 1.5 or so.
The thermal conductivity K is a measure of how readily a material transmits
heat. It has units of power per length per temperature difference. The power
transmitted through a material is P = K
A
∆T , where A is the cross sectional area, L
L
is the transverse length, and ∆T is the temperature. Often the temperature
dependence of the thermal conductivity cannot be neglected and then K must be
t2
computed from K = ∫ k ( t )dt , where k(t) is the differential thermal conductivity. In
t1
the determination of the heat leak through the members, a ∆T was assumed and the
desired power transmitted was to be low enough that it would not boil away the
Helium from the dewar during the flight. Given the heat of vaporization HV and
density ρ of a material, and the volume V, the hold time, which is the amount of time
before all of the cryogenic liquid boils away, is t H =
V ⋅ Q vap
ρ⋅P
.
Another consideration in designing parts of the load bearing instrument core
was the appropriate bolt size for a given fastening point. The bolts needed adequate
strength to take the transmitted load in shear, but would become significant additions
166
Appendix A. Mechanical and Thermal Design Considerations
to the weight and cost if too big. A load in shear is a load that is applied across a
cross section, rather than transverse to it. The maximum load in shear before failure
can be approximated by L max = A ⋅ S US , where A is the cross sectional area and SUS is
the ultimate tensile strength in shear. An approximation S US ≈ .75 ⋅ S UT can be used
if a reasonable safety factor is included.
Once parts were designed, they were expressed in shop drawings for
machinists to actually make out of metal. Shop drawings must contain every
dimension necessary to fully specify a part. For parts that are sufficiently simple, the
part can be specified by a standard 3-view shop drawing, in which three views
corresponding to three orthogonal planes are shown. Fig. A-5 shows an example of a
standard 3-view shop drawing used to specify a part in the instrument core, that for
the ‘anti-knuckle’ part, along with a photograph of a completed one.
More complicated parts require more drawing views to fully specify. Fig. A-6
shows the shop drawing set for the 3 GHz horn collar pictured in Fig. A-3, which
required six pages to fully specify. The components that I designed requiring the
most specification were the 3 GHz and 30# horns, which required 48 and 68 pages,
respectively. In all, I designed, made CAD drawings of, produced shop drawings for,
solicited bids on, and saw through to completion 63 different mechanical parts that
formed the instrument core, aperture plane, and support truss of the ARCADE 2
instrument as flown in 2005. The lazy susan, mate collar, and exterior dewar frame
167
Appendix A. Mechanical and Thermal Design Considerations
were mechanically designed by others, and comprised an additional 50 or so different
mechanical parts.
Fig. A-5. Shop drawing and photograph of completed ‘Anti-Knuckle’ part, illustrating a
standard 3-view shop drawing.
168
Appendix A. Mechanical and Thermal Design Considerations
This and following pages: Fig. A-6. Shop drawing of 3 Collar part.
169
Appendix A. Mechanical and Thermal Design Considerations
170
Appendix A. Mechanical and Thermal Design Considerations
171
APPENDIX B: THE ARCADE 2 TARGET
The ARCADE 2 external calibrator target is one of the blackest microwave
objects over a wide frequency range ever made. I manufactured and assembled the
298 cones which comprised the radiometric elements of the target, and so even
though the target was not involved in the measurement presented in this dissertation,
it will be of the utmost importance in future results from the ARCADE 2 instrument
and deserves an examination.
Fig. B-1 is a photograph of the target as completed. Fig. B-2 is a photograph
of the back side of the target as flown, in situ on the lazy susan, showing attachments
for the cones and wires for thermometer signals. Fig. B-3 shows target in situ in the
lazy susan from the underside, which provides a nice perspective on what the horns
‘see.’
172
Appendix B. The ARCADE 2 Target
Fig. B-1. Photograph of the ARCADE 2 target, radiometric side.
Fig. B-2. Photograph of the ARCADE 2 target, back side, in place on the lazy susan.
173
Appendix B. The ARCADE 2 Target
Fig. B-3. Photograph of the ARCADE 2 target, radiometric side, in place on the
lazy susan.
The target for ARCADE 1 was made of Eccosorb, and featured rows of
ridges. It was decided that this target allowed unacceptably high thermal gradients to
build up along the length of the target, so the design of the ARCADE 2 target called
for cones, so that the tips of the emitting surfaces are not in direct contact and
therefore better thermally isolated from each other.
The cones of the target are Steelcast, as described in Chapter 6, under header
V-B, cast around an aluminum core. The aluminum core is for thermal conductivity
to maintain each cone at a close to uniform temperature. However, to have the
aluminum core extend too close to the tip of the cones would make the cones more
reflective, so in a compromise between the need for blackness and the need for
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Appendix B. The ARCADE 2 Target
thermal conductivity, a .03” diameter wire extends from the tip of the aluminum core
upward closer to the tip of the cone. A close up of a cone is shown in Fig. B-4.
The blackness of the target at microwave frequencies is a combination of the
properties of the Steelcast, the shape of the cones, and the fact that the horns view the
target at an oblique angle. The ARCADE 2 target, when viewed with the ARCADE 2
horns, is at least 55 dB black over the range from 5 GHz to 90 GHz, and more than 40
dB black at 3.4 GHz. This means that at 3.4 GHz, the reflected power is less than
1/10000 of the incident power, and in the rest of the ARCACE 2 bands reflected
power is less than 1/300000 of the incident power. Table B-1 shows the measured
attenuation of reflections of the ARCADE 2 target for the instrument frequency
bands. These attenuations are measured with a vector network analyzer, and the
reflections from the target are separated from the much larger reflections in the horn
and circular-to-rectangular waveguide transition by moving the calibrator relative to
the feed horn so that the reflection from the target has a phase dependence (Fixsen et
al., 2006).
Table B-1. Measured attenuation of reflections of the ARCADE
2 target when viewed with ARCADE 2 band horns, as measured
with a vector network analyzer. From Fixsen et al., 2006.
Standard
Frequency
Reflected power
Waveguide band (GHz)
attenuation (dB)
WR 284
3.4
42.4
WR 187
5.6
55.5
WR 112
8.3
68.6
WR 90
9.8
62.7
WR 28
30
55.6
WR 10
90
56.6
175
Appendix B. The ARCADE 2 Target
Fig. B-4. Photograph of a target cone.
The procedure for casting the target cones was as follows. First Silicone RTV
molds were prepared in the shape of the final cones, in the method of Chapter 6
section V-B. Cut sections of .03” wire ware epoxied into a tiny cup on the tip of the
aluminum cores with Eccobond, an electrically conductive silver epoxy. The cone
cores were then bolted to a bar which aligns with the mold and contains holes to
allow excess Steelcast and out-gassing acetone to flow out, as in Fig. B-5. The
surface of the bar and the mold were coated with mold release, with special care to
get a fine coating of mold release into the very tip, and the 25% Steelcast mixture was
176
Appendix B. The ARCADE 2 Target
prepared, using the acetone thinning method of Chapter 6 section V-B. A small
amount of liquid Steelcast was poured into the mold and the mold was agitated to
force the viscous Steelcast to flow into the cone tip. The mold is then evacuated in a
vacuum oven to remove air inclusions that may be at or near the tips. The remainder
of the mold is filled, agitated to allow air inclusions to rise to the surface, and the bar
with cores is pushed into the mold and screwed in place. A cone mold is shown in
Fig. B-6. I found that a mold could handle 5 to 7 casting batches before becoming
too brittle.
Fig. B-5. Photograph of a batch of cone cores ready for casting.
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Appendix B. The ARCADE 2 Target
Fig. B-6. Photograph of mold for casting Steelcast over cone cores to form
target cones, along with metal positives around which the mold was formed.
In order to monitor the temperature of the target, I fabricated some 50 cones
with ruthenium-oxide resistance thermometers located at various places within.
These were prepared in a special manner, with a hole cut in the core in which the
manganin thermometer wires could pass. Tefflon tube was run through the core, and
hollow threaded rod around the Tefflon was screwed into the base of the core to
provide an attachment mechanism. The manganin wires were run through the Tefflon
tube, and the wire-pro pin ends of the wires were heat shrunk to the end of the tube,
creating a robust structure, in contrast to very easily breakable bare manganin cryowires. In a delicate operation, I epoxied the thermometer chip to the cone core in the
desired location, and for those cones with thermometers not on the core, I cast ‘caps’
of Steelcast with a special mold on which the thermometers were epoxied. Fig. B-9
178
Appendix B. The ARCADE 2 Target
Fig. B-7. Photograph of Instrumented target cone core with Ruthenium-Oxide
resistance thermometer at tip of wire.
shows the locations of thermometers in the instrumented cones. Most of the
instrumented cones contained one thermometer, but five contained two.
The assembled target for the 2005 flight, as shown in Fig. B-9, consisted of
the cones affixed a plate, the ‘slip plate,’ which was contiguous with a ‘slip’
surrounding the target cones. A insulative stack of alternating layers of woven
fiberglass sheet and aluminum was between the slip plate and the target back, which
was contiguous with a ‘skirt’ surrounding the slip. The target liquid helium tank was
then stood off from the target back, and resistance heaters on the target back were in
place to provide heat to control the temperature of the target in the neighborhood of 2
to 3 K.
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Appendix B. The ARCADE 2 Target
Fig. B-8. Locations of thermometers in instrumented cones, in
cross section. The profile of the cone core and tip wire, as well
as the extra ‘cap’ layer, are visible.
Fig. B-9. Schematic of target as assembled for the 2005 flight.
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Appendix B. The ARCADE 2 Target
During the 2005 flight, the target was unable to cool to the desired
temperatures, with the cones remaining above 4K except for a brief period. Fig. B-11
shows temperatures in the target during a portion of the 2005 flight. By examining
the in-flight thermometry data, I was able to determine that the source of heating for
the target was both from above, with convection and radiation from warmer
components, and from the sides, where the target approached close to the edge of the
lazy susan and was warmed conductively and convectively from the warm dewar wall
and outside, and possibly from nitrogen condensation. The importance of this side
heating is especially noticeable in the brief era of the flight where the main boil-off
heaters in the dewar were turned on, sending a stream of cold gas past the edge of the
lazy susan and allowing the target temperature to dramatically drop, as can be seen in
Fig. B-11. It is clear that the aperture plane was not a source of heating for the target
and in fact acted as a colder heat sink.
Based on this analysis, a new design for the 2006 flight featured a new skirt
with an integral liquid helium tank, to ensure that external heat loads would always
fall on a surface that was sunk to the bath temperature. We also put liquid helium
tanks ed by superfluid pumps from the bath in contact with the aperture plane in order
to ensure that the aperture plane was isothermal. In the 2006 flight, the target was
successfully cooled and successfully temperature controlled, verifying the new target
thermal design. There were base to tip thermal gradients in the cones, however, due
to the 1.4 K aperture plane acting as a large thermal sink. This was somewhat
181
Appendix B. The ARCADE 2 Target
expected, with the thinking being that a cold aperture plane would assure that the
target could achieve sufficiently cold temperatures, while the 2006 flight would
provide engineering data leading to an optimal configuration in a future flight.. A
final iteration of the design will feature temperature control of the aperture plane to
around 2.7 K, achieved through the use of heaters and thermal standoffs from the
liquid helium tanks cooling the aperture plane.
Fig. B-10. Temperatures in the target during selected records from the 2005
flight. The tip ‘C’ thermometer of a doubly instrumented CR cone (red), the
base ‘R’ thermometer (blue), the target slip exterior (dashed), the target back
(solid), the aperture plane (dash-dot), and the target tank (light dotted). The
main boiloff heaters were turned on near record 14000.
182
APPENDIX C: 2006 FLIGHT AND
FUTURE PROSPECTS
ARCADE 2 has flown again in July 2006 with a redesigned lazy susan
movement mechanism, and a new target thermal setup as discussed in Appendix B.
In this flight we were able to successfully control the target and move the lazy susan
to achieve many views of both the sky and target at all channels. All radiometers
save the one at 5 GHz functioned to yield useful data.
The data analysis effort from the 2006 flight of ARCADE 2 is ongoing, and a
discussion of that effort or the flight or instrument performance in detail is not within
the scope of this work. However, a few remarks are in order. As shown in Fig. C-1,
we were able to control the target temperature so that the radiometer output when
viewing the target bracketed that when viewing the sky. With frequent views of the
external calibrator at known temperatures within used to model the various
instrument couplings, switch and amplifier temperature issues will likely be irrelevant
and 1/f drifts will much less relevant than in the 2005 flight. Additionally, we were
able to tip the instrument in pitch to directly measure the change in emission from
flight train components, and so a value for TSKY in many sky pixels will be achieved
at frequencies ranging from 3 to 90 GHz at possibly the tens of milliKelvin and
possibly the several millKelvin level.
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Appendix C. 2006 Flight and Future Prospects
Fig. C-1. Section of 3 GHz low channel radiometer output from the 2006 flight.
Shown are a view of the target and two views of the sky in the time ordered data.
Plot courtesy of Dale Fixsen.
From the 2006 flight there will likely be three remaining primary sources of
uncertainty in TCMB in the major science channels at 3, 8, and 10 GHz. One will be
gradients in the target temperature. The target displayed an almost 600 mK gradient
from the base of the cones to the tip, with the tip colder than the base. This was due
to the aperture plane, being crashed to the liquid helium bath temperature of 1.4 K as
planned, acting as a large thermal sink. With the target being less than isothermal, the
coupling of the radiometer output to the temperatures within the target will be less
184
Appendix C. 2006 Flight and Future Prospects
straightforwardly linear, leading to increased uncertainty in the instrument couplings.
As discussed in Appendix B, this gradient due to the cold aperture plane was
anticipated somewhat, and the effect could be ameliorated in future flights if the
aperture plane were temperature controlled to around 2.7 K through the use of heaters
and thermal standoffs from the liquid helium tanks cooling the aperture plane.
A second major source of uncertainty will be 1/f noise. In the 2006 flight,
most of the internal loads did not respond to temperature control and were sunk to the
bath temperature. This was possibly due to superfluid liquid splashing or dripping
onto the load casings. With the radiometers run 1.3 K off null, the effect of switching
does not exactly cancel all gain drifts, as discussed in Chapter 6. There will then be
some significant unmodelable component of the radiometer output contributing to the
uncertainty in TSKY. It is not out of the question that better design could allow the
loads to be properly temperature controlled in future flights.
The third major source of uncertainty in TCMB will be subtraction of the
Galaxy. At 3 GHz, the Galactic zero level is around 40 mK, but the Galactic signal is
subject to the uncertainty in, and in the scaling of, Haslam and WMAP Galactic
emission maps, as discussed in Chapters 4 and 9. This could easily limit the precision
of TCMB to several milliKelvin, and this effect cannot be ameliorated in any way by
improving the instrument performance.
An additional future flight of the ARCADE 2 experiment tweaked with a
temperature controlled aperture plane and successfully temperature controlled internal
185
Appendix C. 2006 Flight and Future Prospects
will loads may achieve TCMB to several milliKelvin at 3 GHz and 5 GHz and will
likely further constrain the free-free signal than at present. However, given the limits
imposed by the Galactic signal at these frequencies and below, further efforts to
constrain deviations from blackbody in the CMB may need to focus on improving
existing constraints at higher frequencies where the Galactic signal is weaker, even
improving on existing results at the FIRAS frequencies.
186
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