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Microwave observations of the southern sky from the TopHat experiment: The cosmic microwave background and the Magellanic Clouds

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THE U NIV ERSITY OF CHICAGO
MICROWAVE OBSERVATIONS OF THE SOUTHERN SKY FROM THE
TOPHAT EXPERIM ENT: THE COSMIC MICROWAVE BACKGROUND
A ND THE MAGELLANIC CLOUDS
A DISSERTATION SUBM ITTED TO
THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES
IN CANDIDACY FO R THE DEGREE OF
D O C TO R OF PHILOSOPHY
DEPARTM ENT OF PHYSICS
BY
JEFFERY J. BEZAIRE
CHICAGO, ILLINOIS
JUNE 2003
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UMI Number: 3088716
UMI
UMI Microform 3088716
Copyright 2003 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
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Copyright © 2003 by Jeffery J. Bezaire
All rights reserved
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In a very real way, the two people m ost responsible for this work are my parents.
From my earliest memories my mother has always instilled in me a love of reading,
thinking, and learning, and she passed along to me her fascination with nature,
science, and the cosmos.
My father taught me the value of working hard and
always encouraged me to pursue my own interests and aspirations even though
they were very different from his own. Thank you both so much. I dedicate this
work to you, and to my new son Eric. I promise to try to pass these values on to
him.
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ACKNOWLEDGEMENTS
I would first like to thank my cohorts and accomplices James Aguirre and Tom
Crawford, w ith whom I’ve gone to the ends of the Earth (Palestine, Texas and
Greenbelt, Maryland) and even on to Antarctica. I have enjoyed the tim es we
have spent together. An inordinate amount of thanks are due to Grant W ilson,
who helped shape three graduate students into experim ental physicists and who
at tim es seemed to alm ost singlehandedly keep TopHat progressing through sheer
force o f will. And of course, we are all indebted to Stephan Meyer, our advisor
and mentor, whose sense of fascination, curiousity, and delight set the tone of the
entire lab, and is the way physics should be done.
At Goddard Space Flight Center I was fortunate to work with two of the
kindest and sm artest people I have ever m et, Dave Cottingham and Dale Fixsen.
It is rare that one’s approach to thinking is changed by individuals, but that is one
of the results of my working with Dave and Dale on TopHat over the last several
years. Special thanks are due to Dave and his wife Christine for opening their
home to three wayward graduate students and a postdoc for weeks, and weeks,
and weeks at a tim e, making life on the road with TopHat much more bearable.
Thanks also to Ed Cheng, for getting the support and facilities needed at GSFC
to ensure TopH at’s com pletion, and for being our boom ing voice in the cacaphony
that som etim es is NASA. Thank you to Bob Silverberg, Peter Tim be, and Sean
Cordone, and everyone who has worked on TopHat over the years.
I would like to thank the National Scientific Balloon Facility, which took on
considerable extra work and risk in developing the capability of flying a large topmounted payload, and also Julian Borrill for help with his M ADCAP software,
access to, and support with the NERSC supercomputing resources.
iv
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V
Finally, and certainly most of all, I would like to thank my wife Fionna for
being understanding, tolerant, and supportive for all these years, and the many
months spent apart.
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TABLE OF CONTENTS
ACKNOW LEDGEM ENTS
iv
LIST OF FIGURES
ix
LIST OF TABLES
Part I
xiii
Introduction and Experim ental Overview
1
C H A PTER 1 OVERVIEW
1.1 In tr o d u ctio n ............................................................................................................
1.2 Top Package O v e r v ie w ......................................................................................
1.2.1 Telescope O p t i c s ...................................................................................
1.2.2 C r y o s t a t .....................................................................................................
1.2.3 Dewar O p t i c s ..........................................................................................
1.2.4 Detectors ..................................................................................................
1.2.5 Telescope R otation S y s te m ..................................................................
1.2.6 Electronics, Communications, and PowerD istribution . . . .
1.2.7 Pointing Sensors ...................................................................................
1.2.8 Thermal System ...................................................................................
1.2.9 Physical Constraints for Top-Mounted Package .......................
1.3 B ottom Gondola O v e r v ie w ...............................................................................
1.3.1 Power Generation and Distribution S y s te m .................................
1.3.2 Electronics, Communications, and D ata S t o r a g e .......................
1.3.3 B ottom Gondola S e n s o r s .....................................................................
1.3.4 Thermal System ...................................................................................
1.4 Observing S t r a t e g y ..............................................................................................
1.5 Flight O v e r v ie w ....................................................................................................
1.5.1 Launch D y n a m ic s ...................................................................................
1.5.2 “Ascent Event” .......................................................................................
1.5.3 T i l t ................................................................................................................
1.5.4 Observations at Float A ltitu d e ..........................................................
1.5.5 Flight P a t h ..............................................................................................
2
2
3
4
4
6
6
8
9
10
13
13
13
15
16
17
17
18
18
18
20
24
27
29
Part II
Hardware
C H A PTER 2
32
ROTATION SYSTEM
33
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vii
CH A PTER 3
THERM AL DESIGN
3.1
Thermal Environment .....................................................................................
3.2
B ottom Gondola Thermal D e s i g n ...............................................................
3.3
Telescope Thermal D e s i g n .............................................................................
3.3.1
Dew ar/Prim ary M ir r o r ......................................................................
3.3.2
Secondary Mirror S u p p o rts...............................................................
3.3.3
Top E le c tr o n ic s ....................................................................................
35
35
36
38
39
42
43
C H A PTER 4
IN-FLIGHT NOISE
4.1
Broadband N o i s e ...............................................................................................
4.2
Bearing “Pops” ...................................................................................................
4.3
Spin-Synchronous S i g n a l .................................................................................
4.4
Correlated Noise Between C h a n n e ls ............................................................
47
47
48
49
57
Part III
C H A PTER 5
Analysis
ANALYSIS OVERVIEW
59
60
C H A PTER 6
PREPROCESSING LOOP
6.1
Building the Pointing Matrix ......................................................................
6.1.1
Glint Sensor Phase Angle D e te r m in a tio n ...................................
6.1.2
Glint Sensor Sun A ltitude D eterm in a tio n ...................................
6.2
Pointing Model Parameters ..........................................................................
6.3
Pointing Model F i t ............................................................................................
65
65
71
72
74
74
C H A PTER 7
M APM AKING
7.1
General Derivation of Skymap from Time-Ordered D ata ...................
7.2
Fixsen’s E x p a n s io n ............................................................................................
7.3
Building the Weight M a t r i x ..........................................................................
7.4
Multigrid Mapmaking .....................................................................................
78
78
80
82
87
C H A PTER 8
M ADCAP M AP-M AKING
8.1
M ADCAP Input ...............................................................................................
8.2
M odifications to M ADCAP ..........................................................................
8.3
B est-Fit M a p s ......................................................................................................
8.4
M ap Level Internal C on sistan cy C h e c k s ......................................................
91
91
94
96
96
C H A PTER 9
D UST REMOVAL
108
9.1
Resolution C o a r s e n in g ....................................................................................... 108
9.2
Dust M o d e llin g ......................................................................................................108
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C H A PTER 10 M ADCAP PO W ER SPECTRUM ESTIMATION
114
10.1 Overview of Quadratic Estim ator Method for Power Spectrum Es­
tim ation ..................................................................................................................... 114
10.2 M ADCAP I n p u t s ..................................................................................................116
10.3 Power Spectrum Level Internal Consistancy C h e c k s .............................. 117
10.4 D is c u s s io n ................................................................................................................123
CH A PTER 11 MAGELLANIC CLOUDS
128
11.1 In tr o d u ctio n ............................................................................................................ 128
11.2 Integrated Differential P h o to m e te r y ..............................................................131
11.3 Source R e g i o n s ..................................................................................................... 132
11.4 Consistancy T e s t s ................................................................................................. 135
11.5 DIRBE D ata .........................................................................................................135
11.6 Calibration F i t ..................................................................................................... 137
11.7 Calibrated F l u x e s ................................................................................................. 141
11.8 D is c u s s io n ................................................................................................................141
CH A PTER 12 CONCLUSIONS
149
A PPE N D IX A CALCULATION OF COLOUR CORRECTIONS
151
REFERENCES
154
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LIST OF FIGURES
1.1
1.2
1.3
1.4
1.5
Assembly drawing of top package (dimensions in inches) ..................
Cross-section of the TopHat cryostat.............................................................
TopHat bandpasses, each normalized to unit peak transmission. . .
Glint sensor geom etry (both axes are in d e g r e e s)...................................
The TopHat observing strategy traces out a 24° diameter circle ev­
ery 16s. As the sky rotates about the SCP over the day, the observ­
ing circle sweeps out a 48° cap centered on the SCP. Each day of
observations repeats this sky coverage to produce an independant
m ap.............................................................................................................................
1.6 TopHat launch sequence. Top-left: the tow balloon lifts the tele­
scope and main balloon above the spool truck. Top-right: after main
balloon inflation the tow balloon is released, leaving the telescope
perched on top of the main balloon (bottom -left). Bottom -right: af­
ter main spool is released and gondola is released the balloon begins
ascent w ith telescope still on top and support gondola hanging below.
1.7 The net acceleration m agnitude and lateral acceleration m agnitude
of the top package during the spool release of the second TopHat
test flight...................................................................................................................
1.8 The tilt angle of the top package shows strong altitude dependance.
1.9 The spin axis tilt angle depends linearly on the balloon radius. The
balloon radius here is calculated as the inverse cube root of the
ambient pressure (ie. assum ing a closed, isothermal ideal gas sphere
for the balloon). The red line is the best fit linear m odel......................
1.10 The residual tilt angle from the linear model of Figure 1.9 varies
sinusoidally with azimuth angle between the sun and the spin axis.
The red curve is the sine of the azimuth angle between the sun and
the spin axis.............................................................................................................
1.11 Expected sky coverage (calculated assuming no tilt) is shown on the
left and the actual sky coverage is shown on the right............................
1.12 The 28 day TopHat balloon flight path. The gaps in the flight path
are due to interm ittent telem etry from the balloon..................................
5
7
8
12
19
21
22
25
26
27
28
31
3.1
Dewar pressure jacket tem perature.................................................................
41
4.1
4.2
Typical noise power spectrum ...........................................................................
Noise power at various frequencies varies over tim e. Black, purple,
blue, green, and red lines are l/1 6 H z, 0.25Hz, 0.5Hz, 3Hz, and 6Hz
respectively. The dashed curve is the temperature of the upper
honeycomb deck (and presumably the bearing) on an arbitrary scale
to show correlation with increased noise. The upper honeycomb
deck temperature varies between —3°C and 33°C.....................................
47
ix
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X
4.3 A typical distribution of bearing “pops”. The angular distribution
changes after each thermal cycle. The pops virtually disappear when
the bearing is warm..............................................................................................
4.4 Bearing “pop” rate vs tem perature...................................................................
4.5 First harmonic am plitude for each channel vs time. N ote that galaxy
crossings occur around hours 5, 30, 55, and 80..........................................
4.6 First harmonic phase (relative to tilt direction) for each channel vs
tim e. N ote that galaxy crossings occur around hours 5, 30, 55, and
80.................................................................................................................................
4.7 Spin synchronous signal shows very little dependance on chopped
atmospheric column depth .................................................................................
4.8 Spin synchronous signal shows strong correlation with tilt am plitude.
4.9 Spin synchronous signal shows even stronger correlation with the
product of tilt am plitude and mirror tem perature....................................
5.1
5.2
6.1
6.2
6.3
6.4
7.1
8.1
The Preprocessing Loop flowchart. D ata is cleaned and noise and
pointing m odel parameters are fit iteratively to produce cleaned
tim estream , pointing matrix, and noise model that can be handed
to M ADCAP for full analysis. The shaded boxes correspond to the
parts of the pipeline fully described in this thesis.....................................
The M ADCAP pipeline flowchart. The input tim estream s, pointing
matrix, and noise models are taken from the output of the prepro­
cessing loop depicted in Figure 5.1. The shaded boxes correspond
to parts of the pipeline fully described in this th esis...............................
A spherical triangle w ith interior angles a, b, c and sides A, B, C .
Pointing Model G e o m e t r y ...............................................................................
Pointing m odel fit x 2 contours. This slice through the (9sunx, 9beam)
subspace is evaluated with (4>beam, Quit) held at their best-fit values
and has contours spaced by l a .........................................................................
Pointing m odel fit %2 contours. This slice through the {4>beam, 9tut)
subspace is evaluated with (9sunx,9 beam) held at their best-fit values
and has contours spaced by lcr.........................................................................
49
50
51
52
53
54
56
61
62
66
67
77
77
Multigrid mapmaking converged maps at each resolution level. The
map is for day 2, channel 4, and took ~ 2 minutes to make on a
400MHz P entiu m class com puter. T he galaxy at th e b o tto m o f the
map appears saturated due to the plot range chosen here, but is
properly resolved in the m aps...........................................................................
90
Channel 1 (150 GHz) best-fit map using data from entire flight. Plot
scale saturates galaxy signal to show structure over the rest of the
s k y ..............................................................................................................................
97
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8.2
8.7
8.8
8.9
Channel 2 (210 GHz) best-fit map using data from entire flight. Plot
scale saturates galaxy signal to show structure over the rest of the
sky............................................................................................................................... 98
Channel 3 (380 GHz) best-fit map using data from entire flight. Plot
scale saturates galaxy signal to show structure over the rest of the
sky............................................................................................................................... 99
Channel 4 (430 GHz) best-fit map made using data from entire
flight. Plot scale saturates galaxy signal to show structure over the
rest of the sky. ..................................................................................................... 100
Channel 5 (640 GHz) best fit map made using data from entire
flight. Plot scale saturates galaxy signal to show structure over the
rest o f the sky. ..................................................................................................... 101
Channel 6 (dark channel) best-fit map made using data from entire
flight.............................................................................................................................. 102
Epoch A, Epoch B sum and difference maps for channels 1 and 2. . 105
Epoch A, Epoch B sum and difference maps for channels 3 and 4. . 106
Epoch A, Epoch B sum and difference maps for channels 5 and 6. . 107
9.1
9.2
9.3
Channel 1 before and after dust subtraction and dipole removal. .
Channel 2 before and after dust subtraction and dipole removal. .
Channel 6 before and after dust subtraction and dipole removal. .
8.3
8.4
8.5
8.6
. I ll
. 112
. 113
10.1 Angular power spectra for channel 1 CMB sum and difference maps. 117
10.2 Angular power spectra for channel 2 CMB sum and difference maps. 118
10.3 Angular power spectra for channel 6 sum and difference maps.
. . . 118
10.4 A -B difference power spectra made with a variety of sky cuts. . . . 120
10.5 Angular power spectra for channel 1 difference maps made using
two independant divisions of the data. The y 2 and P T E are for
the m odel that the two power spectra are realizations of the same
underlying noise power spectrum ...................................................................... 122
10.6 Angular power spectra for channel 2 difference maps made using
two independant divisions of the data. The y 2 and PT E are for
the m odel that the two power spectra are realizations of the same
underlying noise power spectrum ...................................................................... 122
10.7 Angular power spectra for channel 6 difference maps made using
two independant divisions of the data. The y 2 and PT E are for
the m odel that the two power spectra are realizations of the same
underlying noise power spectrum ...................................................................... 123
10.8 Angular power spectrum for channel 1 CMB map with correction
applied to pixel-pixel noise covariance matrix. Red error bars in­
clude uncertainty in correction term. N ote that the I = 7 bin is
off-scale high with a value of 41000T22000 .............................................. 124
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xii
10.9 Angular power spectrum for channel 2 CMB map w ith correction
applied to pixel-pixel noise covariance matrix. Red error bars in­
clude uncertainty in correction term ................................................................. 124
lO.lOAngular power spectrum for channel 6 CMB map with correction
applied to pixel-pixel noise covariance matrix. Red error bars in­
clude uncertainty in correction term ................................................................. 125
10.11 Angular power spectra for the linear combination of channels 1 and
2 sensitive to CMB (CMB Channel) and the linear com bination of
channels 1 and 2 orthogonal to CMB (Null Channel)...............................127
11.1 TopHat channel 5 map showing the on-source and off-source regions
used in calculating the LMC, SMC, and blank field fluxes, as well as
the dust regions used for calibration. The on-source and off-source
regions for 30-Doradus are shown in the inset. In calculating the
flux for the LMC we om it the 30-Doradus on-source region (the
inner circle in the inset and the blackened circle in the main figure). 134
11.2 Best-fit dust m odels in the five galactic D ust Regions used to cali­
brate TopHat channels 3, 4, and 5. Residuals are shown below each
plot and the five m odel spectra are plotted together in the lower
right panel................................................................................................................... 140
11.3 Best-fit models for the LMC, SMC, and 30-Doradus. The residuals
for each region are shown below each plot. The models for the three
source regions are plotted together in the lower right panel.................... 143
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LIST OF TABLES
1.1
1.2
A breakdown of the weight of the top package...........................................
Telescope parameters for each observation day...........................................
14
29
6.1
Pointing m odel parameters and uncertainties. The parameter un­
certainty projected onto the sky for 9sunx and gtut depend on the
tilt am plitude and sun position and flight averages are quoted here.
76
8.1
8.2
Reduced x 2 for m odel that tim estream data is sky signal best-fit map
plus signal from spin synchronous signal model plus noise consistant
w ith noise model. Each channel has approximately 2.5 x 106 degrees
of freedom....................................................................................................................103
Reduced x 2 for test that Epoch A, Epoch B sum and difference
maps are consistant with noise. Each x 2 has approxim ately 9000
degrees of freedom.................................................................................................... 104
11.1 Locations of regions used in the flux analysis. All coordinates are
J2000.......................................................................................................................... 133
11.2 x 2 ! est ° f the null hypothesis for the blank field......................................... 135
11.3 x 2 t est ° f the consistancy of each region between Epochs....................... 136
11.4 DIRBE gains and errors....................................................................................... 137
11.5 D ust Region dust m odel parameters from the calibration fit................. 141
11.6 Calibrated fluxes and errors for the LMC, SMC, and 30-Doradus. . 142
11.7 Spectral model fit results for LMC, SMC, and 30-Doradus.................... 142
11.8 Correlation matrices for the m odel fits to the LMC, SMC, and 30Doradus........................................................................................................................ 142
11.9 Comparison with previous surface brightness values for the Magel­
lanic Clouds................................................................................................................ 144
ll.lO R esu lts of fit to the alternate m odel of Equation 11.6............................... 147
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Part I
Introduction and Experimental
Overview
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C H A PT E R 1
OVERVIEW
1.1
Introduction
TopHat is a balloon-borne instrument designed to survey a significant portion
of the sky at far-infrared and subm illim eter wavelengths and measure the degree
angular scale anisotropies in the Cosmic Microwave Background (CM B). Built by
a collaboration of scientists and engineers at the University of Chicago, N A SA ’s
Goddard Space Flight Center (GSFC), the University of W isconsin, the Danish
Space Research Institute, the University of California at Davis, and the Bartol
Research Institute, it was launched from McMurdo Station, Antarctica by the
N ational Scientific Balloon Facility (NSBF) on January 4 2001. TopHat collected
data in five frequency bands for three days, repeatedly observing an ~ 60° diameter
cap centered on the South Celestial Pole (SCP).
TopHat is a farily com plicated instrument whose various parts were designed
and built by a number of people over many years. Here I provide a description of
the entire instrument, at various levels of detail, to allow the unfamiliar reader to
understand the experiment as a whole. My contributions to the instrument recieve
a more detailed treatm ent, and include portions of the rotation system , the thermal
design of the telescope, and the calibration and ground testing of the pointing
sensors. The analysis was conducted by a smaller group of people, and again I
describe the entire analysis pipeline to provide context, while going into detail
in describing the parts of the analysis to which I made significant contributions.
In particular, I developed the physical m odel of the spin synchronous signal, the
pointing m odel and the pointing m odel fit, and the fast mapmaking routine used
2
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3
in our preprocessing loop.
This dissertation is divided into three parts. In Part I, I provide a com plete
overview of the experiment, and an overview of the flight. In Part II, I give a
detailed description of the parts of the hardware for which I had a significant role
in designing, building, or testing. In Part III, I give a detailed description of the
data analysis pipeline, again em phasizing the parts of the analysis for which I was
primarily responsible.
1.2
Top Package Overview
The TopHat experiment consists of two parts, a top package mounted onto the
apex fitting at the very top of the balloon, and a support gondola hanging in
the traditional payload location below the balloon. The top package consists of
the telescope, dewar, telescope support electronics, pointing sensors, and power
distribution system . The bottom gondola provides power generation for the en­
tire experiment, disk drive data archiving, and interface w ith the N S B F ’s SIP for
telem etry and commanding. W ires sewn into the gores of the balloon allow power
transmission, telemetry, and com m anding between the bottom gondola and the
top package. In this section I describe all the major system s of the top package.
Figure 1.1 gives a schem atic drawing of the top package. The bottom of the
figure shows the lower honeycomb deck, which is bolted to the balloon’s apex plate.
Bolted to the top of the lower honeycomb deck, and shown in red, is the bearing
assembly. Housing a 26 inch diameter bearing, this bearing assembly allows the
entire telescope assembly to rotate about the vertical with respect to the fixed
lower honeycomb deck. Bolted to the top of the bearing assembly is the upper
honeycomb deck, which acts as the main platform for the rotating part of the
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4
experiment. Bolted to the top honeycomb deck is a conical sun-shield designed to
shield the telescope side-lobes from the sun and Earth as well as reduce the thermal
heating of the instrument. W ithin the sun-shield, and standing off from the upper
honeycomb deck on G10 legs is the primary mirror. An on-axis Cassegrain design,
the primary is lm in diameter, w ith a 0.122m diameter secondary mirror suspended
by Kevlar threads between the secondary support posts. The cryostat is strapped
to the back of the primary mirror, with its horn protruding through a hole in
the center of the primary mirror.
Not shown in this image for clarity are the
telescope support electronics, which consist of two card cages m ounted on the
upper honeycomb deck beneath the primary mirror. Also not shown in this image
is the motor drive system which was m ounted on the lower honeycomb deck inside
the bearing assembly.
1.2.1
Telescope Optics
The TopHat telescope is a lm on-axis Cassegrain design.
The primary mirror
was machined from a single piece of aluminum, its back surface honeycombed to
provide a lightweight, strong structure, its front surface machined to a slightly
potato-chipped paraboloid with x and y focal lengths of 435.18mm and 434.35mm
respectively. The secondary mirror is a section of hyperboloid 122mm in diameter,
with a 19mm diameter hole in the center to avoid reflecting the dewar horn image
back onto itself.
1.2.2
Cryostat
The TopHat dewar is perhaps the m ost remarkable new technology developed for
TopHat. The dewar (see Figure 1.2) is an internally pumped ^He refrigerator with
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5
82.4
\o
\o
Sun'harth Shield
1 (0
S u n & Glint
Sensors
►I
/
t\
S ec o n d a r y
Support P o sts
ar
Support
E le c tr o n ic s.
Bearing and
Motor Drive
H/.H
S eco n d a ry
Mirror
le le s io p e A sserrioly
X
Primary
Mirror
Cryostat
JLSl
Balloon Load
T ap es
Interior of Balloon
Figure 1.1 Assem bly drawing of top package (dimensions in inches)
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6
supporting ^He and liquid nitrogen (LN) reservoirs. W hile this type of dewar is
not new, the extrem ely small size and long hold tim e of the TopHat dewar make it
som ewhat unusual. The dewar is a mere 35 cm tall and 26 cm in diameter, weighs
10 kg when full of cryogens, and can m aintain a cold stage tem perature of 0.27K for
more than 10 days under float conditions. The extremely small size of the dewar
is in large part due to the development of helical fill tubes that provide a large
thermal conduction path in a short physical height. The insertion and removal of
a helical helium transfer tube, combined with pumping on the vent tube to get an
adequate helium transfer (and the myriad opportunities to generate ice plugs in the
fill lines) make operating the TopHat dewar a task not for the feint of heart. For a
com plete description of the dewar design, fabrication, and performance please see
Fixsen et al. (2001).
1.2.3
Dewar Optics
The dewar optics include the horn, etendu (14mm^sr) defining back-to-back W in­
ston cones, dichroic beam splitters, band defining filters, and various infrared block­
ers.
The five optical channels (hereafter denoted channels 1-5) have frequency
bands centered at 150, 210, 380, 430, and 640 GHz. A sixth channel (denoted
channel 6) blocked off from all incom ing radiation serves as a dark channel for
monitoring microphonic and electrical noise. Figure 1.3 shows the optical band­
passes as measured by Fourier transform spectroscopy.
1.2.4
Detectors
The TopHat detectors are m onolithic silicon bolom eters with ion-im planted ther­
mistors and were constructed at G SFC ’s Detector Development Laboratory. Each
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7
Tophat Long-Holdtime Cryostat
Condensing Dewar
Fill Tube
Condensing Dewar
Vent Tube
Vacuum
Port
4He Fill/Vent Tube
Nitrogen^
Fill/Vent Tube
Iso-thermal Shield
Pressure
Ja c k e t----------- >
3Helium
Space
Super­
insulation -----Zeolite
Pump
Nitrogen^
Can
4Helium
Can
Anti-buckling
Plates
Helium
Level-Sensor
Condensing
Dewar
Window
3
n
25K Stage
Alclads-------
Black Poly
Goretex >
Work Volume
2.8K
View
p0rt
Fluorogold
Suspension
Straps
Zotefoam
Preamplifier
Figure 1.2 Cross-section of the TopHat cryostat.
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8
bolom eter consists of a ~ 1mm diameter absorbing disk suspended from the sup­
port frame by four thin legs. The frame provides the thermal link to the ~ 270mK
bath temperature. The legs provide a weak thermal link from the bolom eter disk
to the frame, so that radiation absorbed by the bolometer disk raises the bolometer
temperature above the bath temperature. A thermistor im planted on the bolom e­
ter disk measures the changes in the bolom eter temperature and therefore the
changes in the incident power absorbed by the detector.
1.2.5
Telescope Rotation System
As seen in Figure 1.1, the entire telescope and sun-shield sits upon a bearing and
spins about the (nom inally) vertical spin axis. The drive motor is fixed to the lower
honeycomb deck and turns the telescope (via a tim ing belt and two gears) at a
constant rate of one rotation per 16 telem etry frames (approximately 16 seconds).
The telescope can be rotated in either direction.
The drive system was a continuous source of trouble during the development of
TopHat. Numerous designs were developed only to fail during thermovac testing.
1
.o
G
C
/3
C
/2
0 .8
sG
O
5
c^5
0.6
S
-H
B
S
0. 2
0.0
O
200
400
TS
600
800
(GPiz)
Figure 1.3 TopHat bandpasses, each normalized to unit peak transmission.
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9
The final design worked successfully in terms of m aintaining a constant spin rate
w ith no slipping or stopping, but still produced erratic “pops” and “groans” when
cold that were picked up microphonically by the detectors (see Section 4.2 for
details). See Part II for a more com plete description of the rotation system .
1.2.6
Electronics, Communications, and Power Distribution
The top package electronics are divided into two card cages.
The VM E cage
(named after the VME backplane it houses) contains all of the digital electronics.
The VXI cage (again named after its backplane) contains the analog electronics
for the telescope housekeeping and pointing sensors.
Commands are sent from the bottom gondola to the top package via a pair of
redundant opto-isolated RS485 signals up the balloon wires. The telem etry frame
produced by the top computer is transm itted down to the bottom com puter via a
separate opto-isolated RS422 line down the balloon.
Power for the entire instrument is generated on the bottom gondola (see Sec­
tion 1.3.1). To reduce ohmic losses, power is transm itted up to the top package at
a high bus voltage (nom inally 120V) and down converted on the top package to the
various voltages required to operate the telescope. Once at the top package, the
power, commanding, and telem etry lines must pass between the stationary base
and the rotating telescope. A 36 circuit slip ring located on the telescope spin axis
accomplishes this. The slip ring circuits were tested in a vacuum chamber at 2 Torr
for several m onths carrying 1 amp at 130V. W hile no arcing was ever observed, the
bus voltage com ing from the bottom gondola was later determined to be possibly
as high as 180V while operating on solar panels (120V was the bus voltage when
operating on batteries). For safety’s sake, as well as for space considerations, the
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10
bulk of the power converters were placed on the lower honeycomb deck, and no
voltages above 28V were passed through the slip ring.
1.2.7
Pointing Sensors
Since the telescope is not actively pointed, but just spins at constant angular
speed relative to the balloon, the pointing must be reconstructed after flight from
a variety of pointing sensors. See Section 6.1 for a com plete description of how the
pointing m odel is constructed from the pointing sensor signals.
A tilt-m eter is mounted on the upper honeycomb deck, 154mm from the spin
axis along the telescope’s x-axis (i.e.. towards the low point of the mirror. The
tilt-m eter is an Applied Geomagnetics Model 711-1 two axis tilt-m eter and each
axis is sampled at 16Hz. It’s x-axis is pointed in approximately the same direction
as the telescope’s x-axis (i.e. radially from where the tilt-m eter is m ounted), while
it ’s y-axis points in the azimuth direction. The tilt-m eter is sensitive to tilting
of the upper honeycomb normal relative to the local vertical. The tilt-m eter is
also sensitive to linear accelerations (linear accelerations are indistinguishable from
gravitational fields). These include balloon pendulation, accelerations about the
spin axis, and overall accelerations of the balloon as it moves through wind shears.
The tilt-m eter is insensitive to rotations about the vertical.
A gyroscope is mounted on the upper honeycomb deck with its x and y axes
approximately parallel to those of the tilt-m eter. The gyro is a surplus Pershing
II tw o-axis rate gyroscope, w ith each axis sam pled at 16Hz. T h e gyro is sensitive
to rotations of the upper honeycomb normal vector about the x and y axes. It is
insensitive to linear accelerations and to rotations about the honeycomb normal.
A m agnetom eter is mounted on the outside of the sun-shield, as high as possible
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11
to get it far from the m agnetic m aterials in the motor and bearing but low enough
to avoid the tow balloon m ounting hardware. It is an Applied Physics System
APS534 3 axis miniature fluxgate m agnetom eter and each axis is sampled at 16 Hz.
The m agnetom eter signal has proved to be not useful for pointing reconstruction.
The telescope itself has a non-negligible m agnetic field (presumably from iron in
the stationary motor and bearing). Coupled with the unknown field strength and
direction of the Earth’s m agnetic field for the Antarctic flight (which takes the
balloon very close to the south m agnetic pole), this has made the m agnetom eter
signal useless. Fortunately it was a redundant pointing sensor. If Grant W ilson is
reading this, I owe you a can.
The slip ring was equipped with an optical encoder with 512 counts per rotation.
The encoder signal latched an 8kHz clock, and the latched value was sampled at
16Hz, differenced from the last sample, and called the encoder signal. The encoder
signal is basically the tim e between successive encoder counts.
It is useful for
determining if the motor is slipping poles (due to the current setting being too
low).
One of the assum ptions of the pointing model is constant angular speed
between the lower and upper honeycomb decks, and the encoder signal shows this
was true.
A sun sensor consists of a pair of sm all photovoltaic cells m ounted vertically on
opposite sides of the sun shield and differenced, yielding a signal proportional to
the cosine of the azimuth angle between the sun and the sun sensor normal vector.
There are two sets of sun sensors, one pair aproximately on the telescope’s + / x-axis and one pair on approximately the telescope’s + / - y-axis. The sun sensors
sit within a baffle that blocks their view below the horizon. The baffle is intended
to block reflections of the sun off the balloon and petal assembly from hitting the
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12
sun sensors. The inside surface of the baffle is painted flat black.
The glint sensors are four sm all optical telescopes m ounted on the sun shield
adjacent to each sun sensor. Each glint sensor consists of a small ( ~ 2mm diameter)
lens w ith a mask at its focus and an integrating cavity with photodiode behind the
mask. The mask has three long, narrow slits in it, allowing the sunlight to pass
through to the photodiode only when the sun is at the sky image of one of the
slits. Figure 1.4 shows the measured response of a typical glint sensor to the sun’s
position relative to the boresight of the glint sensor.
Glint Sensor Beammap
b
20
1
'
■V ,
, V--T—r
&
<D
3
-2 0
-2 0
-1 0
0
10
20
Azimuth from boresight (degrees)
Figure 1.4 Glint sensor geom etry (both axes are in degrees)
As the telescope spins, the sun crosses the mask image on the sky at constant
altitude and causes three seperate glint signals in the photodiode, one for each of
the three slits. The central slit is vertical, giving a measurement of the telescopes
azimuth relative to the sun. The tim ings of the first and third glints relative to
the second glint depend on the altitude of the sun, and should be a measure of the
sun’s altitude in the telescope’s frame of reference.
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13
1.2.8
Thermal System
A rather elaborate top package thermal system was required to keep the electronics
at safe operating tem peratures while at the same tim e m inim izing thermal loading
on the dewar and m aintaining a stable thermal environment for the telescope
optics. See Section 3.3 for a detailed discussion.
1.2.9
Physical Constraints fo r Top-Mounted Package
Riding on the top of the balloon gives the TopHat telescope an unobstructed view
of the sky overhead. It also places severe weight restrictions on the instrument.
The design of the top package was driven by these weight considerations. The
entire top package weighs 247.51b. The center of mass is a mere 26.5 inches above
the attachm ent point on the balloon apex plate.
For comparison, the cryostat
alone for the BOOM ERanG LDB experiment weighs 5501b and stands 61 inches
tall (Masi et al., 1999).
Table 1.1 gives a breakdown of the weight of the top
package components.
1.3
B ottom Gondola Overview
The bottom gondola provides power generation for the entire instrument, onboard
disk storage for the flight data, command and telem etry interface to the top pack­
age, and interfacing with N SB F ’s SIP for commanding, telem etry downlink, and
GPS data.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 1.1 A breakdown of the weight of the top package.
Component
Weight (lbs)
sun shield
primary mirror and legs
thermal balnketing and dewar mount
secondary mirror and support struts
upper honeycomb deck
lower honeycomb deck
bearing, housing, and gear
radiator panels and lower sunshield struts
em pty card cages
electronics cards
power distribution boxes
cables
heat pipes
tilt-m eter, m agnetom eter, gyro, slip ring
dewar (empty)
cryogens
amplifier
motor
m iscelleneous (i.e. unaccounted for)
63
41
8
1
12
5
12.5
13.5
9.5
8
9
8
4
3.75
17
5
1.5
4
21.75
Total
247.5
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15
1.3.1
Power Generation and Distribution System
Each side of the gondola is equipped with six Solarex M SX-Lite solar panels.
Each panel nom inally generates 30W at 15V under full solar illum ination at room
temperature. On each side, the six panels are wired in series to provide a voltage
of ~ 9 0 V when illum inated. The four sides are wired in parallel (with protective
diodes) to provide a solar panel bus voltage of ~ 90V providing up to 260W
(depending on sun elevation) in total from the sides of the gondola that are facing
the sun.
In addition to the solar panel power system , there is a battery power system for
ground testing, backup in case of solar panel failure, and for providing supplem ental
power when the sun is too low to provide sufficient power to the solar panels to
run the entire experiment. The battery system is composed o f two battery boxes
wired in parallel for redundancy (again with protective diodes). Each box contains
three stacks of 26 2.7V lithium batteries supplied by NSBF and modified to have
their internal diodes and fuses removed. In each box, the two stacks are wired
in parallel to produce a box voltage of 70V. The two battery boxes are wired in
parallel to produce a battery bus voltage of 70V capable of delivering 360 amphours of current.
The solar panel bus and each battery box can be individually commanded on or
off. The solar panel bus and the battery bus are wired in parallel (with protective
diodes) to power the Low Voltage Power System (LVPS). If the solar panels are not
producing sufficient current to power th e entire experim ent (and the solar panels
and battery boxes are commanded on), the solar panel bus voltage will get pulled
down to the battery bus voltage and additional current will be drawn from the
batteries to supplement the solar panels.
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16
The LVPS powers the bottom gondola power distribution box, which provides
all the voltages needed on the bottom gondola. It also feeds the High Voltage
Power System (HVPS) which adds a 50V boost to the LVPS bus voltage and is
sent up the balloon to the top package.
1.3.2
Electronics, Communications, and Data Storage
All the electronics on the bottom gondola are contained in a pressure vessel with
an approximately 1 atmosphere dry nitrogen atmosphere. The bottom computer
recieves the telem etry frame com ing down from the top package, fills in data pro­
vided by the bottom gondola, and sends the telem etry to four destinations; the
Line O f Sight (LOS) radio link, the Tracking and D ata Relay Satellite System
(TD RSS), on board hard disks, and out over ethernet during ground testing. The
bottom computer also provides com m anding to the top package, and aquires GPS
data from the N SB F ’s SIP.
The LOS link is a high speed radio link usable while the balloon is within
line of sight of McMurdo (the first day or so of the flight). It can also be used
when underflying an aircraft below the balloon. Our full telem etry stream occupies
16.4 k ilob its/sec of the LOS’s available 300 kilobits/sec bandwidth. We can also
perform a high speed dump of the on board disks at up to the full 300 kilobits/sec
speed.
The TDRSS satellite link provides telem etry and com m anding coverage over
th e entire flight.
T h e dow nstream band w idth is only 4 k ilo b its/se c , so our full
telem etry stream cannot be handled over the TDRSS link.
We constructed a
reduced telem etry frame for TDRSS, that includes an extrem ely reduced version
of the dataset that would still allow full scientific analysis of the data. The TDRSS
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17
link didn’t function properly during flight, so I won’t go into the details of the
reduced TD RSS telem etry frame.
The onboard disks are two standard commercial SCSI drives, each 2 GB in size.
The TDRSS telem etry data is w ritten to duplicate files on each hard drive. The
drives were recovered after the flight and the data recovered from them . All of the
analysis was carried out on data taken from the flight hard drives.
1.3.3
B ottom Gondola Sensors
The bottom gondola carried a variety of sensors, m ost of which were housekeeping
sensors. Voltages and currents were monitored on each solar panel side, battery
box, power bus, and power distribution box voltage. Temperature sensors m oni­
tored the tem perature of solar panels, battery boxes, hard drives, and electronics.
In addition to the housekeeping sensors, the bottom gondola had a two axis
tilt meter and two pressure sensors. The tilt meter monitors pitch and roll of the
bottom gondola. The high range pressure gauge has a dynamic range of 0 Torr to
lOOOTorr. The low pressure gauge has a range of 0 to 10 Torr.
The bottom computer also gets GPS latitude, longitude, altitude, and tim e
from the N SB F ’s SIP.
1.3.4
Thermal System
The bottom gondola presents a much simpler thermal design problem than the top
package, in that the electronics are all contained in a pressure vessel and the only
concern is m aintaining safe operating tem perature for each gondola component.
The solar panels did provide som ething of a challenge however. See Section 3.2
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18
1.4
Observing Strategy
Long Duration Balloon (LDB) flights launch from McMurdo Station Antarctica,
which is at latitude 78° S, or 12° from the South Pole. The TopHat telescope is
designed to spin about the vertical bearing axis, com pleting one rotation every 16s.
The telescope beam is 12° from the spin axis, so that each rotation sweeps out a
small circle on the sky 24° in diameter centered on the local zenith (see Figure 1.5.
During the rotation the Earth spins about its axis moving the spin axis slightly less
than 1 arcminute on the sky. During the course of one sidereal day the sky rotates
above the telescope and the observing circle sweeps around the South Celestial
Pole (SCP) covering a cap 48° in diameter centered on the SCP. Each observing
day yields another com plete and independant map of the TopHat sky coverage.
During the austral summer, vortex winds circle the South Pole, keeping the
balloon at approxim ately constant latitude as it circles Antarctica.
1.5
Flight Overview
TopHat was launched from McMurdo Station Antarctica at 6:55 U T on January
4, 2001.
After approximately four hours of ascent, the balloon reached a float
altitude of 125,000 feet.
1.5.1
Launch Dynamics
Since th e 110 kg telescop e is m ounted on top of th e balloon, inflating th e balloon
becomes more of a challenge than usual for the N SB F launch team . A smaller tow
balloon is attached to the top of the sun-shield and inflated to lift the telescope and
uninflated main balloon above the spool truck. The main balloon bubble is then
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19
TopHat Sky Coverage After 16 Seconds
TopHat Sky Coverage After 16 M
inutes
TopHat Sky Coverage After 6 Hours
TopHat Sky Coverage After 1 D
ay
Figure 1.5 The TopHat observing strategy traces out a 24° diam eter circle every
16s. As the sky rotates about the SCP over the day, the observing circle sweeps out
a 48° cap centered on the SCP. Each day of observations repeats this sky coverage
to produce an independant map.
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20
inflated, at which point the tow balloon is released leaving the telescope sitting
freely on top of the main balloon bubble. From this point the launch procedure
is the same as for a standard balloon. The spool truck releases the spool and the
balloon bubble rises somewhat violently until it is over top of the launch vehicle.
The launch vehicle drives around the launch pad to match speed with the balloon,
getting the gondola directly under the balloon and then releases the gondola, letting
the balloon start its ascent. Figure 1.6 shows the launch sequence.
Just how violent the spool release would be for the top mounted package was
unknown and a major concern during TopHat’s development. N SB F conducted
two test flights for TopHat, one of which was sim ply to practice launch procedures,
the other to practice launch procedures and qualify the bottom gondola. In both
test flights a top package mockup was flown that reproduced the shape, weight,
and moments of inertia of the real top package. A three axis accelerometer was
flown on the mockup to monitor the top package dynamics at spool release. The
detailed dynam ics of the spool release varied considerably in each of the two test
flights and the Antarctic flight, though the peak loads were similar. Figure 1.7
shows the accelerations measured during the spool release of the second TopHat
test flight. Peak acceleration loads of ~ 2 .5 g were experienced in each test flight,
with lateral acceleration loads peaking at about 1.5g, well within the lOg and 5g
design strengths. W hile accelerometers were not available for the Antarctic flight,
the spool release appeared considerably more gentle than either of the test flights.
1.5.2
“A scent E ven t”
During the ascent to float altitude, at approximately 9:30 UT on January 5, when
the balloon was at an altitude o f approxim ately 100,000 feet, there was an un-
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21
Figure 1.6 TopHat launch sequence. Top-left: the tow balloon lifts the telescope
and main balloon above the spool truck. Top-right: after main balloon inflation the
tow balloon is released, leaving the telescope perched on top of the main balloon
(bottom -left). Bottom-right: after main spool is released and gondola is released
the balloon begins ascent with telescope still on top and support gondola hanging
below.
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22
L at e ra l A c c e l e r a t i o n During Spool Release
2.0........................ F........... ............................... ....................................
0
10
20
30
Time in S e c o n d s
40
50
M a g n i t u d e o f Net A c c e l e r a t i o n During Spo ol Release
3 . 0 b..........................
;
2.5 D
0
4
10
20
30
Time in S e c o n d s
40
50
Figure 1.7 The net acceleration m agnitude and lateral acceleration m agnitude of
the top package during the spool release of the second TopHat test flight.
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23
explained “event” in the five optical channels. All five optical channels went to
their negative rails within a single 64Hz sample (once the acausal digital filter is
taken into account). They remained railed for approximately three seconds, and
then recovered with a tim e constant of about one second. The DC levels for each
optical channel dropped by 1-2 mV between the sample preceeding this event and
the sample following it (the DC levels are only sampled at 1Hz). Channel 6 showed
a small AC signal consistant with cross-talk in the amplifier, but otherwise was
quiet. No other signal on the telescope showed anything unusual during this time.
The fact that the event was seen in all the optical channels but not in the dark
channel rules out a problem with the bias voltage and the readout electronics which
are common to all six channels. The DC level change in each channel is consistant
with a 5% - 20% decrease in loading, with the lowest frequency channels changing
the m ost. If it really is a change in optical loading, the suddenness of the event
rules out a tem perature change in an optical component. Any movement of an
optical element inside the dewar, or o f the dewar relative to the primary mirror
should be seen as microphonic noise in the dark channel.
Another possibility is m otion of the secondary mirror.
The tension in the
kevlar threads supporting the secondary was so high that any movement of the
secondary would be very fast. W hile it is very easy to envision failure m odes that
would send the secondary flying off into space or punching a hole in the primary
mirror, it is very difficult to concieve of a failure mode that would result in the
secondary m oving only a small amount. The pointing m odel best-fit parameters
(see Section 6.3) show the beam in approxim ately the expected location relative
to the telescope spin axis and the glint sensor azimuth.
Lateral m otion of the
secondary or tipping o f the secondary would move the beam on the sky.
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Only
24
m otions towards and away from the primary leave the beam position unchanged,
and to get the secondary to only move towards or away from the primary involves
m otion of all three sets of kevlar threads. If a single kevlar thread were to slip
slightly in its mount, the dominant secondary m otion would be a com bination of
lateral m otion and tipping.
A final possibility is a deformation of the dewar window. At ground level the
pressure difference between the ambient air and the dewar vaccuum jacket causes
the dewar window to bow in considerably. As the balloon ascends, this pressure
difference disappears and the dewar window relaxes back away from the dewar
horn. If this relaxation does not occur gradually, but rather the window holds its
bowed shape and then suddenly snaps into its relaxed shape, a fast optical signal
could be generated.
1.5.3
Tilt
As the balloon made its ascent it became apparent that the entire top package was
tipped over several degrees. As seen in Figure 1.8, at float the tip angle varied
between 3° and 6° depending on the altitude of the balloon . This large a tip angle
was not expected.
The direction o f the tilt was fixed to the balloon and pointed approxim ately
towards the side of the balloon where the wires are sewn into the balloon gores.
A rigid sphere m odel of the balloon w ith the weight of the wires suspended on
one side reveals an expected tilt of about 1.5° but it seems likely that a flexible
balloon would tilt more as the helium filled the bubble formed at the top of the
balloon by a tilt. The wires that were flown were larger (to satisfy N SB F flight
testing requirements) and more numerous (for redundancy purposes) than origi­
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25
nally planned. The original design also called for the wires to be sewn into the
balloon gores in two locations on opposite sides of the balloon so as to balance
The tilt angle appears to be a linear function of the balloon radius (taking the
radius to vary like the inverse cube root of the ambient pressure), as can be seen
in Figure 1.9. W hen the best-fit linear m odel is removed from the tilt angle, the
residuals still show some interesting structure. The dominant remaining variation
in tilt am plitude has a sinusoidal dependance on the azimuth angle between the
sun and the spin axis. It would appear that the side of the balloon facing the sun is
heated more than the opposite side and differential thermal expansion between the
two sides causes the balloon to tilt by approxim ately 0.3° . The balloon material
is polyethelene, which has a coefficient of thermal expansion of about 0.002°C- 1 ,
considerably higher than m ost m etals. A 0.3° tilt implies a (0.3°/360°) 0.1% dif­
ferential thermal expansion, which would require a 5°C tem perature difference
between the sun facing side and anti-sun side of the balloon. This is not an unTilt Angie v e r s u s Balloon Altitude
8
_CD
4
cn
c
<
P
2
0
0
10
20
30
40
Altitude ( k m )
Figure 1.8 The tilt angle of the top package shows strong altitude dependance.
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26
reasonable tem perature difference. This tilt would have been present even in the
absence of the tilt induced by the balloon wires, and is several tim es larger than
we had been expecting.
T ilt A n g le v e r s u s B a ll o o n
R a d iu s
5 .0
4 .5
4.0
3.5
3 .0
2 .5
1.00
1. 02
1.04
R a d iu s
1.06
(a rb itra ry
1.08
1.10
u n its )
Figure 1.9 The spin axis tilt angle depends linearly on the balloon radius. The
balloon radius here is calculated as the inverse cube root of the ambient pressure
(ie. assum ing a closed, isothermal ideal gas sphere for the balloon). The red line
is the best fit linear model.
Clearly the levelness of the balloon top was not given sufficient consideration
in the design of TopHat. Certain (apparently incorrect) calculations had implied
that the m ost the balloon could be out of level would be about 0.1°, but we could
have measured the tilt of the top plate in the two test flights if we had flown the
flight tilt-m eter. The accelerometers flown on the test flights had d.c. drifts that
made slow variations in tilt angle im possible to measure (indeed the accelerometers
were intended to measure launch dynamics, not float levelness) which we knew
about before the test flights. There was no reason not to fly the tilt-m eter on the
test flights, and indeed in hindsight it was a dramatic oversight not to. The tilt
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27
produced a large spin-synchronous signal that was extremely difficult to m odel out
of the data (see Section 4.3).
The large tilt also changed the sky coverage of the experiment, since the spin
axis was no longer pointed towards the zenith, allowing the telescope to observe
~ 5° closer to the horizon. The expected and actual sky coverage for the flight is
shown in Figure 1.11. The tilt increases our sky coverage from a 48° cap to ~ 58°
cap centered on the SCP and makes the coverage considerably less uniform.
1.5.4
Observations at Float Altitude
Upon reaching float altitude optim al glint sensor threshold settings were deter­
mined, load curves were taken to determine optim al detector bias settings, and the
minimum working motor current was determined. CMB observations commenced
-| q r 1 <'
T ilt A n g le R e s id u a l v e r s u s S u n A z i m u t h
1 '■ t " 1"■' r"'
I 1 1 1 1 1 1 1 1 1 I 1 ..................
m
CD
CD
p
-
1.0
0
100
A z im u th
B e tw e e n
200
Sun a nd
300
400
S p in A x is ( d e g r e e s )
Figure 1.10 The residual tilt angle from the linear m odel of Figure 1.9 varies
sinusoidally with azimuth angle between the sun and the spin axis. The red curve
is the sine of the azimuth angle between the sun and the spin axis.
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28
at 14hl0m UTC on January 4, 2001 with the start of TopHat observation sidereal
day 1. Since the vortex winds carry the telescope towards the west, one sidereal
day for the telescope is longer than twenty-four hours. Day 1 observations ended
at 15h27m UTC on January 5.
Load curves were taken to check the bias settings (optim al bias had not changed
from the previous day). For the second day of observations the bias voltage polar­
ity was switched. Any signals resulting from power deposited on the bolom eters
reverses sign with a bias polarity switch, while electrically coupled signals do not.
The telescope rotation direction was reversed between day 1 and day 2 to search
for tim ing offsets. Day 2 observations went from 15h46m UTC on January 5 until
17h27m UTC on January 6.
Load curves were again taken between day 2 and day 3 observations and the
Sky Coverage A
ssum
ing Zero Tilt
(In
te
g
ra
tio
ntim
e
)1
/g
Sky Coverage W
ith Actual Tilt
(In
te
g
ra
tio
ntim
e
),/a
Figure 1.11 Expected sky coverage (calculated assuming no tilt) is shown on the
left and the actual sky coverage is shown on the right.
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29
bias voltage polarity was again switched. Day 3 observations went from 18h02m
UTC January 6 until 19h04m UTC January 7. At this point we had com pletely
lost the TD RSS connection and were unable to command the telescope or monitor
telemetry. As a result day 4 observations had the same bias polarity and spin di­
rection as day3. Day 4 observations went from 19h04m UTC January 7 until 14h00
UTC on January 8 at which point the 4He had run out and the pot tem perature
had started to increase.
Table 1.2 Telescope parameters for each observation day.
Day
1
2
3
4
Bias Polarity
+
-
+
+
1.5.5
Spin Direction
1
T
T
T
Flight Path
Although the dewar ran out of helium after four days of observations, the balloon
was not in a recoverable location at that tim e, and was allowed to proceed on
its flight path around the continent. The polar vortex winds slowed considerably
as the flight progressed, until they collapsed com pletely when the balloon was
about 3 /4 of the way around the continent and the winds carried the balloon
in a random fashion w ith a generally southerly direction (see Figure 1.12). The
balloon was finally term inated after a record 28 days at float altitude and the
gondola descended on parachute and landed just South of the Ross Ice Shelf on
a 40° slope at 2800 feet altitude in the foothills of the Queen Maude mountain
range, closer to the South Pole than to McMurdo Station. The top package, which
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30
falls to Earth still attached to the now deflated balloon, drifted in the opposite
direction from the gondola and was lost from sight by the recovery crew as they
followed the gondola’s descent. The recovery pilot managed to land the plane on
a glacier within hiking distance of the gondola and the recovery crew was able to
recover the flight data disks from the gondola (thank you Alex!). The top package,
with telescope and dewar, was never seen again.
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31
o’
IS O ’
Figure 1.12 The 28 day TopHat balloon flight path. The gaps in the flight path
are due to interm ittent telem etry from the balloon.
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Part II
Hardware
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C H A PT E R 2
ROTATION SYSTEM
The rotation system consists of the bearing, upper and lower bearing housings,
motor, gears, and drive belt. The bearing itself is a Kaydon Reali-Slim K D250XP6,
type X, Class 6 bearing.
It is a four point contact bearing, w ith a 25.5 inch
pitch diameter. The balls are 0.250 inches in diameter, are separated by a brass
race, and number 153 in total. The bearing is connected to the lower honeycomb
deck by the lower bearing housing, and to the upper honeycomb deck by the
upper bearing housing. W ithin each bearing housing, the bearing is constrained
vertically at twenty-four spring-loaded point contacts to allow some deformability
in the bearing housing as the bearing rotates.
Radially, the bearing housings
provide enough clearance to accom odate the differential thermal expansion between
the steel bearing and the aluminum housings.
The bearing is lubricated with
Christo-Lube 1000RP grease, which has a low viscosity that is fairly temperature
independant.
Attached to the lower honeycomb deck is the drive motor, a Warner KM L091F07
stepper motor. Mounted on the m otor’s drive shaft is the pinion pulley, a Stock
Drive Parts A6A16-020DF3706 pulley. The pinion has twenty teeth and a pitch
diameter of 0.509 inches. Attached to the upper honeycomb deck is the custombuilt main pulley, with 448 teeth and an 11.408 inch pitch diameter. A tim ing belt
connects the two pulleys. The tim ing belt is Stock Drive Parts #A -6B 16-515037
belt with 515 teeth, pitch length of 41.2 inches, 3 /8 inch width, constructed with
a urethane body and kevlar re-enforcing cord. A pair of idler pulleys guides the
tim ing belt onto the main pulley, providing 270° engagement of the tim ing belt on
the pinion.
33
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34
The motor is driven with a sine wave current profile to provide a constant
stepping rate that is synchronized to the detector sampling. Synchronous step­
ping provides phase stability between drive system microphonics and the detector
sampling, resulting in very narrow drive system noise lines in the detector power
spectra which can be effectively notch filtered. The telescope com pletes one ro­
tation every 16 telem etry subframes (approxim ately 16s). The current am plitude
is commandable to allow adjustment of the maximum torque deliverable by the
motor. The drive system can turn the telescope in both the spin up and spin down
directions.
Mounted at the spin axis on the lower honeycomb deck is a slip ring which
transfers power and telem etry between the stationary and rotating parts of the
telescope. The slip ring is a Michigan Scientific model SR 36M /E 512 36 pin slip
ring with an optical encoder.
The slip ring bearing is lubricated (by Michigan
Scientific) with Krytox 240AZ low tem perature vacuum grease (MoS2 dry lubri­
cant proved unacceptable and required the replacement of the bearings after the
lubricant clumped, causing rough bearing action).
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C H A PT E R 3
THERM AL DESIG N
The high-altitude Antarctic environment offers a significantly more challenging
thermal m anagem ent problem for balloon payloads than an overnight m id-latitude
flight does. The challenge in an overnight m id-latitude flight is to keep the elec­
tronics warm in an environment of constant (and predictable) thermal radiation
from the Earth and essentially zero thermal radiation from the sky (very little of
a CMB experiment is warmed by the CMB!). An Antarctic LDB flight, on the
other hand, must contend with the radiation from the sun, solar radiation reflected
off the Earth’s surface, and thermal radiation from the Earth. All three of these
power sources are variable in an Antarctic flight, depending on the flight path the
balloon takes around the continent and the tim e of day.
3.1
Thermal Environment
During an Antarctic flight (around January 1) the sun is close to declination
—23.5°.
McMurdo Station is at an approximate latitude of 78° South, though
circumpolar flight paths have taken the balloon as far North as 75° South. Taking
this as a worst case scenario, we find the sun’s elevation can vary between 8.5°
and 38.5° over each day of an Antarctic flight. At float altitude (unlike at ground
level), the sun’s intensity doesn’t vary as a function of it ’s elevation due to the
lack of atmospheric absorption. However, the angle of incidence on the experi­
ment changes considerably, changing the power loading on particular parts of the
experiment by factors of several.
The solar radiation reflected off the surface of the Earth is the m ost highly
variable com ponent of the incom ing power on the experiment. It can vary from
35
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36
being nearly negligible to being ~50% of the total incident power. Its intensity
depends on the elevation of the sun (reflected power is nearly negligible when the
sun is 8.5° above the horizon) and the albedo of the surface under the balloon.
Surface albedo can vary from 0.8 when the balloon is over snow to as low as 0.2
when the balloon is over open ocean.
Finally, the Earth’s infrared emission can also vary considerably as the balloon
travels over the high Antarctic plateau (surface temperatures of —40°C) or open
ocean (surface temperature around 0°C). This radiation is also different from the
solar radiation in that it peaks in the infrared as opposed to the visible. The surface
coatings of the experiment tend to be far more absorbent in the infrared than the
optical (so as to reject solar radiation). These properties make the payload more
sensitive to changes in Earth IR emissions than the power levels alone would imply.
3.2
B ottom Gondola Thermal Design
The bottom gondola’s thermal design focused on two goals. The first was to keep
the electronics inside the pressure vessel at a safe temperature. The second was to
keep the solar panels within operable temperature ranges. The thermal analysis
and design m odifications were performed by G SFC ’s Rob Chalmers. I summarize
it here for the sake of completeness.
The bottom computer, housekeeping electronics, and data hard drives were
all contained inside the pressure vessel at ~ 1 atmosphere pressure. This greatly
simplifies their thermal management. The pressure vessel was equipped with commandable fans to provide a variable thermal link between the electronics and the
pressure vessel walls. The overall power incident on the pressure vessel was dom i­
nated by the incident sunlight, so the pressure vessel walls were essentially a fixed
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37
temperature bath to link the electronics to.
The solar panels provided a much more challenging thermal design problem.
The actual cause of the problem lies in the power generation and distribution sys­
tem , which was designed and built before any thermal analysis was performed.
For fixed incident sunlight, our solar panels actually had higher efficiency with
increased temperature. Naively we would want to run our solar panels hot (within
the mechanical lim its of the panels), and indeed the NSBF runs their solar panels
at temperatures around 90°C. To efficiently transmit power to the top package,
the TopHat power system uses a high bus voltage (> 120V), providing a fairly
high lower lim it to the acceptable solar panel output voltage (considerably higher
than N S B F ’s). The problem for TopHat is that while the power output of the
solar panels increases with temperature, the maximum output voltage decreases
(the greater increase in output current still results in overall increased power). The
power generation and distribution system was designed with the solar panels and
batteries as parallel sources, dioded so that the batteries provide supplemental
power to the experiment whenever the solar panel bus voltage is pulled down to
the battery voltage. This design was intended to handle the midnight period of
each day when the sun dips low in the sky and power generated by the solar pan­
els is insufficient to fully power the experiment. At sufficiently high temperatures
(> 75°C), however, the open circuit voltage of the solar panels is below the battery
voltage, so that even though the solar panels could be fully powering the experi­
ment, their output voltage is so low that the batteries are providing full power to
the instrument.
The thermal design solution to this problem was to mount the solar panels on
the bottom gondola at a fairly steep angle to lim it the amount of solar heating they
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38
would be exposed to, as well as to mount them on over-sized radiator panels to
try to increase radiative cooling. The obvious problem with this solution is that it
further reduces the amount of electrical power the solar panels are able to generate,
particularly when the sun is low on the horizon. In fact, subsequent power analysis
showed that the solar array would be unable to provide full power to the instrument
during approxim ately 11 hours of the day, requiring the battery com plem ent to
total 360Amp-hours, sufficient to fully power the instrument on batteries alone for
5 days! In the end, it appears that the thermal m odel was conservatively pessim istic
in its assum ptions (as the modeler stated when he constructed the m odels - indeed
they were intended to bracket the worst possible extreme thermal environments),
and the solar panel temperatures ran considerably cooler than m odel predictions,
providing full power to the instrument for 75% of the flight and partial power
during the 25% of the flight when the sky was low in the sky.
3.3
Telescope Thermal Design
The thermal design of the top package provides a much more challenging and
interesting problem. There were several design goals:
— keep the electronics in safe operating temperature range throughout the flight
— keep the primary and secondary mirrors at stable temperatures
- keep the dewar shell as close to —30° C as possible
- minim ize the spin synchronous temperature fluctuations of telescope com po­
nents in the telescope side-lobes
Since the electronics and dewar shell were in close physical proximity to each other,
and wanted to be at very different temperatures, it was decided to partition the
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39
top package into two separate thermal system s. A layer of superinsulation-covered
polyester blanketing separates the hot electronics from the cold dewar/primary
mirror.
W ith a strong thermal barrier between the electronics and the dewar/primary
mirror, we have now truly separated the two thermal problems from each other
and can treat them independently. We will first look at the dewar/prim ary mirror
system.
3.3.1
D ew ar/P rim ary Mirror
The hold tim e of the LN stage of the dewar is dom inated by radiation loading from
the of the outer shell of the dewar. To extend the hold tim e of the LN stage from 5
days (at room temperature) to 12 days, it is neccessary to reduce the tem perature
of the dewar shell. A lower lim it to the dewar shell temperature is determined
by the o-rings used to make the vacuum seal. The o-rings were made of EDPM ,
which remains elastic down to tem peratures of —40°C. A minimum dewar shell
temperature of —30°C was chosen to give 10°C margin to the vacuum seal. During
m ost of the day, the sun is kept off of the mirrors by the sun shield. The sun
shield consists of a foam core with inner and outer skins of aluminum epoxied to
the foam. The inside skin was coated w ith silvered Teflon (A gFEP) tape (optical
properties discussed in more detail below). The reflective interior surface of the
sun shield was in the shape of a truncated 30 degree cone, so that when the sun was
less than 30 degrees above the horizon all rays of sunlight reflect out of the cone
after a maxim um of two bounces. The inside surface of the sun shield was painted
flat black below the level at which the 30 degree sun would shine, so that stray
sunlight would be absorbed by the walls of the sun shield rather than reflecting
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40
down the cone to hit the mirrors.
The incident power on the primary mirror is then lim ited to the infrared emis­
sion from the sun shield and any power dissipation from the dewar (the dewar was
physically strapped to the primary mirror, providing a fair thermal connection)
and amplifier (a couple of W atts at m ost). Since m ost of the primary mirror’s field
of view is in fact cold sky (COBE tells us it is ~2 .7 K ) the primary mirror has
the potential to be used as a wonderful radiator panel for the dewar. We had in
fact considered using the primary mirror as a radiator for all the top electronics as
well, but the need to extend the hold tim e of the LN stage of the dewar superceded
that possibility. The front surface of the primary mirror was coated with a thin
layer of SiOx (the x refers to some m ixture of silicon monoxide and silicon diox­
ide - the exact com bination of which is unknown and depends on the oxygenating
environment of the vacuum deposition chamber). The SiOx layer has an infrared
em issivity of ~ 0 .7 5 , while being transparent to optical light as well as to in-band
sub-mm radiation. The lm mirror with this highly emissive coating is capable of
radiating 120W to space at room tem perature (assuming half of its field of view to
be cold sky). This would be enough to cool the electronics, and is certainly enough
to keep the dewar cold.
In fact, the primary mirror could potentially be too good a radiator for the
dewar. Depending on the altitude of the sun (and therefore the tem perature of
the inside skin of the light shield), the primary mirror was modeled to get as cold
as —100°C, which is too cold for the dewar o-rings. To guard against this failure
mode a heater is installed on the dewar shell whose power output was proportional
to the tem perature of the dewar shell below -27°C. The heater is turned on at
—27°C and was at full power of 12W at —30°C. This system worked perfectly
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41
during flight, as can be seen in Figure 3.1.
Dewar P r e s s u r e J a c k e t T e m p e r a t u r e
-10
o
0)
o
CD
CD - 2 5
E
CD
I—
30
35 t
0
20
40
60
Time ( h r s )
80
100
120
Figure 3.1 Dewar pressure jacket temperature.
The primary mirror had sufficient thermal mass and sufficiently constant input
power (on rotation tim e scales) that its spin synchronous tem perature fluctuations
were undetectable (fluctuation am plitude less than lm K ) so that spin synchronous
signal due to mirror temperature fluctuations are less than 10/rK assum ing a high
sub-mm em issivity of 0.01 for the mirror, and more likely less than 1/iK for a more
reasonable sub-mm em issivity of 0.001 for the mirror.
The primary mirror exchanges thermal radiation with the cold sky and the
inside surface of the sun-shield. Therefore, to minimize the tem perature of the
primary mirror, we should minim ize the temperature of the inside surface of the
sun-shield. As previously mentioned, the inside surface of the sun-shield need also
be reflective in the visible and microwave so as to reflect sunlight out of the cone
without hitting the primary mirror. The inside surface of the sun-shield is coated
w ith silvered Teflon (A gFEP) tape acquired from Sheldon Corporation. The tape
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42
is 5mil thick Teflon onto which silver is vapour-deposited, followed by a thin layer
of pressure-sensitive adhesive (PSA ). Integrated over the solar spectrum , the tape
has an absorbtivity of ~ 0.12, while being highly specularly reflective.
In the
infrared the Teflon front surface of the tape has an em m issivity of ~ 0.86. The
thermal properties of the tape are perfect for our application inside the sun-shield,
providing a cold, optically and sub-mm reflective surface.
3.3.2
Secondary M irror Supports
The secondary mirror was suspended between three support posts on six kevlar
threads, one thread going to the bottom and one thread going to the top of each
support post. W hen the secondary is properly strung up, there is sufficient tension
in the kevlar to bend the secondary support posts in slightly at the top. Kevlar
has a negative coefficient of thermal expansion, so that it stretches when it cools.
This expansion gets taken up by reduced bending of the secondary support posts.
Since the posts are bolted at the bottom to the primary mirror, all of the bending
m otion happens at the top of the posts, and the secondary mirror moves away from
the primary. This causes the telescope to change focus as the tem perature changes.
To elim inate this problem, the mount point for the kevlar threads at the bottom
of the secondary support posts were modified to include a spring whose spring
constant matches the effective bending spring constant at the top of the support
post. The thermal expansion of the kevlar then gets taken up equally by the top
o f the p ost bending and th e b o tto m p ost m o u n t’s spring relaxing, resu ltin g in very
little movement of the secondary. Tests performed between 20°C and —25°C in
the walk-in freezer at the University of Chicago show that the secondary move­
ment is 0.005m m /°C . W hile we don’t have temperature data for the secondary
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43
mirror during the flight, the primary mirror (which the secondary should track
in temperature) varied between —50°C and —13°C. The secondary was originally
positioned to be at optim al focus at a tem perature of —20° C, so that the secondary
focus position change during flight was at m ost —0.15mm to + 0.10m m from the
design focus. This small a change in focus produces an unmeasureable change in
the beam.
3.3.3
Top Electronics
Since they are not in a pressure vessel, the electronics on the top package do not
have the benefit of convective cooling.
The electronics on the top package are
housed in two electronics racks referred to by their backplanes. The VME card
cage houses all the digital electronics for the top package and consumes ~ 20W
of power. The VXI card cage houses the analog electronics and consumes ~ 12W
of power. The top power distribution boxes consume less than 10W each, which
given their wide acceptable operating tem perature range, was deemed insufficient
to warrant thermal management. The VME cage, however, contains commercial
com ponents rated for operation only between 0°C and 70°C.
The top electronics thermal management system was designed around the use of
heat pipes. Developed for this very application in satellite system s, heat pipes have
traditionally been very expensive, custom -made devices. Recently, manufacturers
such as Thermacore (from whom we purchased our heat pipes) have been massproducing sm afl heat pipes for coolin g laptop C P U s w ith no fan (thus exten d in g
battery life).
In its m ost basic form, a heat pipe is a sealed, hollow copper tube that has
been evacuated and then backfilled with a small amount of working fluid. If one
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44
end of the pipe is hotter than the other then the vapour pressure at the hot end
will be higher than at the cool end. Since the working fluid vapour is the only gas
in the pipe, the difference in vapour pressure between the two ends translates to
an absolute pressure difference, causing sonic speed gas flow from the hot end of
the pipe to the cold end. The vapour pressure at the cold end of the pipe now
exceeds the saturated vapour pressure of the fluid and condensation occurs at the
cold end of the pipe. If the condensed fluid is able to return to the hot end of the
pipe (via gravity for instance), then a fluid/vapour loop is achieved with net heat
flow from the hot end of the pipe to the cold end.
Actual production heat pipes are slightly more com plicated. Instead of being
hollow copper tube, the inside surface of the tube is sintered copper. This dramat­
ically increases the surface area of the copper/fluid interface, increasing heat flow
efficiency. It also allows the liquid from the cold end of the pipe to return to the
hot end of the pipe via capillary flow, so that the pipe can be used in the abscense
of gravity (in a satellite), or against the direction of gravity (as some of the pipes
were used in TopHat).
The working fluid in a heat pipe determines many of the heat p ipe’s m ost im­
portant qualities. Fluid with a high heat of vaporization yields a high heat flow per
fluid molecule in the heat loop, and therefore a high effective thermal conductivity
for the pipe. Water has a very high heat of vaporization and makes an excellent
working fluid for this reason. The triple point temperature of the working fluid
determines another im portant quality of the heat pipe - its freeze-out tem pera­
ture. If any part of the heat pipe falls below the triple-point tem perature of the
working fluid, the gas will freeze out at that part of the pipe and eventually all
the fluid in the pipe will solidify there. W hen this happens, the heat pipe stops
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45
working entirely and the effective conductivity of the pipe drops to that of the
copper walls. W hen the temperature is brought back up above the triple point
the pipe begins working again and is not damaged in any way, but the heat pipe
essentially turns off when any part of the heat pipe falls below the triple point
temperature. For TopHat, we used this as a feature of the heat pipes instead of
a drawback. Heat pipes with water (freeze-out temperature 0°C) and heat pipes
with m ethanol (freeze-out tem perature —70°C) were used in parallel with each
other. This gives variable effective thermal conductivity between the electronics
and the radiator panels, protecting against overcooling (a concern going through
the tropopause) and overheating (a concern at float altitude when operating in
the sunlight). The efficiency of the heat pipes also increases slightly as the op­
erating tem perature increases above the triple-point temperature (lower viscosity
improves capillary flow). These effects combine to form a thermal link between the
electronics and the radiators that is weakest when the electronics (hot end) are at
0°C, then increases by a factor of 10 or so when the radiator panel (cold end) is at
0°C. The other appealing characteristic of heat pipes is that they are com pletely
passive.
The consume no power and have no moving parts.
This makes them
especially appealing for an experiment with a m inim al power budget and detectors
that are extrem ely sensitive to microphonics.
The conical sun-shield seperates into two pieces at the height of the primary
mirror. The lower portion is made of an aluminum u-bar frame covered with a skin
of sheet aluminum (22 mils thick). The skin is made of four panels that each wrap
1 /4 of the way around the instrument. The two panels that were closest to the two
electronics cages were used as radiator panels. The outside of the radiator panels
was covered with silvered Teflon tape (see Section 3.3.1 for optical properties) to
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46
minimize their tem peratures and m axim ize their radiating ability. Water filled
heat pipes were attached to the back surface of each radiator panel to spread heat
out across the radiator panels. The radiator panels are heat sunk to the thick
aluminum u-bar frame only by being bolted to it, so that the radiator panels are
easily removed to access the electronics. Water and m ethanol filled pipes run in
parallel from the u-bar frame to heat sinks at the front of each card cage. From
this heat sink, heat pipes run up the front of the card cage to cool the card face
plates. Another set of heat pipes runs to a heat sink at the bottom rear of the
card cage. From this heat sink, heat pipes run up the full length of the backplane.
The idea was that the individual cards could be cooled through the backplane, and
that individual chips that generate large amounts of power would have additional
heat sinking to that card’s front face plate. A power audit indicated that the CPU
and the digital 48 industry packs were the only individual com ponents consuming
concentrated amounts of power (1.2W for the CPU, and 5W for the IPs). Short
copper braids were carefully epoxied to these chips and heat sunk to the card face
plates.
The extensive thermovac testing schedule that diagnosed the drive system prob­
lems allowed us to thouroughly test the cooling system . The only uncertainties
we had about the in-flight performance concerned passage through the tropopause
(where cold dense air and wind could overcool the electronics directly) and whether
the thermal m odel prediction for the radiator panel tem peratures was accurate.
The ascent through the tropopause was pretty much a non-event from the point
of view of the top electronics. W hile the sun-shield dropped 45C and the radiator
panels dropped 20C in about 20 minutes, the cpu temperature drops a mere 2.5C.
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C H A PT E R 4
IN-FLIGHT NOISE
4.1
Broadband Noise
As expected, the broadband noise had a largely white component and a 1 /f com­
ponent. Figure 4.1 shows a typical noise power spectrum (in this case taken from
channel 2 during the first 400 rotations of day 1).
£ h a n n e l 2 Noise Power S p e c t r u m ( 4 0 0 r ots of day 1)
10- 2 E
,
,
,
,
,
,
,
,
,
,
,
,
,
,
.
j
X
(N
S.
10 - 3
10
_4
o
<D
CL
GO
<D
$
O
10
-5
Q_
(D
if)
10
10
-7
0
2
4
F r e q u e n c y (Hz)
6
8
Figure 4.1 Typical noise power spectrum.
W hat was unexpected was the fact that the noise was not stationary. There was
a diurnal change in the am plitude and shape of the noise that correlates strongly
with the temperature of the upper honeycomb deck (and presumably the bearing).
Figure 4.2 shows how the noise at several frequencies varied over the course of the
flight, along w ith the tem perature of the upper honeycomb deck.
Notice that the noise in the lowest frequencies doesn’t vary substantially, but
that the white noise level roughly doubles when the bearing is coldest (around
47
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48
0°C). During lab tests in the fridge at Chicago, the bearing always ran much more
sm oothly warm than it did cold (as cold as —30°C) in the fridge.
4.2
Bearing “Pops”
W hen it got cold the bearing (or bearing housing, or honeycomb decks... we never
could identify the source) made “popping” noises that were fairly well correlated
with bearing angular position. W hat is believed to be microphonic pickup of these
bearing pops can be seen in the data. There are very short events that are not
fit well by a cosm ic ray hit m odel (ie. delta function power on the bolomter) and
are often seen sim ultaneously in m ultiple channels. They also correlate with the
angular position of the bearing, as can be seen in Figure 4.3.
Channel 2 Noise vs Time
0.08
>
c
in
^ 0.06
0
c
CD
13
1
0.04
c
<D
5
0.02
CD
00
*
0.00
0
20
40
60
80
100
Time ( h o u r s )
Figure 4.2 Noise power at various frequencies varies over tim e. Black, purple, blue,
green, and red lines are l/1 6 H z , 0.25Hz, 0.5Hz, 3Hz, and 6Hz respectively. The
dashed curve is the temperature of the upper honeycomb deck (and presumably
the bearing) on an arbitrary scale to show correlation with increased noise. The
upper honeycomb deck tem perature varies between —3°C and 33°C.
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49
Also consistant w ith pre-flight measurements is the temperature dependence of
the bearing pop rate. Figure 4.4 shows a marked rise in bearing pop rate below
15°C.
4.3 Spin-Synchronous Signal
The tim estream signal in every channel is dom inated by a spin-synchronous signal
that has nothing to do with the sky. The signal is m ainly at the first harmonic of
the spin frequency, with an am plitude that varies over tim e (see Figure 4.5) and
a phase that is fixed relative to the tilt direction for channels 1-4 (see Figure 4.6).
The phase of the signal in channels 1-4 changes by 180° when the detector bias
polarity was flipped at the beginning of day 2 and again at the beginning of day
3. This is a strong indication that the signal in these channels is due to power
deposited on the bolometers. The fact that channel 6 (the dark channel) displays
Bearing Pop Angular Distribution -
Day 5 Channel 2
800
600
§
o
400 -
o
200
0
200
300
100
Be ar in g Ang ul ar P osi t i on ( d e g r e e s )
400
Figure 4.3 A typical distribution of bearing “pops”. The angular distribution
changes after each thermal cycle. The pops virtually disappear when the bear­
ing is warm.
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50
very different behaviour (for both am plitude and phase) suggests that the dominant
spin synchronous signal in the optical channels has an optical source, and that there
may be subdom inant spin synchronous signals in the optical channels that have
the same source as the signal in channel 6. The behaviour of channel 5 may be
due to these two sources being of comparable size.
The m ost obvious candidate for an optical spin synchronous signal is atm o­
sphere.
The spin axis is tipped at a variable angle to the vertical, so that the
atmospheric column depth seen by the telescope varies sinusoidally at the spin
frequency. The tilt angle and atmospheric column depth above the balloon vary
diurnally, which could explain the diurnal am plitude variations of the spin syn­
chronous signal. Unfortunately, the phase of the spin synchronous signal is alm ost
180° out of phase with an atmosphere signal. The spin synchronous signal’s m ax­
imum (in terms of power on the bolomters) occurs close to when the telescope is
pointed the highest in the sky (and hence through the least atm osphere). FurBearing P o p s Fr e qu en c y vs T e m p e r a t u r e
£. 2 . 5
cn
m
Q
_
0
5
10
15
20
25
30
35
U p p e r H o n e y c o m b Deck T e m p e r a t u r e ( d e g C)
Figure 4.4 Bearing “pop” rate vs temperature.
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51
Ch 2 -
Ch 1 — 1 st H arm onic A m plitude vs Time
1 st H arm onic A m plitude vs Time
A m plitude
3
2
1
0 .5
0
0
0.0
0
20
Ch 3 -
60
80
40
Tim e ( h rs )
1 st H arm onic A m plitude vs Time
100
20
40
60
80
Tim e (h r s )
Ch 4 — 1 st H arm onic A m plitude vs Time
100
4
3
3
2
2
1
0
0
20
40
60
80
Tim e ( h rs )
Ch 5 — 1 st H arm onic A m plitude vs Time
0
0
100
20
Ch 6 -
3
5
2
x>
2
3
1
1
40
60
80
Tim e (h r s )
1 st H arm onic A m plitude vs Time
100
0) 0
CL
0
0
1
\ :
n
20
40
60
Tim e (h rs )
80
100
0
20
40
60
Tim e (h r s )
80
100
Figure 4.5 First harmonic am plitude for each channel vs tim e. N ote that galaxy
crossings occur around hours 5, 30, 55, and 80.
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52
Ch 1
Ch 2 — 1 st H arm onic P h a s e R elative to Tilt
1st H arm onic P h a s e R elative to Tilt
400
300
3
Ch 3 -
40
60
Tim e (h r s )
1 st H arm onic P h a s e R elative to Tilt
5• ....... t
1
200
100
0
20
Ch 4 -
40
60
80
Tim e (h r s )
1 st H arm onic P h a s e R elative to Tilt
100
40
60
80
Tim e (h r s )
1 st H arm onic P h a s e R elative to Tilt
100
400
•
300
3
20
80
40
60
Tim e (h r s )
Ch 5 — 1 st H arm onic P h a s e R elative to Tilt
200
100
0
20
Ch 6
400
400
(0
0a>
) 300
<D
^ 200
<
(0i>
O
3
200
10 0
o
_
0
20
60
40
Tim e (h rs )
80
100
0
20
40
60
Tim e ( h r s )
80
100
Figure 4.6 First harmonic phase (relative to tilt direction) for each channel vs time.
N ote that galaxy crossings occur around hours 5, 30, 55, and 80.
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53
thermore, the am plitude of the spin synchronous signal due to atmosphere should
vary linearly with the chopped atmospheric column depth during a rotation. The
ambient pressure is proportional to the atmospheric column depth above the bal­
loon, so we can combine the pressure data and the tilt-m eter data to calculate
the chopped atmospheric column depth for each rotation. Figure 4.7 shows that
the am plitude of the spin synchronous signal is not proportional to the chopped
atmospheric column depth, confirming that the signal is not caused primarily by
atmosphere.
Ch 2 -
Ch 1 — 1 st H arm onic Amplitude vs A tm o sp h ere
<u
x>
3
1 st H arm onic A m plitude vs A tm o sp h ere
3
1.5
■Do 2
<D
1.0
.'i-'
rj
Q_
£
< 0 .5
oL
0 .0
0 .0 8
0.10
0 .1 4
0 .1 6
0.12
C h o p p ed c o lu m n d e p th (Torr)
Ch 4 — 1 st H arm onic A m plitude vs A tm o sp h ere
0 .0 6
0 .0 6
0 .0 8
0 .1 0
0 .1 2
0 .1 4
0 .1 6
C h o p p ed c o lu m n d e p th (T orr)
Ch 3 - 1 st H arm onic A m plitude vs A tm o sp h ere
4
3
2
1
oL
0L
0 .1 6
0 .0 8
0 .1 0
0 .1 2
0 .1 4
C h o p p ed c o lu m n d e p th (Torr)
Ch 5 — 1 st H arm onic A m plitude vs A tm o sp h ere
0 .0 6
0 .0 8
0 .1 0
0 .1 2
0 .1 4
0 .1 6
C h o p p ed c o lu m n d e p th (T orr)
Ch 6 - 1 st H arm onic A m plitude vs A tm o sp h ere
0 .0 6
-
r Tit
3
<D
TJ
3
Q
2
1 1
'r
0
0 .0 6
0 .0 8
0 .1 0
0 .1 2
0 .1 4
C h o p p e d c o lu m n d e p th (Torr)
0 .1 6
0 .0 6
0 .0 8
0 .1 0
0 .1 2
0 .1 4
C h o p p ed c o lu m n d e p th (Torr)
0 .1 6
Figure 4.7 Spin synchronous signal shows very little dependance on chopped at­
mospheric column depth.
There does appear to be a strong correlation between the tilt am plitude and the
spin-synchronous signal (see Figure 4.8), though there still appears to be interesting
structure in the residuals to a linear tilt model.
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54
A m plitude
Ch 1 -
Ch 2 — 1 st H arm onic A m plitude vs Tilt Angle
1 st H arm onic A m plitude vs Tilt Angle
-o 2
3
Q_
I 1
0 .5
0
0.0
2
3
4
5
2
6
Tilt a n g le ( d e g r e e s )
Ch 3 — 1 s t H arm onic A m plitude vs Tilt Angle
Ch 4 -
3
4
5
Tilt a n g le ( d e g r e e s )
1 st H arm onic A m plitude vs Tilt Angie
6
3
6
4
3
.
"a
3O
2
3
i 2
E
<
0
0
2
1
3
4
5
6
2
Tilt a n g le ( d e g r e e s )
Ch 5 — 1 st H arm onic A m plitude vs Tilt Angle
3
3
2
-o 2
CD
4
5
Tilt a n g le ( d e g r e e s )
Ch 6 — 1 st H arm onic A m plitude vs Tilt Angle
0
3
"
o
_
I 1
0
0
2
3
4
Tilt a n g le (d e g re e s )
5
6
2
3
4
Tilt a n g le ( d e g r e e s )
5
6
Figure 4.8 Spin synchronous signal shows strong correlation w ith tilt amplitude.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
55
Optical power on the detectors that is proportional to the tilt, in phase with the
tilt, and not due to atmosphere is likely due to an element of the telescope optics
moving transverse to the optical axis during the course of the rotation. Thermal
emmission from the mirrors is a major com ponent of the total optical loading of
the instrument, so that spin synchronous changes in mirror tem perature, or spin
synchronous changes in mirror illum ination will generate a spin synchronous opti­
cal signal. Temperature sensors on the primary mirror show no spin synchronous
temperature variation. Spin synchronous changes in mirror illum ination are pro­
portional to the change in illum ination area (which can easily be made to be
proportional to tilt angle) and also proportional to the absolute tem perature of
the mirrors.
Figure 4.9 dem onstrates that much of the residual structure from
Figure 4.8 disappears when the mirror temperature is included in the model.
The best physical m odel for the source of the spin-synchronous signal involves
a small, spin synchronous movement of the dewar optics relative to the dewar
shell, and since the dewar shell is strapped to the primary, relative to the mirrors.
W hen the spin axis is vertical, the dewar’s optical axis is (like the telescope’s) 12°
from the vertical. The internal support straps strongly constrain the m otion of
the dewar optics in the dewar’s radial directions (ie. in the direction of the optical
axis and in the horizontal direction normal to the dewar’s optical axis) (see Fixsen
et al. (2001)). T hey provide a much weaker constraint on m otion along the long
axis of the dewar. W ith the spin axis vertical, the long axis of the dewar is 78°
from the vertical and has a constant projection onto the local gravity vector. When
the spin axis is tipped over an angle 6, however, the long axis of the dewar has a
variable projection onto the gravity vector as the telescope rotates (from 78° — 9 to
78° + 0). If the dewar supports are spring-like (which they will be to first order),
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56
Ch 2 — 1 st H arm onic A m plitude vs Tilt Angle * Mirror Temp
3
Ch 1 — 1 st H arm onic A m plitude vs Tilt Angle * Mirror Tem p
>
"aO
a
|
Q_
0 .5
0
0.0
400
Ch 3 -
3
400
600
800
1000
1200
1400
1600
Tilt a n g le * M irror T e m p e ra tu re
1 st H arm onic Amplitude vs Tilt Angle * Mirror Temp
Ch 4 4
600
800
1000
1200
1400
1600
Tilt a n g le * M irror T e m p e ra tu re
1 st H arm onic A m plitude vs Tilt Angle * Mirror Temp
TD
>
2
i
2
<
1
E
1
0
0
400
400
600
800
1000
1200
1400
1600
Tilt a n g le * M irror T e m p e ra tu re
Ch 5 — 1 st H arm onic Amplitude vs Tilt Angle * Mirror Temp
3
600
800
1000
1200
1400
1600
Tilt a n g le * M irror T e m p e ra tu re
Ch 6 — 1 st H arm onic A m plitude vs Tilt Angle * Mirror Temp
3
a> 20
t>
2
"ci.
1 1
1
0
0
400
600
800
1000 1200
1400
Tilt a n g le * M irror T e m p e ra tu re
1600
400
600
800
1000
1200
1400
Tilt a n g le * M irror T e m p e r a tu r e
1600
Figure 4.9 Spin synchronous signal shows even stronger correlation with the prod­
uct of tilt am plitude and mirror temperature.
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57
then the m otion of the dewar optics in the long dewar axis direction will go like
0 cos ( u t+ 5 ) , where u is the spin frequency and u jt+ S = 0 when the long dewar axis
is pointed towards the tilt (which will also be in phase with the tilt meter, since the
tilt meter axis and the long dewar axis are approximately aligned). As the dewar
optics moves back and forth relative to the mirrors, the horn beam moves back and
forth slightly across the secondary, and the reflected beam moves back and forth
slightly across the primary. The primary is underfilled by the beam, so that the
total illum ination of the primary w on’t change by slightly moving the beam on it.
The secondary, on the other hand, is overilluminated, with some of the geometric
rays com ing out of the dewar horn missing the secondary com pletely (this isn ’t a
problem because they then fall on the cold sky - primary spillover would fall onto
hot structural elements of the telescope). Moving the horn beam slightly on the
secondary therefore slightly changes the total solid angle of the beam that hits the
hot secondary. The horn beam is effectively doing a small chop between the hot
secondary and the cold sky.
Channel 5 remains somewhat of a mystery. The higher spin harmonics in each
channel show evidence of much smaller spin synchronous signal, though they don’t
show the same am plitude variations as the first harmonic.
4.4
Correlated N oise Between Channels
There appears to be noise that is correlated between the channels. This poses a
problem for constructing a dust m odel to be removed from the CMB channels, as
the m odel assumes that maps at different frequencies have uncorrelated noise (see
Section 9.2). The correlated noise is likely microphonic in origin. We use the dark
channel (channel 6) as a monitor of this noise source and construct a correlated
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58
noise m odel for each channel using channel 6 as a tem plate. The effective transfer
function between the source of the correlated noise and each detector is different
(detectors are mounted in different locations, dewar wiring differs from detector to
detector). We therefore construct a noise tem plate for each channel from channel
6 frequency by frequency. First we F F T each rotation, generating 256 frequency
bins per rotation and approxim ately 5000 rotations per day. For each frequency
bin, we then fit each day’s worth of data in the optical channel to channel 6
m ultiplied by an am plitude and phase. Once best-fit am plitudes and phases for all
256 frequency bins are found, we build a m odel noise tim estream by F F T ’ing back
to the tim e domain and subtract this noise m odel from the optical channel. This
process is repeated for each channel on each day independantly. The fit is carried
out on residual tim e streams that have had the best-fit m odel of the sky and the
spin synchronous signal removed from them. The correlated noise m odel is then
removed from the original tim estream and the process iterated (see Part III).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Part III
Analysis
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C H A PT E R 5
ANALYSIS OVERVIEW
The analysis pipeline can be broken up into two broad sections. The first is called
the Preprocessing Loop and is roughly outlined in the flowchart found in Fig­
ure 5.1. The preprocessing loop cleans the raw tim estream of glitches, deconvolves
the instrum ent transfer function, fits pointing m odel parameters and noise m odel
parameters iteratively to produce cleaned, whitened data tim estream s, a pointing
matrix, and noise m odels that can be used as input to M ADCAP for full analysis.
The second section of the analysis pipeline uses the M ADCAP software and dust
modelling software with the inputs generated by the preprocessing loop to gener­
ate maps, covariance matrices, a dust tem plate, a CMB map and power spectrum.
This pipeline section is depicted in Figure 5.2. The shaded boxes in these two
flowcharts correspond to parts of the analysis pipeline that are fully described in
this thesis
The analysis is broken into two pipelines with two different mapm aking tech­
niques at their cores. The preprocessing loop utilizes a very fast “multigrid” mapmaking technique (see Section 7.1) which can quickly produce an optim al map on
a small computer, but cannot produce the pixel covariance m atrix associated with
the map. Since the pixel covariance is required for power spectrum estim ation, the
M ADCAP pipeline uses a brute force inversion to get the pixel covariance m atrix
and optim al map. The inversion is so com putationally intensive that it can only
be performed a few tim es on a supercomputer, requiring that we develop the fast
mapmaking procedure used in the preprocessin loop.
To understand the rather com plex analysis pipeline it is insightful to begin at
the end and work backwards.
The ultim ate goal of the analysis is to estim ate
60
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61
'
Raw
Timestream
Data
Derail, Deglitch,
and Whiten
Noise
Timestream
'Deglitched,
Whitened
Timestream
Data
Build Pointing
Matrix
Fit Timestream
Noise Model
I
Cleaned,
Deglitched,
Whitened
Timestream
Data i
Fit Spin
Synchronous
Model and
Map
Compare wih
Last Iteration
Converged?
Y es
No
Do Pointing Fit
No
Compare wih
La3t Iteration
Converged?
Yes
d a ta
p roduct
p ro c e ss
Pointing
Matrix
Figure 5.1 The Preprocessing Loop flowchart. D ata is cleaned and noise and point­
ing m odel parameters are fit iteratively to produce cleaned tim estream , pointing
matrix, and noise m odel that can be handed to M ADCAP for full analysis. The
shaded boxes correspond to the parts of the pipeline fully described in this thesis.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
62
Deglitched
Whitened
Tlmestream
Timestream
Pointing
Matrix
Noise
Model
/
/
/
/
/ /
/
S>™ hT
t Mo? ' '
Templates
MADCAP
Mapmaking
Channel
3,4,5
Maps
Channel
3,4,5
/
Covariance /
Matrices /
/
/ Channel
2 Map
/
Channel 2
Covariance
Matrix
Build Dust
Model
Remove Dust
Model from
Channel 2
Dust
Map
Template
Dust
Template
Covariance
Matrix
Beam
/
/
Noise
Window
Covariance
Function / /
Matrix
I j
MADCAP Power
Spectrum
Estimation
CMB
Power
Spectrum
Fisher
Matrix
d a ta
pro d u ct
p ro c e ss
Figure 5.2 The M ADCAP pipeline flowchart. The input tim estream s, pointing
matrix, and noise m odels are taken from the output of the preprocessing loop
depicted in Figure 5.1. The shaded boxes correspond to parts of the pipeline fully
described in this thesis.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
63
the CMB power spectrum.
The M ADCAP package im plem ents the Quadratic
Estim ator M ethod of Bond et al. (1998) and can generate an estim ate of the
CMB power spectrum given a CMB map and pixel-noise covariance matrix. This
calculation involves many m ultiplications and inversions of very large matrices,
and can only be done a few tim es on the IBM SP supercomputer at NERSC.
Generating a CMB map and covariance m atrix free of dust contam ination in­
volves generating a dust m odel and covariance m atrix from channels 3, 4, and 5
and then projecting this m odel from channels 1 and 2. This procedure requires
maps and covariance matrices for all five channels, and again is carried out on the
NERSC supercomputer.
M A D C A P’s mapmaking procedure is able to generate maps and covariance ma­
trices for each channel given a deglitched, whitened tim estream , pointing matrix,
tim estream noise m odel, and design tem plate vectors for the spin synchronous sig­
nal model. The M ADCAP mapmaking procedure involves the inversion of a very
large m atrix and also requires execution on the supercomputer.
The presence of the pixel-pixel noise covariance matrices in all the calcula­
tions in Figure 5.2 requires that they each be performed a sm all number of times.
Going from the raw tim e stream data to data ready to be input to M A D C A P’s
mapmaking procedure requires many iterations to find the unknown parameters in
the pointing m odel (see Section 6.1), distinguish cosmic ray strikes and bad data
from sky signal, and properly estim ate the tim e-stream noise in the presence of
sky signal and spin synchronous signal. Efficiently finding iterative solutions to
these problems required that we build the preprocessing loop shown in Figure 5.1.
The key feature of the preprocessing loop is that none of its steps require the
inversion or m atrix-m atrix m ultiplication of large matrices. The multigrid map-
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64
making algorithm makes use of a clever expansion to replace a m atrix inversion
with m ultiple m atrix-vector m ultiplications (see Section 7.2), allowing map solu­
tions (w ithout covariance matrices) to be generated on a m odest desktop computer
in a few minutes.
The preprocessing loop serves sim ply to produce cleaned, whitened tim e-stream
data, pointing matrix, and tim e-stream noise m odels for input to the M ADCAP
software that will generate full solutions with covariance matrices. O f course the
fast multigrid mapmaking procedure was also extremely valuable in our study of
the spin synchronous signal, which occupied the majority of the tim e spent on the
analysis even though the spin synchronous m odel appears as a sm all com ponent
of the flowchart here.
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C H A PT E R 6
PR EPR O C ESSIN G LOOP
The preprocessing loop is depicted in Figure 5.1. The procedures in the shaded
boxes are sections of the analysis pipeline for which I contributed substantially and
are fully described in the following sections.
6.1
Building the Pointing M atrix
Reconstructing the beam right ascension and declination from the pointing sensor
data involves a bit of ellaborate spherical trigonometry. Recall that a spherical
triangle is a triangle on the surface of a sphere whose sides are great circle arcs.
For a spherical triangle with interior angles A, B, C and sides a, b, c as depicted
in Figure 6.1 then the following trigonom etric identities are useful:
sin a
sin 6 sin e
Sme Rule: —— - — —— —= ———
sin A
sin B sin C
Cosine Rule for sides: cos c =
(6.1)
cos a cos b + sin a sin b cos C
(6.2)
Cosine rule for angles: cos C = — cos A cos B 4- sin A sin B cos c
(6.3)
The geom etry for the TopHat pointing m odel is seen in Figure 6.2 and consists
of four spherical triangles. The vertices of the triangles are the sun, the telescope
spin axis, the beam center, the zenith, and the “effective zenith”. The effective
65
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66
zenith points opposite the apparent direction of the local gravity vector i f in
the balloon’s frame of reference.
If the balloon is moving at constant velocity
across the Earth’s surface, the zenith and the effective zenith coincide. W hen the
balloon accelerates (when travelling through wind shears for instance) the linear
acceleration i t appears in the tilt-m eter data as if the local gravity vector were
i f ' = i f — i t . The effective zenith points in the direction opposite i f ' .
The spherical triangles in Figure 6.2 are labelled 1 through 4, and it is in this
order that they are solved to get the coordinates of the beam center. Triangle 1
consists of:
Zenith
The ra and dec of the zenith are determined by the GPS latitude,
longitude, date, and tim e signals. These are sampled at 1Hz.
Sun
The ra and dec of the sun are calculated from the GPS date and tim e
using sunradec.pro (an IDL routine from the IDL astronomy library).
4z
Figure 6.1 A spherical triangle with interior angles a, b, c and sides A, B, C
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 6.2 Pointing Model Geometry
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68
These are sampled at 1Hz.
Z sun
The zenith distance of the sun (Z sun ) is easily calculated from the
zenith and sun positions calculated above.
a
The azimuth difference between the sun and the effective zenith. The
azimuth of the effective zenith is opposite the direction of the balloon’s
acceleration. The balloon’s acceleration is determined by locally fitting
the GPS latitude and longitude to polynom ial functions of tim e and
differentiating twice to get accelerations.
5
The displacement angle between the zenith and the effective zenith.
This angle is given by arctan |= f | where ~ct is the balloon’s horizontal
linear acceleration determined by locally fitting the GPS latitude and
longitude to polynom ial functions of tim e and differentiating twice to
get accelerations. 6 is typically a few arcminutes.
Z'sun
The effective zenith distance of the sun. Solved for using
cos Z'sun = cos Z sun cos 5 + sin Z sun sin S sin a
/3
(6.4)
Solved for using
.
sin Z sun sin a
( 6 -5 )
and
cos (3 =
cos Z sun - cos 5 cos Z'sun
G in 5
AG
i n Z'sun
7J
sm
sin
(6.6)
Triangle 2 locates the spin axis relative to the sun and the effective zenith.
consists of:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
It
69
Z'sun
From triangle 1 above
9
The angle between the spin axis and the effective zenith (also called
the tilt angle). 9 ranges between 3 and 6 degrees throughout the flight.
It is determined by taking
t i l t x 2 + t i l t y 2 after removing offsets from
tiltx and tilty.
0
The azimuth angle
(about the spin axis) between the sun and the
effective zenith. Measured by comparing the phase of the glint sensor
firings to the tilt-m eter m axim a (see Section 6.1.1).
8 sun
The sun altitude in the telescope frame of reference (ie the angle be­
tween the spin axis and the su n). Supposed to be measurable w ith the
glint sensors (see Section 6.1.2), but can also be solved for using
sin a sin 0 cos Z ' cos 9 — cos a cos 0
^
cos S sun = ------1 — sm Z'sun sin 9 sm a sm 0
7
6.7
Solved for using
sin a sin 0 cos Z '
cos 1
cos 9 — cos a cos 0
— a -----1 - s m a sm 0 sm Z ' sm
9
(6-8)
and
si: S sun sin 0
sin
7—^ -----sin Z,sun
ss m
— —
m 7'■v =
, .
(6.9)
Triangle 3 p osition s th e beam relative to th e spin axis and th e effective zenith. It
consists of:
9
From triangle 2
06eam
Telescope opening angle, nom inally 12°.
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70
p
The telescope rotation angle is determined by taking arctan ( t i l t y / t i l t x ) .
Z'beom
The effective zenith distance of the beam. It is solved for using
cos Z'beam = cos 6 cos (pbeafrn + sin 9 sin (j)heam COS P
v
(6.10)
Solved for using
COS (fibeam
c o s ^ = ----------- . a
COS 6 COS Z b
y ,---------------
sm 0 sm Z heam
{
^
(6-11)
and
sm v =
sin (pbearn sin p
(6 .12)
S in Z'beam
and finally triangle 4 locates the beam relative to the true zenith and therefore to
a known RA and Dec. Triangle 4 consists of:
Z beam
From t a n g l e 3
S
From triangle 1
X
Solved for using
X = 2 7 T - / ? - 7 - jz
Zbeam
The beam zenith distance is solved for using
cos Zbeam = cos 5 cos Z'beam + sin S sin Z beam cos x
£
(6.13)
(6.14)
Solved for using
COS £ =
COS Z beam ~~ COS ^ COS Zbeam
sin 5 sin Z beam
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/g ^
71
and
s in £ =
(6 16)
S1H
% beam
Once the beam zenith distance and azimuth relative to the sun are known it is a
simple m atter to generate the RA and Dec of the beam.
6.1.1
Glint Sensor Phase Angle D eterm ination
Each glint sensor has one vertical slit in the middle of its field of view, and two
parallel 45 degree slits that form an N shape on the sky (see Figure 1.4). The
central slit of each glint sensor is used to determine the rotation phase angle of
the telescope w ith respect to the sun. The rotation angle between glint sensors
was calibrated during a ground test on December 2 6 ^ , 2000. There is an overall
rotation angle offset, which is allowed to vary as the pointing parameter 9sunx. It
should be an angle close to zero.
The center tim e of each central glint is calculated from the glint sensor data
(with a few thousandths of a second precision) and a list of (tim e,rotation phase
angle) ordered pairs is constructed. In m ost rotations all four glint sensors see
three glints and provide a rotation phase angle measurement. W hen the sun is
high in the sky, and the telescope spin axis is tilted towards the sun, som e or all
of the glint sensors fail to image the sun in all three slits. These events are not
used to generate (tim e,rotation phase angle) pairs. If no glint sensor images the
sun in a rotation then the maximum of the su nx sensor is used in place of glint
sensor 0 (the offset between sun x maximum and glint sensor 0 was measured in
the ground test and shouldn’t change, since the + sunx sensor and glint sensor 0
are mounted on the same bracket). Once the com plete list of (tim e, rotation phase
angle) is generated, cubic spline interpolation is used to generate rotation phase
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72
angles for all sample tim es. N ote that the central slits on the glint sensors are not
quite vertical. This leads to a slight sun-altitude dependence in the rotation phase
angle for each glint sensor. We evaluate the pointing m odel iteratively, assuming
first that the sun altitude in the telescope frame is the same as the sun altitude
in the zenith frame. After the spin axis location has been determined, the sun
altitude in the telescope frame is calculated and the glint rotation phase angles
are recalculated. These new rotation phase angles are then used to recalculate the
rest of the pointing model.
6.1.2
Glint Sensor Sun Altitude D eterm ination
The glint sensors are also able to measure the altitude of the sun in the telescope’s
frame of reference. As the telescope rotates, the sun crosses the N in Figure 1.4
more or less horizontally. The tim e between the three glints depends on the altitude
of the sun. If the tim es of the three glints are given by t\ , t 2, and t 3 then the glint
“crossing ratio” R c defined as
turns out to be a nearly linear function of sun altitude. The R c - altitude rela­
tionship for each glint sensor was calibrated from the December 2 6 ^ ground test
data. A second order polynom ial fits the data well, leaving randomly distributed
residuals w ith RMS am plitude ~ 5 arcminutes. The ground calibration was done
with the telescope spin up.
If the rotation direction is reversed, then the first
and third glints swap roles, and the same calibration can be used with 1 — R c as
argument instead of R c .
The in-flight glint altitude data is not internally consistant. Each glint sensor
provides an altitude measurement each rotation, and the four altitude measure­
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73
ments do not agree with each other (at the 1.5° level). The difference in altitude
measurements in the four glint sensors changes over tim e and appears to be highly
correlated with the tem perature of the sun shield. In particular, when the sun is
very high and the upper portion of the inside surface of the sun shield is illum i­
nated there appear rather sudden changes in the relative altitude measurements of
the glint sensors. W hen the sun shield is illum inated like this there is a significant
temperature gradient between the inner skin of the sun shield and the outer skin of
the sunshield, as well as a significant tem perature gradient along the height of the
inner skin. Differential thermal expansion is likely causing the sunshield to deform
assym etrically and changing the relative pointing of the four glint sensors. For
this reason we do not use the glint sensor altitude measurements in the pointing
model.
The glint sensor azimuth measurements do not appear to be affected by this
shield deformation, indicating the deformations m ainly deform the sunshield skins
vertically and radially, which moves the glint sensor pointing vertically, but not
with any significant horizontal m otion. Throughout the flight, the relative tim ­
ing of the four glint sensor central glints is consistant w ith the pre-flight ground
measurements, even showing the same slight dependence on sun altitude due to
the mask slits not being vertical. There does appear to be a slight constant az­
imuth offset in one of the glint sensors (about 6°) that could have been caused by
that glint sensor assembly being bumped between the ground test on December 26
and attaining float altitude on January 4. The offset is easily measured from the
in-flight data. We feel confident that the glint sensor azimuth measurements can
safely be used in the pointing model.
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74
6.2
Pointing M odel Param eters
The pointing m odel has four a priori unknown parameters. They are
0sunx
The azimuth angle between the boresight of glint sensor zero and the
tilt meter x-axis. This parameter should nom inally be zero, as the tilt
meter axis and the glint sensor axis are aligned to w ithin a few degrees.
@beam
The azimuth angle between the boresight of glint sensor zero and the
beam.
This parameter should also nom inally be zero, as the beam
optical axis is aligned w ith the glint sensor axis to within a few degrees.
<f>beam
The beam opening angle is the angle between the spin
axis and the
beam center. This parameter should nom inally be 12°.
gtilt
The gain of the tilt-m eter. The tilt meter was calibrated before flight,
but not with the accuracy required to handle the large anomolous tilt.
This parameter should therefore be close to 1.
In principle, the tilt-m eter x and y axes could have seperate gains in the pointing
model. The large, slowly varying tilt allows us to cross calibrate the x and y axes
of the tilt-m eter directly from the raw tilt-m eter data so that the single parameter
gtiit can be used sim ultaneously for both axes.
6.3
Pointing M odel Fit
To find the best fit pointing m odel parameters we find the pointing m odel param­
eters p* = (Osunx, Obeam, (f>beam, gtiit) that minimize the mapmaking y 2
X 2 ( l ? ) = (A ( j ? ) s ('p ’) ) T N -1 A (-j?) s ( j ? )
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(6.17)
75
where A (~jt) is the pointing m atrix and s ("]?) is the best fit map for the pointing
parameters jf, which is given by Equation 7.7. See Section 7.1 for a com plete
description of the mapmaking process.
The data used in this fit must first have the best fit spin synchronous signal
model removed from it. We perform the pointing fit on a small subset of the tim e
ordered data, selecting only rotations that cut through the galaxy. The reasons for
this cut are twofold. First, reducing the number of tim estream points reduces the
tim e taken to evaluate Equation 6.17 for each point in the parameter space search.
Second, alm ost all of the signal is in the galaxy, so that by restricting ourselves
to rotations cutting through the galaxy we keep all of the signal but reduce the
number of degrees of freedom in the fit, making the contours in y 2space much
cleaner.
Since Equation 6.17 is nonlinear in
a search of the parameter space is re­
quired to find the best fit parameters. The x 2 space has several local minima,
neccessitating a fairly thorough grid search. It turns out that the ( 0 s u n x ,9 b e a m )
subspace is fairly uncorrelated from the
(0beaTO,gtut)
subspace (which makes intu-
titive sense because the 0’s produce tw ist of the observing circle about the spin
axis, while tpbeam and g tui move the observing circle on the sky w ithout tw isting it.
We therefore start with the parameters at their nominal values of (0,0,12,1) and
iteratively construct two dimensional grids in the (0sunx,6beam) and {(frbeam, g m t)
subspaces with a grid spacing in each parameter corresponding to ~ 1.2 arcminutes on the sky, until a stable minimum is found (the convergence is reasonably
fast).
The pointing fit can be done on each channel independantly, and the best fit
parameters are found to be identical in each case (channel 4 is used in the final
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76
fit, as it has the highest signal to noise). This indicates that transfer function
problems are not affecting the pointing fit, and that the beams all share a common
center (as was seen during the pre-flight beam m apping).
The pointing fit can also be done on each day individually. This has the ad­
vantage that the spin direction changes between days. The projection onto the sky
of tim ing offsets in the data (which are degenerate with azimuth angle offsets in a
single spin direction) change sign when the telescope spin direction changes (while
azimuth angle offsets don’t change sign), allowing us to detect tim ing offsets in the
data that had not been previously accounted for. The final pointing m odel fits are
done with galaxy crossing rotations from all four days sim ultaneously to improve
signal to noise.
Figure 6.3 shows the x 2 contours around the best-fit pointing m odel parame­
ters. Table 6.1 lists the best fit values and uncertainties for the pointing m odel
parameters.
Table 6.1 Pointing m odel parameters and uncertainties. The parameter uncertainty
projected onto the sky for 9sunx and gtat depend on the tilt am plitude and sun
position and flight averages are quoted here.
9 su n x
- 1 .6 °
I
0
CO
9 beam
4*beam
11.78°
1.008
9 tilt
Uncertainty
U0
B est fit value
p
Parameter
0.3°
0.08°
0.012
Uncertainty projected onto sky
~ 3.6 arcminutes
3.6 arcminutes
4.8 arcminutes
~ 3.6 arcminutes
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77
Pointing Fit* 2 Slice
•1
■2
-3
-4
-5
-6
■2
-3
0
•1
1
ft*,™ (degrees)
Figure 6.3 Pointing m odel fit x 2 contours. This slice through the (Osunx,0beam)
subspace is evaluated with ((pbeam, 9tut) held at their best-fit values and has contours
spaced by lcr.
Pointing Fit^ 2 Slice
1.06
1.04
1.02
do 1.00
0.98
0.96
0.94
11.2
11.4
11.6
11.8
12.0
12.2
^beam(degrees)
Figure 6.4 Pointing m odel fit y 2 contours. This slice through the (4>beam>gtiit) sub­
space is evaluated with (9sunx, Obeam) held at their best-fit values and has contours
spaced by lcr.
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C H A PT E R 7
M A PM A K IN G
7.1
General Derivation of Skymap from Time-Ordered
D ata
For each detector channel, we have a stream of time-ordered data (TO D ) dt that
we would like to turn into a map. Given the sky coordinates (9t , ij)t) for each tim eordered data point, and dividing the sky into J\fpix pixels, we can construct the
pointing m atrix A that projects the skymap into the tim e stream.
l if
ep
Atp = <(
(7.1)
0 otherwise
where the subscript t runs over all time-ordered data points and the subscript p
runs over all pixels. The tim e ordered data can now be written as
d t = A tpSp + n t
(7-2)
where nt is the tim e-stream noise, and s p is the sky map convolved by the (assumed
symmetric) instrument beam and pixelized into N p pixels. If the tim e-stream noise
is Gaussian, the noise probability function can be written as
P(n) = ( 2 7 r ) - ^ e x p - i (nTN _1n + Th [InN])
(7.3)
Li
where N is the tim e-tim e noise correlation m atrix given by
Ntt' = (n tn £ )
78
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(7.4)
79
and the () indicate an ensemble average over all possible realizations of the tim e­
stream noise. We can use equation (7.2) to substitute for n in equation (7.3) to
get a probability distribution for dt, given the underlying sky map sp
P (d |s ) = ( 2 7 r ) " ^ e x p |- ^ ( ( d - A s ) T N ( d - A s) + TV [InN])
j
(7.5)
Using Baye’s Theorem, with uniform priors for sp, Equation (7.5) is proportional
to the likelihood of the sky map sp given the time-ordered data dt. The log of this
likelihood can be w ritten as
£ = _ ! ( ( d - A s)T N (d - A s) + TV [In N ])
(7.6)
M aximizing with respect to sp gives the best-fit sky map sp
sp = (A t N -1 A) _1 A t N “ M
(7.7)
Using equation (7.2) to elim inate d, we get
sp =
(A t N -1 A) 1 At N _1 (A sp + nt)
(7-8)
which dem onstrates that the best-fit map is the true sky map plus some pixel noise
given by
n p = (A t N "1A ) “ 1 A TN ~ 1nt
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(7.9)
80
Assum ing that the pixel noise and the sky map are uncorrelated,
( s s T ) = ( s s T ) + ( n n T)
(7-10)
where the pixel-pixel noise covariance N ppiis given by
( n n T)
( (A t N “ 1A ) -1 A ^ N ^ n ^ N ” 1A (At N _1A ) _1 ^
(A t N _1A) _1 A t N - x (n t n £ ) N _1A (A TN “ XA ) _1
( a t n - 1a ) “ 1 a t n - 1n n ~ 1a ( a t n - 1a ) “ 1
( a t n - 1a ) “ 1
(7.11)
To determine the CMB power spectrum from equation (7.10) we need the best fit
map from equation (7.7) and the pixel-pixel noise covariance m atrix from equa­
tion (7.11). Nppi is an N ViX^ N PiX matrix, and building it requires us to invert a
matrix of this size, which is a tim e consuming, memory intensive calculation for
~ 40 ,000 pixel maps.
For many purposes (determining the best-fit pointing for
example) we only require the best-fit map, which naively requires us to first build
Nppi and so is equally tim e consuming. We now derive a fast iterative technique
for determining the best fit map that does not require a large m atrix inversion.
7.2
F ixsen’s Expansion
If we define the pixel-pixel weight m atrix to be
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81
=
A TN -1 A
(7.12)
then equation (7.7) becomes
= W ~ xA N _1d
(7.13)
We renormalize W by m ultiplying it by A"1/ 2 where A is a diagonal m atrix whose
diagonal elem ents are the diagonal elem ents of W .
§p =
=
A“ 1 /2 (A -1/2W^A“ 1/2) _ 1 A -1/2A N _1d
(7.14)
\ ~ 1/2 ( I - M ) - 1 b
(7.15)
where M = \ ~ 1/2W X ~ Y' 2 - I is a m atrix with zero on the diagonal and small
off-diagonal elem ents (assuming that W is m ostly diagonal, which should be true
for TopHat) and b= A“ 1/ 2A N _1d is the noise weighted, coadded map,normalized
by A-1 / 2. Since M
is small, we can use a Taylor expansion for the inversion to get
Sp
OO
= A“ 1/2] T M n&
n—0
=
zn + 1
z0
A-1 / 2.Zoo , where
(7.16)
(7-17)
= M z n + b , and
(7-18)
= b
(7.19)
So we have replaced an 0(J\fpix) m atrix inversion with an infinite number of 0 (A fpix)
m ultiplications of a m atrix on a vector. If M is small enough, however, we can
truncate the recursion (and therefore the sum) after a small number of iterations
to get a very good approximation to sp. Furthermore, M tends to be sparse (~10%
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82
for TopHat), so that the memory requirements for storage are substantially less
than for the entire covariance m atrix N (though still too large for the computers
available at the tim e), and the m ultiplication can be speeded up considerably by
proper cleverness. I describe this cleverness (only some of which is due to me much is due to Dale Fixsen) below.
7.3
Building the Weight M atrix
First we consider how to efficiently build the pixel weight m atrix W . Recall from
equation (7.12)
W = At N - xA
(7.20)
The pointing m atrix A has a particularly simple structure. Each row has a single
non-zero column, corresponding to the pixel that was being observed during that
row’s tim e sample. Furthermore, each non-zero element is a 1, since we are gener­
ating a map of the sky convolved with the instrument beam, and assum ing the gain
of the instrum ent is constant. If the instrument noise is stationary and guassian,
then the tim e-tim e noise covariance m atrix N -1 can be determined sim ply from
the tim e-stream noise power spectrum as follows, taking T to be the Fast Fourier
Transform (FFT) operator over all tim e.
N a,
=
( n ( t ) n ( t '))
(7.21)
(7.22)
.7 W -1
(7.23)
(7.24)
(7.25)
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83
(7.26)
(7.27)
where n (u>) is the tim e-stream noise power spectrum. We can therefore calculate
a row of N a,1 by Fourier transform as follows
N -,1 =
(7.28)
( B / . T - 1) 1
(7.29)
=
r
(u{uf
S ^ y 1r
(7.30)
(7.31)
and since the noise is taken to be stationary, N^,1 is circulant (the n + 1th row is
equal to the n th row shifted to the right by one position), and fully characterized
by one row.
A row in N^,1 is formally nonzero for all lags A t = t/ — t, but
asym ptotically approaches zero rather quickly for lags long compared to the 1 /f
knee of the instrument (~0.5H z).
The TopHat observing strategy enables full
reconstruction of the skymap over the entire observation area using only frequencies
equal to the spin frequency (1 /1 6 Hz) and higher (any signals present in frequencies
below the spin frequency are also contained in frequencies above the spin frequency,
excepting the dc level to which we are entirely insensitive).
explicitly truncate
We can therefore
at lags longer than one rotation and still fully reconstruct
the skymap. We will lose some signal-to-noise, since we are throwing away data at
low frequencies, but these low frequencies are where the noise is biggest and most
difficult to properly estim ate. As we will see, the mapmaking speed scales like the
square of the number of nonzero lags in N^,1, so this slight loss of signal-to-noise
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84
is more than com pensated for by mapmaking efficiency.
Unfortunately our detector noise was not stationary during the flight (see Sec­
tion 4.1).
It varied diurnally both in am plitude and in spectral shape, closely
tracking the tem perature of the bearing (presumably due to increased microphonic
noise). The variations in the noise power spectrum were on tim escales long com­
pared to a rotation (and therefore long compared to the support of N tt/) and we
can approximate the noise as being piecewise stationary, and N",1 block circulant
with each block being calculated as above using the tim e-stream noise power spec­
trum for that particular stationary piece. We again lose some signal-to-noise with
this approximation because we give zero weight to difference measurements made
across the boundary between stationary pieces. We will now take this approxima­
tion to an extreme and make each rotation a stationary piece. The noise power
spectrum may be the same in adjacent rotations, but we take
to be block cir­
culant with each block being one rotation long. Formally, if f is the F F T operator
on one rotation, then we define our quasi-Fourier operator F to be block diagonal,
with f on each diagonal block.
[f]
0
0
...
0
0
[f]
0
...
0
0
0
[f]
•••
0
0
[f]
0
0
I refer to this as a “quasi-Fourier” operator because it operates on a tim e-dom ain
vector to give a vector partway between the tim e and frequency domains. The F
operator takes each individual rotation in a tim e-dom ain vector and F F T ’s it in
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85
place, with the resulting vector being a rotation-ordered set of frequency domain
data. This operator will be convenient in calculating W = A TN -1 A because the
pointing m atrix A is sparse in the tim e domain, while the noise m atrix N _1 is
sparse in the frequency domain. The quasi-Fourier operator F transforms them
into a basis in which they are each still relatively sparse, allowing for efficient
calculation of the contraction.
We can now rewrite equation (7.12) as
W
=
A t N - xA
(7.33)
=
A t ( F - 1F N F " 1F ) _ 1 A
(7.34)
=
A t F _1 (F N F -1 ) -1 FA
(7.35)
=
A t F - 1N “ 1FA
(7.36)
where the rotation by rotation frequency weight m atrix N ” 1 is diagonal and has
one over the square of the noise power spectrum for each rotation on its diagonal.
F -1 N -1 F is block diagonal, with each block being one rotation wide. If we rewrite
this as a three indexed object Gr>ij, with the first index specifying the rotation
number, and the other two indices specifying the location within that rotation’s
submatrix, and we rewrite the pointing m atrix as a three indexed object A r jp,with
the first index specifying rotation number, the second index specifying sample
number within that rotation, and the third index specifying pixel number, we can
rewrite equation (7.36) as
T lrots n f r e q n f r e q
W pp'
=
y]
r = 1 i= 1
^ r ,ip G r ,ij^ -r,jp '
j=1
where n rots is the number of rotations and n f req is the number of frequencies in the
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86
one rotation F F T operator f. For 16 Hz sam pling and a one day map, Ufreq = 256
and n rots = 5000. Since our noise is piecewise stationary over about 10 pieces per
day, there are only about 10 different Grjij submatrices that can all be precalculated
and stored in memory. Since A r .ip is 1 for p — p ri and 0 otherwise, we can construct
W ppi using the following algorithm
set WpP>= 0
for each rotation r
for each sample i in rotation r
for each sample j in rotation r
^
Pr,iPr,j =
^ P r .iP r j
^ r ,ij
and building Wpp> takes n rot x n f2req = n tod x n freq floating point additions.
We explicitly set the d.c. weight to zero for each rotation since the experiment
is a.c. coupled. This introduces a singularity in W (the d.c. mode of the weight
m atrix is zero) that we must take into account when evaluating W ~ l via equa­
tion (7.18). After each iteration, we add an offset to z n + 1 so that the pixel values
in z n + 1 sum to zero.
Even though W pp> is fairly sparse (only about 10% of it ’s elem ents are non­
zero), it was still too big to hold in memory on the computers we were using for
the bulk of the analysis. Equation 7.18 actually only requires us to com pute the
product of the m atrix M on a vector, not com pute the m atrix M itself. We can
adapt the algorithm above to calculate y, the product of M with a vector x as
follows
set Xp = 0
for each rotation r
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87
for each sample i in rotation r
/V r,i “ ^ P r ,i
^ r ,i,i
set yv — 0
for each rotation r
for each sample i in rotation r
for each sample j %i in rotation r
G r , i , j % p r ^j
V P r.i
~
V p r,i “T
\ 1/2
APr ,i
,1 /2
Pr ,j
This operation requires O (n rot x n j req — n tod x rifreq) operations for each m ulti­
plication of the m atrix on a vector, as opposed to the O (0.1ripix) operations for the
m ultiplication of a 10% sparse m atrix on a vector. For a single day of TopHat data,
ntod — l-5 x l0 6, n f req = 256, n pix — 45000 so that the new algorithm is requires
~ 4 x l0 8 operations compared to ~ 2 x l0 8 operations for the sparse m atrix m ulti­
plication. This factor of two loss in performance is com pensated for by the memory
requirements dropping from ~ 1 .2 G B for storing the sparse m atrix to ~24M B for
storing the tim estream data and pointing, making the algorithm accessible to a
quite m odest computer.
7.4
M ultigrid M apmaking
Another order of m agnitude speed up in the mapmaking process can be gained
by careful exam ination of the convergence properties of Equation 7.7. The spa­
tial m odes of the map that converge the slowest are the modes with the lowest
weight in the weight m atrix W . In the case of TopHat, the 1 / / noise makes the
lowest tim estream frequencies have the lowest weight. If the map coverage is well
connected, with pixels crossed in several directions at different observation times,
then the low spatial frequency modes have the lowest weight in the weight m atrix
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88
W . The fine structure in the maps quickly converges in a few 10’s of iterations,
while the large angle structure (such as the dipole) takes hundreds of iterations to
converge.
Borrowing a technique from the M APCUM BA developers (Dore et al., 2001),
we exploit these convergence characteristics by regridding the map onto a smaller
set of coarse pixels.
The full resolution map is pixelized using the HEALPIX
(Gorski et al., 1999) ring scheme with nside=256. The pixels are approxim ately
14 arcminutes on a side, and number about 45 000 in our sky coverage. One of
the advantages of using HEALPIX is that it is a hierarchical pixelization scheme.
Each pixel in the nside = 2n pixelization contains four pixels from the nside =
2n+1 pixelization and the HEALPIX package provides a simple m apping between
the two pixelizations.
To coarsen a map by one HEALPIX level we take each
coarse pixel to be the mean of the 4 corresponding pixels in the fine map. To
refine a map by one HEALPIX level we take each fine pixel to have the value of
it ’s corresponding pixel in the coarse map.
Coarsening the pixelization by one HEALPIX step cuts the pixel resolution in
half and correspondingly reduces our required tim estream resolution by a factor of
two. Therefore with each coarsening of the map pixelization we also sm ooth and
subsample the tim estream to make a coarsened tim estream that is half the size of
the uncoarsened tim estream . We never need to refine a tim estream , since we start
with the highest resolution tim estream and coarsen it to the required tem poral
resolution. The “multigrid mapmaking” algorithm then goes like
— calculate b(nside = 256) from d{nside = 256) and A ( n s id e = 256)
— coarsen three tim es to get b(nside = 32), d (n sid e = 32) and A {nside = 32)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- iterate Equation 7.18 for n s id e = 32 resolution to convergence to get z ^ n s i d e
32)
- refine z ^ n s i d e — 32) one tim e and use as starting map z 0(nside = 64)
- calculate d(n sid e = 64) and A {nside = 64) by coarsening from n s id e = 256
- iterate Equation 7.18 for n s id e = 64 resolution to convergence to get z ^ n s i d e
64)
- refine z ^ n s i d e = 64) one tim e and use as starting map z0(nside = 128)
- calculate d{n side = 128) and A {nside = 128) by coarsening from n s id e —
256
- iterate Equation 7.18 for n s id e =
z ^ n s i d e = 128)
128 resolution to convergence to get
- refine z ^ n s i d e = 128) one tim e and use as starting map z 0(nside = 256)
- iterate Equation 7.18 for n s id e = 256 resolution to convergence
Since iterating Equation 7.18 scales like O ( n rol x W/re9), each coarsening reduces
the execution tim e of one iteration by a factor of 4, so that the n s id e = 32 iterations
take
as long as the n s id e = 256 iterations. The low-weight, large angular scale
features in the map can be iterated on hundreds of tim es at low resolution in the
n s id e = 32 iteration, with each higher resolution step requiring fewer iterations.
The n s id e = 256 resolution step typically requiring only a few tens of iterations
to converge.
Figure 7.1 shows the progression of fully converged maps at each
resolution step in a typical multigrid mapmaking run.
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90
N
S
ID
E - 32 : 14 arcminute pixels
N
S
ID
E = 64 : 28 arcminute pixels
2 .5 0 e -0 2 V olts
2 .5 0 e -0 2 Volts
N
S
ID
E = 128 : 1degree pixels
- 2 .5 0 e - 0 2 H H i '
.wmm 2 .5 0 e -0 2 V olts
N
S
ID
E = 256 : 2 degree pixels
- 2 .5 0 e - 0 2 mmmm
vmm 2 .5 0 e -0 2 V olts
Figure 7.1 Multigrid mapmaking converged maps at each resolution level. The
map is for day 2, channel 4, and took ~ 2 minutes to make on a 400MHz Pentium
class computer. The galaxy at the bottom of the map appears saturated due to
the plot range chosen here, but is properly resolved in the maps.
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C H A PT E R 8
M A D C A P M A P-M A K IN G
The M ADCAP package mapmaking procedure (Borrill, 1999) performs a bruteforce evaluation of Equation 7.11 and Equation 7.7 to get the best-fit map and
pixel-pixel covariance matrix.
8.1
M A D C A P Input
M ADCAP takes as input a tim estream noise m odel, tim estream detector data, and
tim estream pointing data. The noise m odel must be supplied in the form of rows of
the tim e-tim e noise weight m atrix N^,1, the inverse of the m atrix in Equation 7.4.
The tim e-tim e noise weight m atrix is assumed to be piecewise stationary (ie. the
noise correlations in a stationary piece depend only on the lag t —t') and sym m etric
(A,,,1 = Nj,,1). We calculate a row of the tim e-tim e noise weight m atrix from our
noise power spectrum as follows, where F is the F F T operator:
N tf1 =
=
( n i“ J ) 1
( 8 . 1)
F _1 ( F n t n j F - 1) ’ 1 F
( 8 .2 )
(8.3)
I ' (" i< W ) ‘ l
(8.4)
F -'h w F
(8.5)
so that each row of N tf} is sim ply the Fourier transform of the inverse noise power
spectrum. A small offset (10~4 of the white noise level) was added to the d.c. weight
(which you may recall was explicitly zero in the multigrid mapmaking scheme) to
91
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92
keep Npp/ from being singular and im possible to invert. The offset was chosen to be
small enough that it does not affect any modes in Npp/ other than the d.c. offset.
As defined in Equation 7.1, each row of the pointing matrix contains a single 1 in
the column of the pixel observed during the tim e sample corresponding to that row.
M ADCAP allows us to expand this definition of the pointing m atrix to include socalled m etapixels. We can add columns to the pointing m atrix that correspond to
sources of tim e stream signal in addition to sky signal. For instance, a single, fixedamplitude, fixed-phase spin-synchronous signal could be added to the tim e stream
model by adding a column to the pointing m atrix that consisted of a sine wave
w ith a period of one rotation. Equation 7.7 would then give the best fit m etapixel
map, consisting of the best fit sky pixels and the sim ultaneously best-fit spinsynchronous am plitude. Equation 7.11 would then give the m etapixel-m etapixel
correlation matrix, consisting of the pixel-pixel correlations, the spin synchronous
amplitude variance, and the correlation between the spin-synchronous signal and
the sky pixels. We can then marginalize over the spin-synchronous m etapixel by
projecting the m etapixel map and m etpixel-m etapixel covariance m atrix onto the
sky pixel subspace.
In practice, we add an extremely aggressive set of spin synchronous signal
m etapixels to the pointing matrix. Each spin harmonic from the 1st to the 32n^ is
modelled by a set of slowly varying b-splines m ultiplied by a sine wave and a cosine
wave at the harmonic frequency. The best-fit m etapixels are the am plitudes of each
b-spline. For harmonics 1-3, the b-splines have knot spacings of 50 rotations (~ 1 3
minutes), and the sine and cosine com ponents are independent so that the ampli­
tude and phase of the spin synchronous signal at these harmonics is allowed to vary
on tim escales longer than 50 rotations. Knot spacing shorter than this causes sig­
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93
nificant correlations between the sky pixels and the spin synchronous m etapixels,
since the balloon m otion doesn’t move the observing circle an appreciable amount
on the sky during the 100 rotations that each spline is active. Spin harmonics 4-6,
having significantly narrower line widths in the residual tim estream s, have knot
spacings of 100 rotations, while harmonics 7-32 have knot spacings of 300 rota­
tions. This compromise reduces the overall number of m etapixels to a managable
number, which is im portant since the calculations scale like r?vix. Approxim ately
7000 m etapixels are added to each channel to fit the spin-synchronous signal model
over the entire flight.
We also add m etapixels to the pointing m atrix to handle bad data. The pre­
processing loop identifies two distinct forms of bad data samples, unm odelable
glitches and railing events. Unm odelable glitches are short duration events that
are not well fit by a cosmic ray profile of a delta function of power on the detec­
tor convolved with the instrument transfer function. They are likely microphonic
“bearing pops” (see Section 4.2) and typically only last one 16Hz sample after the
transfer function deconvolution (well fit glitches last only one 64Hz sam ple). To
properly handle these unm odellable glitches we should assign each unm odellable
glitch its own m etapixel, and have the pointing m atrix row for that tim e sample
have all zeroes in the sky pixels and a single 1 in that glitch’s m etapixel column.
Then the am plitudes of all the unm odellable glitches would be sim ultaneously fit
with the rest of the tim estream model and could be marginalized over afterwards.
Unfortunately there are typically ~2 5 0 0 0 unm odellable glitches in each channel
which would drive the total number of m etapixels in each fit close to 80000, which
is beyond our com puting resources. Instead we chose to sim ply leave the point­
ing m atrix unchanged for each unm odellable glitch sample and sim ply interpolate
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94
the tim estream data through that one sample. This should have negligable effect
of the analysis, particularly since we find virtually no sky signal at spatial scales
corresponding to a single tim estream sam ple width.
The other bad data samples flagged during the preprocessing loop are railing
events.
W hen a glitch causes some part of the readout electronics to saturate,
the tranfer function deconvolution fails.
The railing event and some region of
data surrounding it (typically one or two rotations) must be removed from the
tim estream and replaced by a constrained realization of the tim estream noise and
spin synchrounous signal before the transfer function can be deconvolved. The final
tim estream therefore has fake data in the railing event rotations that has nothing
to do with the true sky signal and m ust not be used in fitting for the sky pixels.
We assign two railing event m etapixels to each rotation that is contam inated by
a railing event. For the first half of the rotation the first m etapixel is observed
(and no sky p ixels), while during the last half of the rotation the second m etapixel
is observed. Two m etapixels are used because the tim e-tim e noise weight m atrix
described above is nonzero for half a rotation forward in tim e and half a rotation
backwards in tim e about each sample. Using a single m etapixel would assign spu­
rious noise correlation between the last sky pixel observed before the railing event
and the first sky pixel observed after the railing event via their correlation with
the single railing event m etapixel. Using two m etapixels reduces this correlation
by adding a second level of transitivity.
8.2
M odifications to M A D C A P
One subtle but im portant change was made to the M ADCAP source code for use in
analyzing the TopHat data. M ADCAP allows us to specify the tim e-tim e weight
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95
function in a piecewise stationary manner. An am biguity arises in how to best
handle the interface between stationary sections. One approach would be to just
change the weight function used as we cross the boundary. The problem with this
approach is that the resulting
will not be symmetric for t and t' within one
rotation, but on opposite sides of a boundary. M ADCAP utilizes linear algebra
routines that depend on the matrices being symmetric, so this approach is ruled
out. W hat M ADCAP actually does is to truncate the tim e-tim e weight function
at the boundaries between sections. The resultant N ft,' is block diagonal, but not
block circulant. This causes a problem for the TopHat data because our tim e-tim e
weight function is explicitly designed to integrate to near zero (actually to ICC4
of the white level as described above). For the half rotation just before and the
half rotation just after a section boundary, truncating the tim e-tim e weight func­
tion at the boundary causes the effective tim e-tim e weight function to integrate
to a rather large positive number. W ith M A D C A P’s truncation in place, as we
approach a boundary the d.c. measurement gets a nonzero weight. The result is
that a handful of rotations (one for each piecewise stationary boundary) receives
a nonzero weighting for its d.c. measurement and stands out as a stripe in the
map. There seems to me to be no way around this striping for an a.c. coupled
experiment (which virtually all CMB experiments analyzed with M ADCAP are),
and this would seem to me to be a (small) bug in the M ADCAP software. Unfor­
tunately, there is no “right” way to fix this problem, but for us a less obtrusively
bad solution is to make the stationary sections actually be block circulant by edge
wrapping rather than edge truncating. Thus each row in the tim e-tim e weight
matrix explicity sums to near zero and our a.c. coupling is preserved. The disad­
vantage to this solution is that the first and last rotation in each piecewise section
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96
are now artificially correlated by the edge wrapping, but this is a small effect.
8.3
B est-F it Maps
Final best-fit maps made using data from the entire flight are presented for each
channel in Figures 8.1 - 8.6. The maps are uncalibrated (map units are Volts) and
the plot range has been selected to show structure in the high galactic latitude
dust at the expense of saturating the galaxy signal.
8.4
M ap Level Internal Consistancy Checks
One of the m ost useful features of the TopHat observing strategy is that the ex­
periment observes the entire sky coverage each day of the flight, allowing us to
make many independant maps and compare them to root out system atic effects.
In the sections that follow, we divide the tim e stream data into two halves, de­
noted Epoch A and Epoch B, and construct best-fit maps and covariance matrices
independantly for the two Epochs. We can then use comparisons of the two maps
as an internal consistancy check of our data analysis to this point.
One of the difficult things about analyzing a dataset of this sort is being able
to tell if the tim e-stream m odel is good enough, and if not, where it is defficient.
A first order test is whether the tim e-stream y 2 indicates a good fit of the model
to the data.
A poor y 2 is somewhat difficult to interpret for 2.5 x 106 degrees of freedom. The
X2
is made up of the pointing matrix, the noise model, and the spin synchronous
signal m odel. A poor y 2 can indicate a problem with any of these three com­
ponents, or could indicate other, unmodelled signal left in the tim e stream. How
each of these potential problems contam inate the best-fit map and pixel covariance
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97
Channel 1
3 . 0 0 e —0 2 V o lts
Figure 8.1 Channel 1 (150 GHz) best-fit map using data from entire flight. Plot
scale saturates galaxy signal to show structure over the rest of the sky.
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98
Channel 2
3 .0 0 e-0 2
3 . 0 0 e —0 2 V o l t s
Figure 8.2 Channel 2 (210 GHz) best-fit map using data from entire flight. Plot
scale saturates galaxy signal to show structure over the rest of the sky.
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99
Channel 3
-2 .0 0 e-0 2
2 . 0 0 e —0 2 V o lts
Figure 8.3 Channel 3 (380 GHz) best-fit map using data from entire flight. Plot
scale saturates galaxy signal to show structure over the rest of the sky.
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100
Channel 4
J**4■p'f*
-2 .0 0 e-0 2
2 . 0 0 e —0 2 V o l t s
Figure 8.4 Channel 4 (430 GHz) best-fit map made using data from entire flight.
Plot scale saturates galaxy signal to show structure over the rest of the sky.
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101
Channel 5
-2 .0 0 e-0 2
2 . 0 0 e - 0 2 V o lts
Figure 8.5 Channel 5 (640 GHz) best fit map made using data from entire flight.
Plot scale saturates galaxy signal to show structure over the rest of the sky.
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102
Channel 6
a>r'
V J**'
^
«*},;, ’ -- •»ift- ■
V‘*,k ' *
-2 .0 0 e-0 2
M
r’ m
^
2 . 0 0 e —0 2 V o l t s
Figure 8.6 Channel 6 (dark channel) best-fit map made using data from entire
flight.
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103
matrix is not at all clear. Since all that we need for successful power spectrum esti­
m ation is accurate maps and covariance matrices, tests on the residual tim estream
aren’t as useful as tests on the maps and covariance matrices.
The A -B difference x 2 tests whether the difference between the Epoch A and
Epoch B maps is consistant with the two pixel covariance matrices and is defined
as
X2 = (si —s2)T {Ni + N 2 ) 1 (si —S2 )
(8-6)
where si and S2 are the individual epoch maps and IVi and A 2 are the pixel covari­
ance matrices of the individual epoch maps. The maps have a 5° galactic latitude
cut, and the LMC, SMC, and Chameleon Nebulae cut because small pointing er­
rors cause large day-to-day differences in very high contrast regions of the map.
Since these regions are cut in the power spectrum estim ation step, we cut them
here. Figure 8.7 and Figure 8.9 show the sum and difference maps for each channel.
Table 8.2 lists the %2 for the sum and difference maps for each channel. Several
things are im m ediately obvious from the table and figures. First is that the sum
maps contain appreciable sky signal (as hoped for and expected). Second, there are
Table 8.1 Reduced y 2 for m odel that tim estream data is sky signal best-fit map
plus signal from spin synchronous signal m odel plus noise consistant with noise
model. Each channel has approximately 2.5 x 106 degrees of freedom.
Channel
Epoch A x 2
Epoch B x 2
1
2
3
4
5
6
1.073
1.036
1.007
1.017
1.008
1.105
1.083
1.043
1.010
1.021
1.011
1.148
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104
large angular scale modes in the difference maps that depend only on declination
that seem inconsistant with noise. More troublingly, there appear to be features in
the difference maps that are correlated between channels. Also interesting is that
the channel 6 difference map appears featureless, while the channel 6 sum map
appears to have declination dependant structure. This implies there is unmodelled
non-sky signal that has some projection onto the sky and is correlated from day
to day.
Table 8.2 Reduced y 2 for test that Epoch A, Epoch B sum and difference maps
are consistant with noise. Each y 2 has approximately 9000 degrees of freedom.
Channel
Difference y 2
Sum y 2
1
2
3
4
5
6
1.58
1.22
1.13
1.27
1.17
1.16
2.95
3.72
3.73
3.47
3.10
1.26
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105
C h a n n e l 1 : E p o ch A — E p o ch B
C h a n n e l 1 : E poch A + E p o c h B
6 . 0 0 e - 0 2 V o lts
4 .0 0 e —0 2 V o lts
Channel 2
:
Epoch A- Epoch B
2 . 0 0 e - 0 2 V o lts
Channel 2 : Epoch A+ Epoch B
—4 .0 0 e —02
4 .0 0 e —0 2 V olts
Figure 8.7 Epoch A, Epoch B sum and difference maps for channels 1 and 2.
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106
C h a n n e l 3 : E p o ch A - E p o ch B
—1 . 0 0 e - 0 2
1 .0 0 e - 0 2 V olts
Channel 4 : Epoch A- Epoch B
-2 .0 0 e - 0 2
2 . 0 0 e - 0 2 V o lts
C h a n n e l 3 : E p o ch A + E p o c h B
2 . 0 0 e - 0 2 V olts
Channel 4 : Epoch A+ Epoch B
2 .0 0 e —0 2 V olts
Figure 8.8 Epoch A, Epoch B sum and difference maps for channels 3 and 4.
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107
C h a n n e l 5 : E p o ch A - E p o c h B
- 2 .0 0 e - 0 2
2 . 0 0 e - 0 2 V o lts
2 .0 0 e —0 2 V o lts
Channel 6 : Epoch A- Epoch
—4 .0 0 e —02
C h a n n e l 5 : E p o ch A + E p o ch B
4 .0 0 e —0 2 V olts
Channel 6 : Epoch A+ Epoch B
-2 .0 0 e - 0 2
2 . 0 0 e - 0 2 V olts
Figure 8.9 Epoch A, Epoch B sum and difference maps for channels 5 and 6.
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C H A PT E R 9
D U ST REMOVAL
9.1
R esolution Coarsening
Once solutions for m etapixel maps and covariance matrices are found, the maps and
covariance matrices are projected onto the sky pixel subspace marginalizing over
all the non-sky m etapixels. The pixels used in the mapmaking step are HEALPIX
nside=256 pixels, approximately 14 arcminutes on a side, and number about 45000.
We also perform sky cuts to marginalize over sky pixels within 5° of the galaxy,
the LMC, the SMC, and the Chameleon Nebulae. This reduces the number of
sky pixels to '3 8 0 0 0 . The M ADCAP power spectrum estim ation routines involve
many m atrix inversions and m ultiplications that scale like n ^ix, and 38000 pixels
is beyond our available com puting resources, even on the IBM SP supercomputer
at NERSC. We therefore coarsen our maps and covariance matrices to HEALPIX
nside=128 resolution (28 arcminute pixels) by averaging the four nside=256 pixels
conatined in each nside=128 pixel. N side=128 pixels for which fewer than all four
nside=256 pixels were observed (near the edges of the map) are marginalized over.
This procedure reduces the total number of sky pixels used in the subsequent dust
m odelling and power spectrum estim ation to about 9300.
9.2
D ust M odelling
N ext we must separate the dust signal from the CMB signal in the channel 1 and
channel 2 maps. Since the CMB dipole appears in channels 1 and 2 and is partly
aligned with the galactic dust gradient we must marginalize over the dipole modes
in all channels to keep the dipole signal from biasing the fit to the galactic dust
108
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109
gradient. Channels 3, 4, and 5 are used to generate a spatial dust tem plate by
minim izing the following y 2 function
X2 = 1.9"" - £>“ 5 " )
(.9 "" - D " B " )
(9.1)
where a and (3 run over sky pixels and /v, and u run over channel numbers 3, 4,
and 5. Here S a,i are the best fit sky maps, D a is the spatial dust tem plate being
fit for, B '1 is a dust spectral dependance being fit for, and
is the pixel-
pixel-channel-channel covariance matrix. Vnpliu is assumed to be zero for n ^ v
(i.e. no noise correlations between channels) with
given by the pixel-pixel
covariance m atrix for channel n produced by MADCAP, marginalized over, and
resolution coarsened as described previously.
Since Equation 9.1 is nonlinear in the m odel parameters it is solved iteratively
by assuming a solution for B ^ , finding the best-fit spatial tem plate D a , assuming
this spatial tem plate and finding the best-fit
convergence.
and iterating this procedure to
There is a degeneracy between the overall scaling of D a and the
overall scaling of B
which we break by requiring B ^ B ^ = 1. The solution to
Equation 9.1 is given by
£3“
=
B" =
1 C'-'V-mx.v -B'"’''
[r99;,.x,„r/TVV.xr,,,..9"""'
(9.2)
(9.3)
and th e form al errors on th e dust m odel param eters are given by
c o v (D af
=
[B“Vag^ B T l
(9.4)
c o v (B "f
=
[ ir v ^ r fiY 1
(9.5)
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110
Once the dust m odel spatial tem plate and it ’s covariance m atrix is solved for,
we can fit channel 1, 2, and 6 to the dust m odel each with a single free parameter
B l (i = 1, 2, 6) which is the projection of channel i along the dust m odel tem plate.
B { = [DaVa m D p] - 1 D a'Va)p/iiS ^
(9.6)
Subtracting this dust com ponent from the channel i map we get a CMB map T f
and covariance m atrix
Ta
N f
derived from channel i.
=
S i * _ B iD a
=
V al3u + (B * )2 cov ( D af - cov (B*) D aD 0
(9.8)
which are used in the subsequent power spectrum estim ation. N ote that the sub­
script i denotes which channel (1, 2, or 6) the CMB map has been derived from.
Figures 9.1 - 9.3 show channels 1, 2, and 6 before and after dust subtraction and
dipole removal.
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I ll
Channel 1
1 .2 0 e —01 V o lts
Channel 1, Dust Subtracted, Dipoles Removed
-855
1 .2 0 e —0 2 V o lts
Figure 9.1 Channel 1 before and after dust subtraction and dipole removal.
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112
Channel 2
6 . 0 0 e —0 2 V o lts
Channel 2, Dust Subtracted, Dipoles Removed
i.*
—3 .0 0 e —0 3
3 .0 0 e —0 3 V o lts
Figure 9.2 Channel 2 before and after dust subtraction and dipole removal.
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113
Channel 6
—4 . 0 0 e —0 2 ■ in II
’<mm 4 . 0 0 e - 0 2 V o lts
Channel 6, Dust Subtracted, Dipoles Removed
6h X
**
?V.''a
—6 . 0 0 e —0 3 mmsmm-
-m m I 6 . 0 0 e - 0 3 V o lts
Figure 9.3 Channel 6 before and after dust subtraction and dipole removal.
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C H A PT E R 10
M A D C A P PO W ER SPEC TR U M ESTIM ATION
10.1
Overview of Quadratic Estim ator M ethod for Power
Spectrum Estim ation
M ADCAP provides an im plem entation of the Bond et al. (1998) quadratic estim a­
tor m ethod for power spectrum estim ation. The CMB tem perature signal on the
sky can decomposed into spherical harmonics as
s(6,<j)) = Y 2 alrnBlYlm(e,<f>)
( 10 . 1)
l,m
where Bi is the spherical harmonic transform of the telescope beam (assumed to
be azimuthally sym m etric) and pixelization scheme.
If the tem perature anisotropies are statistically isotropic (as inflation m odels
predict), then the variance of the m ultipole moments is independant of m, and is
given by
( 10 . 2 )
and for the case of Gaussian anisotropies (again as inflation m odels predict), the
Ci spectrum com pletely defines the anisotropies. Thus we seek to find an estim ate
of the Ci spectrum from our CMB map and covariance matrix.
If our pixelized CMB map Tp is a com bination of sky signal sp and noise n p
which are both assumed to be independant random Gaussian processes, then the
total pixel-pixel map correlations are
D ppf
=
( TpTp,)
114
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(10.3)
115
(s„sj) + ( rv»J)
(10.4)
Spp' “I- Npp'
(10.5)
and the probability distribution of the map given a particular Ci spectrum (and
therefor Spp/) is given by
P ( T p \ Ci )
=
(27rrnpW2exp j - i (T ^ D ^ T ) + T r [ l n D ] |
=
( 2 t r p p“ /2 exp j - 1 ( T t (S + N )-1 T ) + Tr [In (S + N)] [10.7)
(10.6)
If we assume a uniform prior for the probability distribution of the Q spectrum,
then this is also the likelihood of the Ci spectrum given the map. We must then find
the Ci that m axim ize Equation 10.6. Since there exists no closed form solution for
this maximum, we use Newton-Raphson iteration to find the zero of the derivative
of the log of this likelihood function.
Close to a maximum, this function can
be approximated as quadratic. Evaluating the first and second derivative of the
function at a point allows us to calculate the location of the maximum of the
(local) quadratic approximation to the likelihood. If the likelihood were actually
quadratic, this would be the correct solution. Since the likelihood is not actually
quadratic, the new point not be at the true maximum, but if our starting point was
sufficiently close to the true maximum and the likelihood contours are reasonably
well behaved, it will be closer to the maximum than the original point. Iterating
allows us to get arbitrarily close to the true maximum (assum ing the likelihood
function is strongly and singly peaked). Evaluating the second derivative of the
likelihood function at the maximum allows us to estim ate formal errors on the
best-fit power spectrum.
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116
Note that evaluating the first and second derivative of the log-likelihood func­
tion are com putationally intensive tasks. The matrices S and N are n pix x n PiX and
calculating the derivative w ith respect to each m ultipole ends up making the whole
process an O (n ilRrations^k-bins^i7l) operation, which is ~ 3 0 0 x more com putation­
ally intensive than the mapmaking process if done at the same pixel resolution.
10.2
M A D C A P Inputs
M ADCAP takes as input to the power spectrum estim ation routines the CMB
map and pixel-pixel covariance m atrix from Equation 9.7 and Equation 9.8, a
beam and pixel window function, and the 1-space binning we wish to fit the
power spectrum with.
For the beam and pixel window function we take the
azimuthally-sym m etrized beam window function measured in pre-flight tests at
the GEMAC facility at GSFC m ultiplied by the pixel window function for the
HEALPIX nside=128 pixelization scheme. Since we don’t observe the entire sky,
our estim ates of individual m ultipoles will be correlated (since the spherical har­
monics aren’t orthoganol on a sm all patch of the sky). As a result we can only
generate reasonably uncorellated estim ates of 1-space bins with the bin spacing
Al >
where 6 is the angular scale of our map. W hile our m ap’s diameter of
60° naively implies we should be able to use bins A l = 6, we in fact find we need
bins of A = 14 to reduce the bin-bin correlations sufficiently to keep the numerical
routines stable (partly due to the holes in our map from sky cuts). W ithin each
bin, we assume the power spectrum to have flat band power in 1(1 + 1)C/.
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117
10.3
Power Spectrum Level Internal Consistancy Checks
To have any confidence in our CMB power spectrum result we need to have some
sort of internal consistancy checks on the power spectrum we obtain. To this end,
we have divided the tim e stream data into two epochs (A and B) and perform the
full M ADCAP analysis on them independantly com plete with independant mapmaking and dust modelling. The power spectra obtained from each epoch should
be consistant with each other. Since the power spectrum error bars generated by
M ADCAP include cosmic variance terms (and our two epochs are sam pling the
same sky), a better test is to take the Epoch A and B CMB maps and covari­
ance matrices and generate an A-B difference map and covariance matrix. Since
the sky signal in the A and B maps should be identical, the A-B difference map
should contain only noise consistant with the A-B covariance matrix, and a power
spectrum estim ate generated from the A-B map and covariance m atrix should be
consistant with zero (cosmic variance terms are zero in the case of zero signal).
Channel
6x10
*
5X10
1
A n g u la r
Pow er S pectra
4 E
4 :
A
Epoch A + Epoch
o Eooch A ■ Epoch
4x10
CM
3X10
2x10
1 x1 0
4
4 4
A.
100
200
Multipole
300
400
I
Figure 10.1 Angular power spectra for channel 1 CMB sum and difference maps.
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118
Channel 2 Angular Power S p e c t r a
4000
^
3000
S
2000
=4
1000
A
Epoch A + Epoch B
a
or
-1000 o
-2000L
o
100
200
300
Multipole
400
Z
Figure 10.2 Angular power spectra for channel 2 CMB sum and difference maps.
Channel
6
A n g u la r
P ow er S pectra
A
Epoch A + Epoch
o A
100
200
Multipole
300
40 0
Z
Figure 10.3 Angular power spectra for channel 6 sum and difference maps.
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119
As is clearly shown in Figures 10.1, 10.2, and 10.3, the difference power spectra
are not consistant with zero for any of the dust subtracted CMB maps.
Fur­
thermore, the difference power spectra are not small compared to the sum power
spectra. Either there is non-sky signal still contam inating the maps (with different
contam ination in each epoch), or the noise m odel used to construct the pixel-pixel
noise covariance m atrix is underestim ating the actual noise level in the experiment.
We repeat the power spectrum estim ation w ith a variety of sky cuts to see if partic­
ular parts of the map are causing the problem. Problems w ith the dust m odelling,
or residual contam ination from the galaxy should affect the bottom half of the map
more than the top half of the map (since there is significantly more dust in the
bottom half of the map). Another logical sky cut to make is a declination cut that
removes the outside edges of the map. The central pixels of the map get crossed
from many different directions during the flight and in many different orientations
of the telescope, whereas the pixels near the edges of the map are only observed
when the telescope is pointing towards the horizon, with each observation being in
nearly the same orientation, allowing spin synchronous signals to add coherently.
Figure 10.4 shows the A-B difference power spectra obtained with these sky cuts
made. There is no evidence that the contam ination is strongly concentrated in any
particular region of the sky map.
The fact that the x 2 for the A-B difference maps are worse than the tim estream
model x 2 im plies the problem is not just a sim ple scaling of the noise level (which
would scale both x 2 values uniformly), but rather some noise source that projects
onto the sky more efficiently than it projects into the timestream. This is charac­
teristic of a noise source localized in frequency to some number of spin harmonics,
and is not at all a surprising result. The spin-synchronous signal, the sky signal,
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120
and random noise all come in at the spin harmonics, and disentangling them is
very difficult. The fact that the contam ination does not appear to be localized on
the sky implies that the contam ination is com ing into the map in a fairly random
way (ie it is not phased with the telescope rotation). A plausible explanation for
this contam ination is that some com ponent of the spin synchronous m odel is not
sufficiently flexible to fully model the variations in the spin synchronous signal.
Unfortunately adding significantly more flexability (via tighter knot spacing and
more m etapixels) is com putationally unfeasible. However, if the current spin syn­
chronous m odel, with its variable am plitude and phase, is fitting out the bulk of
the spin synchronous signal then the residual spin synchronous signal should be
fairly random in phase and amplitude. Indeed, the channel 6 A + B and A-B power
spectra are in good agreement w ith each other, indicating the contam ination in
the Epoch A and Epoch B maps are not particularly correlated, but appear to
C h a n n e l 2 D i f f e r e n c e Map A n q u l a r P o w e r S p e c t r a
1 i i | . i , i , , , , , | , , , , , i , , , | j
2000
Full M a p
^
lo p half
1000
—
C e n tr a l 10 d e g ra d iu s ;
V
(N
\
tT
0
+
S
-1 0 0 0
-2 0 0 0
0
100
200
M ultipole
300
400
I
Figure 10.4 A-B difference power spectra made w ith a variety of sky cuts.
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121
be drawn from the same underlying source distribution. If we assume that the
contam ination in the channel 1 and channel 2 derived CMB maps is caused by the
same mechanism (although possibly with a different source distribution) as the
contam ination in the channel 6 derived map, then we can infer that the channel 1
and channel 2 difference map power spectra are each realizations of the contam ina­
tion noise source power spectra that are contam inating those channels’ sum power
spectra as well. In other words, for each channel we can attem pt to use the differ­
ence power spectrum measurement to generate a correction to the pixel-pixel noise
covariance m atrix to account for the contam ination, and then use this corrected
pixel-pixel noise covariance m atrix in estim ating the power spectrum of the sum
map. The correction we add to the pixel-pixel noise covariance m atrix is of the
form
+ £
^
r
c t,f,p ‘ t w )
<10-8)
where C f f f is the difference power spectrum, P i(x) is the Legendre polynom ial
and Xpp' is the cosine of the angle between pixels p and // . Here we are estim ating
the underlying contam inating noise power spectrum from a single realization (that
of the A-B m ap ). We can generate another realization by dividing the tim estream
dataset in a different way, taking Epoch I to be the first and last quarter of the
tim estream data and Epoch II to be the second and third quarters. It is easy to
show that the I - II noise is independant of the A - B noise, and that this second
jackknife dataset gives us another realization of the underlying contam ination noise
power spectrum.
Figures 10.5 - 10.7 shows the A - B
power spectra and the
I - II power spectra and the y 2 of the m odel that the two power spectra are
realizations of the same underlying distribution. We use a best-fit to the underlying
contam ination power spectrum based on the A - B
and I - II power spectra in
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122
Equation 10.8).
Channel
1 Angular Power S p e c t r a
3 x 1 0 4 E~
A
O
Epoch I - Epoch II
nooh
A
Reduced
£ 2x10
-
!• p o c h
x2 =
R
1-26 51 8 :
CN
°
1X104
4
100
200
300
400
I
Multipole
Figure 10.5 Angular power spectra for channel 1 difference maps made using two
independant divisions of the data. The %2 and PT E are for the m odel that the
two power spectra are realizations of the same underlying noise power spectrum.
Channel
2
A n g u la r
Pow er S pectra
2000
A
Epoch I - Epoch
poc
Reduced x
a.
I'M!'
■P<
= 0.7 9 2 9 0 6 -
0 .7 8 8 8 2 !)
1000
op
+
--
m
i
-1000h
-2 00 0 L
0
100
200
Multipole
300
400
I
Figure 10.6 Angular power spectra for channel 2 difference maps made using two
independant divisions of the data. The x 2 and PT E are for the m odel that the
two power spectra are realizations of the same underlying noise power spectrum.
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123
Corrected sum power spectra are now made for each of channel 1, 2, and
6 ’s CMB maps. The corrected power spectra are shown in Figures 10.8 - 10.10.
Channel 6 is now consistant with zero. Channel 1 and channel 2 derived CMB
power spectra are now reasonably consistant with each other (m ainly due to the
very large error bars on the channel 1 points) if the lowest m ultipole bin is discarded
(which is reasonable given the size of the channel 1 correction in that bin). The
channel 2 power spectrum shows no evidence of a harmonic peak. It is consistant
with flat power of 1000/cK2
10.4
Discussion
Although our corrected channel 6 power spectrum is now consistant with zero
and the corrected channel 1 and 2 power spectra are fairly consistant with each
other and show non-zero anisotropy power, the dramatic disparity between our
Channel
6
A n g u la r
Power
S pectra
1.0
CM
>
E
0. 8
A
Epoch I - Epoch II
o i'poch A
Re duced
OJ
E p o c h 13
= 1.1 9 0 1 2 I _
0. 6
0. 4
+
At
0. 2
0. 0
-0.2
1 00
200
Multipole
300
400
I
Figure 10.7 Angular power spectra for channel 6 difference maps made using two
independant divisions of the data. The x 2 and PT E are for the m odel that the
two power spectra are realizations of the same underlying noise power spectrum.
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124
Corrected Channel
1 Angular Power S p e c tra
5000
4000
£
Corrected Sum
o Epoch A + Epoch B
3000
2000
iS *
+
1000
0
-1000
-2000
100
200
Multipole
400
300
L
Figure 10.8 Angular power spectrum for channel 1 CMB map with correction
applied to pixel-pixel noise covariance matrix. Red error bars include uncertainty
in correction term. Note that the I = 7 bin is off-scale high with a value of
41000±22000
C orrected
Channel
2
A n g u la r
Power
S pectra
5000
4000
tc
CM
cC
Cor rected Sum
o Epoch A + Epoch B
3000
2000
1000
+
0
Sr
-1000
-2000
100
200
Mu l t i p o l e
300
400
I
Figure 10.9 Angular power spectrum for channel 2 CMB map with correction
applied to pixel-pixel noise covariance matrix. Red error bars include uncertainty
in correction term.
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125
power spectra and other measurements (Halverson et al. (2001); Netterfield et al.
(2001); Hanany et al. (2000); Hinshaw et al. (2003)) requires us to take pause
and appraise our confidence in our result before contem plating further analysis
(like cosm ological parameter estim ation). A last internal consistancy check can be
made between channel 1 and channel 2. These two channels should be measuring
not just the same CMB power spectrum, but the same CMB sky. We can make a
linear com bination of the channel 1 and channel 2 maps that is proportional to the
CMB fluctuations (the CMB channel), and another linear com bination orthogonal
to the CMB (the null channel). If the non-zero power spectra in Figures 10.8 and
10.9 are measuring CMB anisotropy then the CMB channel should have a similar
looking power spectrum while the null channel should have a power spectrum
consistant w ith zero. Figure lO .llsh ow s the power spectra of the CMB channel
and the null channel, and they each show similar non-zero anisotropy power. This
C orrected
Channel
6
A n g u la r
Pow er S pectra
0.4
S'
C o rre c te d S u m
o E p o c h A 4- E p o c h B
X
0.2
eg
0.0
\
cT
"
+
-
0.2
-0 .4
0
100
200
Multipole
300
400
I
Figure 10.10 Angular power spectrum for channel 6 CMB map w ith correction
applied to pixel-pixel noise covariance matrix. Red error bars include uncertainty
in correction term.
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126
clearly dem onstrates that the source of the anisotropy power in channel 1 and
channel 2 is not the CMB. We are left with no choice but to conclude that a CMB
measurement from this dataset is not possible.
The fundamental problem is that we have a tim e varying spin synchronous sig­
nal with an am plitude of order 3mK in channel 2, which is two orders of m agnitude
larger than the CMB anisotropy signal we are trying to measure. Rem oving this
system atic signal to better than 1% has proven to be very difficult. The 5° tilt of
the balloon top plate definitely contributed to this very large am plitude, but even
a tilt of 0.5° (closer to our pre-flight expectations) would have left the spin syn­
chronous signal an order of m agnitude larger than the CMB signal. Two aspects
of the telescope design conspired to cause this flaw in the experiment. Firstly, the
beam out of the dewar horn slightly over-illuminates the secondary mirror. This
was knowingly done in an attem pt to minimize the width of the beam on the sky
to give greater sensitivity to higher m ultipole moments. The beam spilling over
the secondary was deemed to not be a problem with TopHat’s on-axis optical de­
sign because the spill-over falls on the cold, uniform sky. Secondly, the internal
dewar stages are supported by tensioned radial straps, while m otion in the z-axis
of the dewar was only constrained by stretching these straps out of their plane in
a “drumhead” mode. These two design aspects interact when the dewar is slightly
tilted so that the internal dewar optics (and the dewar horn) move slightly in the
dewar’s z direction. This moves the beam slightly relative to the secondary mirro,
and changes the amount of beam spilling over the secondary. Under-illum inating
the secondary and/or additional z-axis strapping of the dewar stages would have
drastically reduced the spin synchronous signal.
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127
A n g u la r
Power S pectrum
3000
20 00
CM
1000
!o
+
oCMB Channe:
000
0
50
100
Multipole
150
200
I
Figure 10.11 Angular power spectra for the linear com bination of channels 1 and
2 sensitive to CMB (CMB Channel) and the linear combination of channels 1 and
2 orthogonal to CMB (Null Channel).
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C H A PT E R 11
MAGELLANIC CLOUDS
11.1
Introduction
W hile the TopHat data are too contam inated to extract a reliable CMB measure­
ment, there is a considerable amount of information about the foreground dust
that is at a much higher signal-to-noise level than the CMB and can be accurately
measured even in the presence of residual contam ination. Interest in interstellar
dust emission has been increasing recently among cosm ologists.
D ust emission
from our own galaxy is a primary source of contam ination for Cosmic Microwave
Background measurements (Masi et al., 2001; Jaffe et al., 2003). A ssociation of
the observed Cosmic Infrared Background (CIB) with dust emission from high
redshift galaxies (Puget et al. 1999; Scott et al. 2000) and the recognition of the
potential use of this source as a probe of structure formation (Guiderdoni et al.
(1998); Blain et al. (1999); Haiman & Knox (2000); Knox et al. (2000)) has placed
a high priority on understanding the properties of extragalactic dust.
Properties of interstellar dust have largely been determined by unltraviolet
(UV), optical, and near to infrared (NIR - MIR) extinction measurements in our
own and other galaxies (M athis, 1990). D ust grain size, com position, and density
can be m odelled from extinction measurements (Mathis et al., 1977; Draine & Lee,
1984; Li & Draine, 2001). At frequencies below 3000GHz, the dust emission is
believed to be dom inated by thermal greybody emission from dust grains heated
by the interstellar radiation field (IRF). D ust emission m odels over the TopHat
frequency bands therefore depend on the spectrum of the IRF and the em issivity
of the dust grains.
128
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129
Em issivity can be a function of frequency, grain size, and grain com position.
Hildebrand (1983) and Draine & Lee (1984) (hereafter DL84) argue that for dust
grains much smaller than the wavelength of interest, the emission cross section is
proportional to the volume of the grain. The em issivity per unit mass of dust is
then independant of the grain size over wavelengths long compared to the grain size.
DL84 argue that in this same regime (grains much smaller than wavelength) electric
dipole radiation will dom inate and the dust em issivity will scale like the square of
the frequency. The sim plest dust m odel then has thermal emission m ultiplied by
a dust mass and a v 2 emissivity. Later theoretical work (Tielens & Allam andola,
1987) and laboratory measurements (Agladze et al., 1996) have shown that the
em issivity power law index can vary with grain com position and range from v l to
v 2-7 even in the small grain size regime. (Agladze et al., 1996) have even observed
tem perature dependant power law indices for certain types of amorphous silicates.
The em ission spectrum can be further com plicated if different grain types lie along
a single line of sight, even if the tem perature distribution is constant between the
different dust populations.
Dust grains in the diffuse interstellar medium are predicted to be in the 10K
- 20K tem perature range (DL84). Very small grains can be transiently heated to
substantially higher tem peratures (hundreds of Kelvins) by absorbing a single UV
photon, subsequently cooling by re-em itting in the NIR and MIR (Sellgren, 1984).
Li & Draine (2001) estim ate the maximum grain size at which this effect should
be noticable (~25nm ) and estim ate re-emission from these sm all grains to be an
im portant contributor to radiation at 5000GHz, but small compared to thermal
emission from larger, cooler grains at frequencies below 3000GHz. We therefore do
not expect transiently heated dust grains to be an im portant contributor to our
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130
measurements.
There exist few multiband, continuum measurements of extragalactic dust emis­
sion in the IR and sub-mm. Dunne & Eales (2001) have combined data from the
Infrared Astronomy satellite (IRAS) at 5000GHz and 3000GHz with 670GHz and
350GHz data from the Sub-m illim eter Common User Bolom eter Array (SCUBA)
and other sub-mm measurements in the literature to characterize the dust em is­
sion from 32 nearby galaxies. In an earlier study, Dunne et al. (2000) used the
IRAS data and the SCUBA 350GHz data to derive temperatures of 35.6 ± 4.9K
and em issivity indices of 1.3 ± 0.2 for a larger sample of 104 galaxies. Adding
the second SCUBA point in the new study they find that the data are better fit
by a two com ponent dust m odel w ith a hot component with tem perature of 31K
- 60K and a cold com ponent with tem perature of 18K - 32K. The authors note
that these determ inations are not definitive, both due to the low number of data
points for m ost of the sources (only
10
sources had measurements in more than
four spectral bands, and only one in more than six bands - each dust com ponent in
a model generally has 3 unknown param eters), and due to the inherent degeneracy
between a broadened tem perature distribution and a shallower em issivity power
law. Having observations in many spectral bands is clearly key to being able to
distinguish m ultiple dust com ponents and their emission properties.
Here we combine four frequency bands from TopHat (channels 2-5) with three
bands from the Diffuse Infrared Background Experiment (DIRBE) on the Cos­
mic Background Explorer (COBE) satellite for an overall frequency span of 2453000GHz. We make total flux measurements of the two galaxies closest to our
own: the Large Magellanic Cloud (LMC) and the Small Magellanic Cloud (SMC),
as well as the active star-forming region inside the LMC known as 30-Doradus (also
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131
known as the Tarantula Nebula). We report calibrated spectra of all three regions.
We fit single com ponent greybody emission m odels to the calibrated spectra and
report best-fit optical depth, temperature, and em issivity power-law index for each
region.
11.2
Integrated Differential Photom etery
TopHat has no absolute reference for the power com ing from the sky. It is sensi­
tive only to power differences between different positions on the sky. The formal
uncertainty on any quantity with a nonzero projection onto a constant sky flux is
infinite. Therefore all quantities we wish to estim ate from the TopHat maps must
be purely differential. The integrated flux in a set of (on-source) pixels minus the
integrated flux in a surrounding set of (off-source) pixels with equal solid angle is
such a differential quantity. It is also a convenient quantity to measure the flux
from an extended source that is observed through a relatively uniform, optically
thin foreground. The differential measurement gives the flux from the source alone,
since the foreground contributes equally to the on-source and off-source regions.
We have seen that residual contam ination is m ost prominant in the large angu­
lar scale modes of the maps, and in modes that are only a function of declination.
We therefore choose source regions that are fairly small compared to the overall
size of the map (although they are still quite large compared to typical photom etric
regions) to avoid large angular scale contam ination. We choose off-source regions
that have the same number of pixels at a given declination as the on-source re­
gion so that the differential flux measurement is insensitive to modes that are only
functions of declination. We arrange the off-source pixels sym m etrically in right
ascension about the on-source region so that the differential flux measurement is
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132
insensitive (to first order) to other large scale modes that we do not want to include
in the measurement, such as the CMB dipole.
Finally, the m ost com pact sources in our maps have a full width at half m axi­
mum of about 1°. Since our sky coverage does not contain any sufficiently bright
point sources we are unable to obtain in-flight beam maps. To make the flux anal­
ysis independant of the details of the beam shape (and the different beam shapes
in different channels), we chose source regions that are significantly larger than the
1°
angular scale set by our m ost com pact sources.
Once the on-source and off-source pixels have been chosen, we construct a
weighting vector w that is
1
in every on-source pixel,
-1
in every off-source pixel,
and 0 in all other pixels. For a given map m and noise covariance m atrix N , we
calculate the quantity
S = w Tm
(11-1)
with a corresponding error variance of
cr| =
w t
N w
(11-2)
as our integrated differential flux measurement.
We calculate flux measurments for TopHat channels 2-5 (245, 400, 460, and 630
GHz) as well as for the DIRBE bands
8
, 9, and 10 to cover the frequency range
245 - 3000GHz.
11.3
Source Regions
Regions of interest in our field that are particularly well suited to the analysis
described above include the LMC, the SMC, and 30-Doradus (the Tarantula Neb­
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133
ula). These are all extragalactic objects observed through an optically thin galactic
foreground. We can therefore measure the total flux com ing from these objects
while subtracting the galactic foreground component.
In addition to the LMC, SMC, and 30-Doradus source regions we have selected
five regions of appreciable galactic dust emission out of the galactic plane which
we use to calibrate TopHat channels 3, 4, and 5 and one “blank” region. The blank
region was chosen such that its differential flux measurements in the DIRBE bands
are consistant w ith zero. Since we expect the TopHat bands to be dom inated by
thermal dust emission correlated with the DIRBE bands, the blank field serves as a
consisntancy check on the m ethod for the TopHat bands. The remaining regions are
numbered 1-5, with regions 3, 4, and 5 corresponding to the Chameleon Nebulae.
The location and solid angle of each source region are listed in Table 11.1. The
on-source and off-source pixels for each region are shown in Figure 11.1.
Table 11.1 Locations of regions used in the flux analysis. All coordinates are J2000.
Region
RA Center
Dec Center
Solid Angle (ster)
D ust 1
D ust 2
Dust 3
Dust 4
Dust 5
Blank
LMC
SMC
30-Dor
7h7m52s
19ft30m0s
H h4m24s
12h49m5s
12/l54m0s
23/T 6 TO24s
5h18m28s
0ft52m16s
5/l39m28s
-78°50'6"
—80° 7'11"
—77°32'57"
-79 °5 6 T 1 "
—77°10/53"
-m°2&?>7"
-68°29'36"
-72°56'32"
—69°3'4"
0.01596
0.01596
0.00118
0.00113
0.00118
0.00644
0.01481
0.00412
0 .0 0 1 2 0
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134
Figure 11.1 TopHat channel 5 map showing the on-source and off-source regions
used in calculating the LMC, SMC, and blank field fluxes, as well as the dust
regions used for calibration. The on-source and off-source regions for 30-Doradus
are shown in the inset. In calculating the flux for the LMC we om it the 30-Doradus
on-source region (the inner circle in the inset and the blackened circle in the main
figure).
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135
11.4
Consistancy Tests
We probe the consistancy of the TopHat flux measurements with a variety of y 2
tests, again making use of the two independant Epoch maps for each channel.
First we ask whether the blank field has flux consistant with zero in each Epoch.
The results o f this test are presented in Table 11.2 which lists the y 2 /d .o .f and the
probability to exceed(PT E) this value. We find that the combined blank field flux
measurements for the TopHat channels are consistant w ith zero in both Epochs.
Table 11.2 y 2 test of the null hypothesis for the blank field.
Epoch
y 2 /d .o .f
PT E
I
II
Sum
5.165/4
1.438/4
6.603/8
0.27
0.84
0.58
N ext we test whether the flux measurements for each region are consistant
between Epochs. The results of this test are shown in Table 11.3. We find that
the results are consistant between Epochs and so use a weighted average of the
two Epoch flux measurements in all subsequent analysis.
11.5
DIRBE D ata
Having dem onstrated the internal consistancy of the TopHat data, we proceed to
combine it with DIRBE data to extend the range of our spectral coverage. The
DIRBE observations are the closest in frequency to our own, extend in frequency
from 1250GHz, and with a beam of 0.7°, are well suited to measurements on the
angular scales of the Magellanic Clouds.
We use the Zodi-Subtracted Mission
Average (ZSMA) intensity and standard deviation data from bands
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8
, 9, and 10,
136
with nominal band centers of 1250, 2140, and 3000 GHz.
The DIRBE maps exist in the COBE quadcude pixelization and must be repixelized for proper comparison to the TopHat maps. We repixelize the maps by
resampling them w ith the HEALPix 14' pixelization which oversamples the DIRBE
beam. We then calculate the fluxes for each region in the same manner as for the
TopHat maps.
We note that, due to the differential nature of our analysis, the errors in flux
due to the DIRBE absolute offset uncertainty and zodical subtraction uncertainty
are negligible.
DIRBE calibration errors are im portant, however, and m ust be
considered in addition to the random errors in the ZSMA maps. Our estim ate of
the DIRBE gain calibration error is based on the work of Hauser et al. (1998) and
the cross-calibration of the DIRBE gains using FIRAS (Fixsen et al., 1997). The
gains and errors found by the various authors are given in Table 11.4. How these
gains are used in the analysis is described in Section 11.6.
Table 11.3 x 2 test of the consistancy of each region between Epochs.
Region
y 2 /d .o .f
PT E
D ust 1
D ust 2
D ust 3
D ust 4
D ust 5
Blank
8 .556/4
4 .155/4
4 .803/4
1.051/4
4 .344/4
4 .699/4
0.07
0.39
0.31
0.90
0.36
0.32
LMC
SMC
30-Dor
3.757/4
4 .001/4
0 .901/4
0.44
0.41
0.92
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137
11.6
Calibration Fit
The CMB dipole is observed in the TopHat channel 2 (245 GHz) map with high
signal-to-noise and can be used as a calibration source for that channel. We use
the best measurement of COBE (Fixsen & Mather, 2002) for the dipole am plitude
and direction, and apply a correction for the Earth’s m otion about the sun. The
error in the calibration is small compared to the flux measurement errors, and so
we ignore it for the rest of the analysis.
The CMB dipole is not detectable in the higher frequency TopHat channels ,
and so another calibration technique must be used. Unfortunately there are no
known calibration sources in the 400 - 640 GHz range at angular scales of 1°. We
instead use a m odel of the high-latitude galactic dust measured in the five Dust
Regions to interpolate between TopHat’s calibrated 245GHz channel 2 and the
calibrated DIRBE bands at 1250, 2140, and 3000 GHz. This interpolation is used
to calibrate TopHat channels 3, 4, and 5 (400, 460, and 640 GHz) in the galactic
Dust Regions. This calibration is then applied to the flux measurements in the
LMC, SMC, and 30-Doradus.
The spectral interpolation is made based on the following model of the dust.
Each galactic Dust Region is assumed to be optically thin at each frequency, and
Table 11.4 DIRBE gains and errors.
Reference
3000GHz
2140GHz
2140GHz
1250GHz
1240GHz
Gain
® g a in
Gain
@ g a in
Gain
6 g a in
3000GHz
Hauser et al., 1998
1.00
0.135
1.00
0.106
1.00
0.116
Fixsen et al., 1997
1.25
0.150
1.04
0.02
1.06
0.02
This work, as prior
1.00
0.135
1.04
0.02
1.06
0.02
This work, best fit
1.21
1.03
1.05
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138
have uniform dust temperature and optical properties throughout. The integrated
flux from each region is m odelled to be given by
Fv
=
J dQdlpd (O, I) KmB v (T )
=
ACItuB u (T)
(11.3)
where ru is the mean optical depth along all lines of sight through the source
region at frequency v, B V(T) is the Planck blackbody brightness at frequency v
and tem perature T , Km is the dust opacity in cm 2 /g , pd is the dust mass density,
and the integral is taken over lines of sight and solid angle. We assume a power-law
em issivity to for the dust so that
(11.4)
J dlpd(l)Km (u0)
t { v q ) ( v / Vq)*
{ v / v q ) 01
,
where v$ = 600 GHz.
We then sim ultaneously fit the measured fluxes from the five Dust Regions to
the dust m odel in Equation 11.3, allowing each region to have its own spectrum
with free parameters T, a, and r(z/0), and each of the three unknown TopHat
calibrations and the DIRBE calibrations are allowed to vary but are constrained
to be identical for each of the five Dust Regions. We assume Gaussian priors for the
DIRBE calibrations as given by the best com bination of the Hauser et al. (1998)
and Fixsen et al. (1997) values and shown in Table 11.4. The best-fit m odel is
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139
found by m inim izing the following x 2 function
(F (i,j)/K (i,j)-c (j)F (i,j)\2
2
~ h h A
^
'
fc (j)-c(j)\2
M
w
r i r J
’
(
5)
where i runs over the five Dust Regions and j runs over the seven frequency
bands, F ( i , j ) is the uncalibrated flux measurement for region i in frequency band
j , F ( i , j ) is the m odel flux, c{j) is the calibration, a F ( i , j ) is the uncertainty in the
uncalibrated flux, c ( j ) is the nominal value of the calibration in the DIRBE band
j , and a ( c ( j) ) is the uncertainty on that value. K ( i , j ) is the colour correction for
frequency band j , given the m odel spectrum for region i. The colour correction is
a scaling needed to correct for the fact that the effective center frequency for each
band depends on the assumed source spectrum (due to the fact that the frequency
bands are wide enough that the source spectra vary appreciably w ithin a band).
The m ethod for calculating the colour correction is explained in Section TBD .
The fit is done iteratively, using the colour correction from the previous iteration’s
spectrum. There are a total of 35 data points, and 21 free parameters in the fit.
The best-fit m odel has a %2 /d .o .f = 3 6 /14, with most of the excess x 2 coming
from the TopHat points. The residuals appear randomly distributed, however, and
fitting the data to a two com ponent dust m odel does not significantly improve the
X 2-
We conclude that the TopHat errors are likely underestim ated (which we saw
evidence for in the CMB analysis as well), and to be conservative in our error
estim ate on our fitted calibrations we double the TopHat errors used in the fit.
This improves the x 2 /d -°-f for the one component dust m odel to 2 0/14, which has
a PT E of 0.13. The best-fit dust m odel and its residuals for each Dust Region are
shown in Figure 11.2. The best-fit dust m odel parameters for each D ust Region
are listed in Table 11.5, along with the square root of the diagonal elem ents of the
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140
covariance matrix. N ote that the parameters have significant correlations between
them. We take the TopHat and DIRBE calibration parameters from this fit and
marginalize over the dust model parameters. In the analysis that follows we use
the full covariance m atrix for the calibration parameters.
100.0
100.0
100.0
10.0
10.0
10.0
1.0
1.0
X
p
Eb
D
ust Region 2
D
ust Region 1
Dust Region 3
o.i
o.i
o.i
-2
<>
300
1000
u (GHz)
-2
300
3 000
1000
3 000
v (GHz)
100.00
10.0
10.00
10.0
x
p
E
Dust Region 4
?/
Dust Region 5
0.1
0.1
1.0
0.5
1.0
0.5
- 0 .5
- 0 .5
0.10
M odel S p e c t r a :
-----------...............
— —
-----------
0.0
0.01
re g io n
re g io n
re g io n
re g io n
re g io n
1
2
3
4
5
«s.
30 0
1000
v (GHz)
3000
30 0
1000
u (GHz)
3000
30 0
1000
3000
v (GHz)
Figure 11.2 Best-fit dust m odels in the five galactic Dust Regions used to calibrate
TopHat channels 3, 4, and 5. Residuals are shown below each plot and the five
model spectra are plotted together in the lower right panel.
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141
11.7
Calibrated Fluxes
Using the calibrations obtained using the five D ust Regions in Section 11.6 we
can calculate calibrated fluxes and associated errors for the LMC, SMC, and 30Doradus source regions. The errors in the calibrations are uncorelated with the
flux measurement errors, so that the covariance of the calibrated fluxes is the sum
of the covariance of the flux measurements and the covariance of the calibrations.
Table 11.6 lists the calibrated fluxes and errors for the LMC, SMC, and 30-Doradus.
The errors quoted are the square root of the diagonal elem ents of the combined
covariance matrix.
These fluxes and their full covariance matrices are then fit
separately for each region to the spectral m odel of Equation 11.3. The best-fit
values, diagonal errors, and %2 /d .o .f for each region are listed in Table 11.7. N ote
that the fitted parameters are highly correlated, with the correlation m atrix for the
fits given in Table 11.8. The data, best-fit models, and residuals for each region
are shown in Figure 11.3.
11.8
Discussion
We have measured the integrated flux relative to background of the LMC (mi­
nus 30-Doradus), SMC, and 30-Doradus in 7 frequency bands ranging from 245
Table 11.5 Dust Region dust m odel parameters from the calibration fit.
Region
Dust
Dust
D ust
D ust
Dust
1
2
3
4
5
T(K )
19.4
15.0
14.5
14.8
14.7
o t
{ K )
rM
( x lO - 5)
0.83
Or (XlO-5 )
a
1.54
0 .1 2
2 .0 2
0.13
0.16
0.15
0.15
0 .6
1 .0
0.14
0.19
0.7
1 1 .0
2 .0
0 .6
4.4
0 .8
1.98
2.06
0 .6
8 .0
1.5
2 .0 0
1 .0
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142
Table 11.6 Calibrated fluxes and errors for the LMC, SMC, and 30-Doradus.
Instrument
TopHat
DIRBE
Frequency
30-Dor
LMC
SMC
(GHz)
F (kJy)
TpfkJy)
F (kJy)
0 > (k Jy)
F (kJy)
<xF (kJy)
245
1.63
0.17
0.27
0.03
0.30
0.07
400
7.12
0.53
1.13
0.09
0.87
0.14
460
9.60
0.77
1.68
0.14
1.33
0.23
630
27.59
2.27
4.23
0.36
2.82
0.62
1250
112.3
2.3
24.99
0.54
11.12
0.45
2140
177.7
3.7
46.29
1.00
18.89
0.78
130.1
17.6
39.56
5.35
15.87
2.14
3000
Table 11.7 Spectral m odel fit results for LMC, SMC, and 30-Doradus.
Region
T ( K)
aT {K)
r{y0) (xlO -5 )
<Tr (xlO -5 )
a
LMC
SMC
30-Dor
24.4
29.5
25.5
1.9
2.9
1 .0
0 .2
0.31
2 .2
2 .1
0.05
0.4
1.40
0.96
1.58
0.08
0.15
0.08
X2/ d.o.f
PTE
4.07/4
1.80/4
1.92/4
0.40
0.77
0.75
Table 11.8 Correlation matrices for the m odel fits to the LMC, SMC, and 30Doradus.
Region
Parameter
T
LMC
T
1 .0 0 0 0
r (vo)
a
30-Dor
T
1 .0 0 0 0
T
t ( vo)
a
a
-0.7719
0.5862
1 .0 0 0 0
1 .0 0 0 0
Tiyo)
a
SMC
^o)
-0.9646
-0.9629
1 .0 0 0 0
-0.6881
0.4770
1 .0 0 0 0
1 .0 0 0 0
-0.8817
1 .0 0 0 0
-0.7230
0.3631
1 .0 0 0 0
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143
100.0
100.0
1.0
1.0
S
M
C
L
M
C
0.1
0.1
-2
300
1000
3000
v (GHz)
v (GHz)
100.0
10.0
1.0
30-Doradus
M odel S p e c tr a :
o.io
o.i
LMC
6
4
2
"3 0
•3 - 2
3 0 -D o ra d u s
0.01
300
1000
v (GHz)
3000
300
1000
3000
Figure 11.3 Best-fit models for the LMC, SMC, and 30-Doradus. The residuals for
each region are shown below each plot. The m odels for the three source regions
are plotted together in the lower right panel.
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- 3000GHz. The measurements from 245GHz to 630GHz are new results made
from the maps of the southern sky made by the TopHat experiment, the first to
measure these regions at these frequencies on degree angular scales. The only pub­
lished measurements of integrated continuum fluxes of these regions in the sub-mm
are from Andreani et al. (1990), using tim estream data from single scans of a ~ 1°
beam across the LMC and SMC in two very wide frequency bands (Azq ~ 35GHz,
A u2 ~ 240GHz) with effective band centers of zq = 145GHz and u2 — 260GHz.
Their reported surface brightness is compared w ith our results in Table 11.9. We
cannot explain the orders of m agnitude discrepancy between our results and theirs,
though we note the very large uncertainties on their results. We also note that
the predictions of Finkbeiner et al. (1999) are in much closer agreement to our
measurements.
Table 11.9 Comparison with previous surface brightness values for the Magellanic
Clouds.
All surface brightnesses given in units of 10~ 18W cm _ 2 sr_ 1 //m ~1.
Reference
Andreani et al. (1990)
Andreani et al. (1990)
Finkbeiner et al. (1999)
This work
Center Frequency
(GHz)
145
260
260
245
Surface Brightness
LMC + 30 Dor
198±59
1220T530
75
24±2
SMC
174±51
905T440
25
1 4± 3
We have fit the calibrated TopHat and DIRBE measurements to a single com­
ponent dust emission m odel consisting of a blackbody spectrum m ultiplied by a
power law em issivity and found that a single temperature and power law index for
each region fits the data adequately. A two component dust emission m odel does
not produce a significantly better fit. Stanimirovic et al. (2000) analyzed DIRBE
data using their own foreground subtraction process to produce integrated fluxes
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145
for the SMC, and found that their fluxes determined from DIRBE bands 7,
8
, 9,
and 10 (1250GHz - 5000GHz) did not appear, by eye (without a formal fit), to
follow a one component model. Dunne & Eales (2001) found that a sample of
32 nearby galaxies observed in the frequency range 350 - 5000 GHz have spectra
better described by a two com ponent model than a single com ponent one. We
also find that if we extend our frequency range by including the DIRBE 5000GHz
channel our data are no longer fit well by a single component model. However, over
the frequency range of 245 - 3000 GHz a single component dust m odel appears to
be sufficient for the LMC, SMC, and 30-Doradus.
If we interpret the results of this fit to be an accurate physical description of
a single dust population in each region rather than just a convenient parameter­
ization, then we can draw some general conclusions about the properties of the
dust and IRF in these three regions. Before doing so, however, we note that there
are other physically plausible m odel spectra that can also provide a good fit to
the data. For example, as pointed out by Reach et al. (1995) in fitting FIRAS
data to a greybody thermal emission model, the integrated emission from a con­
tinuous distribution of dust tem peratures with uniform em issivity power-law index
is difficult to distinguish from a single tem perature dust component with a shal­
lower em issivity power-law index. A plausible physical m odel to fit our data to is
that of a continuous temperature distribution of greybody em itters ranging from
Tdust = T c m b up to some maximum tem perature Tdust = Tmax, all with an em is­
sivity power-law index of a = 2, as predicted by the simplest physical models. For
convenience we choose a power-law dust temperature distribution d N / d T oc T~@,
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146
which gives the following model for the integrated flux:
pTm ax
Fv oc /
T ^ B v { T ) { u / u 0f d T
(11.6)
JTcmb
The %2 /d .o .f and best-fit Tmax and (5 for our three regions are listed in Table 11.10.
This alternate m odel is bolstered as well by evidence in previous measurements
of significant dust temperature variations within the SMC. Stanim irovic et al.
(2000) report SMC dust tem peratures of 23K < Tdust < 45K based on IRAS
5000/3000G H z flux ratios. The absolute m agnitude of these values is somewhat
questionable because of offset and calibration issues in the IRAS data, as well
as the fact that the 5000GHz fluxes may be sensitive to t ransient, UV heating of
very sm all dust grains, but the variations of this quantity within the SMC implies
a non-uniform IRF, which should result in variable temperature of larger grains
within the SMC as well.
Our single tem perature com ponent dust m odel (Equation 11.3) gave best-fit
temperatures of (2 4 .4 ± 1 .9 )K for the LMC (minus 30-Doradus), (25± 2.2)K for 30Doradus alone, and (29.5± 2.9)K for the SMC. The alternate, broad tem perature
distribution m odel of Equation 11.6 gives similar temperatures for the best-fit
Tmax- The general result that the SMC is warmer than the LMC was also seen by
Sauvage et al. (1990), who account for this by noting that the lower dust to gas
ratio in the SMC results in more UV photons per dust grain. These same authors,
however, found the 5000/3000G H z ratio in 30-Doradus to be significantly larger
than the rest of the LMC, while we find the two regions to have tem peratures
within l a of each other.
As in the SMC, the 5000/3000G H z derived relative
tem peratures are sampling the very small, transiently UV heated dust grains, but
the more intense IRF in 30-Doradus should result in increased large grain dust
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temperature as well. The situation is perhaps better explained by the m odel of
constant power-law em issivity index and a distribution of dust grain temperatures.
In this m odel 30-Doradus has a comparable Tmax as the rest of the LMC, but a
shallower tem perature power law index, indicating a larger fraction of the total
dust mass in 30-Doradus is at high tem peratures than is the case in the LMC.
This agrees with intuition which suggests that the hotter dust would tend to be
found nearby hot, early-type stars such as those found in 30-Doradus.
Table 11.10 Results of fit to the alternate m odel of Equation 11.6.
Region
T
J- max
P
X2 /d .o .f
PTE
LMC
SMC
30-Dor
25.5
30.0
27.0
1 .8
4 .9 1 /4
1.30/4
4 .9 2 /4
0.30
3.0
1 .0
0 .8 6
0.29
Using either model to interpret our measurements, we can ask how the dust in
the LMC and SMC compares with the dust in our own galaxy. FIRAS measured
the galactic emission over the whole sky in the frequency range of 30-3000GHz with
a 7° beam (too large to resolve our regions). Several groups have attem pted to
characterize the galactic dust in various regions of the sky using the FIRAS data..
Reach et al. (1995) break the FIRAS coverage into 23 regions in the galactic plane
and seven regions at galactic latitudes above
10°
and fit these regions to a variety
of dust models. They found that in a two com ponent dust m odel with em issivity
power-law index of both com ponents fixed at a =
2
, the hotter com ponent ranged
in tem perature from 18.6K < Tdust < 24.7K in the galactic plane and 16.8K <
Tdust < 18.3K for the high latitude regions. Finkbeiner et al. (1999) fit sm oothed
IRAS and DIRBE data to FIRAS in regions above 7° galactic latitude, excluding
the Magellanic clouds and HII regions in Orion and Orphiuchus.
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Their best-
148
fit model was a two component greybody fit with floating em issivity power-law
indices for each com ponent. The mean tem peratures of the two com ponents were
(Ti) = 9.4K and (T2) = 16.2K.
We find that the temperature of the dust in
the Magellanic clouds (or the temperature of the hottest dust in the tem perature
distribution models) to be on the high side of all of these galactic temperatures.
The tem perature we derive for the SMC is significantly hotter than any of the
galactic measurments, while the tem peratures we derive for the LMC and 30Doradus are comparable to the hottest regions seen by Reach et al. (1995) in the
galactic plane.
We derive effective em issivity power-law indices of 1.40 ± 0.08 for the LMC
(minus 30-Doradus), 1.58 ± 0.08 for 30-Doradus alone, and 0.96 ± 0.15 for the
SMC. Reach et al. (1995) found a similar range of effective power-law indices
(0.92 < a < 1.60) in fitting FIRAS data to a one component m odel with a as a
free parameter. They find slightly better fits when they use a two tem perature
component m odel with fixed a = 2. Pollack et al. (1994), in m odelling the IR
emission from stellar accretion disks at ~ 100K predict an em issivity power-law
index of 1.5 for frequencies below 500GHz and an index of 2.6 for higher frequencies.
According to the authors, this is due to the changes in relative contributions from
so-called astronomical silicates (which the authors claim should have an index of 1 )
and organic species. In fitting extrapolated and sm oothed IRAS and DIRBE data
to FIRAS measurements, Finkbeiner et al. (1999) found that a two com ponent
m odel close to these values (aq = 1.67, a 2 = 2.7 with equal power from the two
com ponents at around 500GHz) was the best-fit to high-latitude regions of the
galaxy. However, in our three extragalactic regions we see no evidence of a change
in em issivity power-law index over the observed frequency range (245 - 3000GHz).
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C H A PT E R 12
CONCLUSIONS
The TopHat experiment observed a ~ 60° diameter cap over the South Celestial
Pole with sub-degree resolution in five frequency bands ranging from 150 - 640
GHz , the first measurements of this part of the sky at these frequencies and angu­
lar scales. A lthough the telescope was designed specifically to reduce system atic
signal, an unexpected large tiltin g of the instrument on the balloon top induced
an extremely large spin synchronous instrument signal, several orders of m agni­
tude larger than the anticipated CMB signal. The heavily cross-linked observing
strategy, repeated independant daily observations of the entire sky coverage, and
extensive instrument monitoring sensors (including thermometers, position sensors,
and dark channel) have allowed us to generate a physical m odel of the instrument
signal which removes a large portion of the spin synchronous signal. Unfortunately
the residual instrum ental signal contam inates the maps too heavily to make inter­
nally consistant CMB power spectrum measurements. However, the contam ination
in the maps is small compared to the dust signal in bright regions of the sky, and
the TopHat maps provide a unique opportunity to measure the dust emission spec­
trum at several frequency bands, integrated over fairly large portions of the sky.
We have performed these measurements for the LMC (excluding 30-Doradus), the
SMC, and 30-Doradus alone. The differential nature of the analysis measures the
total flux com ing from each extragalactic region relative to the uniform galactic
foreground, so that the derived spectra for each object are not contam inated by
contributions from dust in our galaxy. We have performed a similar analysis on
DIRBE data to extend the range of our spectral coverage up to 3000GHz. The
high number of frequency bands in our spectral range allows us to fit the spectrum
149
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150
of each region to a variety of physical dust emission models. We find that the spec­
tra are consistent with a simple physical m odel of a single tem perature-component
greybody emission m odel with a power-law emissivity, with a two com ponent dust
model not significantly improving the fit.
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A P P E N D IX A
CALCULATION OF COLOUR CORRECTIONS
If an experiment has finite bandwidth (t(v) 7 ^ S(v — vc)) one must assume a source
spectrum in order to report a source surface brightness at a single frequency. The
power detected from that source is assumed to be
P in = r j A n
J
I 0( u) t(u )d v
(A .l)
where rj and AVt are the optical efficiency and throughout of the instrument, I q( v ) is
the assumed surface brightness of the source, and t(u) is the bandpass transmission
normalized to 1.0 at its peak. The effective band center uc is usually chosen so
that
J I 0(u)t(u)du
Jt(v)dv
(
}
The band centers for TopHat are calculated assuming a Rayleigh-Jeans (RJ)
source spectrum
I o ( v c)
=
Ir
j (vc)
=
1A
r2kT^
cl
(A .3)
where r is the optical depth of the source and k is Boltzm ann’s constant. Since
the detected power is assumed to be
r
iP
Pin = rjAQ / r 2 k T — t{ v )d v
151
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(A .4)
152
we can write
^
“ riASl f v 2t ( v ) d v V^
<A '6>
If we assume a differentsource spectrum, for example a greybody with power-law
emissivity, the assumed input power is
=
Pin
=
T]AVL
r]AVt
J
J
I GB{u)t(u)du
(A . 6)
T ( v o ) ( i / / v 0) a B l/( T ) t ( v ) d i '
and the source spectrum inferred from the detected power is
I c b M
=
(v„)(VJ V„ T B ^ ( T )
r
Pin
ijAV. f v aB „ ( T )t ( u ) d i
f
f
u2t( u) du
(A .7)
-> £ B „ ,( T )
2
- v r 2B , c( T ) I , u W c)
v aB w{ T ) t{ v) d v
I r M )
f t ( v0) ( v /
v0)aB u{ T ) t { v ) d v _
J v~lt(v)dv
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(A . 8 )
(A.9)
153
For and arbitrary experiment with bandpass t( v) that reports its surface bright­
ness measurements assuming a spectrum I q{u ), the surface brightness assum ing a
different source spectrum h { u ) is given by
IM )
=
=
(A . 1 0 )
h { v c) K
h { y c)
A(z'c)
h{vc)
f lQ(v)t{v)dv
-l
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