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Interstellar dust thermal emission at millimeter and microwave wavelengths

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I N T E R S T E L L A R D U S T T H E R M A L E M I S S I O N AT M I L L I M E T E R
AND MICROWAVE WAVELENGTHS
by
Zhuohan Liang
A dissertation submitted to The Johns Hopkins University in conformity with the requirements for
the degree of Doctor of Philosophy.
Baltimore, Maryland
August, 2011
© Zhuohan Liang 2011
All rights reserved
UMI Number: 3492589
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Abstract
Interstellar dust grains are particles of size between a few to hundreds of
nanometers, mostly made up of carbon and silicon, found in the vast space between
stars within a galaxy. They are important because dust plays a major role in cycling
matter and energy between stars and the interstellar medium. Models for interstellar
dust thermal emission are fit to a set of 214-channel dust spectra at 60-3000 GHz.
Data consist of a new and improved version of dust spectra derived from the measurements of the Far Infrared Absolute Spectrophotometer of the COsmic Background
Explorer satellite, sky maps at 100 //m, 140 /jm and 240 jivn measured by the Diffuse
Infrared Background Experiment, also onboard the COBE satellite, and the 94 GHz
dust map measured by the Wilkinson Microwave Anisotropy Probe satellite. A singlecomponent model with its emissivity spectral index fixed at 1.7 is the best among
all dust models tested. It fits 88% of the sky with a Xdof — 1-13 at 210 degrees of
freedom. Within this sky region, temperatures of the dust grains are predicted to be
between 16.4 K and 25.1 K, and optical depths are between 1.3 x 10~6 and 5.1 x 10~4.
The uncertainties of the dust temperature are < 10%, while the uncertainties of the
optical depth are < 63%. In comparison with the two-component ai j 2 = 2.0 dust
model by Reach et al. (1995b) and the Finkbeiner, Davis & Schlegel (1999, FDS)
Model # 8 , the a — 1.7 model is shown to be better able to trace out dust spectral
variations over the entire FIRAS frequency coverage in sky regions where these two
models are valid. Currently, uncertainties of the best-fit parameters are limited by
FIRAS angular resolution and noise, and the angular resolution of the model inherits
that of the FIRAS. When data of better quality become available, such as from the
Planck mission, this one-component a = 1.7 (<$Tdust/7dust < 10%) model can be used
ii
to check future dust models.
Thesis Defense Committee: Professor Charles Bennett (advisor), Professor Holland
Ford, Professor Bruce Barnett, Professor Darrell Strobel, and Dr. Scott Friedman.
m
Acknowledgements
Graduate school is a period of rapid growth for a young person. For all that
I have learned in the last six years, I thank Professor Bennett for the opportunity to
work and learn under his guidance. His acute sense in identifying important research
questions, tireless pursuit of their answers, and the very high standard he sets for
himself have inspired me and taught me to appreciate the complexity of modern day
research problems and the tremendous amount of effort that goes into tackling any
one of them. I started work on this interstellar dust project with no prior experience
in the field. In order to obtain the results in this thesis, I am grateful for numerous
helpful discussions with Dr. Janet Weiland, Dr. Dale Fixsen, Dr. Ben Gold, Dr.
David Larson, and Dr. Domenico Tocchini-Valentini.
For the successful completion of this thesis, I also thank the IT staff for
maintaining the health of the computing systems and the administrative staff of the
department in keeping me on track in the graduate program.
Coming to Hopkins, I am grateful for the friendship and support of Shiping,
Urmila, Ching-Wa, Soyoung, Justice, Ivy, Isa, Roxana, Frank, Tingyong, Divya, John,
Yuan, Ivelisse, Waqas, Nuala, Bridget, and Arpit.
I would not be where I am without the nameless but not unremembered acts
of kindness of many, many people. In particular, I thank Mr. Citron for instilling in
me the confidence to pursue physics. I thank Ms. Stalonas and the late Ms. Callahan
for teaching me English, a language that I hardly spoke on the first day I arrived at
their classes. I am greatly indebted to Professor Watson, Professor Clark, Professor
Holmes, Professor Gage, and Dr. Valone for their patience, encouragement and for
showing me the excitement of learning. I thank Zhang Laoshi, Stephanie Hranjac,
Nelly, and Natalee for their support, encouragement and for teaching me through
iv
their exemplary attitude in life.
Grandma has been my greatest resource for advice and solace. She has the
magical ability of remembering all my friends (Yes, all of them, from elementary
school to graduate school!) even though she can't remember what she just read on
the newspaper five minutes ago. 2Yee has taken care of me over the years while
my parents are away. Her selflessness and thoughtfulness guide me in making every
important decision. Kowfu steps in for my parents and takes care of my needs even
before attending to the needs of his own children. Grandpa, 3Yee, Kowmo, 2YeeJiong,
3YeeJiong are my staunch supporters, always there whenever I want to talk or need
their help.
My greatest fortune is to have my loving and supportive parents. A question
that I have been asking myself for years is this: When I don't agree with a course
of action, can I still give others the attention, support and, most importantly, the
freedom to choose what they want to do? Mom and Dad have showed me that it is
possible, and I am much happier knowing their answer.
V
Contents
Abstract
ii
Acknowledgements
iv
List of Figures
ix
List of Tables
xiii
1
Introduction
1.1 What Is Interstellar Dust?
1.2 Origin of Diffuse Galactic Far-infrared and Sub-millimeter Emission
1.3 A Foreground in Cosmic Microwave Background Experiments
1.4 Theoretical Models
1.5 Relation between Temperature and Emissivity Index
1.6 Empirical All-sky Models
1.7 Plan for the Thesis
2
Instruments and Data Sets
2.1 COBE DIRBE
2.2 COBE FIRAS
2.3 WMAP
9
9
9
10
3
Data Preparation
3.1 Deducing FIRAS Dust Spectral Maps
3.1.1 Dipole
3.1.2 Zodiacal Light
3.1.3 Emission Lines
3.1.4 Cosmic Infrared Background
3.2 Beam Difference
3.3 Map Projection and Spatial Resolution
3.4 Gradient Correction
3.5 Color Correction
3.6 DIRBE Uncertainties
3.7 Temperature-intensity Conversion
12
12
12
14
14
14
16
17
18
18
18
19
4
Overview of Model Fitting Strategy
20
vi
1
1
3
4
5
6
7
7
5
One-component Dust Models for 7° Sky Regions
5.1 a = 2.0 Model
5.2 Other Fixed-a Models
5.3 Summary
22
22
32
33
6
One-component Dust Models for Averaged Sky Regions
6.1 Fitting Averaged Spectra of Latitudinal Rings
6.2 Fitting the Averaged Spectrum of the High-latitude Sky
6.3 Fitting Averaged Spectra of High-latitude Bands
6.4 Fitting Averaged Spectra of Longitudinal Regions
6.5 Fitting Averaged Spectra of Different-size Sky Regions
6.5.1 a = 2.0 Model
6.5.2 Other Fixed-a Models
6.6 Summary
38
38
41
41
45
51
51
57
58
7
More Complex Dust Models for Galactic-Plane Spectra
7.1 One-component Free-a Model
7.2 Two-component a1>2 = 2.0 Model
7.3 Two-component Models with Other Fixed Values of a\ and OLI
7.4 Three-component 0:1,2,3 = 2.0 Model
7.5 Summary
62
62
65
66
72
72
8
Understanding the Galactic-plane Fits
8.1 Highest y2 Channels Concentrate around 900 GHz and 2000 GHz
8.2 Synchrotron Emission Could Not Have Caused High x2
8.3 Highest x 2 Channels Do Not Have the Largest Deviations between Data and Model
8.4 Emission Lines and Their Asymmetric Profiles Are Partially Responsible for High \ 2
8.5 Re-fit Model without Channels at 800 - 1000 GHz and 1500 - 2000 GHz
8.6 Re-fit Model without Dichroic Channels
8.7 Re-fit Model without Channels in the Vicinity of Emission Lines
8.8 Frequency-Frequency Covariance
8.9 Summary
76
76
76
78
78
81
81
84
89
91
9
Summary and Discussion
9.1 Summary of Dust Models in This Thesis
9.2 Comparison with Contemporary All-sky Dust Models
9.2.1 Reach et al. Model
92
92
93
93
9.2.2
FDS Model # 8
94
10 Conclusion
99
Appendix
100
A One-component Dust Models for 7° Sky Regions
A.l a = 1.4 Model
A.2 a = 1.5 Model
A.3 a = 1.6 Model
A.4 a = 1.7 Model
A.5 a = 1.8 Model
A.6 a = 1.9 Model
A.7 a = 2.1 Model
A.8 a = 2.2 Model
100
101
105
109
113
117
121
125
129
vn
A.9 a = 2.3 Model
133
B One-component Fixed-a Models for Averaged Sky Regions
137
C More Complex Dust Models for Galactic-Plane Spectra
C.l One-component Free-a Model
C.2 Two-component alj2 = 2.0 Model
C.3 Three-component ai,2,3 = 2.0 Model
146
147
149
152
Bibliography
153
Vita
166
viii
List of Figures
1.1
Mollweide projection of t h e Milky Way in galactic coordinates
3.1
3.2
3.3
Examples of t h e new F I R A S d u s t spectra
Locations of highest C + emission represented by 100 F I R A S 7° pixels
CIB models by Hauser et al. (1998), Fixsen et al. (1998) and Finkbeiner et al. (2000).
5.1
5.2
Xdof of t h e best-fit one-component a = 2.0 models for individual pixel spectra. . . .
Best-fit values of Td u s t, T, C + intensity and N + intensity, a n d their uncertainties from
fitting one-component a = 2.0 model to individual pixel spectra
5.3 Correlations among Tdust, T, C + intensity a n d N + intensity of t h e best-fit onecomponent a = 2.0 models for individual pixel spectra
5.4 Examples of best-fit one-component a = 2.0 models for individual dust spectra. . . .
5.5 Xdof v s - Galactic latitude of t h e best-fit one-component a = 2.0 model for individual
pixel spectra
5.6 T h e Xdof distribution of t h e best-fit one-component a = 2.0 model for individual pixel
spectra
5.7 Distribution of t h e best-fit Td us t from fitting one-component a = 2.0 model to spectra
of individual pixels
5.8 Percentage error of t h e best-fit Td us t from fitting one-component a = 2.0 model to
spectra of 7° pixels
5.9 Xdof and best-fit parameters as a function of a for a mid-latitude pixel
5.10 Xdof a n d best-fit parameters as a function of a for a high-latitude pixel
5.11 68% a n d 95% probability contours of best-fit Tdust and best-fit a for FIRAS pixels
# 1 and # 3 6 7 2
5.12 Sky maps of Xdof' a a n o - ^dustl value at each pixel comes from t h e model t h a t gives
minimum Xdof among all tested a models
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
Xdof a n d Tdust of t h e best-fit one-component fixed-a models for different latitude-ring
division of the sky
Xdof, STdust/Tdust
a n d <5Tdust/Tdust of t h e minimum-Xd o f model a t each ring as a
function of Galactic latitude
WMAP's, t e m p e r a t u r e analysis mask in COBE quad-cube res6
Xdof a s a function of a
Demonstration of dividing t h e KQ75 unmasked sky into latitude bands
Xdof °f t h e one-component a = 1.8 a n d a = 2.0 fits to latitudinal band average spectra.
<5Tdust/Tdust and 5T/T of t h e best-fit one-component a = 1.8 models to t h e average
spectra at each latitudinal band
Xdof a n d <5Tdust/Tdust from t h e a = 1.8 and a = 2.0
fits
IX
8
13
15
16
23
24
26
27
28
29
30
31
34
35
36
37
39
42
43
44
46
47
48
49
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
Sky map of Tdust from the best-fit one-component a = 1.8 models for longitudinal
averaged spectra
50
Distributions of x20f; Xdof v s - Galactic latitude
53
Distributions of Tdust of all-sky one-component a = 2.0 models that satisfy <5Tdust/Tdust <
20%, 10%, 5% and 2.5%, respectively.
54
Distributions of r of all-sky one-component a = 2.0 models that satisfy <5TdUst/7dust <
20%, 10%, 5% and 2.5% respectively.
55
Sky directions where best-fit x 2 do not monotonically increase as averaging region of
the fit is reduced
57
Distributions of Xd0f for all-sky one-component fixed-o models that satisfy <5Tdust/TdUst <
10%; % of sky that has a fit whose Xdof e x c e e d s the probability cutoff
59
All-sky maps of Xdof ^dust and r of the best-fit one-component a = 1.7 model that
satisfies ^Tdust/Tdust < 10% with the least amount of averaging
60
All-sky maps of variable sky averaging, STdust/Tdust and ST/T of the best-fit onecomponent a = 1.7 model that satisfies <5Xdust/Tdust < 10% with the least amount of
averaging
61
7.1
Sky maps of Xdof > a-> ^dust; T and uncertainties of the best-fit one-component free-a
models for 7° Galactic-plane spectra
63
'•*•
Xdof, free a
VS
- Xdof, one-comp a = 2 0
"^
7.3 Xdof °f two-component ai,2 = 2.0 fits to Galactic plane spectra; Xtwo-comp ai 2=2 v s v2
67
Aone-comp o.=1
7.4
7.5
7.6
7.7
7.8
8.1
8.2
8.3
Best-fit parameters from fitting two-component 01,2 = 2.0 models to Galactic plane
spectra
Tj ow vs. Thigh and T\OW VS. Thigh of the best-fit two-component 011,2 = 2.0 models for
275 Galactic-plane spectra
Xdof °f n t s that use two-component models with different fixed values of a i and 0-2
for the spectra of FIRAS Pixels 4899 and 5023
Sky maps of Xdof a n d the two emissivity spectral indices from the best-fit twocomponent models with fixed values of ct\ and 02 for the 275 Galactic plane spectra.
Sky maps of the two dust temperatures from the best-fit two-component models with
fixed values of OL\ and 02 for the 275 Galactic plane spectra
2
er
x P frequency channel using best-fit two-component a.\^ = 2.0 models
Difference between model and data scaled by standard errors
Highest x 2 channels in plots of difference between data and model for FIRAS Pixel
4192 and 4339
8.4 High x 2 channels near frequency of the C + line
8.5 Scatter plot of Xdof fr°m fitting two-component 0^2 = 2.0 model to spectra without
channels at 800 - 1000 GHz and 1500 - 2000 GHz vs. xLf from f u l 1 spectral fit. . .
8.6 Xdof fr°m fitting two-component 0-1,2 = 2.0 model to spectra that do not include
dichroic channels vs. Xdof fr°m fitting two-component 01,2 = 2.0 model to the full
214-channel spectra
8.7 Best-fit two-component 01,2 = 2.0 models from full-spectral fit and no dichroic crossover channel
fit
8.8 Frequency channels that are within specified frequency ranges of FIRAS emission lines.
8.9 Xdof from fitting spectra that do not include frequency channels at or near emission
lines vs. xlof fr°m fitting the full 214-channel spectra
8.10 Changes in Xdof a s a result of not fitting frequency channels near emission lines. . .
8.11 FIRAS covariance between one channel and all other channels for spectra of Pixel
4170 and 359
x
68
71
73
74
75
77
79
80
82
83
84
85
86
87
88
90
9.1
9.2
9.3
Predictions for average dust spectra in three sky regions by the two-component models
of Reach et al. (1995b) and the one-component a = 1.7 (<5TdUst/Tdust < 10%) model.
Three dust spectra and their corresponding model predictions by FDS Model # 8 and
t h e one-component a = 1.7 (<5Tdust/TdUst < 10%) model
All-sky maps of Xdof of FDS Model # 8 and the one-component a = 1.7 (<5Tdust/Tdust <
10%) model in Galactic coordinate Mollweide projection
A . l Xdof °f the best-fit one-component a = 1.4 models for individual pixel spectra. . . .
A.2 Best-fit Tdust, T and their uncertainties from fitting one-component a = 1.4 model to
individual pixel spectra
A.3 Best-fit C + intensity, N + intensity and their uncertainties from fitting one-component
a = 1.4 model to individual pixel spectra
A.4 Correlations among Tdust, T, C + intensity and N + intensity of the best-fit onecomponent a = 1.4 models for individual pixel spectra
A.5 Xdof °f the best-fit one-component a = 1.5 models for individual pixel spectra. . . .
A.6 Best-fit Td u s t, T and their uncertainties from fitting one-component a = 1.5 model to
individual pixel spectra
A.7 Best-fit C + intensity, N+ intensity and their uncertainties from fitting one-component
a = 1.5 model t o individual pixel spectra
A.8 Correlations among Td u s t, r , C + intensity and N + intensity of the best-fit onecomponent a = 1.5 models for individual pixel spectra
A.9 Xdof °f the best-fit one-component a = 1.6 models for individual pixel spectra. . . .
A.10 Best-fit Tdust, T and their uncertainties from fitting one-component a = 1.6 model t o
individual pixel spectra
A. 11 Best-fit C + intensity, N + intensity and their uncertainties from fitting one-component
a = 1.6 model to individual pixel spectra
A. 12 Correlations among Xd us t, T, C + intensity and N + intensity of t h e best-fit onecomponent a = 1.6 models for individual pixel spectra
A.13 x^of °f the best-fit one-component a = 1.7 models for individual pixel spectra. . . .
A.14 Best-fit Td u s t, T and their uncertainties from fitting one-component a = 1.7 model to
individual pixel spectra
A.15 Best-fit C + intensity, N + intensity and their uncertainties from fitting one-component
a = 1.7 model to individual pixel spectra
A.16 Correlations among Tdust, T, C + intensity and N + intensity of the best-fit onecomponent a = 1.7 models for individual pixel spectra
A. 17 Xdof °f the best-fit one-component a = 1.8 models for individual pixel spectra. . . .
A.18 Best-fit Xdust, T and their uncertainties from fitting one-component a = 1.8 model to
individual pixel spectra
A.19 Best-fit C + intensity, N + intensity and their uncertainties from fitting one-component
a = 1.8 model to individual pixel spectra
A.20 Correlations among Tdust, T, C + intensity and N + intensity of the best-fit onecomponent a = 1.8 models for individual pixel spectra
A.21 Xdof of t h e best-fit one-component a = 1.9 models for individual pixel spectra. . . .
A.22 Best-fit Tdust, T and their uncertainties from fitting one-component a = 1.9 model to
individual pixel spectra
A.23 Best-fit C + intensity, N + intensity and their uncertainties from fitting one-component
a = 1.9 model to individual pixel spectra
A.24 Correlations among Tdust, T, C + intensity and N + intensity of the best-fit onecomponent a = 1.9 models for individual pixel spectra
A.25 Xdof °f t h e best-fit one-component a = 2.1 models for individual pixel spectra. . . .
XI
95
97
98
101
102
103
104
105
106
107
108
109
110
Ill
112
113
114
115
116
117
118
119
120
121
122
123
124
125
A.26 Best-fit Td u s t, T and their uncertainties from fitting one-component a = 2.1 model to
individual pixel spectra
A.27 Best-fit C + intensity, N + intensity and their uncertainties from fitting one-component
a = 2.1 model to individual pixel spectra
A.28 Correlations among Tdust, T, C + intensity and N + intensity of the best-fit onecomponent a = 2.1 models for individual pixel spectra
A.29 Xj o f of the best-fit one-component a = 2.2 models for individual pixel spectra. . . .
A.30 Best-fit Tdust, T and their uncertainties from fitting one-component a = 2.2 model to
individual pixel spectra
A.31 Best-fit C + intensity, N + intensity and their uncertainties from fitting one-component
a = 2.2 model to individual pixel spectra
A.32 Correlations among Td u s t, T, C+ intensity and N + intensity of the best-fit onecomponent a = 2.2 models for individual pixel spectra
A.33 Xdof °f the best-fit one-component a = 2.3 models for individual pixel spectra. . . .
A.34 Best-fit Td u s t, T and their uncertainties from fitting one-component a = 2.3 model t o
individual pixel spectra
A.35 Best-fit C + intensity, N + intensity and their uncertainties from fitting one-component
a = 2.3 model to individual pixel spectra
A.36 Correlations among Td u s t, T, C + intensity and N + intensity of the best-fit onecomponent a = 2.3 models for individual pixel spectra
All-sky maps of Xdof' ^dust and r of the best-fit one-component a = 2.0 model t h a t
satisfies 5Tdust/Tdust
< 20%
B.2 All-sky maps of variable sky averaging, <5Tdust/TdUst and ST/T of t h e best-fit onecomponent a = 2.0 model t h a t satisfies STdust/Tdust
< 20%
B.3 All-sky maps of Xdof > ^dust and r of t h e best-fit one-component a = 2.0 model t h a t
satisfies 5Td u s t/T d u s t < 10%
B.4 All-sky maps of variable sky averaging, <5Tdust/^dust and ST/T of the best-fit onecomponent a = 2.0 model t h a t satisfies <5TdUst/2dust < 10%
B.5 All-sky maps of Xdof > ^dust and r of the best-fit one-component a = 2.0 model t h a t
satisfies <5T dus t/Td ust < 5%
B.6 All-sky maps of variable sky averaging, <JTd us t/Td us t and 5T/T of the best-fit onecomponent a = 2.0 model t h a t satisfies (STdust/^dust < 5%
B.7 All-sky maps of Xdof ^dust and r of the best-fit one-component a = 2.0 model t h a t
satisfies <5T dus t/T du st < 2.5%
B.8 All-sky maps of variable sky averaging, STdust/Tdust and 5T/T of t h e best-fit onecomponent a = 2.0 model t h a t satisfies <WdUst/2dust < 2.5%
126
127
128
129
130
131
132
133
134
135
136
B.l
C.l
C.2
C.3
C.4
C.5
C.6
Sample 7° spectra and their best-fit one-component
Sample 7° spectra and their best-fit one-component
Examples of best-fit two-component a = 2.0 models
Examples of best-fit two-component a = 2.0 models
Examples of best-fit two-component a = 2.0 models
Examples of best-fit three-component 0:1,2,3 = 2.0
spectra
xii
free-a models
free-a models
and their corresponding spectra.
and their corresponding spectra.
and their corresponding spectra.
models and their corresponding
138
139
140
141
142
143
144
145
147
148
149
150
151
152
List of Tables
2.1
Spectral Coverage of DIRBE, FIRAS k WMAP
6.1
6.2
% sky fit by one-component a = 2.0 models in different region sizes
56
Best-fit one-component a = 2.0 models for the average spectra centered at Pixel 4062. 56
xm
11
For Mom and Dad
XIV
Chapter 1
Introduction
1.1
W h a t Is Interstellar Dust?
Interstellar dust grains are particles with dimensions between a few to hundreds of nanome-
ters. They consist mostly of carbon and silicon, and are found in the vast space between stars within
a galaxy. Below, I discuss our current understanding of the Milky Way's interstellar dust and its
relation with other constituents of the Galaxy.
The Milky Way Galaxy is a system of about 10 11 stars that has been assembled over the last
13.2 x 109 years. Within this system, space between stars is referred to as the interstellar medium
(ISM). The ISM is not empty but filled with gas, dust grains, relativistic particles, magnetic fields
and photons. The Milky Way has four structures: the bulge, the disk, the halo, and the dark halo.
The Galactic bulge is about 6 kpc across, 4 kpc in height, and shaped like a football. It is dominated
by old stars, though young stars, gas and dust are also found here. The Galactic disk has a diameter
of about 30 kpc, and a scale height of ~ 300 pc. It has almost all the gas and young stars, a few
old stars and some dust. Here, stars and gas move in approximately circular orbits in the Galactic
plane and form spiral arms. The Galactic halo is roughly spherical with a scale length of about 20
kpc. It consists of very old stars and globular clusters. Here, the concentration of gas and dust is
low, and the halo has no obvious sub-structure. Finally, the dark halo spans over a radius > 200
kpc. This vast space mainly consists of dark matter, which takes up 80 — 90% of the total mass of
the Galaxy. Whereas star formation takes place in the bulge and in the disk, at a rate of ~ 5 M Q in
total every year, the halo has not formed any new stars in the last 1010 years (Kriigel 2003; Chaisson
& McMillan 2010).
Interstellar dust plays an important role in the cycling of matter between stars and the
ISM. Dust grains form mainly as high-temperature condensates in the wind of old stars, such as red
giants and planetary nebulae, and during nova and supernovae explosions. During their lifetime, dust
grains grow in size when gas atoms condense onto their surface or when they collide into each other
1
and form larger grains. On average, dust grains linger in the ISM for about a billion years before
being destroyed. Processes that cause the demise of dust grains include: evaporation (heated above
their condensation temperature due to getting too close to luminous stars or being present at star
formation), sputtering (portions of a grain being ejected after collision with a gas particle, such as in
shocks associated with supernova remnants), grain-grain collision, shattering, and photodesorption
(ejection of grain atoms by photons). Among these, star formation is the most destructive process
for dust grains, followed by shocks of supernova remnants (Kriigel 2003).
In terms of mass, interstellar dust takes up a tiny portion of the total mass of the Milky
Way. Its total mass is estimated to be ~ 3 x 1O 7 M 0 , as opposed to ~ 5 x 1O 9 M 0 for the total gas
mass, o r ~ 7 x 10 11 M Q for total mass of the Milky Way. However, dust grains convert 30% of the
total power radiated from stars into infrared wavelengths. The total power radiated by the Milky
Way comes from stars and is estimated to be 3.6 x 10 10 L Q . The vast majority of starlight leaves the
Milky Way after being absorbed by dust grains and re-radiated in the infrared (> 10 /im) (Kriigel
2003).
Three types of dust grains have been identified in the ISM: Polycyclic Aromatic Hydrocarbons (PAHs) with a characteristic size of a few nm and the dominant emitters at A < 12 /im; very
small grains (VSGs) with sizes of a few tens of nm, which radiate at 25 fim < A < 60 /xm; and large
grains with sizes between a few tens of nm to 0.1 /im emit at A > 100/im. Due to their small size,
each PAH or VSG has internal energy less than or equal to the energy of a typical stellar photon.
When such a photon is absorbed, it causes a spike in the grain's temperature. The smaller the
grain, the higher the maximum temperature. In comparison, large grains have more thermal mass,
so individual photon absorption does not as substantially change the energy content of a large dust
grain. These grains are thus in thermal equilibrium with the local interstellar radiation field, and
maintain a steady-state temperature over time. (Mathis et al. 1977; Draine & Lee 1984; Mathis &
Whiffen 1989; Kim et al. 1994; Mathis 1996; Hong & Greenberg 1978; Greenberg 1989; Greenberg &
Li 1996; Li & Greenberg 1997; Dwek et al. 1997; Lagache et al. 1998; Draine 2003; Li 2004; Zubko,
Dwek & Arendt 2004; Paladini et al. 2007). Radiation from the large grains accounts for ~65% of
the total emitted power of all dust grains.
The interaction between dust and stellar radiation has been a key source of our knowledge
about such particles. For example, distant stars appear obscured, so we expect that interstellar
dust grains have sizes comparable to or larger than the wavelengths of stellar photons, making them
effective absorbers of the photons. The spectra of stars appear reddened, so we expect that dust
grains scatter and absorb more short-wavelength radiation than long-wavelength radiation. Light
from distant stars is found to be polarized, so we know that dust particles are non-spherical. As a
result of these processes, the diffuse far infrared emission is the manifestation of dust radiating away
the absorbed shorter wavelength energy, originally produced by stars.
2
1.2
Origin of Diffuse Galactic Far-infrared and Sub-millimeter
Emission
The identification of the Galaxy's diffuse emission with the thermal emission of interstellar
dust is an arduous task. It involves identifying energy sources that power the observed thermal
spectrum and showing the energy is transferred. Our current understanding in this area has been
built upon the the work cited below.
Mezger, Mathis & Panagia (1982) and Mathis et al. (1983) fit spectra of diffuse emission
from the Galactic plane using the Mathis, Rumpl & Nordsieck (1977) dust extinction model for
grain sizes between 0.1 and 1 /im, and find that the interstellar radiation field (ISRF) between 0.09
and 8 /u.m is dominated by stellar radiation modified by dust scattering, and that the ISRF between
8 and 1000 /xm is dominated by re-emitted radiation from dust grains. They further identify two
main contributors to the diffuse emission at wavelengths > 20 /im: dust heated by O stars (60%),
and dust associated with diffuse atomic intercloud gas heated by the general interstellar radiation
field (40%), and conclude that dust associated with clouds containing no luminous sources of heating
such as OB stars contributes less than 7% of the total diffuse far-infrared/sub-millimeter emission.
OB stars are hot, massive stars of spectral types O or B. They emit copious amounts of ultraviolet
radiation, which ionizes the surrounding interstellar gas of the giant molecular cloud.
Cox, Kriiegel & Mezger (1986) extend the previous models to fit all observations then
available, which were all from the Galactic disk. They show that the combined spectrum has two
maxima: one at ~ 100 ^m and the other at wavelengths between 4 and 20 /j,m. The total luminosity
of the re-radiated dust emission from the Galactic disk is ~ 1.5 x 10 1 0 L Q or about 40% of the total
stellar luminosity. The best-fit dust model contains four components:
•
A cold dust (T ~ 15—25 K) component associated with atomic hydrogen and a very cold
dust (< T > ~ 14 K) component associated with molecular hydrogen and located inside quiescent
molecular clouds. These two components dominate the sub-millimeter part of the spectrum, and
are heated by the general ISRF generated by both young and old stellar populations. The total
luminosity of these two components accounts for ~ 37% of the total infrared luminosity of the
Galaxy.
•
A warm dust (T ~ 30 — 40 K) component associated with ionized gas in extended low
density H II regions heated by O and B stars. The total warm dust luminosity accounts for ~ 50%
of the total Galactic IR luminosity.
•
A hot dust (T ~ 250 - 500 K) component consisting of VSGs and PAHs (radius > 5 A)
heated by the general ISRF and normal grains (radius > 0.1//m) heated by M and K giants. This
hot dust component accounts for ~ 13% of the total Galactic luminosity.
The first all-sky infrared measurements taken by the Infrared Astronomical Satellite
(IRAS,
IRAS Explanatory Supplement (1988)) made possible the study of the matter distribution outside
3
of the Galactic disk. Boulanger & Perault (1988) show that Galactic emission exists at Galactic
latitude |6| > 10° at 12, 25, 60 and 100 /xm, and that the emission comes mainly from dust heated
by the ISRF associated with atomic gas and diffuse ionized gas. More specifically, they show that
the Galactic latitude profiles of the IRAS all-sky measurements at 60 and 100 //m follow a cosecant
law from 10° to the poles. This means that dust emission decreases from low to high latitudes in
the manner as would be expected for a plane-parallel configuration.
Using measurements at 140 and 240 [im taken by the Diffuse Infrared Background Experiment (DIRBE, DIRBE Explanatory Supplement (1998)) of the COsmic Background
Explorer
(COBE, Mather (1982)) satellite, Sodroski et al. (1994) study interstellar dust properties in the
Galactic plane region (\b\ < 10°), and find that large dust grains emitting at equilibrium temperature dominate the emission in the two wavelength bands. They show a longitudinal and latitudinal
gradient in dust temperature and that dust temperature decreases with increasing Galactocentric
distance. They conclude that 60 — 75% of far-infrared luminosity comes from cold dust (~ 17 — 22
K) that is associated with diffuse H I clouds; another 15 — 30% of the total luminosity comes from
cold dust (~ 19 K) that is associated with molecular gas; and finally less than 10% of the total
luminosity comes from warm dust (~ 29 K) in extended low-density H II regions.
1.3
A Foreground in Cosmic Microwave Background Experiments
In recent years, models of thermal dust emission in the far-infrared and millimeter wave-
lengths have become an important tool in cosmic microwave background (CMB) anisotropy studies
because they are needed to help remove this major foreground contaminant.
CMB radiation is the cool remnant of the hot big bang. It fills the entire universe and can
be observed today at an average temperature of about 2.7 K. Studies of its angular variation provide
information obtainable in no other way about the global geometry and expansion of the universe.
However, measuring variations in this weak signal is difficult, and the elimination of contaminants
in the data needs to be done accurately. For example, the Wilkinson Microwave Anisotropy Probe
(WMAP) satellite measured the full sky in five frequency bands: 23, 33, 41, 61 and 94 GHz, where
dust contributes substantially to its highest frequency measurements. As a result, the dust signal
must be removed from data before any cosmological analysis begins.
The WMAP team had applied three different methods to model the foregrounds (Barnes
et al. 2003; Hinshaw et al. 2007). These include using the internal linear combination (ILC) method
to make a CMB map, applying the Maximum Entropy Method (MEM) to model foreground components, and subtracting external templates from WMAP data to remove foregrounds. Among the
three, only the template fitting method has been used to produce data for cosmological analysis. The
other two methods have been excluded because of their very complicated noise properties. There-
4
fore, to remove the thermal dust contribution the WMAP Team subtracted a component fit to the
template from the Finkbeiner, Davis & Schlegel (1999) (FDS) classic study of interstellar dust in
the far-infrared (Bennett et al. 2003b; Hinshaw et al. 2007; Gold et al. 2009).
1.4
Theoretical Models
Consider the simplest example of a spherical dust grain of radius a in an interstellar radia-
tion field that has energy density uv. (For a comprehensive review, interested readers may want to
consult Bohren & Huffman (1983); Draine (2003); Kriigel (2003); Li (2005b).)
The rate of energy increase by photon absorption can be written as
- fI
T*aSCU„dv,
Wahs=
lbs — /
Jo
where c is the speed of light, r^ bs is the mass absorption coefficient at frequency ^, also known as
opacity, and uw dv is photon energy between frequencies v and v + dv.
The emissivity, eu, is the emission of energy per unit wavelength per unit time per unit
solid angle and per unit mass. The isotropic emission over all directions equals 47re„, and the rate
at which dust grains lose energy by photon emission can be written as
i>oo
W r a d = / 47re„di/.
(1.1)
Jo
Because of thermal equilibrium, Kirchhoff's law relates the emissivity, e„, with the mass absorption
coefficient, r a b s ,
S
e„ = rf
where BU(T) = ~^-
hv/kT_i
B„(T),
is the Planck function. When we substitute the expression for ev into
Eq.1.1, we get
/•OO
/ 47rr, a b s ^(T d u s t )di/.
Jo
Again using thermal equilibrium, power absorbed = power radiated,
Wrad=
/
T?>scuvdv=
Jo
The mass absorption coefficient, T
4^ a b s J B 1 ,(r d u s t )d J A
/
(1.2)
Jo
abs
, depends on a dust grain's size, shape, geometry and compo-
sition. Since we cannot directly sample the interstellar dust, assumptions must be made about its
values. For dust emission in the far-infrared, T abs is often approximated as
rfs
=
TO W
o)a,
(1-3)
where v§ is the normalization frequency specified at model fitting, and a is the power-law spectral
index. (Note that a is sometimes referred to as (3 in the literature.) Using the Kramers-Kronig
dispersion relation, one can derive the limit
/
Jo
r^dv
<
5
3ir2F/p,
where p is the mass density of a dust grain, and F is the orientationally-averaged polarizability
relative to the polarizability of an equal-volume conducting sphere, depending only on a grain's
shape and the static dielectric constant of its material (Li 2005b). This relation leads to a lower
limit on a, such that a > 1 for v —> 0.
In cases where the wavelength of the radiation is much larger than grain size, the square
relation between T^bs and frequency, i.e., r* bs oc i/2, applies to both crystalline and amorphous
materials, with the exception of amorphous layered materials and very small amorphous grains, for
which T^bs oc v1. Variations in the structure of the latter two cases give rise to 1 < a < 2 (Li 2005b;
Tielens 2005).
Substituting Eq.1.3 into Eq.1.2 and assuming that an isothermal source is the cause of the
measured spectrum, we arrive at an expression that can be compared directly with the observed
intensity:
h=TQ{v/y0)aBu{Tdust)
(1.4)
In the model-fitting portion of this thesis, equation (1.4) is fit to the data and the best-fit value
of Tdust is extracted. In some models, a has a pre-defined value and in others it is considered a
variable.
1.5
Relation between Temperature and Emissivity Index
Shetty et al. (2009) explored the effects of noise on spectral model fitting and the corre-
lation between a and Tdust- They find that the accuracy, of parameter estimation depends on the
wavelength range where spectral information is available for model fitting. They conclude that the
a — Tdust distributions of fits using a model with one fixed a, a model with temperature dependent
a, and a model with two fixed a's have similar patterns. Results from Shetty et al. emphasize
the importance of using adequate spectral information when fitting dust models. At 100 — 600 /zm,
spectral measurements sample both the Rayleigh-Jeans tail and the peak of a ~ 20 K modified
blackbody spectrum, so noise has little effect on the fit. On the other hand, this same wavelength
range corresponds to just the Rayleigh-Jeans tail of a warmer source. Since the slopes of the longwavelength regions of blackbody functions at different temperatures are the same, a fit to such a
limited range of spectral measurements cannot uniquely determine temperature of the source.
Because it is difficult to constrain both the emissivity index and dust temperature in the
same model, I have tested models that set the emissivity index to fixed values in Chapter 5 and
Chapter 6.
6
1.6
Empirical All-sky Models
Among major efforts to derive an all-sky dust model from spectral measurements, Reach
et al. (1995b) use dust spectra derived from sky measurements between 60 — 3000 GHz of the Far
Infrared Absolute Spectrophotometer (FIRAS, FIRAS Explanatory Supplement (1997)) instrument
onboard the COBE satellite at frequencies between 60 — 3000 GHz to constrain dust models with
emissivity proportional to v2. They find that dust emission is best described by a three-component
dust model: a warm (16 — 21 K) and a cold (4 — 7 K) component that are present everywhere in the
sky, and an intermediate temperature (10 — 14 K) component that exists only at Inner Galaxy.
Boulanger et al. (1996) derive another set of dust spectra using FIRAS measurements at
100 /xm to 1 mm and a v2 emissivity dust model. They find that the average spectrum of dust
associated with H I gas has a temperature at 17.5 K.
Using high and low-frequency band dust spectra provided by the FIRAS Team, the 100 /im
dust emission map produced by Schlegel et al. (1998), and a filtered DIRBE 7ioo/-^240 flux ratio map,
Finkbeiner et al. (1999) find that a best-fit model (their Model # 8 ) consists of two dust components:
a cold component following a v1
2 70
component following a v
67
emissivity law with temperature at 7.7 — 13.1 K, and a warm
emissivity law with temperature at 13.6 — 21.2 K.
This list of results provides a glimpse of the diverse findings in the study of thermal dust
emission in the far-infrared and submillimeter. Even when the majority of their data come from the
same source, the best-fit dust models differ both in the number of dust components and in the range
of dust temperature. What these variations show is that the derived dust properties depend as much
on the fitting method and the functional form of the model as on the data. In the following chapters,
I summarize results from an independent and comprehensive model building exercise to identify a
model for thermal dust emission that is supported by the best data available at the present.
1.7
Plan for the Thesis
This thesis updates the latest all-sky model, FDS Model # 8 , with newer measurements
from the WMAP and a new set of FIRAS dust spectra which has the
TOMB
removal corrected.
The approach taken here accounts for both spectral and spatial variations of the emission spectrum
during model fitting. The previous five sections have provided theoretical basis of thermal dust emission, including an introduction to the physical parameters and a summary of their current estimates.
Chapter 2 presents data from the Cosmic Background Explorer (COBE) satellite's DIRBE and FIRAS experiments and data from the Wilkinson Microwave Anisotropy Probe (WMAP, Bennett et
al. (2003a)) satellite mission. Chapter 3 details procedures that were taken to synchronize calibrations of the different data sets. Chapter 4 gives an overview of model fitting strategies taken for this
thesis. Chapters 5-8 present results from fitting a variety of dust models to the data, analysis of
the results, and discussion on each model's strengths and weaknesses. Chapter 9 summarizes results
7
Figure 1.1: Mollweide projection of the Galaxy in Galactic coordinate. The map center is the
Galactic center. Upper and lower ends of the minor axis are the Galactic poles at +90° and —90°
latitudes respectively. The major axis of the ellipse represents the Galactic plane; the left and right
ends of the major axis represent +180° and —180° longitudes respectively. This map serves as a
reference for all-sky maps presented in this thesis. It was produced by the WMAP Science Team
(Bennett et al. 2003a). The small blue and purple circles represent strong microwave point sources.
of the best-fit model obtained in previous chapters and compares it with those in contemporary
literature. Finally, Chapter 10 concludes the thesis and suggests how to use the results.
Throughout this thesis, all-sky maps are presented in Galactic coordinate Mollweide projection with the Galactic center at the center and longitude increasing to the left. Figure 1.1 is
provided as a reference for this type of map.
8
Chapter 2
Instruments and Data Sets
2.1
COBE DIRBE
The DIRBE instrument was a cryogenically cooled 10-band absolute photometer designed
to measure the spectrum and angular distribution of the diffuse infrared background. It had a
0°7 beam and covered the wavelength range from 1.25 to 240 /Ltm. During its lifetime, the DIRBE
achieved a sensitivity of 10~ 9 W m _ 2 sr~ 1 at most wavelengths (Mather 1982; Boggess et al. 1992;
Mather et al. 1993; Silverberg et al. 1993; DIRBE Explanatory Supplement 1998).
Among the many goals that DIRBE achieved, it detected and provided the hitherto most
stringent constraint on the isotropic extragalactic cosmic infrared background (CIB) (Dwek et al.
1998a; Hauser et al. 1998; Dwek et al. 1998b; Finkbeiner et al. 2000; Wright & Reese 2000; Wright
2001; Odegard et al. 2007). It also made possible the modeling of the scattered light and thermal
emission from interplanetary dust (IPD) (Reach et al. 1995a; Reach, Franz, Kelsall, & Weiland
1996; Reach 1997; Kelsall et al. 1998), and the Galactic infrared emission (Sodroski et al. 1994;
Freudenreich et al. 1994; Dwek et al. 1997; Sodroski et al. 1997; Wright 1998; Schlegel et al. 1998;
Reach, Wall & Odegard 1998; Arendt et al. 1998).
This thesis uses the 1997 "Pass 3b" Zodi-Subtracted Mission Average (ZSMA) Maps at
bands 100 fxm, 140 fim and 240 /jm. These maps measure the Galactic and extragalactic diffuse
infrared emission, and have been calibrated to remove the zodiacal light (zodi or IPD) emission.
They are available at the Legacy Archive for Microwave Background Data Analysis (LAMBDA) 1 .
2.2
COBE FIRAS
The FIRAS instrument was a polarizing Michelson interferometer designed to precisely
measure the difference between the CMB and a blackbody spectrum. The FIRAS had a 7° beam
x
The LAMBDA Web site is http://www.lambda.gsfc.nasa.gov/
9
and covered the frequency range from 1—97 c m - 1 at 0.45 c m - 1 resolution. The Pass 4 final data
release consisted of sets of 6067 pixels (99% of the full sky) for low- and high-frequency bands, with
each pixel associated with a 210-channel spectrum (Mather 1982; Boggess et al. 1992; Bennett et al.
1993; Mather et al. 1993; Fixsen et al. 1994a, 1997b; FIRAS Explanatory Supplement 1997; Fixsen
& Mather 2002).
The FIRAS project was crucial to the study of cosmology and the content of the interstellar
and interplanetary media. It confirmed the thermal spectrum of the CMB (Gush et al. 1990; Mather
et al. 1990; Wright et al. 1990; Shafer et al. 1991; Cheng et al. 1991; Mather et al. 1994; Fixsen et
al. 1994b; Wright 1994; Fixsen et al. 1996, 1997a; Mather et al. 1999), mapped the major spectral
lines in the far-infrared sky (Wright et al. 1991; Bennett et al. 1994; Wright 1997; Fixsen, Bennett,
& Mather 1999), extracted the Galactic dust distribution (Wright et al. 1991; Wright 1993; Reach
et al. 1995b; Finkbeiner et al. 1999; Fixsen & Dwek 2002), and supported the detection of the CIB
(Puget et al. 1996; Fixsen et al. 1998; Hauser 2001).
A set of dust spectral maps is derived from the Destriped Sky Spectra Maps, and the
procedures are described in Section 3.1. These 210 6063-pixel maps comprise the main body of
spectral information used in this thesis.
Six types of uncertainties have been characterized by the FIRAS Team. Recipes for applying each type of uncertainty can be found in Fixsen et al. (1994b); FIRAS Explanatory Supplement
(1997); Mather et al. (1999). A shortcut is provided by the FIRAS Team for applications that examine data in either frequency space or pixel space, as in each case only one dominating systematic
is relevant. This shortcut does not apply to this work because the new model needs to respond to
both spectral and spatial variations of dust emission. Therefore, the following analysis includes all
six FIRAS uncertainties (The short names in the parenthesis correspond to the physical quantities
that participate in the calculation and are not acronyms of their English names): detector noise
(D), emissivity gain uncertainties (PEP), bolometer parameter gain uncertainties (JCJ), internal
calibrator temperature errors (PUP), absolute temperature errors (PTP), and destriper errors (/3).
Section 7.10 of FIRAS Explanatory Supplement (1997) provides very helpful instructions on how to
apply these errors. For example, the D and /3 matrices vary only among pixels, while PEP, JCJ, PUP
and P T P matrices differ for different frequencies. Interested readers are referred to the Supplement
for detailed information.
2.3
WMAP
The Wilkinson Microwave Anisotropy Probe was designed to determine the geometry, con-
tent and evolution of the universe by measuring temperature anisotropy of the CMB radiation. It
consisted of two back-to-back offset Gregorian telescopes, and used 20 high electron mobility transistor (HEMT) based differential radiometers to measure the brightness difference between two lines
of sight that were 141° apart. At five frequency bands: 23, 33, 41, 61 and 94 GHz, the
10
WMAP
Table 2.1: Spectral Coverage of DIRBE, FIRAS & WMAP
DIRBE
FIRAS WMAP
A (/^m)
100
140
240 103 - 4407
3189
1/A (cm" 1 )
100
71
42
2-97
3
v (GHz) 2998 2141 1249
68-2911
94
made full sky measurements, which were analyzed by the data processing pipeline and form 13'
FWHM HEALPix 2 pixelization maps. The spin motion of the observatory and its scanning strategy
symmetrized the WMAP beams. Beam sizes were estimated using square-root of the beam solid
angle. In order of increasing frequencies they are: 0.88, 0.66, 0.51, 0.35 and 0.22° (Jarosik et al.
2003; Page et al. 2003; Barnes et al. 2003; Hinshaw et al. 2003; Jarosik et al. 2007; WMAP Five-Year
Explanatory Supplement 2008; Hinshaw et al. 2009; Hill et al. 2009).
The WMAP has helped establish the six-parameter ACDM model as the standard model
in cosmology. Using the first five years of WMAP data, Gold et al. (2009) studied different types of
diffused foreground emission and confirmed the validity of the foreground removal technique used for
WMAP's cosmological studies; Wright et al. (2009) found 390 flat-spectrum radio point sources in
the extragalactic sky; Nolta et al. (2009) derived the temperature and polarization angular spectra
of the CMB. Dunkley et al. (2009) reported the best-fit parameters of the model using WMAP data
only, and Komatsu et al. (2009) combined WMAP data with distance measurements from Type la
supernovae and Baryon Acoustic Oscillations in the distribution of galaxies to constrain models of
inflation and dark energy. Using WMAP data alone, a flat universe consists of 21.4% dark matter,
4.4% baryons, and 74.2% dark energy.
A dust temperature map (at 94 GHz), derived from the "base model" in WMAP's Five-Year
foreground modeling analysis by Gold et al. (2009), is used in this thesis. In the same study, different
models were tested to account for diffused foreground emission at different WMAP bands. Main
ingredients in those models were: nonthermal synchrotron, thermal bremsstrahlung, and thermal
dust. Some models included additional elements such as steepening synchrotron and/or spinning
dust. Their likelihood analysis showed that basic model with just three main foreground components
was sufficient to subtract out foregrounds from sky maps at high Galactic latitudes.
2
For definition and applications of the HEALPix projection, refer to Gorski, Hivon, & Wandelt (1999), Gorski et
al. (2005), and Calabretta & Roukema (2006).
11
Chapter 3
Data Preparation
The goal of this thesis is to find a dust model that is sensitive to spectral variations in both
frequency and direction using data from the (DIRBE, FIRAS, and WMAP) experiments. Since
these three experiments have different hardware constraints and calibration standards, to use them
in the same fit, different aspects of these maps need to be unified to ensure that measurements are
compared on an equal footing.
The following procedures on beam differences, map projections and spatial resolutions,
DIRBE-FIRAS absolute calibrations, and temperature-flux conversion are applied to both signal
and noise maps. Procedures on zodiacal light ("zodi") zero-point corrections and FIRAS systematic
errors are applicable to noise maps only.
3.1
Deducing FIRAS Dust Spectral Maps
The FIRAS dust spectra on LAMBDA have a "jump" between the low- and high-band data.
This is due to an inconsistent use of the CMB monopole temperature. In this thesis a new set of dust
spectra is deduced from the Destriped Sky Spectra Maps, which are the final products of FIRAS's
calibration process. The following describes procedures to take out a 2.7278 K blackbody for the
CMB, a dipole of the Earth's motion with respect to the CMB, a zodi model, and a contribution
from the cosmic infrared background (CIB).
Examples of the new dust spectra are plotted in red in Figure 3.1. For comparison, corresponding dust spectra provided by the FIRAS Team are plotted in purple.
3.1.1
Dipole
A WMAP-determined dipole (Hinshaw et al. 2009) is removed from the destriped spectra.
Specifically, T C M B = 2.7278 K, T dipoIe = 3.355 mK and (l,b) = (263°99,48°26).
12
Average dust spectrum of region 0 < I < 9 0 , - 6 0 < b < - 3 0
1000
100
Frequency (GHz)
Average dust spectrum of region 135< I < 1 8 0 , 10< b < 3 0
1000.000
100.000
10.000
>,
1.000
c
0 100
1000
100
Frequency (GHz)
Average dust spectrum of region 180< I < 2 2 5 , 10< b < 3 0
1000.000
100.000
10.000
1000
100
Frequency (GHz)
Figure 3.1: Plotted in red are new dust spectra and their standard deviations; in purple is dust
intensity derived by the FIRAS team; in dark green is total sky intensity measured by FIRAS; in
blue is the 2.7278 K CMB spectrum; and in cyan is dipole measured by the WMAP.
13
3.1.2
Zodiacal Light
Zodiacal light is the thermal emission and scattered light from interplanetary dust in our
solar system. A time-dependent parametric model for its emission was derived in Kelsall et al.
(1998), and the FIRAS Team extended those results to the entire FIRAS frequency coverage (FIRAS
Explanatory Supplement 1997). Derivation of the zodi model for FIRAS hinges on the fact that
FIRAS measurements overlap with DIRBE bands at 240 /im and 140 /im. Therefore, the DIRBE
model predictions for these two bands were coadded and fitted with a power law emissivity model
and extrapolated to frequencies covered by FIRAS. The zodiacal light model used here is among
the FIRAS data products on LAMBDA. For further details of the model derivation, see the FIRAS
Explanatory Supplement (1997).
3.1.3
Emission Lines
FIRAS detected 18 molecular and atomic lines emitted by interstellar gas. Since the FIRAS
frequency resolution is much larger than the width of each of these lines, each line profile is effectively
FIRAS's instrument response to a delta function. Among these 18 detected emission lines, not all
of them have a discernible presence all over the sky. Most notably are the [C II] and [N II] lines,
for whom there is a distinct gradient of intensity from the center of the Galaxy to higher latitudes.
Other emission lines, though they were detected, are weak in most of the sky except at the Inner
Galaxy. Here, the Inner Galaxy is referred to the ±90°-Galactic-longitude region of the Galactic
disk in an all-sky map of the Galaxy in Mollweide projection.
The FIRAS line intensity maps on LAMBDA were derived from the remaining signal of
destriped spectra after removing CMB, dipole, zodi, and their galactic dust spectra. Since it is the
goal of this thesis to fit the dust spectrum and not use a previous model, the FIRAS line intensity
maps cannot be used here to remove emission line contribution. Instead, intensities of [C II] and
[N II] emission are fit as parts of the overall model in the following work.
Figure 3.2 shows locations in the sky where [C II] emission is strongest (red pixels). Spectra
taken at these regions are also affected by strong [N II], CO and CI emission. As a result, modeling
spectral variation will be more complicated.
3.1.4
Cosmic Infrared Background
The isotropic CIB signal was removed from sky spectra using results from three studies:
For DIRBE measurements at 140 /im and 240 /im, the CIB is removed at 15 and 13 n W / m 2 / s r " 1
respectively according to Hauser et al. (1998). For the DIRBE band at 100 /xm, the CIB is removed
at 25 n W / m 2 / s r _ 1 , as given in Finkbeiner et al. (2000). Notice that the Finkbeiner prediction is
within the upper and lower limits estimated by the DIRBE Team. To remove the CIB from FIRAS
sky spectra, the CIB model in Fixsen et al. (1998) is used.
14
L o c a t i o n s of s t r o n g e s t [CII] e m i s s i o n c o v e r e d by 100 FIRAS p i x e l s
Figure 3.2: Locations of highest [C II] emission represented by 100 FIRAS 7° pixels (red). This
map is in Galactic coordinate Mollweide projection with the Galactic center at the center and
longitude increasing to the left. The isolated group of red pixels on the left corresponds to the
Cygnus Region, and the group of pixels on the right corresponds to teh Carina Nebula.
15
CIB Model by Fixsen et al. (1998)
1
1.000 —
1
1
1
1
1"
•
1
.^—"'** '
•
r
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5
^ i - \
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^ *
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^
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\_
\-
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~
~
~
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•7s'
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~
' . • ' /
0.001
' \
4?
" ^ 0.100 —
—3
"
"
"w
c
a>
1
•
"
V)
1
Hauser et a l . (1998)
Finkbeiner et al. (2000)
-
-
1
1_
L
i
*
•
i
•
•
•
100
i
i_
.
i
•
1000
Frequency (GHz)
Figure 3.3: CIB predictions by Hauser et al. (1998), Fixsen et al. (1998) and Finkbeiner et al.
(2000). The solid black curve stands for CIB model in Fixsen et al. (1998). The dash-dot curves
are the one-sigma error of the model. Red crosses are estimates of Hauser et al. for DIRBE bands
at 140 /xm and 240 /zm; the error bars stand for one- and two-sigma error reported in their study.
An estimate of the CIB at 100 /xm given by Finkbeiner et al. (2000) is represented by the blue
cross, and the one and two sigma error bars are in the same and similar colors.
Figure 3.3 shows CIB predictions from these three studies. The solid black curve stands
for CIB model in Fixsen et al. (1998). The dash-dot curves are the one-sigma error of the model.
Red crosses are estimates of Hauser et al. for DIRBE bands at 140 /xm and 240 /an; the error bars
stand for one- and two-sigma error reported in their study. An estimate of the CIB at 100 /xm given
by Finkbeiner et al. (2000) is represented by the blue cross, and the one and two sigma error bars
are in the same and similar colors.
3.2
B e a m Difference
The three instruments that produced the data used in this study had different beam pat-
terns. For example, the FIRAS used a quasi-optical multimode horn antenna to collect radiation
from a 7° field of view (Mather 1986). The horn was designed in a trumpet bell shape to reduce
16
response to off-axis radiation. As a result, when the beam profile was measured on the ground and
in flight, it was found to have very low sidelobes over the two decades of frequency measured by
FIRAS. The central portion of the beam (9 < 3°5) is approximated by a top hat since any slight
azimuthal asymmetry should have been smoothed out by the rotation of the instrument along its
own axis (Mather et al. 1993) during its operation.
On the other hand, DIRBE was built with a goal to reject stray light to measure the absolute
spectrum and angular distribution of the CIB. This goal was met by using a series of optical elements
and baffle protections, among which the last field stop set the 0°7 x 0°7 instantaneous field of view
for all spectral bands (Silverberg et al. 1993; DIRBE Explanatory Supplement 1998). To construct
the Zodi-Subtracted Mission Average (ZSMA) maps, the DIRBE Team calculated the zodiacal light
intensity using the Interplanetary Dust (IPD) model by Kelsall et al. (1998), and subtracted it off
from each weekly measurement. The remaining signal was averaged over time. In this way, the
ZSMA maps preserve the original 0°7 x 0°7 angular resolution of the sky observation.
The dust map used here from the WMAP was one of the products derived from Markov
chain Monte Carlo fitting of temperature and polarization data (Gold et al. 2009). Since their
analysis used the band-averaged maps that were smoothed by a 1° Gaussian beam, the dust map
have the same angular resolution.
The FIRAS beam is the lowest common angular resolution achievable among all three data
sets, and the higher resolution DIRBE and WMAP maps need to be convolved with the FIRAS
beam to make them all 7° maps. Additionally, since Fixsen et al. (1997b) found that the FIRAS
beam was elongated in the scan direction by 2°4, that pattern is matched in the degraded DIRBE
and WMAP maps by convolving those data with an effective FIRAS beam.
3.3
Map Projection and Spatial Resolution
Both DIRBE and FIRAS maps are organized in COBE quadrilateralized spherical cube
format (quad-cube, Chan & O'Neill (1975), O'Neill & Laubscher (1976), White & Stemwedel (1992),
and Calabretta & Greisen (2002)). While DIRBE maps are in quad-cube resolution level 9 (res9,
19'.43 per pixel), FIRAS maps are in quad-cube res6 (2°59 per pixel). Different from FIRAS and
DIRBE, the WMAP dust map is in HEALPix (Gorski, Hivon, & Wandelt (1999), Gorski et al.
(2005), and Calabretta & Roukema (2006)) res6 (54'.97 per pixel). One way to reconcile these
different formats and spatial resolutions is to carry out analysis in COBE quad-cube res6. This
decision is made to retain maximum amount of information contained in the original data sets and
achieve the highest common resolution possible.
As a result, DIRBE maps were re-binned to res6; the WMAP dust map was first converted
into a quad-cube res9 map and then re-binned to res6. During WMAP's dust map conversion, care
was taken to ensure that no excessive artificial noise was introduced to the final map. Comparing
the original HEALPix-projection map with the re-binned quad-cube-projection map, 98.6% of the
17
49,152-coordinate pairs sampled give no difference between the quad-cube and the HEALPix values.
The difference ranges between 0-0.0059 mK, which amounts to a maximum of 0.11% noise increase
for the original HEALPix map.
3.4
Gradient Correction
The FIRAS dust maps are based on coadding interferograms, so their values are generally
not at the defined center of map pixels. This difference requires an additional correction step to
prepare the quad-cube res6 DIRBE and WMAP maps. Details of this technique are described in
Fixsen et al. (1997b). In summary, a second-degree surface function is fit to the intensity and location
information of a pixel and its immediate neighbors in one of the converted maps. This function is
then used to predict emission at the FIRAS mean position for that particular pixel. Overall, a 5%
rms correction is applied to each of the DIRBE maps at 100/xm, 140/rni and 240 /xm and to the
WMAP dust map.
3.5
Color Correction
In accordance with the IRAS convention (IRAS Explanatory Supplement 1988), DIRBE
photometric measurements were reported in MJy/sr at nominal wavelengths, i.e., assuming the
source spectrum to be v Iv = constant. Since each DIRBE band has a much wider bandwidth than
a FIRAS channel, spectral shape could have changed enough that at the nominal wavelength the
real intensity is significantly different from the normalized intensity. As a result, color corrections
are included in the overall model. This means that model predictions are compared with DIRBE
measurements using the relation Iv(model) = K-IU(DIRBE),
where K is the color correction factor
defined as
TS _ J
(Iv/Iy0)actualRvdv
$(l>0/v)quotedRvdv
'
In this equation, J'(!„/1vo) actual is the specific intensity of the sky normalized to the intensity
at frequency VQ, and Rv is DIRBE relative system response at frequency u, whose values were
documented in DIRBE Explanatory Supplement (1998) Section 5.5.
3.6
DIRBE Uncertainties
The DIRBE photometric system was maintained to ~ 1% accuracy by monitoring the
internal stimulator during 10 months of cryogenic operation and observing the bright stable celestial
sources during normal sky scans. It was absolutely calibrated against Sirius, NGC7027, Uranus, and
Jupiter. Among different types of uncertainties identified by the DIRBE Team, those relevant to
this thesis project are standard deviations of intensity maps, detector gain and offset uncertainties,
18
and zodi model uncertainties (Hauser et al. 1998; Kelsall et al. 1998; Arendt et al. 1998). For bands
8-10, respectively, the detector gains are: 0.135, 0.106 and 0.116 nW m~ 2 sr _ 1 ; detector offsets are
0.81, 5 and 2 nW m~ 2 sr _ 1 ; and zodi model uncertainties are: 6, 2.3 and 0.5 nW m~ 2 s r _ 1 (Arendt
et al. 1998). The process of re-binning the high resolution DIRBE maps into FIRAS resolution
affects only the standard deviations of the original maps. The final uncertainty is the quadrature
sum of the individual noise components.
3.7
Temperature-intensity Conversion
Foreground maps of the WMAP production are reported in antenna temperature, TA, in
mK. On the other hand, maps produced by DIRBE and FIRAS Teams are reported in spectral
intensity, J„, in MJy/sr. In the following analysis, the WMAP dust map is converted into flux
density values following Iv = 2 (^) 2 k TA, where v is the effective frequency (93.5 GHz) of the dust
map (Gold et al. 2009; Jarosik et al. 2003).
19
Chapter 4
Overview of Model Fitting
Strategy
As explained in the Introduction, thermal emission of a dust grain is described by a modified
blackbody function:
Iv = T ev Bv(T<iust),
where Bv(Tdust) is the blackbody spectrum at temperature Tdust, £v = (y/vo)a
is the emissivity
with spectral index a, and r is the optical depth normalized to frequency u0 = 900 GHz. One or
more instances of this expression are used to model dust spectral variation in the following chapters.
In addition to measuring thermal dust emission, the prepared FIRAS spectra retain contributions from [C II] and [N II] emission, due to the lack of precise all-sky templates. As a result,
two emission lines are modeled at the same time with the dust:
I[cii}{v) = [CH] intensity x / [C ii](^),
I[mi]{v) = [Nil] intensity x
f[Nn](is),
where f{v) is the synthetic line profile determined by FIRAS response to a delta-function signal.
Together, the full model has the form
/total = /dust + /[CII] + /[Nil],
(4-1)
and is referred to by the number of dust components it contains. For example, a two-component
dust model means
I{v) = n ei B„(Ti) + r 2 e2 B„(T2) + [CII] intensity x / [ C I I ] + [Nil] intensity x / [ N I I ] .
Each full model is fit to the data by minimizing a three-part x2> with each part corresponding to one of the three data sets:
X
= XDIRBE + XFIRAS + XWMAP,
20
(4-2)
where
Xfnstrument = 2^(-^obs - ^ m d l ) i ( M " ) t j ( 7 o b s - 7 m d l ) j ,
(4.3)
-fobs(^) is the observed spectral intensity, Imdii^) is model prediction, and M is the covariance matrix
of the respective data set.
In the following chapters, I report results from fitting data using four types of models:
one-component fixed-a models (Chapters 5 and 6), one-component free-a model (Section 7.1), twocomponent fixed-ai and fixed-a^ models (Sections 7.2 and 7.3), and finally three-component a.\ =
a.2 = 03 = 2.0 model (Section 7.4). Some model types are composite, such as the one-component
fixed-a models, which consist of distinct models when parameter a assumes the value of a different
positive real number. I report results for only those models where the a are physical. Moreover,
the 6063 quad-cube spectra available for model fitting can be averaged to increase signal-to-noise of
the final spectra. Fitting such spectra gives a different set of best-fit parameters and are considered
distinct models. Chi-square per degree of freedom values, X^of'
are
reported throughout the thesis to
assess the goodness-of-fit. For example, fits that use a one-component fixed-a model have 214 — 4 =
210 degrees of freedom. Similarly, fits that use a two-component fixed-a model have 214 — 6 = 208
degrees of freedom. Due to the number of distinct models considered here, analysis on the strengths
and weaknesses of each model is given immediately after presentation of the fit results. Chapter 9
serves the purpose of inter-model comparison and presentation of the best-model produced in this
thesis.
21
Chapter 5
One-component Dust Models for
7° Sky Regions
5.1
a = 2.0 M o d e l
Analysis
A one-component a = 2.0 model is fit to the spectral data at each 7° pixel as determined
by the FIRAS resolution. Figure 5.1 shows an all-sky map of the Xdof °f
an
6063 fits performed
on the data sets. Figures 5.2 — 5.3 present all-sky maps of the best-fit parameters (Tdust, T, [C II]
intensity, and [N II] intensity) for the valid fits (Xdof — 1-13 f° r 10% probability at 210 degrees of
freedom), their uncertainties and correlations. Examples of dust spectra and their corresponding
best-fit models are presented in Figure 5.4. These include examples of a mid-latitude and a highlatitude spectra, and an example of a spectrum at the Galactic plane.
Interpretation
The fits have acceptable values (Xdof — 1-13) over most of the sky except in the Galactic
plane. Figure 5.5 examines Xdof a s
have Xdof ~ 1>
an(
a
function of Galactic latitude. It shows that most fits at |6| > 10°
a
l fits t \b\ < 10° have a Xdof ^ •*-• F i g u r e 5-6 compares the distribution of Xdof
at
|6| > 10° with the distribution of Xdof f° r t n e entire sky. Both distributions resemble a Gaussian and
center at 0.93. This means that the fits don't have a significant systematic bias, though statistical
errors of the data are slightly overestimated. The model is a good fit to the data over 87% of the
entire sky.
Figure 5.7 shows that majority of the sky has a dust temperature between 15 K and 22 K
when a = 2.0. Additionally, the sky maps in Figure 5.2 show that Tdust is high around both Galactic
22
one_comp, a = 2
23.7
1.1
Figure 5.1: Xdof °f the best-fit one-component a = 2.0 models for individual pixel spectra. This
map is in the Galactic coordinate Mollweide projection. The map center is the Galactic center.
The upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left
and right ends of the major axis represent +180° and —180° longitudes respectively. The center of
the upper left quadrant is approximately the location of the North Ecliptic Pole (NEP), and the
center of the lower right quadrant is approximately the location of the South Ecliptic Pole (SEP).
The lower limit of the color scale is set at x\of
=
1-13, corresponding to a 10% non-chance
occurrence at 210 degrees of freedom. This map shows that a majority of the fits in directions
away from the Galactic plane are good, but the model is rejected in the plane.
23
one_comp, a = 2
<5Tone_comp,
a = 2
one_comp, a
<5Tone„comp,
a = 2
ISO
4^
Figure 5.2: Best-fit values of Tdust (top left), r (top right), [C II] intensity (next graphic panel) and [N II] intensity (next graphic panel), and
their uncertainties from fitting one-component a = 2.0 model to the 7° spectra. These maps are Mollweide projections in Galactic
coordinates. The center of each map represents the Galactic center. The upper and lower ends of the minor axis are +90° and —90° latitudes
respectively, and the left and right ends of the major axis represent +180° and —180° longitudes respectively. Pixels whose fits have high x2
values (Xdof
>
l--^ for 210 degrees of freedom) are masked in white. (Continued in the next graphic panel)
[CII] Intensity,one_comp,
a = 2
Log(1.0e-08
[Nil] Intensityone_comp,
Log(1.0e-09)
(5[CII] l n t e n s i t y o n e _ c o m P i a
a = 2
Log(1.0e-02)
<5[NII] l n t e n s i t y o n e _ c o m p p a
= 2
= 2
to
0.0030
0.050
0.0010
0.020
(additional caption for Figure 5.2) In the error maps, the thread-like pattern starting from the center of the upper left quadrant (North
Ecliptic Pole, NEP) to the center of the lower right quadrant (South Ecliptic Pole, SEP) is the signature of FIRAS detector noise, which
dominates the sum of all errors. This pattern is the result of COBE mission's scan pattern and FIRAS's data calibration. Its emergence
means that uncertainties of these fits are limited by FIRAS detector noise. The group of black pixels that slant from the NEP to the SEP are
positions where FIRAS did not provide data.
r
r
T T one_comp a = 2
-10
0 90
-Oil
'T [Nil] one_comp a - 2
- 0 13
r
T [CII] one_comp a = 2
0 095
- 0 070
~[NII] T one_comp a - 2
0 20
- 0 25
[ClI] T one_comp a = 2
Oil
"[CII] [Nil] one_comp a = 2
Oil
- 0 0050
0 020
Figure 5.3: Correlations among Tdust, T, [C II] intensity and [N II] intensity of the best-fit one-component a = 2.0 models for 7° spectra.
These maps are Mollweide projections in Galactic coordinates with the Galactic center at the center and longitude increasing to the left. The
thread-like pattern starting from the center of the upper left quadrant (North Ecliptic Pole, NEP) to the center of the lower right quadrant
(South Ecliptic Pole, SEP) is the signature of FIRAS detector noise, which dominates the sum of all errors. This pattern is the result of
COBE mission's scan pattern and FIRAS's data calibration. Its emergence means that uncertainties of these fits are limited by FIRAS
detector noise.
FIRAS pix # 2 6 2 8 , glon = 177 97, glat == - 1 1 87
10"
a = 2 00
T = 16 02 ± 1 16e-01
T = 2 51e-04 ± 6 68e-06
y"
= 0 99
= -0 94
10u
^
[Cll] intensity =
1 03e-01
±
6 68e-06
[Nil]
1 89e-16 ±
1 &4#-02
ntensity =
-
10
=
0 01
= - 0 07
= -011
=
0 09
=
0 08
[CI] [Nl]
'T [CI]
r
T [Nil]
r
[CII] T
r
[NI] T
10"4
10"
100
1000
frequency
(GHz)
FIRAS pix # 2 0 8 4 , glon = 195 38, glat = - 4 3 65
10 4
a =
T =
10'
—
10 u
—
2 00
17 8 2 ± 5 1 7 e - 0 1
T =
2 59e-05 ± 3 27e-06
X'one -comp = 0 71
^
3
10
=
T
[Cll] i n t e n s i t y =
[Nil] i n t e n s ty =
- 0 92
^ji-^"
3 93e-02
±
3 27e-06
1 83e-02
±
2 53*^02
^^rf^^f^
_,-'
—
[Cll] [N ,
T [Cll] =
r
T [Nil] =
=
-0
0
r
0
[Cll] T =
r
[NII] T = - 0
r
1 0 ~
S—-
^---^
'^ 4
—
r
-
-
-
0 00
03
04
02
07
-
10"
100
1000
frequency
FIRAS pix § 4 9 2 2 , glon =
(GHz)
28 98, glot =
- 0 11
10"
10'
3
a =
2 00
T =
20 63 ±
T =
1 80e-03
X2o„e ™
P
rT
- 0 91
T
2 03e-02
=
±
[ C l l l intensjJjt^»»<*«55eTt5o"' ±
Tensity =
4 43e-06
4 60e-01
±
«•*•-
4 43e-tiB—^
4 19e-02
= 7 86
10"
0 01
=
0 02
= - 0 09
=
0 04
=
0 07
'[Cll] [Nil]
10~
rT [c„]
rT [N„]
r [ c „] r
» r
4
10"
100
1000
frequency
(GHz)
Figure 5 4: Examples of best-fit one-component a = 2.0 models (red curve) and their
corresponding dust spectra (orange). Each spectrum consists of the FIRAS 210-channel spectrum,
a WMAP measurement at 94 GHz, and DIRBE bands at 100 /xm, 140 /im and 240 /xm.
27
i
i
i
i
i
i
,
1
1
15 -
I
I
I
.
i
-
X
X
-
-
XX
x
X
10
X
X
o
•o
*?
xA*
X
5
-
J1
—
-
xS'fe
li*I*
imm
x •
0
^^ipiMiwW
,
,
0totitH$NK&Rt$tft0*
1
-50
Figure 5.5: Xdof
vs
i
0
Galactic Latitude (°)
50
- Galactic latitude. The value of x^ of is obtained from fitting the
one-component a = 2.0 model to the spectrum at each 7° pixel. This plot shows that most fits for
|6| > 10° have xlof ~ *> a n d
fits f o r
28
\b\ <
10
°
have a
Xdof > 1-
X2dof Distribution
-i
250
1
— i — i — i — i — i — i — i —
r
-i
1
r-
-i
1
r-
200
For Ibl > 10°, 4999 pixels ore selected
Center: 0.93
Width: 0.09
150
The full sky includes 6063 pixels
Center: 0.93
Width: 0.10
100
5262 pixels (87% sky) with v2 ^
Center: 0.93
Width: 0.10
1.13
50 -
J i n r l f 1 ^ rJUfnrfT-nr^-,r~irU-1 J^irrw
0.6
0.8
1.0
1.2
1.4
1.6
X'aoi of o n e - c o m p , a = 2.0
1.8
2.0
Figure 5.6: The Xdof distribution, where xjjof *s calculated at each 7° pixel for the best-fit
one-component a = 2.0 model. The red distribution and best-fit Gaussian include only pixels with
Galactic latitudes |6| > 10°; the green distribution and best-fit Gaussian include all 6063 pixels at
all possible values of b. The best-fit Gaussian to the distribution with Xdof — 1-13 ' s
m
blue.
Values of the parameters are printed in corresponding colors. These distributions show that there
is no apparent systematic bias in the fits. The fact that the Gaussian centers at Xdof
=
0-93 means
that errors in the data are slightly overestimated.
poles and right above the Inner Galaxy. That high Tdust appears in the Inner Galaxy is reasonable
because star formation takes place in the Galactic disk and at the bulge, and star formation is the
most important heat source for dust. The high Tdust near the poles, on the other hand, has to do
with large uncertainties of the data in these regions compared to those measured at lower latitudes.
Figure 5.8 shows percentage error in Tdust for the whole sky, where 0.7% < <5Tdust/Tdust < 114%.
At high latitudes, the pixels have > 10% errors. This happens because the dust emission decreases
toward high latitudes, which leads to low signal-to-noise data in these regions.
Discussion
The above exercise brings into focus the following issues in dust model fitting:
•
In the Galactic plane, the \ 2 of the fits are large. This means that the one-component a = 2.0
model is rejected by the data.
29
Best-fit O n e - c o m p . a = 2 Model
300 7
250 '7
£ 200
^
150
CD
100 7
50 '7
10
15
20
25
Tdust (K)
30
35
Figure 5.7: Distribution of the best-fit Tdust from fitting one-component a = 2.0 model to the 7°
spectra. This plot shows that majority of the fits predict a temperature between 15 and 22 K.
30
1.0
%
21.0
Figure 5.8: Percentage error of the best-fit Tdust from fitting one-component a = 2.0 model to the
7° spectra. The range of uncertainty for Tdust for the entire set of models is
0.7% < STdust/Tdust < 114%. This map is in a Galactic coordinate Mollweide projection. The map
center is at the Galactic center. The upper and lower ends of the minor axis are +90° and —90°
latitudes respectively, and the left and right ends of the major axis represent +180° and —180°
longitudes respectively. The center of the upper left quadrant is approximately the location of the
North Ecliptic Pole (NEP), and the center of the lower right quadrant is approximately the
location of the South Ecliptic Pole (SEP). Pixels whose fits have high \ 2 values (Xdof > l - ^ for
210 degrees of freedom) are masked in white. The group of black pixels that slant from the NEP to
the SEP are positions where FIRAS did not report measurements.
31
•
High-latitude spectra could be averaged to increase the signal-to-noise ratio of the data and
confidence in the best-fit parameters. While averaging sky signals can reduce noise in the data,
averaging also smears out small scale variations and causes information loss.
5.2
Other Fixed-a Models
Analysis
One-component models with different fixed values of a (a £ [1.4 — 2.3] at 0.1 increment)
are fit to the 7° spectra measured from the entire sky. All-sky maps of the best-fit parameters
(Tdust,T, [C II] intensity, and [N II] intensity), associated errors and covariance, and Xdof of the fits
are presented in Appendix A.
Interpretation
Figures 5.9 and 5.10 examine best-fit models with different values of a for two pixels: one
has high signal-to-noise (HSN) and is at a low Galactic latitude, the other has low signal-to-noise
(LSN) and is at a high Galactic latitude. In both cases, the x^ of vs. a plots show that models with
a wide range of different a values can fit the data well. In the HSN case, a curve fit to Xdof
as
a
function of a is a concave up parabola, with the minimum Xdoi ~ 0-^9 at a = 1.8. The difference
between Xdof at a = 1.8 and that at a = 2.0 is only A^dof = 0.01. In the LSN case, best-fit curve to
Xdof
vs
-
a IS a m u c n
flatter parabola over 1.4 < a < 2.3 with the minimum Xd0{ = 0-80 at a = 1.6.
The difference between x\0f at a = 1.8 and a = 2.0 is 0.002.
Although models with different a have only a small difference in Xdof
tne
best-fit T^st
and T are different significantly in the HSN case: At a = 2.0, the best-fit dust temperature is
Tdust ~ 17.5 ± 0.26 K, compared to Tdust ~ 18.5 ± 0.28 K at a = 1.8. This difference in temperature
is larger than the sum of their errors. Similarly, the difference in r of the two a models is larger
than the sum of their errors. In the LSN case, the difference in Tdust and r of the best-fit a = 1.6
and a = 2.0 models are well within the uncertainties of the respective parameters.
These results demonstrate the sensitivity of the fits to measurement errors. The existence
of measurement noise inevitably causes a high degree of degeneracy between the dust spectral index
and the dust temperature in the fits. While it is difficult to break this degeneracy, high signal-tonoise data help. Fitting data with high signal-to-noise results in well constrained parameters, which
means that the choice of an a model can cause statistically significant differences in the predictions
of these parameters. On the other hand, fitting low signal-to-noise spectra results in small difference
in x^ of and large error bars of the best-fit parameters, and so is not possible to differentiate models
with different fixed values of a. This is demonstrated in Figure 5.11, where the 68% and 95%
confidence contours of a HSN fit enclose much smaller regions in the T — a space than those of a
32
LSN fit.
Figures 5.12 shows sky maps of Xdof>
a an
d ^dust that correspond to the minimum-Xj of
model among all models tested at each pixel. Overall, the parameter maps are noisy. This is a result
of not having sufficient signal-to-noise in the data to set tight constraints on the best-fit parameters.
They suggest that some spatial averaging needs to be examined.
5.3
Summary
This chapter has presented results from fitting one-component fixed-a models to spectra of
7° pixels. These models are sufficient to account for thermal dust emission because Xdof < 1-13 (for
210 degrees of freedom) for most of the high-latitude fits. However, a spectrum with a low signalto-noise ratio, which often occurs at high Galactic latitudes, cannot give adequate constraint to
the best-fit parameters and exacerbates the degeneracy between dust temperature and the spectral
index. In order to identify one best model, the signal-to-noise ratio of the data needs to be increased.
33
FIRAS pix § 0 0 0 1 , glon = 63.78, glat = - 1 1 . 5 3
0.94
0.93
0.92
"x
0.91
0.90
0.89
0.88
1.4
1.6
1.8
2.0
2.2
a
FIRAS pix § 0 0 0 1 , glon = 63.78, glat = - 1 1 . 5 3
22
21
20
^
19
>- 18
17
16
15
1.4
1.6
1.8
2.0
2.2
a
FIRAS pix # 0 0 0 1 , glon = 63.78, glat = - 1 1 . 5 3
9x10"5
8x10"5
7x10"5
6x10"5
5x10~ 5
4x10~ 5
1.4
1.6
1.8
2.0
2.2
a
Figure 5.9: X^of > ^dust and r as a function of a. Best-fit parameters are obtained from fitting
one-component models with specified a values to the spectrum measured in the direction
/ = 63°78 and b = —11°53. This set of plots serves as an example of high signal-to-noise fits. The
green curve in each plot fits a parameter's different values as a function of a. In particular, the
plot of Xdof shows that the model with a = 1.8 is a better fit to data than the model with a = 2.0.
The small error bars of Tdust and r show that the choice of an a model can cause statistically
significant differences in the predictions of these parameters.
34
FIRAS pix jf 3672, glon = 254.32, glot = 65.08
1
i
i
I
•
•
-;
0.81
- x ^ " ^
"x 0.80
:
y\
-
0.79
0.78
1.4
1.6
1.8
2.0
2.2
a
FIRAS pix § 3 6 7 2 , glon = 254.32, glat = 65.08
24
-
22
:
1
^^
T
20
1
i
_
!^-
18
1.4
1.6
1.8
2.0
2.2
a
FIRAS pix # 3 6 7 2 , glon = 254.32, glat = 65.08
1.2x10" 5
1.0x10" 5
8.0x10"
6.0x1 0" 6
1.4
1.6
1.8
2.0
2.2
Figure 5.10: Xdof > ^dust and r as a function of a. Best-fit parameters are obtained from fitting
one-component models with specified a values to the spectrum measured in the direction
I = 254°32 and b = 65°08. This set of plots serves as an example of low signal-to-noise fits. The
jreen curve in each plot fits a parameter's different values as a function of a. Because of the small
difference in Xdof
anc
^ l a r g e error bars of the model parameters, fit results cannot differentiate
models with different fixed values of a.
35
min. x2d0f = 0.89
pixel # 0 0 0 1 , glon = 63.78, glat = - 1 1 . 5 3
min. x2d0f = 0.80
pixel § 3672, glon = 254.32, glat = 65.08
Figure 5.11: 68% and 95% probability contours of best-fit Tdust and best-fit a for a high
signal-to-noise spectrum (upper plot) and a low signal-to-noise spectrum (lower plot). The white
cross represents location of the minimum \ 2 in each case. The blue area is the 68% confidence
region, and the green area is the 95% confidence region. These plots demonstrate the effect of
measurement noise on the degeneracy between a and Tdust in spectral model fitting. For the high
signal-to-noise ratio spectrum (upper plot), the 68% confidence region is within 1.66 < a < 1.93
and 17.85 K < Tdust < 19.48 K; for the low signal-to-noise ratio spectrum (lower plot), the 68%
confidence region is a much wider region, at 1.16 < a < 2.20 and 18.48 K < T dus t < 24.72 K.
36
A dof, one component fixed —a
11
22 8
one—component fixed —a
dust, one-component fixed-a
5 0
Figure 5.12: Sky maps of Xdof
a ano
K
25 0
^ ^dust- These maps are Mollweide projections of the Galaxy
in Galactic coordinates. The center of each map is the Galactic center. The upper and lower ends
of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the
major axis represent +180° and —180° longitudes respectively. The value at each pixel is from the
best-fit model that gives minimum Xdo{ among all tested a models (a G [1.4 — 2.6] at 0.1
increment). In the Galactic plane, the best-fit models have Xdof
as m
Sn
as
22.8. Both a and Td ust
maps are noisy, particularly so at high latitudes. In the a and Tdust maps, pixels that correspond
to fits with a x\0f ' e s s than 10% probability are masked in white.
37
Chapter 6
One-component Dust Models for
Averaged Sky Regions
6.1
Fitting Averaged Spectra of Latitudinal Rings
Analysis
To increase the signal-to-noise of the high-latitude data, the full sky is divided into latitude
rings of approximately equal number of pixels. Then, one-component dust models with different
fixed values of a (a £ [1.4, 2.3]) are fit to the averaged spectrum of each ring.
Figure 6.1 presents x\0i and Tdust of the fits. In every sky division at high latitudes,
|6| > 70°, Xdof ~ 1 f° r a ^
a m
°dels. In addition, best-fit Tdust have large error bars that overlap
with each other. At mid latitudes, 45° < |6| < 70°, xlof
nas va ues
l
intermediate between 1 and 2,
except the two-hemisphere case. Models with a = 1.7,1.8 or 1.9 have lower x 2 than other models.
Best-fit Tdust values are constrained such that their error bars between rings don't overlap. At
|6| < 45°, all fixed-a fits are poor because the spectra contain the Galactic plane.
Interpretation
Figure 6.2 presents x\of ^dust/Tdust and 5Tdust/Tdust of the minimum xlof model at each
ring. In the x\0i plot, different ring divisions (except the two-hemisphere case) give a 1 < x\0i < ^
at |6| > 45°. However, at mid and low latitudes (|6| < 45°), all x^ of »
1. This means that
the averaging has achieved a sufficient signal-to-noise ratio to stress the model. None of the ring
divisions tested gives an acceptable fit to the average spectra that include the Galactic plane. The
Wdust /Tdust and ^TdUst/Tdust plots show that every model constrains TaUst better than Tdust by more
than a factor of 2 and that the constraint on both Tdust and r decreases with narrower rings at mid
38
2 G a l a c t i c - L a t i t u d e Rings
50
= 1l.b4
=
a = 1.7
a =s 1 H
a
1,9
rx = V !)
a =7 1
=V V
a = 2.6
=
a
a
(Tt
-50
0
Galactic Latitude (D)
50
4 G a l a c t i c - L a t i t u d e Rings
4 G a l a c t i c - L a t i t u d e Rings
a = 1.4
a = 1.5
a = 1.7
a = 1.8
a = 1.9
a = 20
a = 2.1
a = 2.2
a = 2.3
40
24
a
_._L_.
a
f ~
a
1
J
1
10
:
=
-
-
-
=
-
•
= 1.4
ST
1 5
= 1./
a
1.8
a
= 1.9
1
a
2 o
!
a
a
= 2.1
=
a
2.J
———
*
16
-50
0
Galactic Latitude (")
50
-50
6 G a l a c t i c - L a t i t u d e Rings
a
o
a
a
a
a
a
a
a
\
1
=
'
=
=
=
=
=
=
=
24
:
J
a
22
- ^
j
_
+ J ~ ~-
!T 1 r
-*~D
7
1
IB
1 4 -
„= i7:
a -
1.8 "
a = 1.9 j( - 2 0 .
|
, r~
5 H
1
-50
an<
, r^
L
18
Figure 6.1: Xdof
150
100
6 Galactic--Latitude R ings
1 4
1.5
1.7
1.8
1.9
2.0
2.1
2.2
2.3
?0
i
0
50
Galactic Latitude (")
0
Galactic Latitude (°)
50
0
50
Galactic Latitude (°)
^ ^dust of the best-fit one-component fixed-a (a e [1.4,2.3]) models for different
latitude-ring division of the sky. The Xdof P'°ts show that at high latitudes, |6| > 70°, Xdof ~ 1 f° r
all a models, and the best-fit Tdust have large error bars that overlap with each other. At mid
latitudes, 45° < \b\ < 70°, Xdof
nas va ues
l
intermediate between 1 and 2, except the
two-hemisphere case. Models with a = 1.7,1.8 or 1.9 have lower \ 2 than other models. At these
latitudes, the best-fit Tdust are well constrained such that their error bars don't overlap. All
fixed-a models tested fit the Galactic-plane average spectra (|6| < 45°) poorly.
39
8 Galac ic-Latitude Rings
8 Galactic-Latitude Rings
a a =
1.4
1.5
a
a
a
a
a
a
a
1.7
1.8
1.9
2.0
2.1
2.2
2.3
=
=
=
=
=
=
24
22
:
j
20
18
16
'.
-100
-50
0
. . 50
'
-
Galactic Latitude (°)
0
50
Galactic Latitude (")
10 Galactic-Latitude Rings
10 Galactic-Latitude Rings
-
a
a
a
a
a
a
a
a
a
100
-50
-100
=
=
=
=
=
1.4
1.5
1.7
1.8
1.9
;C
= 2.1
- 2.2
= 2.3
100
24
150
1.4
1 5
22
%
1.7
1.8
t I
- 20
=
IT
a
1.9
'/
2.1
18
2.2
2.3
16
0L_^
-100
-50
0
Galactic Latitude (°)
50
100
-100
12 Galactic-Latitude Rings
-50
0
50
Galactic Latitude (°)
100
150
12 Galactic-Latitude Rings
a = 1.4
a = 1.5
a
a
a
a
a
a
a
-
=
=
=
=
=
=
=
1.7
1.8
1.9
2.0
2.1
2.2
2.3
J
* 20 -
-100
-50
0
Galactic Latitude (•)
50
-100
100
Figure 6.1 (continue)
40
-50
0
50
Galactic Latitude (°)
100
150
and high latitudes.
Discussion
The results of fitting latitude-ring spectra are that the spectral index may prefer a value
other than the conventional value of 2, and that averaging the data at low Galactic latitudes stresses
the dust model due to the high signal-to-noise ratio that results.
6.2
Fitting the Averaged Spectrum of the High-latitude Sky
Chapter 5 has shown that high-latitude spectra need to be averaged, and Section 6.1 has
shown that any averaged spectra that include the Galactic plane are fit poorly by any one-component
fixed-a model. In order to apply separate analysis to the high- and low-latitude spectra,
WMAP
temperature analysis (KQ75) mask is used.
The KQ75 mask is in HEALPix res9, and needs to be converted into COBE quad-cube
res6 like all the other sky maps used in this thesis. Figure 6.3 shows the sky outside the Galactic
mask that is used in the following sections.
Analysis
One-component fixed-a (a £ [1.4,2.6] at 0.1 increment) models are fit to the average
spectra of the entire unmasked sky. The Xdof
va ues
l
of the best-fit models are plotted as a function
of a in Figure 6.4.
Interpretation
Because of the large Xdof
vames
> none of the fixed-a models tested is a good fit to the
average spectrum of the entire unmasked sky. The shape of a concave up parabola reappears in the
Xdof
vs
-
a
Pl°t w ith
a
minimum of 1.83 at a = 1.8. This means that the averaging has achieved a
sufficient signal-to-noise ratio to stress the model.
6.3
Fitting Averaged Spectra of High-latitude Bands
Analysis
The KQ75 unmasked sky is divided into latitude bands (up to 8) of approximately equal
numbers of pixels (Figure 6.5). Spectra in each of these latitude bands are averaged and fit to
one-component a = 1.8 and a = 2.0 models separately. Xdo! °^ best-fit a = 1.8 and a = 2.0 models
41
Min. x2 at diff. Rings by various a's
25
12 rings .
20
-
8 rings
-
4 rings
6 rings
15
1
j
10
-
-
-
:
4-
-50
-100
0
Galactic Latitude (")
1 2 rings
~ "
100
Chosen <5r/ T at diff. Rings by M i n i m u m )(2
Chosen <5T/T at diff. Rings by Minimum \*
:
50
12 rings
12
:
8 rings
10
6 rings
6 rings
:
-
4 - nqo
8
* nnqs
*£. 2-
:
8 rings
'-
fi
—
-
p_
4
:
:
_
"
-
?
•
—
—
—
0
-100
-50
0
Galactic Latitude (°)
50
100
-100
-50
0
Galactic Latitude (°)
50
100
Figure 6.2: Xdof, *Td u s t/r d u s t and *T d u s t/r d u s t of the minimum-x dof model at each ring; models
being compared at each latitude ring are the best-fit one-component models with specified values
of a (a e [1.4,2.3]). Notice that in the xdof Plot>
a n rm
S divisions (except the two-hemisphere
case) give a 1 < x^ of < 2 at |6| > 50°. However, at mid and low latitudes (|6| < 50°), all xj| of 3> 1.
This means that the averaging has achieved a sufficient signal-to-noise ratio to stress the model.
None of the ring divisions tested gives an acceptable fit to the average spectra that include the
Galactic plane. The 5T d u s t /r d ust and <5rdust/Tdust plots show that every model constrains T d u s t
better than T dust by at least a factor of 2. Comparing models at different latitudes, constraints on
both TdUst and r decrease with less averaging.
42
WMAP T e m p e r a t u r e Analysis Mask in Quad Cube r 6
Figure 6.3: Conversion of WMAP's temperature analysis mask (KQ75) in COBE quad-cube res6.
This map is the Mollweide projections of the Galaxy in galactic coordinates. Map center represents
the Galactic center. The upper and lower ends of the minor axis are +90° and —90° latitudes
respectively, and the left and right ends of the major axis represent +180° and —180° longitudes
respectively. Pixels in red are not masked; pixels in blue are masked.
43
WMAP Temperature Analysis Mask m a j o r i t y u n m a s k e d sky
1
'
'
1
'
'
'
[
•
•
'
1
-
4 -
/
7
"x 3
\
\\
\
7:
j
-
/
-
'"X-
-
x-'
, , ! , , ; !
!
1.4
2.0
a
Figure 6.4: Xdof
as a
1
2.2
2.4
2.6
function of a. Values shown in this plot correspond to the best-fit
one-component fixed-a models that were fit to the average spectrum outside of the KQ75 Galactic
mask. The xlof
vs
-
a
P^0* a g a m
nas
the shape of a concave-up parabola. It reaches a minimum of
1.83 at a = 1.8.
44
are compared at each band in Figure 6.6, and the constraints on parameters Tdust and r for the
a = 1.8 model are presented in Figures 6.7.
Interpretation
The x\0f plots in Figure 6.6 show that both a = f .8 and a = 2.0 models fit high-latitude
spectra better than the low-latitude ones. In the 2-hemisphere case, a = 1.8 fits have lower x^ of
for both hemispheres than a = 2.0 fits. With further division of the sky, fits at the highest latitude
bands (those with |6| > 40°) achieve very similar x^ of between a = 2.0 models and a = 1.8 models.
At low latitudes, there is a wider difference between Xdof of a = 1.8 models and Xdof °f the a = 2.0
models: Right above and below the Galactic plane, Xdof have unacceptably high values, an indication
that bands at lower latitudes still need to be sub-divided or that the model is incorrect.
Figure 6.7 shows constraints on Tdust and r for the a = 1.8 fits. These plots show that the
constraint for parameters are greater at low latitudes than at high latitudes. Similar plots for the
a = 2.0 fits show the same tendency with respect to latitude.
6.4
Fitting Averaged Spectra of Longitudinal Regions
Analysis
The last section has shown that latitude bands near the Galactic plane produce high Xd0f
fits. Here, one-component a = 1.8 and a = 2.0 models are separately fit to the average spectra
of ten subdivided sky regions outside WMAP's KQ75 mask. The specification for regions in the
northern hemisphere is as follows:
•
b > 60°
•
0° < b < 60° and -45° < I < 45°
•
0° < b < 60° and 45° < I < 135°
•
0° < b < 60° and 135° < I < 225°
•
0° < b < 60° and 225° < I < 315°.
A reflection of the above division over the Galactic plane is applied to the southern hemisphere,
and Figure 6.9 demonstrates the locations of these 10 regions. x^ of and ^Tdust/^dust for the two a
models are presented in Figure 6.8.
Interpretation
The Xdof P'°t
m
Figures 6.8 shows that both a = 1.8 and a = 2.0 models produce
marginally acceptable fits to regions in the northern hemisphere and at the southern galactic pole.
45
KQ75 unmasked in 2 bands
KQ75 unmasked in 4 bands
KQ75 unmasked in 6 bands
KQ75 unmasked m 8 bands
Figure 6.5: Demonstration of dividing the KQ75 unmasked sky into latitude bands. These maps are Mollweide projections of the Galaxy in
Galactic coordinates with the Galactic center at the center and longitude increasing to the left.
2 Galactic — a :itude Bands
4 Galactic-latitude
3 0 --
3.0
a = 1.8
2 5
a -
-
a = 1.8
:
2 0
2.0 -
-
.
.
1 5
1n
-100
-50
o
50
Galactic Latitude (°)
1.5 -
1.0
100
•100
6 Galactic —la titud e Bands
2.5
a = 2 0
2.5 -
2.0
3.0
Bands
-
a = 1.8
a
/ o
2.5
i
2.0
l
1 5
1 0
1n
-50
0
50
Galactic Latitude (°)
'-
t
l.b
-100
;
:
I
100
8 Galactic —la tituc e Bands
3.0 1
I
-
2.0
-50
0
50
Galactic Latitude (°)
100
•100
:
!
'
a =
1.8
I
a
y J
I
I
l
!
-
-
r^
L=
-
L^
;
•
i
-50
0
50
Galactic Latitude (°)
100
Figure 6.6: Xdof °f ^ n e one-component a = 1.8 and a = 2.0 fits to latitudinal band average spectra.
Each plot represents a different band division of the sky; for locations of the bands, see Figure 6.5.
47
6J/J
for Galactic-latitude Bands
One - c o m p . a
2
"r
6
8
1.5 -
=
1 .8
Wode'l '
bands
Danas
bands
bands
-
&s
1.0 <o
0.5
-
-
.
-
-
.
-100
6T/T
.
.
, ,
-50
0
50
Galactic Latitude (°)
i
100
for Galactic —latitude Bands
6
One -comp. a = 1.8 Mo del
2 bands
4 bands
6 bands
8 bands
5
&? 4
h 3
^5
-100
-50
0
50
Galactic Latitude (°)
100
Figure 6.7: STdust/Tdust and 6T/T of the best-fit one-component a = 1.8 models to the average
spectra at each latitudinal band. These plots show that constraint for both TaUst
at lower latitudes than at higher latitudes.
48
an
d r are greater
X2dof f ° r Longitudinal Regions
2.4
2.2
Region 60° < b < 90°, 0° < I < 360°
2.0
Region 0° < b < 60°, 135° < I < 225°
Region 0° < b < 60°, 45° < I < 135°
V
Region 0° < b < 60°, - 4 5 ° < I < 45°
1 8
-
Region - 6 0 ° < b < 0°, 225° < I < 315°
Region - 6 0 ° < b < 0°, 135° < I < 225°
1.6
Region - 6 0 ° < b < 0°, 45° < I < 135°
Region - 6 0 ° < b < 0°, - 4 5 ° < I < 45°
1.4
Region - 9 0 ° < b < - 6 0 ° , 0° < I < 360°
1.2
1.8
2.0
2.2
2.4
a
i
61/1 for Longitudinal Regions
2.0
i
1.5
•
i
i
,
i
i
Region 60° < b < 90°, 0° < 1 < 360°
~
Region
Region
Region
Region
Region
Region
Region
Region
~
0° <
0° <
0° <
-60°
-60°
-60°
-60°
-90°
b
b
b
<
<
<
<
<
<
<
<
b
b
b
b
b
60°, 135° < 1 < 225°
60°, 45° < i < 1 35°
60°, - 4 5 ° < I < 45°
< 0°, 225° < 1 < 315°
< 0°, 135° < I < 225°
< 0°, 45° < I < 135°
< 0°, - 4 5 ° < I < 45°
< - 6 0 ° , 0° < I < 360°
_
r
1.0
i
0.5
—
$>
3^
i
i
i
1.8
2.0
2.2
,
,
,
i
,
,
2.4
a
Figure 6.8: Xdof
an(
^ ^dust/^dust from the a = 1.8 and a = 2.0 fits. The x^of Pl°t shows that
a = 1.8 models produce lower Xdof than a = 2.0 models in all regions except those at b > 60° and
b < —60°. In the two |6| > 60° regions, uncertainties of the best-fit Tdust are ~ 2%, while the rest
of the regions have ST^ust/Tdust ~ 0.5%.
49
one—comp, a
Figure 6.9: Sky map of Tdust from the best-fit one-component a = 1.8 models for longitudinal
averaged spectra in a Galactic coordinate MoUweide projection. The Galactic center is at the map
center and longitude increases to the left. This map shows that Tdust in the regions right above
and below the inner Galaxy have higher dust temperature than those next to the outer galaxy.
50
The uncertainty of the best-fit Tdust is small, though the two |6| > 60° regions have higher
uncertainties at <5Tdust/Tdust ~ 2% as opposed to the rest of the sky at 5Tdust/Tdust ~ 0.5%.
Figure 6.9 presents the temperature map of the a = 1.8 fits. It shows that Tdust in regions
right above and below the inner Galaxy have higher dust temperatures than those next to the
outer Galaxy. This pattern has its origin in a temperature gradient in the Galactic plane: The
concentration of stars, dust, and gas are higher in the inner disk than elsewhere on the Galactic
disk. As the concentration of heat sources falls off, so does the dust temperature. Therefore, what
these maps show is that temperatures of the mid-latitude regions follow those on the Galactic plane.
The results from fitting a = 2.0 model show a similar trend.
6.5
Fitting Averaged Spectra of Different-size Sky Regions
Overview
Dividing the high-latitude sky into latitudinal or longitudinal regions can improve constraints on the best-fit parameter values because it increases the signal-to-noise ratio. The different
types of division used in previous sections are based on our expectation for how galactic dust is
distributed in different sky directions. Since our knowledge is not complete, the divisions are not
optimal. Variations in the nature of the dust within the regions are neglected.
An averaging scheme is needed to both increase the signal-to-noise of high-latitude spectra
and to best preserve intrinsic variations in the signal from different sky directions. One approach
that satisfies these objectives is to let models fit spectra that are based on different amounts of
averaging. Starting with the base level, where a pixel's own spectrum is used to fit a model. If
the fit does not give well constrained parameters due to an inadequate signal-to-noise ratio, the
procedure goes to fit the average of the original spectrum and its eight immediate neighbors. This
process of using surrounding spectra to form a new average goes on until the derived parameters
are sufficiently constrained. In this way, results from fits done at the base level have an angular
resolution of 6.71 square degrees. At the next level, results have an angular resolution of 60.37
square degrees, and so on.
6.5.1
a = 2.0 Model
Analysis
Figures B.l—B.2 present the best-fit parameters and Xdof °f the one-component a = 2.0
model. These models all satisfy <5TdUst/Tdust < 20% with the least amount of spectral averaging. In
the parameter maps, if a fit has a x^of > 1-13 for 210 degrees of freedom, then the location of the
51
fit is masked in white.
Similarly, Figures B.3—B.8 are the all-sky maps of one-component a = 2.0 models that
satisfy STdust/Tdust
< 10%, 5% and 2.5%, respectively, by averaging regions.
Interpretation
Sky maps in Figures B.1-B.8 show four sets of all-sky results with different upper limits to
the <5Tdust/Tdust values that are good fits to spectra in most of the sky, except those at the Galactic
plane. A more stringent upper limit on the STdust/Tdust values causes Xdof
ar
° u n d the Galactic
poles to increase in value. The upper plot of Figure 6.10 quantifies this change by plotting x^ of as
a function of Galactic latitude. It shows that Xdof °f high-latitude (|6| > 60°) fits have moved from
0.7 — 1 to 0.8 — 1.3. This is another example that increasing the amount of averaging stresses the
dust model and increases the x2 value.
Also in Figure 6.10, the lower plot presents histograms of x^of of fits that have different upper limits on the values of STdust/Tdust- The three histograms for <5Tdust/^dust < 20%, 10% and 5%
peak at 0.93, 0.94 and 0.96 respectively, and they all have a width of 0.1. The histogram for
<57dust/Tdust < 2.5% peaks at 1, has a width of 0.1 and a thick tail in the range 1.2 < \\0f
<
1-4-
The demand for a 2.5% upper limit of the <5TdUst/TdUst clearly stresses the model fit because it has
many more fits exceeding the x 2 cutoff.
The Tdust histograms for the all-sky models are presented in Figure 6.11. Notice the
migration of high temperature points to lower temperature, and the peaks of these histograms to
a value ~ 0.5 K lower. The all-sky model that satisfies (JTdust/^dust < 2.5% experiences the most
dramatic change, making it skew to the left, the side of low temperatures. This means that some of
the high-temperature fits come with large uncertainties, and by limiting the amounts of uncertainties
in the the predicted temperatures settle into the more realistic values.
In contrast to Tdust's move, the distributions for optical depth, shown in Figure 6.12,
react by moving to higher values. This means that models affected by the 5Tdust/7dust constraint
generally have lower temperature and higher optical depth, though still in the optically thin regime.
Ultimately, it shows that low signal-to-noise ratio tends to cause overprediction in dust temperature,
which is compensated in the wrong model by a smaller optical depth.
The upper limit set for the <5TdUst/?dust values causes a variety of regional sizes in each
all-sky model. Sky maps of N p l x show that more stringent requirement on <$TdUst/7dust means lower
angular resolution for models at high latitudes. For STdust/Tdust < 20%, only 0.59% of the total
# of models have a 60.37 square-degree resolution instead of the default 6.71 square degrees. For
a 10% constraint on Tdustj 0.47% of models acquire a spatial resolution of 167.70 square degrees,
7.24% of models acquire a resolution of 60.37 square degrees, and the rest of the models are at 6.71
square degrees. Table 6.1 summarizes the percentages of models in various region sizes for different
limits on STdust/Tdust,- A balance between having adequate constraint on parameters, preserving as
52
O n e - c o m p a = 2.0 models
-i
1
1
r-
no spectral averaging on 7° spectra
spectral averaging till <5T/T ^ 1 0 %
spectral averaging till 61/J ^ 5%
spectral averaging till 61/1 ^ 2.5%
10
**-fcA'n*
,,
.
-, l~$. ^ jifefc '
.^
, v ^ W f T •*'*' .
1 —I
100
-150
I
I
l_
-50
0
Galactic latitude (°)
_l
1
I
l_
50
100
O n e - c o m p a = 2.0 models
250
no spectral averaging on 7° spectra 200
o 150
c
a>
3
spectral averaging till <5T/T ^ 10%
spectral ave'aging till <5T/T $ 5%
spectral averaging till 61/1 ^ 2.5%
i
I " 100
50
0
—.-i--r^.w—^-^hn
1.0
1.5
~in_-
2.0
X dof
Figure 6.10: Upper: Xdof
vs
- Galactic latitude. With a more restrictive ^Tdust/Tdusti values of % j o f
at high latitudes (\b\ > 60°) migrate from the range 0.7 — 1.0 to the range 0.8 — 1.3. Lower:
distributions of X^of • Values of x^ of in these plots come from all-sky one-component a = 2.0
models that satisfy STdust/Tdust < 20%, 10%, 5% and 2.5%, respectively.
53
O n e - c o m p a = 2.0 models
, T .. . ,_,.r._.T.._ i
300
i
| |
i
i
-r •»•*••—i
I
i
i
i
i
i
i
i
!•_
I _
no spectral averaging on 7° spectra
spectral averaging till 61/1
spectral averaging till 61/1
•spectral averaging till 61/1
200
^ 10%
^ 5%
^ 2.5%/
u
c
3
cr
a)
100 -
W
I
L.M..J — ..I
I ,1—
_J
o
I
L„
L...
a. •
II *H
I*.
i l l
—
I
10
1 • _•
30
T(K)
Figure 6.11: Distributions of Td ust of all-sky one-component a = 2.0 models that satisfy
5T d u s t /T d u s t < 20%, 10%, 5% and 2.5%, respectively. Fits that have a x\of > 1-13 are not
included. These distributions show that some high-temperature, less restrictive fits assume lower
temperatures when restriction on <5Tdust/2dUst becomes greater.
54
O n e - c o m p a = 2.0 models
•
i
i
i i i 111
i
i—i—i—i—i
no spectral averaging on 7° spectra
spectral averaging till <5T/T ^ 10%
spectral averaging till <5T/T g 5%
spectral averaging till 6T/T ^ 2.5%
•u
10"
7
10"
10"
10~
4
—I
I
I
L.
10"
Figure 6.12: Distributions of r of all-sky one-component a = 2.0 models that satisfy
6T d u s t /T d u s t < 20%, 10%, 5% and 2.5%, respectively. Fits that have a Xdof > !- 1 3
are not
included. This plot shows that models affected by the STdUst/TdUst constraint generally have higher
temperature and lower optical depth as the constraint is relaxed.
55
Table 6.1: % sky fit by one-component a = 2.0 models in different region sizes
Region size of models (square degrees)
7
60
168
329
544
812 1134 1509 1939 2422
oTdust/Tdust
<20%
98.10*
0.59
//
/
/
/
/
/
/
7.24 0.47
90.93
<10%
/
/
/
/
/
/
/
<5%
72.98 12.81 7.11 4.31 1.12
/
/
/
44.95 19.69 9.98 7.23 4.51 2.73 2.44
1.84
1.43 0.39
< 2.5%
Table 6.2: Best-fit one-component a = 2.0 models for the average spectra centered at Pixel 4062.
FIRAS Pixel 4062 at glat = 52°27 and glon = 9°36
1
9
25
49
81
121
168
^pix
Sky region area (square degrees)
7
60
168
544
812
1134
329
1.011 0.865 0.869 0.953 0.926
1.016
1.094
Xdof
19.57 18.79 18.63 18.41 18.31
18.10
18.03
T d u s t (K)
Tdust/<Wdust
20.27 31.09 44.66 60.82 82.08 106.27 121.19
many valid models as possible at high latitudes, and keeping region sizes low can be achieved at the
STdust/Tdust
< 10% level.
Discussion
The purpose of including more 7° spectra to form an average is to reduce random noise
in the final spectrum, which in turn makes \ 2 more sensitive to any model that does not have the
exact correct functional form or to data errors that are not precisely correct. This hypothesis is
true in an ideal case where sky signal is uniform. However, real sky signal is not uniform, so the
inverse relation between \ 2
an
d spectral averaging is not always one-to-one. When different types
of emitting sources are averaged, the final spectral shape changes and \ 2 could become larger or
smaller depending on which spectral shape dominates. This is a reminder that losing local features
is a drawback of smoothing a map, so a good model needs only a minimal amount of smoothing
if it can maintain an adequate constraint on the best-fit parameters. In this data set, \ 2 does not
change monotonically with more spectral averaging, i.e., sky region area of a model decreases.
Table 6.2 presents an example of the fluctuation in \\oi as models are fit to more and more
highly averaged spectra centered in the same direction. The \\0i ° f t n e 9-spectra fit and 81-spectra
fit are both smaller than those at previous levels.
A more systematic investigation finds that fits at both high and low latitudes exhibit this
type of behavior with the \2• Fluctuations take place not only once, but as many as 7 times among
10 spatial resolutions examined. The sky map in Figure 6.13 shows the number of times \ 2 fluctuates
between high and low values as models are fit to average spectra that include more of the surrounding
pixels. This phenomenon again shows that \2 is n ° t a good indicator of the amounts of constraint
*1.32% of the sky are not calculated due to the lack of FIRAS measurements.
56
§ of times y2s reverse incremental trend as resolution lowers
0 0
7 0
Figure 6.13: Sky directions where best-fit x 2 do not monotonically increase as averaging region of
the fit is reduced. This sky map is in Galactic coordinate Mollweide projection with the Galactic
center at the center and longitude increasing to the left. This map shows the number of times
(corresponding to a linear scale of colors) \ 2 fluctuates between high and low values as models are
fit to average spectra that include more of the surrounding pixels.
on best-fit parameters.
6.5.2
Other Fixed-a Models
Analysis
Similar to what was done to obtain all-sky one-component a = 2.0 models with various
sky average regions, all-sky one-component fixed-a (a G [1.4,2.6]) models with the upper limit of
the STdust/Tdust values set to be 20%, 10%, 5% and 2.5% are obtained separately.
Interpretation
The upper plot in Figure 6.14 shows Xdof distributions for the 13 all-sky fixed-a models
that satisfy <5Tdust/2dUst ^ 10%. It shows that the largest (2.6) and smallest (1.4) value of a cause
more fits to have large x^ of values. A more precise account of Xdof f° r e a c n all-sky model is presented
in the lower plot of the same figure. The all-sky a = 1.7 model is the best-fit model for this data
57
set. Figures 6.15 and 6.16 present the Xdof
6.6
an<
^ best-fit parameters of the model.
Summary
The last five sections have presented results from fitting one-component fixed-a models to
average spectra of sky regions. The findings are summarized below:
•
Fitting average spectra of latitudinal bands or longitudinal regions gives better constraints on
TdUst and r than fitting 7° spectra.
•
x2 °f regional fits are much higher than \ 2 of fits to 7° spectra at those regions. This means
that the larger areas include different types of dust emission spectra, and that the averaging has
achieved a sufficient signal-to-noise ratio, so spectral variation becomes statistically important. As
a result, the amount of spectral averaging needs to be reduced in order to preserve these variation
in the model.
•
a = 1.8 models can lead to lower x^ of values than a = 2.0 models for fits below 60° in Galactic
latitude and outside the Galactic plane. Average spectra at the highest latitudes are better fit by
an a = 2.0 model.
•
Setting an upper limit to an all-sky model's <5TdUst/TdUst values connects spectral averaging to
the signal-to-noise ratio of the data. As a result, the low signal-to-noise spectra, at high latitudes,
are more heavily averaged than spectra that already have high signal-to-noise ratio. This approach
not only minimizes the amount of spectral averaging, but also leads to adequate \ 2 (Xdof < 1.13
with 210 degrees of freedom) in most of the sky outside the Galactic plane.
•
The one-component a = 1.7 model minimizes the x^ of over the highest percentage of sky area.
58
O n e - c o m p . models with (5T/T ^
1.0
1.2
1.4
10%
1.6
1.8
2.0
X dof
O n e - c o m p . models with <5T/T ^
1
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
10%
i
i
i
i
24
ho 22
X
-
"*
X
20
X
18
_c
"
X
X
.*-»
5
>. 16
.*
CO
o
6*
_
—
—
—
1-13
•D
X
~
—
—
_
X
14
X
*
12
X
X
10
.
i
1
1.4
1.6
.
.
x
.
1
.
.
.
1.8
1
2.0
a
.
.
~
-
X
X2do, >
A
1
1
.
2.2
.
.
.
1
2.4
.
.
1
2.6
Figure 6.14: Upper: Distributions of Xdof f° r all-sky one-component fixed-a models (a G [1.4,2.6])
that satisfy 5Tdust/Tdust
< 10%. Lower: % of sky that has a fit whose Xdof exceeds 10% probability
cutoff. The all-sky a = 1.7 model minimizes the x\Qf
59
over tne
highest percentage of sky area.
7, <5T/T S 10%
T, a = 1 7, cST/T S 1C
r, a = 1 7, (5T/T S 10%
Log(2 Oe-06)
Log(2 Oe-03)
Figure 6.15: All-sky maps of Xdof ^dust and r of the best-fit one-component a = 1.7 model that
satisfies STdust/Tdust < 10% with the least amount of averaging. These maps are in Galactic
coordinate Mollweide projection with the Galactic center at the center and longitude increasing to
the left.
60
§ of s p e c t r a , a = 1.7, <5T/T S 10%
Log(l)
Log(361)
(5T/T, a = 1 7, <5T/T §10%
Log(OO)
Log(%)
Log(lOO)
( 5 T / T , a = 1 7, (5T/T 510%
Log(0 a)
Log(%)
Log(62 9)
Figure 6.16: All-sky maps of variable sky averaging, 5Tdust/Tdust
and 5T/T of the best-fit
one-component a. — 1.7 model that satisfies ^Tdust/TdUst < 10% with the least amount of
averaging. These maps are in Galactic coordinate Mollweide projection with the Galactic center at
the center and longitude increasing to the left.
61
Chapter 7
More Complex Dust Models for
Galactic-Plane Spectra
7.1
One-component Free-a Model
Analysis
Chapters 5 and 6 have shown that none of the one-component fixed-a models is a good fit
to the entire set of sky spectra, especially those on the Galactic plane. The next simplest model
that adds an extra degree of freedom to those models is a one-component free-a model. Here, a is
treated as another variable in the overall model, just like Tdust, T, [C II] intensity and [N II] intensity.
This is reasonable to try in the Galactic plane since the high signal levels support the ability to fit
additional model parameters. This five-parameter model is tested on 275 Galactic-plane spectra
that have a Xdof > 2 from fitting one-component a = 2.0 model to the data.
Xdof
Des
t-fit parameters and uncertainties of the fit to each 7° spectrum are summarized
in Figure 7.1. Examples of best-fit models for spectra on the Galactic plane are presented in Figures
C.l and C.2.
Interpretation
A one-component free-a model fits 108 of the 275 Galactic-plane spectra with a Xdof < 2.
The emissivity spectral index a is constrained to be better than 2.7%, dust temperature Tdust is
better than 1.9%, and optical depth r is better than 4.1%.
Close examination of individual spectral fits in Figures C.l and C.2 shows that free-a
model can fit some of the Galactic plane spectra very well, with Xdof ~ 1- F° r fits that have
Xd0f ^ 1) it often is the case that FIRAS low-frequency data points fluctuate excessively compared
62
X2, free a
a, free a
T, free a
T, free a
CO
Log(l.8^^?7
Figure 7.1: Sky maps of XiLf > a' ^dust, T (this graphic panel) and uncertainties (next graphic panel) of the best-fit one-component free-a
models for 7° Galactic-plane spectra, in Galactic coordinate Mollweide projection with the Galactic center at the center and longitude
increasing to the left. 275 pixels along the Galactic plane participate in this fitting exercise because their fit with a one-component a = 2.0
model results in a Xdof — 2-
6a/a,
free a
(5T/T, free a
Log(3.7e-01)
Log(%)
Log(1.9e+00)
Log(7.7e-01)
Figure 7.1 (continue)
Log(%)
Log(4.1e+00)
0)
25 -
|
20 -
/ ^
i
.*
-
CD
E
-
15
o
o
I
a>
c
o
10 _
o
s<
-
5
T3
-
^ ^ T "•
-*"*«*
" , » " • * . .
<jnm* s . " a . " " ^ I _
»
.
-
*
•
-^mMCM, • • '•
0 X"^ , , " , , , , , , !
5
10
0
X
Figure 7.2: Xdof, free a
vs
2
1
15
20
25
dof , o n e - c o m p , a = 2.0
- Xdof, one-comp a=2 0'
This
scatter plot shows that the extra degree of
e
freedom helps reduce Xdof' y ^ many new Xdof values are still larger than 2.
to uncertainties at those points, so any model that predicts a relatively smooth curve over that
portion of the spectrum would not be adequate.
7.2
T w o - c o m p o n e n t a\^ — 2.0 M o d e l
Analysis
A two-component dust model is tested on the 275 spectra that have a Xdof
>
2 from one-
component a = 2.0 fits. This two-component model includes six parameters: two dust temperatures
(Ti and T 2 ), two optical depths ( n and T2), plus [C II] and [N II] intensities. The power-law
emissivity spectral index for each dust component is set to be 2.0.
Best-fit parameters and their uncertainties are presented in Figure 7.4. The warm dust
component is predicted to have temperatures, Thigh temp; between 16 — 23 K with uncertainties
at 0.08 — 0.97%. The cold dust component is predicted to have temperatures, T\ow temp, between
0.3 - 6.5 K with uncertainties at 2 - 244%.
Sample plots of the best-fit two-component models are presented in Appendix C.2. The
first group of three spectra are examples of fits that have Xdof ~ 1 • The second group shows that
65
models can fit most of the channels well except those in the FIRAS low-frequency band, particularly
at 100 — 600 GHz. For these fits, 1 < Xdof < 3. Group three contains some of the largest Xdof
nts:
Poor fits at channels between 100 — 600 GHz continue to be a problem. The models also miss fitting
the data at frequency v > 1000 GHz.
Interpretation
X^of of fits at Inner Galaxy (119 pixels) are still larger than 2. This is demonstrated by the
sky map of Xdof
m
F i g u r e 7-3. Also in Figure 7.3, the scatter plot shows that all 275 Galactic-plane
fits experience a decrease in Xdof upon switching from one-component to two-component models.
Temperature maps in Figure 7.4 show that Thigh temp is very well constrained; on the
contrary, T\ow temp is not. At the Galactic center, there is a correlation between fits with Xdof S> 1
and fits with T\ow temp at a much lower value than the average CMB temperature. Moving away
from the Galactic center, Xdof decreases to below 2 and T\ow
temp
rises above the averaged CMB
temperature. For these fits, both 71 ow t e m p and T\OW t e m p are well constrained.
To check whether the two dust components are related, correlation coefficient between the
two dust temperatures is found to be —0.56. A linear fit to the scatter plot of T\ow
temp
vs. Thigh temp
(Figure 7.5), taking into account errors in both parameters, has a slope of —0.43. The large x 2 of
the fits means that there is not a linear relation between the two temperatures. Also in Figure 7.5 is
the scatter plot of
TIOW temp
vs. Thigh temp- The correlation coefficient for the optical depths is only
0.089, an indication that the two optical depths are not correlated. A linear fit to points on the plot
has a slope of 7.10, which is due mainly to the absence of constraint in the y-direction in comparison
to the x-direction. These results do not support a one-to-one relation in either dust temperature or
optical depth.
7.3
T w o - c o m p o n e n t M o d e l s w i t h O t h e r Fixed Values of cx.\
a n d e*2
A two-temperature-component 0:1,2 = 2.0 model can be a good fit to about half of the
Galactic-plane spectra, where a one-component model fits poorly. To find models that can fit
the other half of the problematic Galactic plane spectra, two-component models with a± and «2
separately taking on values between 1.3 and 2.8 (a total of 136 models) are fit to each of the 275
spectra. To effectively present results from these fits, XdDf fr°m fitting different models to the same
spectrum are collected into a single plot of Xdof vs. a.\. The second dimension, u<i, is represented
by curves of different colors. Two examples of such plots are presented in Figure 7.6. Out of 275
spectra, 194 of them can be fit to one of these two-component models with a Xdof
<
2- Models
with one a at 2.8 and the other at 1.5 — 1.9 often produce the lowest Xdof The two predicted
temperatures are disjoint, with the warmer component at 14 K < Thigh temp < 17 K, and the cooler
66
X2dof, t w o - c o m p
a12
=
2.0
0.9
1
'
1
1
25
21.4
1
-
o
x
^
20 —
II
CN
,-"
« 15 —
CN
s'
X
X
Q.
•^x
E
o
o
1 10
o
~
—
-
x
X
X
5
*•'
X
^-y^xxx
5
X
—
X
w y ^ * ? - *y . vvvr
0
SwOk
wy^N^*^
i
~
—
—
'
x X
x
7K
i
X2d0„
-
X *
1
1
10
15
one-comp a = 2.0
1
20
25
Figure 7.3: Xdoi of two-component 011,2 = 2.0 fits to Galactic plane spectra. The all-sky map is in
Galactic coordinate Mollweide projection with the Galactic center at the center and longitude
increasing to the left. The scatter plot of xLo-comp ai.2=2
vs
- Xone-comp «=2
shows that
Xdof of all
275 fits decrease upon switching from one-component model to two-component model. 119 fits still
have a \loi > 267
T hig h temp.. t W 0 - C 0 m p
2.0
«1|2
T
io W tem P .. t w o - c o m p
0.3
<5Thi9h/Th,ch. t w o - c o m p a ,
2
=
2.0
ali2 =
K
<5Ti0w/Tiow. t w o - c o m p a 1>2 =
2.0
6.5
2.0
oo
Log(7.9e-02)
Log(%)
Figure 7.4: Best-fit parameters from fitting two-component a ^ = 2.0 models to Galactic plane spectra. These maps are in Galactic
coordinate Mollweide projection with the Galactic center at the center and longitude increasing to the left. 275 Galactic-plane pixels are
shown in this series of maps because fitting one-component a = 2.0 model to their spectra results in a Xdof — 2- In this graphic panel, the
upper two maps show the best-fit temperature of the two dust components, both in a linear scale of colors; the lower two maps provide
percentage errors of the temperature estimates, both in a log scale of colors, (to be continue in the caption of next graphic panel)
T
T
iowtemP.. t w o - c o m p a 1 2 = 2.0
hi 9 hte m p . t w o - c o m p a 1 2 = 2.0
Log ( 2 . 5 ^ ^ 4 ^ ^ ^ ^ ^ ^ ^ ^ ^
Log(a0e-03)
^ T hi9htempA h i g h t e m p ., t w o - c o m p a 1 2 = 2.0
Log(1.8^)T^^^Log(%^^^
Log(?le+00)
Log ( 1 . 7 ^ ) 3 ^ ^ ^ ^ ^ ^ ^ ^ ^
<5TIOW
t e m p / T l 0 W temp .,
Log(?5e-02)
two-comp a,
Log(S^^^^Log(%^^^
2
=
2.0
Log(100.0)
(additional caption for Figure 7.4) In this graphic panel, the upper two maps show the best-fit optical depth of the two dust components,
both in a linear scale of colors; the lower two maps provide percentage errors of the optical depth estimates, both in a log scale of colors. The
error maps show that the high-temperature component is very well constrained but not the low-temperature component. The fit is
unacceptable at the center of the Galaxy since Xdof -^ 1 a n d ^iow temp is much lower than the average CMB temperature. Away from the
Galactic center, Tiow t e m p rises above the averaged CMB temperature, and both T\ov, t e m p and T\OW t e m p become well constrained.
[CII], t w o - c o m p a 1 2 = 2.0
0.16
3.1
<5[CII]/[CII], t w o - c o m p a 1 2 = 2.0
[Nil], t w o - c o m p a 1,2
Log(8.9e-04)
<5[NI!]/[Nll], t w o - c o m p a,
2.0
Log(5.0e-01)
2
= 2.0
—i
o
(additional caption for Figure 7.4) In this graphic panel, the upper two maps show the best-fit [CII] and [Nil] intensities, and the lower two
maps provide percentage errors of the line intensity estimates. All four maps are plotted in a linear scale of colors.
t w o - c o m p , CK 12 =
two —comp, a 1 2
=
2.0
2.0
0.020
0.015
; 0.010
0.005
0.000
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
high temp.
Figure 7.5: T\ow vs. Thigh and T\OW VS. Thigh of the best-fit two-component oti^ = 2.0 models for
275 Galactic-plane spectra. In each plot, a linear function is fit to data points taking into account
uncertainties in both parameters. The slopes and y-intercepts are printed in purple and values of
the functions are drawn in the same color. Correlation coefficients are calculated for both cases
using only values of the parameters. Both plots show large scatter about the linear fit and do not
support a one-to-one relationship between the two dust components.
71
component at 6 K < Tiow temP < 9 K. In addition, both temperatures are very well constrained, with
SThish temp/Thigh temp ~ 1% and d)TIow temp/71ow temp < 7%. Figures 7.7 and 7.8 present sky maps
of Xj of and parameters for these minimum-x^ oi models.
7.4
T h r e e - c o m p o n e n t 0:12,3 = 2.0 M o d e l
Three-component models with a\ = 02 = 03 = 2.0 are fit to Galactic-plane spectra that
have Xdof > ^ from fits using two-component Oi = 02 = 2.0 model. Examples of three-component
fits are presented in Figure C.6.
These examples show that using a three-component model does not improve Xdof of the fits.
Often two of the components share the same temperature, even though the fit routine was executed
with many different initial conditions. These results suggest that the poor fit in the two-component
case cannot be rectified with a third component.
7.5
Summary
This chapter has focused on finding best-fit models to the 275 Galactic-plane spectra that
do not have a good one-component fixed-o model. A one-component free-o model, two-component
models with various fixed spectral indices (1.3 < 011,0:2 < 2.8), and a three-component model with
o>i = a.2 = 0*3 = 2.0 have been fit to these spectra, but none of them is a good fit to all 275 spectra.
The two-component model with Qi = 2.8 and 02 = 1.5 — 1.9 often achieves the lowest \ 2 among all
models tested. Of the 275 Galactic-plane spectra tested, 194 of them have a two-component model
that gives a x\Q{ < 2- This means that 81 spectra, or 1.3% measurements of the entire sky, cannot
be modeled as dust emission in any simple way.
72
Pixel § 4 8 9 9 , glon = 2 0 . 8 1 , glat =
3.83
4.0
«1
«1
3.5
a,
a,
a,
a,
a,
3.0
« i
=
=
=
=
=
=
=
=
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
a, = 2.1
a, = 2.2
= 2.3
« i
a, = 2.4
a, = 2.7
a, = 2.8
i 2.5
2.0
1.5
1.0 _l
1.3
1.4
1.5
1.6
I
I
1.7
1.8
l _
-J
I
L_
1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.1
Pixel § 5 0 2 3 , glon = 4 1 . 5 5 , glat =
4.0
3.5
I
i
Y
i
i
I
-
*-—*—_^___
—-*—-*__^
^fe==
^_
-0.89
1
=
=
=
=
=
=
1
1.3
1.4
1.5
1.6
1.7
1.8
a,
a,
a,
a,
* "
2.0
=
=
=
=
a
a,
I
i
2.1
2.2
2.3
2.4
~
—
~
y
2.8
—
—^_
— ^\^r~~~-^i;—«—_zr
:
I
a,
a,
a
a
a
a
a
a
JL
^
3.0 —
h
2.0
i
-
1 2.5
"x
I
*—*—•
M
—<—
—
_ ..
"—-*——«—____
"
"
*
i
I
I
i
-*
1.5
1.0
i
1.3 1.4
i
i
i
1.5 1.6 1.7 1.8
i
i
i
i
1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8
a,
Figure 7.6: x\of °f
(ai,a2
nts
^h&t use two-component models with different fixed values of a\ and a2
E [1.3, 2.8]) for spectra of FIRAS Pixels 4899 and 5023. Plots for the other 273
Galactic-plane spectra show that models with two a's having the same value is not where the
minimum Xdot ^s- Instead, models with a.\ = 2.8 and a2 = 1.5 — 1.9 often have the lowest Xdot-
73
X dof
T w o - c o m p o n e n t Models for 275 Galactic plane Spectra
a
high temp
io» tempi Two-component Models for 275 Galactic plane Spectra
T w o - c o m p o n e n t Models for 275 Galactic plane Spectra
Figure 7.7: Sky maps of x\0f and the two emissivity spectral indices from the best-fit two-component models with fixed values of ct\ and «2
(ai,oi2 £ [1.3,2.8] separately) for the 275 Galactic plane spectra. These maps are in Galactic coordinate Mollweide projection with the
Galactic center at the center and longitude increasing to the left. 275 Galactic plane pixels are shown here because fitting one-component
a = 2.0 model to their spectra generates a x^ of > 2. Out of 275 spectra, 194 of them have a Xdoi < 2 from the current fit. Models with
a.\ = 2.8 and at = 1-5 — 1.9 often have the lowest \ o\f
1
1
low temp.
Two-component Models for 275 Galactic plane Spectra
high t e m p
Two-component Models for 275 Galactic plane Spectra
° 1 low/ ' low
Two-component Models for 275 Galactic plane Spectra
" ' high/ ' high
T w o - c o m p o n e n t Models for 275 Galactic plane Spectra
^1
Figure 7.8: Sky maps of of the two dust temperatures from the best-fit two-component models with fixed values of a\ and a^
(ai,CK2 £ [1.3,2.8] separately) for the 275 Galactic plane spectra. These maps are in Galactic coordinate Mollweide projection with the
Galactic center at the center and longitude increasing to the left. 275 Galactic plane pixels are shown here because fitting one-component
a = 2.0 model to their spectra generates a Xdoi — 2- Notice that the two predicted temperatures are disjoint, with the warmer component at
14 K < Thigh temp < 17 K, and the cooler component at 6 K < T\ow
temp
< 9 K. Both temperatures are very well constrained.
Chapter 8
Understanding the Galactic-plane
Fits
This chapter presents findings on why fits of the Galactic plane spectra are poorly fit. The
first four sections characterize symptoms of the high-x 2 fits. Sections 8.5—8.7 present a series of
attempts to address known issues in the spectra (None of these attempts is effective). In Section 8.8,
covariance matrices of the high-x 2 spectra are examined. All of them have highly skewed frequencyfrequency covariance, which correspond to known issues in FIRAS calibration of high-temperature
measurements.
8.1
Highest x2 Channels Concentrate around 900 GHz and
2000 GHz
X2 per frequency channel are examined. Examples of these plots are shown in Figure
Figure 8.1, where the highest x 2 frequency channels concentrate at around 900 GHz and 2000 GHz.
Channels with more moderate x 2 > 2 are distributed throughout the entire frequency range.
8.2
Synchrotron Emission Could Not Have Caused High x 2
To check whether synchrotron emission contributes to high x 2 at low-frequencies, signals
in WMAP's
7 th -year synchrotron map are compared with intensity of the FIRAS dust spectra.
WMAP's synchrotron map and its noise map are converted to the same format as FIRAS dust
spectra following the procedures in Section 3.3. Since synchrotron emission has an amplitude of
~ 10~ 3 MJy/sr at 22 GHz, and a spectral index of about —3, the extrapolated signal is minuscule
compared to the dust intensity at the same low frequencies. This means that synchrotron emission
76
FIRAS pix § 4 1 7 0 , glon = 3 2 8 7 8 , glat
=
- 0 60
1
120
100 .
80 "
60
-
« = 2 0 x' ,.o-c„mp = 7 52
T, = 21 08 ± 1 9 5 e - 0 2
T, = 1 72e-03 ± 3 96e-06
T2 = 4 28 ± 1 2 7 e - 0 1
T 2 = 1 21e-02 ± 1 28e-03
[Cll] intensity = 2 42 ± 3 7 1 e - 0 2
[Nil] intensity = 0 36 ± 1 0 3 e - 0 2
X*
-x—X % X x > < x x x x x x xx ; < A > < x< > x^Xxx^ : i ; ^
frequency (GHz)
FIRAS pix § 4 3 4 8 , glon
"
.
1 0 5 , glat
1 46
'
X
_
—
"
a - 2 0 x'l-o-comp = 21 43
T, = 22 26 ± 1 7 5 e - 0 2
T, = 1 97e-03 ± 3 60e-06
T
2 = 2 93 ± 1 4 4 e - 0 1
T
2 = 3 99e-02 ± 4 49e-03
| Cll I intensity = 1 90 ± 4><41e-02
[Nil] ntensity = 0 34 ± 1 3 5 e - 0 2
-
x
X
„
—X
X M X X X X X X X X X x x x xxxx
y ^ l ,
4
^ ^
x
X
x
-
x
x
i-.-^^V ,ill,ii.|^4%MTX
-
-
1000
frequency (GHz)
FIRAS pix § 4 8 7 6 , glon =
-
i
15 2 4 , glat
1
- 2 38
_
X
a = 2 0 y',«, =„™ = 7 03
T, = 20 8 8 ± 2 1 0 e - 0 2
— T, - 1 56e-03 ± 3 88e-06
2 98 ± 3 3 2 e - 0 1
T
- 1 40e~02 ± 3 7 8 e - 0 3
mtensty = 2 22 ± 3 7 3 e - 0 2
[NHJ ntensity = 0 33 ± 1 1 1 e - 0 2
;
—
X
-
.. . . v , , v x „ . . w
X
-
X
*
x^xx
X
X
>X
X, X X
. ^ . . . . . X x x x ^ x ^ x X x %&.
AAX X
~**'
x
x
£x
xx
—
*
-
frequency (GHz)
Figure 8.1: x2 P e r frequency channel using best-fit two-component a\^ = 2.0 models. Values
printed at each plot are the best-fit parameters. Red x 's represent FIRAS channels; green x 's
represent DIRBE bands; and the blue x represents WMAP's W band. These plots show that the
highest x2 channels are around 900 GHz and 2000 GHz. Channels with x2 > 2 distribute
throughout the entire frequency range examined.
77
could not alter the low-frequency spectra enough to cause high x 2 values at those channels.
8.3
Highest \2 Channels Do Not Have the Largest Deviations
between Data and Model
Difference between data and model scaled by standard errors are presented in Figure 8.2.
In each of these plots, notice the negative residuals at channels around 2000 GHz, which correspond
to the frequency range dominated by high-temperature dust emission. Notice also the fluctuations
at channels around 600 GHz. They are expected to be due to transition between FIRAS's high and
low frequency bands.
To examine the issue from a different perspective, plots are made to compare locations of
the highest x 2 channels and those that have largest deviations between data and model. Figure
8.3 shows two of them as examples. In each plot, the black curve that hovers around y = 0 is the
difference between data and the best-fit two-component a.\j, = 2.0 model. Colors of vertical lines
indicate different tiers of highest x 2 channels. For example, lines in red (no more than 5 on each
plot) indicate the highest x 2 channels; the next tier of high x 2 channels (also no more than 5) are
in purple; and the third tier of high x 2 channels are in green. The total number of vertical lines
on each plot indicates the number of channels need to be removed in order to bring Xdof °f the
remaining-channel fit to be below 2. These plots show that the highest x 2 channels often do not
correspond to channels with the largest deviation between data and model.
8.4
Emission Lines and Their Asymmetric Profiles Are Partially Responsible for High x2
Figure 8.3 shows that some of the highest \ 2 channels have to do with emission lines.
For example, in the plot for Pixel 4192, from left to right, the first red channel corresponds to
CO(J = 2 — 1) emission; the next two green channels correspond to CO(J = 3 — 2) emission; and
the next green and purple channels are at the frequency of CO( J = 4 — 3) and [CI] emission. In the
case of Pixel 4339, the first green channel from the left corresponds to CO( J = 2 — 1) emission, and
the next red and green channels correspond to CO( J = 4 — 3) and [CI] emission. It means that line
emissions, other than those from [C II] and [N II], are strong enough at the Galactic plane to affect
X2 of the fits.
Indeed, 52 out of 119 pixels experience one of the highest x 2 due to [CI] emission, whereas
63 out of 119 are affected by CO( J = 2 — 1) emission. Other lines that also affect x 2 of the fits are:
CO(J = 5 - 4 ) , CO(J = 4 - 3 ) , CO(J = 6 - 5 ) , H 2 0 , CO(J = 3 - 2 ) and CO(J = 1 - 0 ) . Ideally,
a complete model should be able to scale the intensity of each of these lines independently. The
78
FIRAS pix # 4 1 7 0 , glon = 3 2 8 . 7 8 , glat =
20
x
-0.60
2
X a = 2,two-comp
CD
10
:x
O
0
—
'--'I
-=
X
K
TK
XX*
v.*
* '
x
WO(
xx
' ' ^ < ^
) 0 < > o c W
^ ^
w
^ «
w
.
vXX
» W ^ ^
X
10
x
-I
20
500
1000
1500
frequency (GHz)
FIRAS pix § 4 3 4 9 , glon = 2.34, glat =
2000
X a=2,tvvo-comp
10
0 ^^W
3000
-0.79
20
-J
2500
_
16.70
:§<
S<"S^>
^si **w>s< <ws ^v*«v* wwWaoOOfrfOtX^CXax
)?
s
***„
j-Witt**—=
-10
-20
500
1000
1500
frequency (GHz)
2000
2500
3000
Figure 8.2: Difference between model and data scaled by standard errors. The models used in these plots are the best-fit two-component
<*i,2 = 2.0 models, obtained from fits that use the full covariance matrix. The orange vertical line indicates the expected frequency of the
[C II] line, the purple vertical lines indicate frequencies of the CO lines, and the blue line is for frequency of the [N II] line. The majority of
the points in each plot are negative which means that the calibration for these spectra are off.
FIRAS pix # 4 1 9 2 , glon
=
331.19,
glat
0.92
l
1000
500
;
0
~
-500
a = 2.0 x2.„o-comP = 8.9 0
need to remove 42 highe st x2 c n a inels to reach x' <
5 highest x' channels
next 5 highest ~x* channe s
remaining 32 highest x* channels
z
-NJWf-
I
-1000
-1500
I
-2000
100
1000
frequency (GHz)
FIRAS pix # 4 3 3 9 , gion
=
357.51,
2.41
glat
1000
500
0
-500
3.e
2-0 x\„
need to remove 17 highest x channels td~ reach x <
7
3 Hgnost x c n o ^ r e s
next 5 nighest x' channe
remaining 7 highest x* channels
-T"^*i
-1000
-1500
-2000
-2500
100
1000
frequency (GHz)
Figure 8.3: Highest x 2 channels in plots of difference between data and model for FIRAS Pixels
4192 and 4339. In each of these plots, the black curve that hovers around y = 0 is the difference
between data and best-fit two-component a\^ = 2.0 model. Colors of vertical lines indicate
different tiers of highest x 2 channels. For example, lines in red (no more than 5 on each plot)
indicate the highest x2 channels; the next tier of high x2 channels (also no more than 5) is in
purple; and the third tier of high x 2 channels is in green. Total number of vertical lines on each
plot indicates channels that need to be removed in order to bring x^of fr°m fits to the remaining
channels to be below 2. These plots show that the highest x 2 channels often do not correspond to
channels with the largest deviation between data and model.
80
drawback for doing so is introducing too many poorly constrained parameters. Clearly, it is not the
most effective handle to reduce the overall x 2 In addition to the presence of emission lines, where contributions were not fit out by the
overall model, strong [C II] and [N II] emission may cause a ripple effect on channels adjacent to
frequencies where these lines are expected. Plots are made to examine fits around [C II] and [N II]
lines in details. For example, Figure 8.4 shows that high \ 2 channels often fall on the shoulders or
knees of the [C II] line profile. In fact, out of 119 plots examined, 117 have one or more highest
X2 channels falling in the effective frequency range of the [C II] line profile. In addition, the black
curve, representing data — model, often forms an 'N' shape that arches over the central frequency
at 1900 GHz. All these mean that fitting out the height of a line profile, as model currently does,
is not adequate. Taking out immediate channels on both sides of the [C II] line, 16 fits achieve
Xdof
<
2- Similarly, after fitting out [N II] intensity, 28 out of 119 pixels still experience high %2 on
the shoulders/knees of the [N II] line profile.
8.5
Re-fit Model without Channels at 800 - 1000 GHz and
1500 - 2000 GHz
The 800 - 1000 GHz and 1500 - 2000 GHz channels are taken out of the fit and the
two-component a\^ = 2.0 models are re-fit to the remaining spectral data.
The scatter plot of xLected channels
vs
- Xaii channels (presented in Figure 8.5) shows that
Xdof decreases slightly for a number of fits, although the new values are still much larger than 2.
This means that channels at frequencies between 800 — 1000 GHz and 1500 — 2000 GHz cannot be
the determining factor of the overall high Xdof •
8.6
Re-fit Model without Dichroic Channels
Another possibility for a localized cause of high x2 are the artifacts from the dichroic filter.
It is the component that separates FIRAS measurements into high and low frequency bands (at
~ 600 GHz). Although a systemic "jump" was removed by procedures described in Chapter 3,
measurements at the two bands' overlapping channels still have higher noise than the rest of FIRAS
channels.
To evaluate the influence of the dichroic's effect on the overall model fitting, two-component
ai,2 = 2.0 models are fit to Galactic plane spectra without measurements at 585-680 GHz. Results
are presented in Figures 8.6 and 8.7
Figure 8.6 is a scatter plot of Xd0f
or tn
e no-dichroic-channel fits vs. Xdof °f the full,
214-channel fits. It shows that new values of Xdof differ little from those obtained from fits that
include the dichroic channels. Figure 8.7 presents examples of the spectral fits, which show that
81
FIRAS pix § 4 1 9 2 , glon = 3 3 1 . 1 9 , glat =
0.92
-200
-400
-600
-
-800
1800
1850
1900
frequency (GHz)
2000
FiRAS pix § 4 3 3 9 , glo
o
' - [ettf-ftne-pfof
Data — ModeT
-100
-200
5 hicrest y ;
r^r^
n
e
next 5 highest x* channels
-300
remaining / hignest x* cnameia
-400
-500
1800
1850
1900
frequency (GHz)
1950
2000
Figure 8.4: High \ 2 channels near frequency of the [C II] line in plots of difference between data
and model for FIRAS Pixels 4192 and 4339. These plots show that high x2 channels often fall on
the shoulders and knees of the [C II] line profile. Out of 119 pixels, 117 have one or more highest
X2 channels falling in the effective frequency range of the [C II] line profile.
82
0
5
10
15
20
25
2
A_ two-comp, all frequencies
Figure 8.5: Scatter plot of Xdof fr°m fitting two-component ai ; 2 = 2.0 model to spectra without
channels at 800 - 1000 GHz and 1500 - 2000 GHz vs. Xdoi from fitting two-component a i j 2 = 2.0
model to the full, 214-channel spectra. This plot shows that removing channels at 800 — 1000 GHz
and 1500 — 2000 GHz lowers values of x^ of slightly for a number of fits, but these new Xdoi
are still high.
83
vames
two —component, a — 2 model
0
5
10
15
20
25
x2
Figure 8.6: Xdof fr°m fitting two-component 0:1,2 = 2.0 model to spectra that do not include
dichroic cross-over channels (580 GHz < v < 680 GHz) vs. xlof fr°m fitting two-component
Qii,2 = 2.0 model to the full, 214-channel spectra. This plot shows that new values of x^of change
little from those obtained from full spectral fits.
best-fit models from the no-dichroic-channel fits completely overlap with best-fit models from the
full, 214-channel fits. The two sets of best-fit parameters are provided in each plot in respective
color. They show that values of the parameters from the two fits differ very little. These results
mean that dichroic cross-over channels do not have a strong influence on the best-fit models and
they are not the main reason for the overall high \2-
8.7
Re-fit Model without Channels in the Vicinity of Emission Lines
To examine the effects of emission lines on model fitting, channels at or near frequencies of
emission lines are first removed from each full spectrum, and then models are re-fit to the remaining
channels. Figure 8.8 shows channels that are removed because they are within a specified frequency
range of an emission line. For example, a 13.6 GHz range means that 36 channels are removed; a
27.2 GHz range means 68 channels are removed; and a 40.8 GHz range corresponds to 96 channels
removed.
Comparison of the new Xdof
w
^ n those obtained previously by fitting the full spectra are
84
-0.60
FIRAS pix # 4 1 7 0 , glon = 3 2 8 . 7 8 , glat =
10000 00
T,
T,
T2
T,
1000 00
=
=
=
=
21.09 ± 1 9 2 e - 0 2
1.72e-03 ± 3.89e-06
4.22 ± 1 3 1 e - 0 l
1 23e-02 ± 1 32e-03
] intensity = 2.42 ± 3 . 7 1 e - 0 2
] intensity = 0.36 ± 1 . 0 3 e - 0 2
- 7 56
100.00
a =
2.00
T, = 21.08 ± 1 9 5 e - 0 2
T, = 1 72e-03 ± 3.96e-06
10 00
1 00 k
T2 = 4.28 ± 1.27e-01
T2 = 1.21e-02 ± 1.28e-03
0.10 5
[CM] intensity = 2.42 ± 3 . 7 1 e - 0 2
[Nil] intensity = 0.36 ± 1 . 0 3 e - 0 2 .
v2
A
= 7 S?
two-comp
^*-
0 0 1 L_
1000
100
frequency (GHz)
FIRAS pix § 4 3 4 9 , g!on =
10000.00
1000.00
=
-E
-
100.00 =
:
10 00
->
2 . 3 4 , glat
:
Zz.C'i J. i b / c 0 2
= 2 0"o 03 <^ 0 1 ? 0 6
2.24 L 3 8 2 e - 0
- 5 0 ' c 02 * y / 1 c- 02
lc j1 intensity
n'p^si'y
,0
con .,
-
^4
-
— I 8 X 4 48e 0 2
42e - 0 9
- 0 2 -
i 6 9o
a =
2.00
;
T, = 2.31 ± 3 . 4 5 e - 0 1
T , = 5 6 3 e - 0 2 ± 1,94e - 0 2
=
_
T 2 = 22.01 ± 1 8 5 e - 0 2
T
2 = 2.04e-03 ± 4.00e-06
1.00 r
=
[C.I] intensity = 1.81 ± 4 48e-- 0 2
[Nil] intensity = 0.32 ± 1.42e-- 0 2
0 10 g ^
001
-0.79
X'two -comp = 16.72
,
=
~
-=
z
—
z
i
100
1000
frequency (GHz)
Figure 8.7: Best-fit two-component 0:1,2 = 2.0 models from full-spectral fit and no dichroic
cross-over channel fit. Spectra used in the no-dichroic-channel fit are in orange (notice that
spectral points at frequencies between 585-680 GHz have been removed); best-fit
no-dichroic-channel model is in red, which is largely overplotted by the blue curve representing
best-fit two-component model of the full spectral fit. Best-fit parameters are printed on each plot
in respective colors. These plots show that dichroic cross-over channels do not have a strong
influence on the best-fit models and that they are not the main reason for the overall high \ 2 •
85
36 Channels Removed for Polling w / i 13 6 GHz of FIRAS Lines
68 Chonnels Removed for Foiling w / i 27 2 GHz of FIRAS Lines
96 Channels Removed for Foiling w / i 40 8 GHz of FIRAS Lires
Figure 8.8: Frequency channels that are within specified frequency ranges of FIRAS emission lines.
The upper plot shows locations of 36 channels that are within 13.6 GHz of emission lines; the
middle plot shows locations of 68 channels that are within 27.2 GHz of emission lines, and the
bottom plot shows 96 channels that are within 40.8 GHz of emission lines.
Removed Channels w /
13 6 GHz of FIRAS Lines
Removed Channels w /
27 2 GHz of FIRAS Lines
Removed Chonnels w /
40 8 GHz of FlRAS Lines
Figure 8.9: Xdof fr°m fitting spectra that do not include frequency channels at or near emission lines vs. Xdof fr°m fitting the full 214-channel
spectra. The left plot represents fits that remove 36 channels within 13.6 GHz of emission lines. 22 out of 119 fits have a new Xdof < 2 and
all fits experience a decrease in Xdof The middle plot represents fits that remove 68 channels within 27.2 GHz of emission lines. 26 fits have
a new Xdof
<
2 and 5 fits experience an increase in Xdof The bottom plot represents fits that remove 96 channels within 40.8 GHz of
emission lines. 24 fits have a new Xdof < 2 and 21 fits experience an increase in Xdof
Figure 8.10: Changes in Xdof
on
go
00
20
00
20
as a resu
l t of not fitting frequency channels near emission lines. In
each map, pixels are color coded into three groups: in blue are fits whose 214-channel fits have a
Xdof
<
2; in green are fits that have a Xdof
<
2 after channels adjacent to emission lines were
removed; in red are fits that retain a x\0{ > 2 even after the lines are removed. The upper map
represents fits that remove 36 channels within 13.6 GHz of emission lines; the operation leaves 97
fits (or 1.6% of the sky) with a \\0{
>
2. The middle map represents fits that remove 68 channels
within 27.2 GHz of emission lines; the operation leaves 93 fits (or 1.5% of the sky) with a Xdof
>
The bottom map represents fits that remove 96 channels within 40.8 GHz of emission lines; it
leaves 95 fits (or 1.6% of the sky) with a \\0i
88
> 2-
^-
presented in Figure 8.9. In the 13.6 GHz case, 22 out of 119 fits have a new Xdoi < 2; in the 27.2
GHz case, 26 fits have a new Xdof < ^i
an
d
m
the 40.8 GHz case, 24 fits have a new XdQf < 2. The
13.6 GHz case is the only block-off range that lowers Xdof f° r
Figure 8.10 demonstrates the changes in xlof
on a s
au
H 9 fits.
^y map. Pixels are color coded into
three groups: in blue are fits that have a Xdof < 2; in green are fits that achieve a x 2 0 f
<
2 once
channels at or near emission lines were removed; in red are fits that have Xdof > ^ whether or not
emission-line channels are removed.
Since none of the above attempts to remove channels at or near emission lines can effectively
reduce Xdof' the emission lines must not be the ultimate cause of a model's poor fit to the data.
8.8
Frequency-Frequency Covariance
The dust emission in the Galactic plane is much stronger and at higher temperatures than
the diffuse emission from dust at high latitudes. According to the FIRAS Explanatory Supplement,
Sec.7.3.2, uncertainties of the data strongly depend on the temperature of the source, and x 2 from
calibration was large for high temperature data.
From my private communication with Dr. Dale Fixsen, a FIRAS Team expert, I learned
that the calibration of the high frequencies required the detector to be in a state that was not realized
in actual observation. As a result, the errors, which account for this difference, are represented by
the "JCJ" errors.
To check if high temperatures have affected the quality of the Galactic-plane data, the
covariance of high-x 2 and x^of ~ 1 spectra are examined and compared at each frequency channel.
Figure 8.11 provides one example from each spectral type, where the plot for FIRAS Pixel 4170
(upper) represents the high-x 2 spectra, and the plot for FIRAS Pixel 359 represents the Xdof ~ 1
spectra. Shown here are the covariances between the channel at 2748 GHz and other channels in the
FIRAS experiment. In the P4170 plot, the overall error exponentially increases from the lowest to
the highest frequencies. This pattern is clearly not Gaussian. In addition, JCJ errors dominate all
other errors. This means that the calibration model expects large discrepancies between calibration
data and actual sky measurements. On the other hand, the P359 plot shows that overall errors
are symmetric around the central frequency (2748 GHz) channel, and the main contributor to the
overall errors is the detector noise. This pattern means that the variance of the data dominates over
uncertainties in the calibration model (represented by JCJ errors), an indication that the calibration
model is performing at its optimal state, and so the data are indeed trustworthy.
Despite all the above discussion, the Inner Galaxy emission is much more complicated than
the outer Galaxy and high latitude emission. Based on the overwhelming evidence, I have concluded
that the Galactic plane spectra cannot be used to accurately characterize dust emission.
89
FIRAS pix ff 4 1 7 0 , glon = 3 2 8 . 7 8 , glat =
-0.60
1.U
•
I
0.5 -
r
0.0
i
between 2748 GHz channel and others
detector + /S
PUP
PEP
scole factor = 29819.14-
0.5
cr
-
PTP
10
•
.
.
i
.
.
i
100
1000
Frequency (GHz)
FIRAS pix ff 0 3 5 9 , glon = 1 2 4 . 4 1 , glat
•
1.0
•
i
•
1.18
i
0.5
0.0
between 2748 GHz channel and others
detector + /?
PUP
PEP
scale factor == 845.32
0.5
•
.
.
i
1,
-
-
-
PTP
10
4\
T-
.
.
i
100
1000
Frequency (GHz)
Figure 8.11: FIRAS covariance between one channel and all other channels for spectra of Pixel
4170 and 359. Pixel 4170 is chosen as an example of high-Xj of spectra; it has a Xdof
=
?-51 when
fit by a two-component ai?2 = 2.0 model. Pixel 359 represents Xdoi ~ 1 spectra; it has a
X^of = 1-09 when fit by a two-component a\^ = 2.0 model.. The total covariance is in black; each
type of the FIRAS errors is in a different color. The scale factor is a multiplication factor that
normalizes the height of the covariance curve in each plot. Comparing these two plots, notice that
in the high=%2 (upper) plot, the overall errors heavily skew toward the highest few frequency
channels, so they are clearly not Gaussian. In addition, JCJ errors dominate all other errors. This
means that the calibration model expects large discrepancies between calibration data and actual
sky measurements. On the other hand, the %^of ~ 1 (lower) plot shows that overall errors are
symmetric around the central frequency, and the main contributor to the overall errors is the
detector noise. This means that variance of the data dominates over uncertainties in the
calibration model (represented by JCJ errors), and so the measurements are more trustworthy.
90
8.9
Summary
In this chapter I have analyzed Galactic-plane spectra that are not well fit by any of the
dust models examined in this thesis. Attempts to identify the cause of high x2 consist of removing
different sets of spectral points that are related to known contaminants to the full 214-channel
spectra, and then models are re-fit to the remaining spectra. None of these attempts was successful
in improving the fits. After I ruled out all possibilities of a localized cause for the high x 2 , 1 examined
the covariance matrices of the high-%2 spectra. To measure the Galactic-plane high temperature
sources, the FIRAS calibration model was stretched beyond its optimal performance. I conclude
that, for this reason, these spectra should not be used to characterize dust emission at the Galactic
plane.
91
Chapter 9
S u m m a r y and Discussion
9.1
Summary of Dust Models in This Thesis
In the last three chapters, I have described how I built models for interstellar dust thermal
emission by comparing model predictions with a set of 6063 214-channel 7° spectra in the frequency
range of 60 — 3000 GHz. In Chapters 5 and 6 I tested one-component dust models with an emissivity
spectral index a set to fixed values between 1.4 and 2.3. These models are fit to 6063 individual
spectra and the averages of subsets of these spectra as defined by longitudinal and latitudinal sky
regions. Analysis of these fits show that models fit the 7°-spectra well, but that the parameters don't
get adequate constraint at high latitudes. In the Galactic plane, the x 2 are too high. Adjusting to
variations in the signal-to-noise ratio of dust spectra, Chapter 6 also builds one-component fixed-a
models that vary in sky region area, while maintaining a minimum constraint on best-fit parameters.
Among all-sky fixed-a (1.4 < a < 2.6) models tested, the one with a = 1.7 is the best fit to the
data, with 6Tdust/Tdust
presents the Xdof
an(
^*
< 10%. Only 11.2% out of the 6063 fits have a Xdof >
ne
L13
- Section 6.5.2
best-fit parameters of the a = 1.7 fits in sky maps.
Chapter 7 presents results of fitting one-component free-a model, two- and three-component
fixed-a models to Galactic-plane spectra. The one-component free-a model is able to fit all but 167
Galactic plane spectra with a Xdof < 2. The two-component models tested include 136 fixed-ai
and fixed-a^ models where each a independently takes on values between 1.3 and 2.8. Despite the
large number of models tested, none of them can fit all Galactic-plane spectra with a Xdof
<
2-
81 spectra (or 1.3% of total sky) at Inner Galaxy still do not have a good dust model. Adding
an extra dust component to the model is shown not to improve the fit. Chapter 7 presents my
investigation into the cause for the poor Galactic-plane fits. After collecting evidence for why the
fits are poor, re-fitting parts of the full, 214-channel spectra to avoid known issues in the data
fails to improve the fit. Subsequently, covariance matrices of the high-x 2 and x do f ~ 1 spectra are
examined and compared, the highly skewed covariance for the high-x 2 spectra shows that the errors
92
are not Gaussian whichis a pre-requisite to apply the x2 test. Therefore, such data are considered
unsuitable to model dust emission. Another issue with using Galactic-plane spectra to model dust
emission is that this emissions is much stronger and much more complex than those from dust at
high latitudes. For Galactic plane spectra that are fit well using one of the two-component models,
the dust temperatures and spectral indices of the best-fit models are presented in Section 7.3. Both
temperatures are constrained to better than 10%, consistent with the high-latitude fits.
9.2
Comparison with Contemporary All-sky Dust Models
In the Introduction, I point out that no two existing all-sky dust models predict the same
number of dust components or the same range of dust temperature. This thesis again uses FIRAS
measurements to derive an all-sky dust model, and again arrives at a conclusion different from any
of the previous results. Therefore, to anyone who needs to use a dust model in the far-infrared or
millimeter wavelengths, this section compares predictions of those dust models with predictions of
the one-component a = 1.7 (£TdUst/7dust ^ 10%) model.
9.2.1
Reach et al. Model
The Reach et al. (1995b) model is a three-component model with all three emissivity
spectral indices fixed at 2.0. It was obtained from spectral fitting of average FIRAS dust spectra
at 120 longitudinal regions on the Galactic plane (\b] > 10°) and 26 regions at high latitudes.
The three components are: a warm component at 16 — 21 K, a cold component at 4 — 7 K, and an
intermediate-temperature component at 10 — 14 K. The warm and cold components are derived from
two-component model fitting, and they are present everywhere in the sky. The third component
is inferred from poor fit between data and the two-component model and is confined to the Inner
Galaxy.
The procedures Reach et al. (1995b) used to deduce dust spectra are similar to those carried
out in this thesis. The difference is with the treatment of spectral contamination from emission lines,
synchrotron emission and free-free emission. While Reach removed channels that are adjacent to
major emission lines or below 150 GHz (to avoid contamination from synchrotron and free-free
emission), here intensities of [C II] and [N II] lines are fit out at the same time as a dust model is fit
to the data, and the data used in this thesis include FIRAS channels at frequency below 150 GHz
and WMAP and DIRBE. The reason why the remaining emission lines, such as CI and CO, have
limited impact on dust model fitting is because these lines are weak, except at Inner Galaxy, and
removing channels in the vicinity of emission lines makes little change to the best-fit dust models,
as shown in Section 8.7. In other words, in regions where line emission is strong, they are not the
dominant contributor to poor fits between model and data; everywhere outside the Inner Galaxy,
the lines are weak, so they have no effect on best-fit dust models, and fits already have a x\Qi ~ 1-
93
For possible synchrotron and free-free contamination of low-frequency FIRAS data, I
showed in Section 8.2 that an extrapolation of synchrotron contribution from 22 GHz, as measured by the WMAP, with a spectral index of —3, to 150 GHz gives a minute fraction of the dust
intensity. Since free-free emission also has a steep spectral index (—2.15), the absolute magnitude
of free-free emission at FIRAS low frequencies is small and could not have changed the scale of the
spectra at those frequencies substantially. For these reasons, synchrotron and free-free emission,
though they are not removed from data that were used to constrain dust models in this thesis, could
not have changed results of the fits significantly.
Compared with the Reach model, the a = 1.7 model (short for one-component a = 1.7
model with <5Tdust/Tdust < 10%) has made a few improvements. For one, throughout the sky, the
a = 1.7 model uses less spectral averaging. Secondly, the a — 1.7 model is a better fit to the data in
regions outside the Inner Galaxy, where it has a good \2 • Evidence supporting this point is shown
in Figure 9.1, which includes one example of Galactic-plane fits and two examples of high-latitude
fit by the Reach model and the a = 1.7 model. In the Galactic-plane fit, both models trace out
spectral points together at frequency below 1200 GHz. At frequency above 2000 GHz, the a = 1.7
model continues to follow the spectral points, while the Reach model overshoots by ~ 100 MJy/sr.
In the high-latitude fits, the a = 1.7 model traces out the full spectrum while the Reach model has
a vertical offset from the data at all frequencies. In fact, all high-latitude models reported in Reach
et al. (1995b) require some, but not uniform, amount of vertical offsets in order to match up with
the data.
Some aspects of Reach model are similar to the findings here. For example, when twocomponent models are used to fit Galactic plane spectra, both studies show that two dust components
are not correlated in temperature, but are correlated in optical depth. The correlation is weak due
to large scatter, and for this reason the two dust components are not written as functions of each
other.
Similar to the finding in Reach et al. (1995b) that a two-component a = 2.0 model is a
poor fit to spectra located within 40° longitude of the Galactic center, Chapter 7 shows that no
two-component model or three-component ai,2,3 = 2.0 model can fit spectra at the Inner Galaxy
well. The reason, however, is not the lack of a third component in the dust model, as Reach suggests,
but limitation of the experiment in calibrating dust emission at the Inner Galaxy.
9.2.2
FDS Model # 8
The FDS Model # 8 has been widely used to remove thermal dust contamination in CMB
measurements. This model has two components: one with a vl
2 70
ture 9.4 K, and the other with a v -
67
emissivity and at mean tempera-
emissivity and at mean temperature 16.2 K.
To determine a best-fit model, FDS defines goodness-of-fit via a linear fit of FIRAS dust
spectra against model predictions at 71% of the high-latitude sky. Excluding channels near emission
94
Average spectrum at 300° < I < 303°, —3° < b < 3°
1 0 0 0 0 .00
1000 .00
100 .00
10 .00
=
E average of 2 o n e - c o m p . or
,:
1.7 models
TT mox = 2 0 . 9 3 ± 0 . 0 3
T T „ „,„ = 2 1 . 2 1 ± 0 . 0 3
7
± 2 14e - 0 6 T ™. =
T T „,„ = 7 . 3 8 e - 0 4
-^e-04±J^tJ^ml
T
1
:
^
-
£
i t i ^
1 two-comp. a,
J^*"
T
~i^<
°^\
0. 10 r
:
2
= 2.0 model
T,, Reoch = 2 2 . 4 8 ± 0 0 6
s^s^
1 0 0 ^r
0. 01
=
-,
i R«.ch = 5 . 1 5 e - 0 4
T2.R„*
T
=
7
-60
5.00e-06
±0.10
2, R«ch = 2 . 8 0 e - 0 3
100
±
±
1.40e-04
-,
':
-,
:
1000
frequency (GHz)
Averag e spectrum at 135° < I < 180°, 10° < b < 30°
1000. uuo
100. 0 0 0
:
overa qe of 125 o n e - c o m p
= 19.19 ± 0.58
TTm!n
r
T
=
a -
1 . 0 8 e - 0 5 ± 1 10e
Tm,n
E
1.7 models
T T m o > = 17.82 ± 0.09
06 T T m „ = 2 . 1 5 e - 0 4 ± 3 . 2 8 e - 0 6
1
10. 0 0 0
1. 0 0 0
r
i^f}
^K^»
0. 100
T
=
-
± °-
i . R~oh = 8 . 3 0 e - 0 6
±
\ ^
~1
1
2.00e-07
T2. R«ch = 5.70 ± 0 . 3 0
T
2. Reo» = 4 . 9 0 e - 0 5 ± 1 . 0 0 e - 0 5
0. 0 1 0
nm
0.001
two-%c©fttp. a 1 2 = 2.0 model
16 80
10
J
l _ ^ _
100
1000
frequency (GHz)
Average spectrum at b > 60°
100 .0000
10. .0000
average of 4 1 5 one t o m p
T, _.. = 22 71 i 1 1 3
1. 0 0 0 0
0. 1 0 0 0
0. 0 1 0 0
0. 0 0 1 0
0 0001
100
1000
frequency (GHz)
Figure 9.1: Predictions for average dust spectra in three sky regions by the two-component models
of Reach et al. (1995b) and the one-component a = 1.7 (<5TdUst/7dUst < 10%) model. The average
spectrum of each specified sky region is in orange. Parameters of the Reach two-component model
are printed in blue, and model predictions are represented by the blue curve. Parameters from the
a = 1.7 model associated with each sky region are printed in red, and the average of model
predictions are plotted in the same color. In the first plot, notice that the blue and red curves trace
out most of the spectral points together at frequency below 1200 GHz. At frequency above 2000
GHz, the red curve continues to follow spectral measurements, while the blue curve overshoots by
~ 100 MJy/sr. The last two plots show that the Reach models have a vertical offset from the data.
95
lines and overlaps of FIRAS high and low frequency bands, 123 linear-fit slopes (or correlation slopes)
are obtained. The minimum deviation between these 123 slopes and unity defines Model # 8 .
The temperature of the two dust components are determined by the sky structure of the
filtered DIRBE -/ioo/-?240 flux ratio map, and the cold component is set to emit a constant fraction
(3.8%) of the power emitted by the warm component.
While this correlation slope method is able to bypass issues with synchronizing absolute
calibrations of different data sets involved, and to retain highly resolved sky structures recorded in
the IRAS 100 fim map, it is not clear how much Model # 8 is affected by the "jump" between FIRAS
low- and high-band measurements since their models are evaluated against those spectra.
To compare how well Model # 8 and the a = 1.7 model fit individual dust spectra, Figure
9.2 presents examples of the corrected FIRAS dust spectra deduced in this thesis at different Galactic latitudes, and their corresponding predictions by Model # 8 and the one-component a = 1.7
(<5Tdust/Tdust < 10%) model. The first plot is an example where predictions of the two models trace
out the FIRAS dust spectra closely. This spectrum is measured at mid latitude. The second and
third spectra, measured from the Galactic plane and high latitude respectively, show that Model#8
over-predicts dust emission, while the a = 1.7 model continues to trace out most of the spectral
points closely. These examples suggest that Model #8's ability to model dust emission over the
sky is not uniform. An assessment is provided in Figure 9.3, where \\oi
*s calculated for Model # 8
using 210-channel FIRAS dust spectra and assuming 210 degrees of freedom. The results, in the
first map, show that Model # 8 fits spectra at mid latitudes well, but those at high latitudes and
surrounding the Galactic plane poorly. For comparison, x\af °f * n e
a =
1-^ mo de-l for 214-channel
DIRBE+FIRAS+WMAP spectra assuming 210 degrees of freedom are shown in the second map.
This map shows that the a = 1.7 model fits spectra of most of the sky well except those at the
Galactic plane. In producing these \ 2 maps, the amounts of spectral and model averaging for the
same line of sight on both maps are made to be the same. For example, at low latitudes, since
Model # 8 has a 6'1 resolution and FIRAS spectra are of 6° 7 each, Model # 8 predictions within the
area of a FIRAS pixel are averaged. Predictions of the a = 1.7 model, on the other hand, already
have a 6°7 resolution, so they can be compared directly with FIRAS spectra. At high latitudes, to
increase the signal-to-noise ratio of the dust spectra, FIRAS spectra and predictions of both models
surrounding each line of sight are averaged respectively before calculating the \2I conclude that the reason Model # 8 predictions do not match well with the FIRAS dust
spectra has to do with the correlation-slope method, which uses one number to represent a model's
goodness-of-fit at each frequency channel for the entire sky. FIRAS data have low signal-to-noise
ratios at high latitudes, so at each frequency only those predictions that are paired with high signalto-noise data (not at high latitudes) are constrained. In addition, the Galactic plane (29% of total
sky) is excluded from FDS model fitting.
96
FIRAS pix # 2 5 8 6 , glon = 162.49, glat =
-21.97
1000.000
a = 1.7
[ C l l ] intensity
5.33e-02
±
100.000 T = 1 8 . 3 5 ± 0 . 1 9
[ N i l ] intensity
3.11e-03
± 5.
f r
= 9.15e-05
± 3.16e-06
1.36e-02
N„.. = 1
10.000
1.000 \-X\o,. «-i 7 = 0.99
Xdof,/0S8
=
1-05
Average of FDS Model # 8 predict! »
in d i r e c t i o n s c o v e r e d by the s o m i
FIRAS pijjcel
0.100
0.010
= 2.70,
= 1.67,
T worm : 1 5 . 3 4 - 1 5 . 5 8
T ,: 8 . 8 9 - 9 . 0 6 K
0.001
100
1000
f r e q u e n c y (GHz)
FIRAS pix # 0 1 0 6 , glon = 7 7 . 7 8 , glat =
100
6.87
1000
f r e q u e n c y (GHz)
FIRAS pix § 1 0 6 9 , glon = 12.84, glat =
-58.45
1000.000^
: a = 1.7
[ C l l ] intensity = 3 . 7 8 e - 0 7
±
4.78e-03
1 0 0 . 0 0 0 ^-T = 2 3 . 0 5 ± 1.67
[ N i l ] intensity = 3 . 8 9 e - 1 6
±
2.06e-03
: T = 1.88e-06 ± 4.28e-07
N„„ = 25
10.000
1.000
rX
2
dof, - 1 7 =
0.88
X dof, FDS8 ** 1-31
0.100
0010
0.001
100
1000
f r e q u e n c y (GHz)
Figure 9 2: Three dust spectra (orange) and their corresponding model predictions by FDS Model
# 8 (green) and the one-component a = 1.7 (<5TdUst/7dUst < 10%) model (red) The first plot shows
that both models closely follow the FIRAS dust spectra. Second and third plots show that
Model#8 over-predicts dust emission, while the a = 1.7 model continues to trace out most of the
spectral points.
97
X2dof of FDS Model #8 for FIRAS
0.60
X2do( of a = 1 . 7 (<5T/TS 10%) Model for DIRBE+FIRAS+WMAP
< 4f-^
;w- •„
-
•
*
^
"
-
-
*•>
s«*_
0.60
1.1
Figure 9.3: All-sky maps of xjjof of FDS Model # 8 and the one-component a =1.7
(<Wdust/Tdust < 10%) model in Galactic coordinate Mollweide projection with the Galactic center
at the center and longitude increasing to the left. xiLf or" Model # 8 is calculated for 210-channel
FIRAS dust spectra assuming 210 degrees of freedom. This map shows that Model # 8 fits spectra
at mid latitudes well, but those at high latitudes and surrounding the Galactic plane poorly. Xdof
of the a = 1.7 model is calculated for 214-channel DIRBE+FIRAS+WMAP spectra assuming 209
degrees of freedom. This map shows that the a = 1.7 model fits spectra of most of the sky well
except those at the Galactic plane. For each x 2 , both data and model within the same region are
averaged before being compared. For example, at low latitudes, since Model # 8 has a 6'1
resolution and FIRAS spectra are of 6°7, Model # 8 predictions within the area of a FIRAS pixel
are averaged. At high latitudes, to increase the signal-to-noise ratio of dust spectra, spectra and
models surrounding each central line of sight are averaged respectively.
98
Chapter 10
Conclusion
In this thesis I have examined models of interstellar dust thermal emission at millimeter
and microwave wavelengths. Starting with deducing a new and improved version of the dust spectra
from the FIRAS calibrated measurements and combining them with measurements taken by DIRBE
and WMAP (Chapter 3), dust models having one to three dust components are fit to some or all of
the 6063 214-channel spectra at fixed or variable sky-region areas. (Chapters 5 — 7).
The one-component a = 1.7 (STdust/Tdust < 10%) model is shown to be the best of all dust
models tested. It is a good fit to the dust spectra over 87.6% of the sky with a x^ of < 1.13 at 210
degrees of freedom. Within this sky region, 16.4 K < T d u s t < 25.1 K, and 1.3 x l O - 6 < r < 5.1 x l O - 4 .
The uncertainties of the dust temperature are < 10% while uncertainties of the optical depth are <
63%. The model has 7° angular resolution and uses spectral averaging near the Galactic poles.
The one-component a = 1.7 (STdust/Tdust < 10%) model is compared with existing dust
models in the far-infrared over 87.6% of the sky where the model is valid. In comparison with the
two-component a i 2 = 2.0 dust model by Reach et al. (1995b), predictions of the one-component
a = 1.7 model are shown to be better able to trace out dust spectral variation over the entire
FIRAS frequency coverage. The a = 1.7 model also uses less spectral averaging than the Reach
model because it is not limited by pre-defined longitudinal and latitudinal sky regions. The other
model considered is FDS Model # 8 , which is shown not to always accurately predict dust emission
measured at high latitudes or near the Galactic plane. In comparison, the a = 1.7 model, though
requiring greater sky-region averaging, can reliably model dust spectral variations over 88% of the
full sky, some regions of which have not been modeled correctly by either of these two models.
Currently, uncertainties of the best-fit parameters are limited by FIRAS angular resolution
and noise, and the angular resolution of the model inherits that of the FIRAS. When data of
better quality become available, such as from the Planck mission, this one-component a = 1.7
(<Wdust/7dust < 10%) model can be used as a consistency check for future dust models.
99
Appendix A
One-component Dust Models for
7° Sky Regions
The following are sky maps of x 2 , best-fit values and uncertainties of T^ust, T, [C II] and
[N II] intensities, and their correlations from fitting a one-component dust model (at different fixed
values of a) to the spectrum at each pixel where FIRAS data are available.
Comparisons of best-fit models across different values of a are shown in Figures 5.9 and
5.10.
100
A.l
a = 1.4 M o d e l
A one_co m p , a
1.4
67.8
Figure A.l: x^ of of the best-fit one-component a = 1.4 models for individual pixel spectra. These
maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map
represents the Galactic center. The upper and lower ends of the minor axis are +90° and —90°
latitudes respectively, and the left and right ends of the major axis represent +180° and —180°
longitudes respectively.
101
one_comp, a - 1 4
one_comp, a
Log(l O e - 0 6 )
<5Tone_comp,
<5T one_comp, a - 1 4
a = 1 4
o
to
Log(OT)
Figure A.2: Best-fit Tdust, T and their uncertainties from fitting one-component a = 1.4 model to individual pixel spectra. These maps are
Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The upper and lower ends of
the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent +180° and —180° longitudes
respectively.
[CII] Intensity,one_comp,
a = 1 4
Log(1.0e+00)
Log(1.0e + 00)
<5[CII] Intensity one _ comp]
[Nil] Intensity,one.comp,
a = 14
<5[NII] lntensity o n e _ c o m p i
a = 1.4
Log(1.0e+00)
a = 14
o
CO
0.0050
0.050
0.0020
0.020
Figure A.3: Best-fit [C II] intensity, [N II] intensity and their uncertainties from fitting one-component a = 1.4 model to individual pixel
spectra. These maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The
upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent
+180° and —180° longitudes respectively.
[C1I] T one_comp a = 14
O
Figure A.4: Correlations among ld u s t , r, [C II] intensity and [N II] intensity of the best-fit one-component a = 1.4 models for individual pixel
spectra. These maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The
upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent
+180° and —180° longitudes respectively.
A.2
a = 1.5 Model
A on e_comp, a
1.5
Figure A.5: xlof of the best-fit one-component a = 1.5 models for individual pixel spectra. These
maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map
represents the Galactic center. The upper and lower ends of the minor axis are +90° and —90°
latitudes respectively, and the left and right ends of the major axis represent +180° and —180°
longitudes respectively.
105
1
' one_comp, a = 1 5
one_comp, a - 1 5
si.one_comp,
<5Tone_comp,
a
a = 15
O
Log(0 1
Figure A.6: Best-fit Tdust, T and their uncertainties from fitting one-component a = 1.5 model to individual pixel spectra. These maps are
MoUweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The upper and lower ends of
the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent +180° and —180° longitudes
respectively.
[CII] Intensity one_comp,
a = 1.5
Log(1.0e-09)
Log(1.0e-0B
(5[CII] lntensity o n e _ c o m p Q
[Nil] Intensity one_comp,
= 15
<5[N!l] lntensity o n e _ c o m p ]
a = 1.5
LogXT70e-02)
a =
,.5
o
0.0050
0.050
0.0020
0.020
Figure A.7: Best-fit [C II] intensity, [N II] intensity and their uncertainties from fitting one-component a = 1.5 model to individual pixel
spectra. These maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The
upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent
+180° and —180° longitudes respectively.
[CII] T one_comp a - 1 5
~T T one_comp
r
[CII] [Nil] one_comp a - 1 5
O
00
Figure A.8: Correlations among Tdust, T, [C II] intensity and [N II] intensity of the best-fit one-component a = 1.5 models for individual pixel
spectra. These maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The
upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent
+180° and —180° longitudes respectively.
A.3
a = 1.6 Model
A one_comp, a =
1.6
Figure A.9: Xdof °f the best-fit one-component a — 1.6 models for individual pixel spectra. These
maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map
represents the Galactic center. The upper and lower ends of the minor axis are +90° and —90°
latitudes respectively, and the left and right ends of the major axis represent +180° and —180°
longitudes respectively.
109
L
one comp, a - 1 6
one_comp, a
Log(l Oe-06)
<5Tone_comp,
Log(Ol)
<5Tone^comp,
a = 16
a = 16
Log(K)
Figure A.10: Best-fit Tdust, T and their uncertainties from fitting one-component a = 1.6 model to individual pixel spectra. These maps are
Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The upper and lower ends of
the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent +180° and —180° longitudes
respectively.
[CII] Intensity one _,.comp,
a - 1.1
Log(l.Oe-OB)
<5[CII] lntensity o n e _ c o m p i a
0.0050
Log(l.Oe-Ol)
=
,.6
0.050
[Nil] Intensity,one_comp,
Log( 1 . 0 ^ ^ 9 )
a = 1 6
Log(1.0e-02)
d[NH] lntensity o n e _ c o m f ) ] a =
0.0020
16
0.020
Figure A.ll: Best-fit [C II] intensity, [N II] intensity and their uncertainties from fitting one-component a = 1.6 model to individual pixel
spectra. These maps are MoUweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The
upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent
+ 180° and —180° longitudes respectively.
1
T [Nil] one_comp a = 1 6
1
[Nil] T one_comp a - 1 6
Figure A.12: Correlations among Tdust, ^ [C II] intensity and [N II] intensity of the best-fit one-component a — 1.6 models for individual
pixel spectra. These maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic
center. The upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis
represent +180° and —180° longitudes respectively.
A.4
a= 1.7 Model
A one_comp, a = 1.7
Figure A.13: x^ of of the best-fit one-component a = 1.7 models for individual pixel spectra. These
maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map
represents the Galactic center. The upper and lower ends of the minor axis are +90° and —90°
latitudes respectively, and the left and right ends of the major axis represent +180° and —180°
longitudes respectively.
113
one_comp, a -
one_comp, a - 1 7
17
Log(l Oe-06)
<5Tone_comp,
<5Tone_comp,
a = 17
a = 17
^:1*^'
Log(OT)
Figure A.14: Best-fit Td ust ,
Log(K)
T an
Log(10 0)
Log(l Oe-06)
d their uncertainties from fitting one-component a = 1.7 model to individual pixel spectra. These maps are
MoUweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The upper and lower ends of
the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent +180° and —180° longitudes
respectively.
[CII] Intensity,one_comp,
a -
1.7
0.0050
a = 1.7
Log( 1 . 0 ^ ) 9 )
Log(1.0e-08)
<5[Cll] lntensity one _ comp]
[Nil] Intensity one_comp,
a = 17
0.050
<5[Nll] lntensity o n e _ c o m p i
0.0020
a = 17
0.020
Figure A.15: Best-fit [C II] intensity, [N II] intensity and their uncertainties from fitting one-component a = 1.7 model to individual pixel
spectra. These maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The
upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent
+180° and —180° longitudes respectively.
r
r
T [Nil]
rf^" y iir^iiiMiy
"[Nil], T one_comp, a
Figure A.16: Correlations among Tdust, T, [C II] intensity and [N II] intensity of the best-fit one-component a = 1.7 models for individual
pixel spectra. These maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic
center. The upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis
represent +180° and —180° longitudes respectively.
A.5
a = 1.8 Model
A one_comp, a =
1.8
Figure A.17: Xdof of the best-fit one-component a = 1.8 models for individual pixel spectra. These
maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map
represents the Galactic center. The upper and lower ends of the minor axis are +90° and —90°
latitudes respectively, and the left and right ends of the major axis represent +180° and —180°
longitudes respectively.
117
one_comp, a = 1 i
"• one_comp, a
Log(1.0e-06)
<5Tone_comp,
MM
<JT,one_comp, a = 1 ;
a
•
* *
•
•< *• *
Log
(oTT
Log
W
Log(10.0)
7^^^^^^^^^PHPF-
Log(1.0e-06)
Log(l Oe-05)
Figure A.18: Best-fit Tdustj T and their uncertainties from fitting one-component a = 1.8 model to individual pixel spectra. These maps are
Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The upper and lower ends of
the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent +180° and —180° longitudes
respectively.
[CII] Intensity one_comp,
Log(l.O^Ts)
<5[Cll] !ntensity o n e _ c o m P ]
a = 1.8
Log(1.0e-01
a =
,8
[Nil] Intensity,one_comp,
Log(1.0e-09)
<5[NII] lntensity o n e _ c o m p ]
a = 1.8
Log(1.0e-02)
a = 18
t «.*;
0.0050
0.050
0.0020
0.020
Figure A.19: Best-fit [C II] intensity, [N II] intensity and their uncertainties from fitting one-component a = 1.8 model to individual pixel
spectra. These maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The
upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent
+ 180° and —180° longitudes respectively.
1
,
r
,*
O'
,
r
T , [Nil]
[CII], T one_comp, a = 1 S
•
^
»
^
'','^^^*^
V
i
-V'-. £
[Clij, [Nil] one_comp, a = 1 8
o
i r T #:
Figure A.20: Correlations among Td ust , r, [C II] intensity and [N II] intensity of the best-fit one-component a = 1.8 models for individual
pixel spectra. These maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic
center. The upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis
represent +180° and —180° longitudes respectively.
A.6
a = 1.9 M o d e l
A one_comp, a =
1.9
Figure A.21: Xdof of the best-fit one-component a = 1.9 models for individual pixel spectra. These
maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map
represents the Galactic center. The upper and lower ends of the minor axis are +90° and —90°
latitudes respectively, and the left and right ends of the major axis represent +180° and —180°
longitudes respectively.
121
' one_comp, a = 1 9
one_comp, a = 1.9
<5T.one_comp,
<5T
a = 1.9
one_comp, a = 1.9
to
to
Log(0.1
Figure A.22: Best-fit Tdust>
T an
d their uncertainties from fitting one-component a = 1.9 model to individual pixel spectra. These maps are
Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The upper and lower ends of
the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent +180° and —180° longitudes
respectively.
[CII] Intensity,one_comp,
a = 1.9
Log(l.Oe-OB)
(5[CII] l n t e n s i t y o n e _ c o m p a =
[Nil] Intensity one_comp,
a = 1.9
Log(1.0e-09)
19
(5[NII] l n t e n s i t y o n e _ c o m p a = ,,
to
0.0050
0.050
0.0020
0.080
Figure A.23: Best-fit [C II] intensity, [N II] intensity and their uncertainties from fitting one-component a = 1.9 model to individual pixel
spectra. These maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The
upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent
+180° and —180° longitudes respectively.
r
T [Nil]
to
Figure A.24: Correlations among Tdust> T, [C II] intensity and [N II] intensity of the best-fit one-component a = 1.9 models for individual
pixel spectra. These maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic
center. The upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis
represent +180° and —180° longitudes respectively.
A.7
a = 2.1 M o d e l
A one_comp, a = 2.
Figure A.25: x^ of of the best-fit one-component a = 2.1 models for individual pixel spectra. These
maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map
represents the Galactic center. The upper and lower ends of the minor axis are +90° and —90°
latitudes respectively, and the left and right ends of the major axis represent +180° and —180°
longitudes respectively.
125
1
one_comp, a = 2 1
one_comp, a = 2 1
to
C5
Figure A.26: Best-fit X"dustj r and their uncertainties from fitting one-component a = 2.1 model to individual pixel spectra. These maps are
MoUweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The upper and lower ends of
the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent +180° and —180° longitudes
respectively.
[CII] Intensity,one_comp,
a = 2.1
[Nil] Intensityone_comp,
a —2
<5[NII] lntensity o n ie__comp,
a = 2.1
Log(l.Oe-OB)
<5[CII] lntensity o n e _ c o m p i a
= 2.1
to
V>
. ^ ^
0.0050
*
%.
U :
0.050
0.0020
0.020
Figure A.27: Best-fit [C II] intensity, [N II] intensity and their uncertainties from fitting one-component a = 2.1 model to individual pixel
spectra. These maps are MoUweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The
upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent
+ 180° and —180° longitudes respectively.
r
T
T one_comp a - 2
to
oo
Figure A.28: Correlations among Tdust, T, [C II] intensity and [N II] intensity of the best-fit one-component a = 2.1 models for individual
pixel spectra. These maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic
center. The upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis
represent +180° and —180° longitudes respectively.
A.8
a = 2.2 M o d e l
A one_comp, a = 2.2
Figure A.29: x^ of of the best-fit one-component a = 2.2 models for individual pixel spectra. These
maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map
represents the Galactic center. The upper and lower ends of the minor axis are +90° and —90°
latitudes respectively, and the left and right ends of the major axis represent +180° and —180°
longitudes respectively.
129
1
one_comp, a
<5To n e _ c o m p ,
one_comp, a = 2 2
= 2 2
<5To n e _ c o m p ,
a = 2.2
a = 2 2
CO
O
Figure A.30: Best-fit TdUst> T
an
d their uncertainties from fitting one-component a = 2.2 model to individual pixel spectra. These maps are
MoUweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The upper and lower ends of
the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent +180° and —180° longitudes
respectively.
[CII] Intensity,one_eomp,
a = 2.2
Log(1.0e-08
6[C\\]
0.0050
[Nil] Intensity,one_comp,
Log(1.0e-09)
lntensityone_compi
a = 2.2
0.050
<5[NII] I n t e n s i t y o n e _ c o m p ,
0.0020
a = 2.2
Log(1.0e-02)
0 = 22
0.020
Figure A.31: Best-fit [C II] intensity, [N II] intensity and their uncertainties from fitting one-component a = 2.2 model to individual pixel
spectra. These maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The
upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent
+180° and —180° longitudes respectively.
r
T [Nil] one.comp a = 22
[CII] [Nil] one eomp a - 2 2
CO
to
Figure A.32: Correlations among Tdust, T, [C II] intensity and [N II] intensity of the best-fit one-component a — 2.2 models for individual
pixel spectra. These maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic
center. The upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis
represent +180° and —180° longitudes respectively.
A.9
a = 2.3 M o d e l
A on e_comp, a
2.3
24.6
Figure A.33: Xdof °f
tne
best-fit one-component a = 2.3 models for individual pixel spectra. These
maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map
represents the Galactic center. The upper and lower ends of the minor axis are +90° and —90°
latitudes respectively, and the left and right ends of the major axis represent +180° and —180°
longitudes respectively.
133
one_comp, a = 2.3
one_comp, a = 2.3
Log(1.0e-06)
<5Tone_comp,
a = 2.3
CO
Log( 1.0^^6)
Figure A.34: Best-fit Td ust , r and their uncertainties from fitting one-component a = 2.3 model to individual pixel spectra. These maps are
Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The upper and lower ends of
the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent +180° and —180° longitudes
respectively.
[CII] Intensity one_comp,
a = 2.3
Log( 1.0^^8)
[Nil] Intensity,one„comp,
Log(1.0e-09)
<5[Cli] lntensity o n
<5[NII] lntensity o n e _ c o m P i a
a = 2.3
Log(1.0e-02)
=
23
co
0.0050
0.050
0.0020
0.020
Figure A.35: Best-fit [C II] intensity, [N II] intensity and their uncertainties from fitting one-component a = 2.3 model to individual pixel
spectra. These maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic center. The
upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis represent
+ 180° and —180° longitudes respectively.
CO
Figure A.36: Correlations among Tdust, T, [C II] intensity and [N II] intensity of the best-fit one-component a = 2.3 models for individual
pixel spectra. These maps are Mollweide projections of the Galaxy in galactic coordinates. Center of each map represents the Galactic
center. The upper and lower ends of the minor axis are +90° and —90° latitudes respectively, and the left and right ends of the major axis
represent +180° and —180° longitudes respectively.
Appendix B
One-component Fixed-a Models
for Averaged Sky Regions
137
X2, a = 2.0, (5T/T S 20%
T, a = 2.0, <5T/T S 20%
T, a = 2.0, <5T/T S 20%
Log(2.0e-03)
Log(2.0e-06)
Figure B.l: All-sky maps of Xdof ^dust
an
d
T
of the best-fit one-component a = 2.0 model that
satisfies (5Tdust/TdUst < 20%, in Galactic coordinate Mollweide projection with the Galactic center
at the center and longitude increasing to the left.
138
# of s p e c t r a , a = 2 0, <5T/T g 20%
Log(l)
Log(361)
<5T/T, a = 2 0, (5T/T S 20%
Log(0 0)
Log(%)
<5T/T, a
Log(0)
= 2 0, 5T/T
Log(%)
Log(20 0)
g
20%
Log(120)
Figure B.2: All-sky maps of model resolution, <5Tdust/7dust and ST/T of the best-fit one-component
a = 2.0 model t h a t satisfies 5TdUSt/TdUst
< 20%, in Galactic coordinate Mollweide projection with
the Galactic center at t h e center and longitude increasing to the left.
139
X2, a = 2.0, <5T/T S 10%
T, a = 2.0, <5T/T S 10%
T, a = 2.0, <5T/T S 10%
Log(S Oe-06)
Log(2.0e-03)
Figure B.3: All-sky maps of x\0i> ^dust and r of the best-fit one-component a = 2.0 model that
satisfies 5XdUst/Tdust < 10%, in Galactic coordinate Mollweide projection with the Galactic center
at the center and longitude increasing to the left.
140
# of s p e c t r a , a = 2.0, <5T/T ^ 10%
Log(l)
Log(361)
<5T/T, a = 2.0, <5T/T g 10%
Log(O.O)
Log(%)
Log(lO.O)
<5T/T, a = 2.0, <5T/T ^ 10%
Log(0.3)
Log(%)
Log(70.0)
Figure B.4: All-sky maps of model resolution, <57d us t/7d ust and 5T/T of the best-fit one-component
a = 2.0 model t h a t satisfies Wdust/Tdust < 10%, in Galactic coordinate Mollweide projection with
the Galactic center at the center and longitude increasing to the left.
141
X2, a = 2.0, <5T/T S 5%
23 7
T, a = 2.0, (5T/T S 5%
5.0
25.0
T,
a = 2.0, 5T/T S 5%
Log(2.0e-06)
Figure B.5: All-sky maps of Xdof' ^dust
Log(2.0e-03)
an
d T of the best-fit one-component a = 2.0 model that
satisfies £TdUst/^dust < 5%, in Galactic coordinate Mollweide projection with the Galactic center at
the center and longitude increasing to the left.
142
# of s p e c t r a , a = 2.0, 5T/T ^ 5%
Log(l)
Log(361)
5T/T, a = 2.0, <5T/T ^ 5%
Log(1.0e-02)
ST/T,
Log(0.2)
Log(%)
Log(5.0e+00)
a = 2.0, <5T/T g 5%
Log(%)
Log(34 4)
Figure B.6: All-sky maps of model resolution, (JXdust/Tdust and ST/T of the best-fit one-component
a = 2.0 model t h a t satisfies 5Tdust/Tdust < 5%, in Galactic coordinate Mollweide projection with
the Galactic center at the center and longitude increasing to the left.
143
X2, a = 2.0, <5T/T S 2.5%
T, a = 2.0, <5T/T S 2.5%
T,
a = 2.0, (5T/T S 2.5%
Log(2.0e-06)
Log(2.0e-03)
Figure B.7: All-sky maps of Xdof > ^dust and r of the best-fit one-component a = 2.0 model that
satisfies ST^ust/Tdust < 2.5%, in Galactic coordinate Mollweide projection with the Galactic center
at the center and longitude increasing to the left.
144
# of s p e c t r a , a = 2 0, <5T/T ^ 2 5%
Log(l)
Log(361)
(5T/T, a = 2 0, <5T/T § 2 5%
Log(l 0 e - 0 2 )
Log(%)
<5T/T, a
Log(0 2)
= 2 0, (5T/T
Log(%)
Figure B.8: All-sky maps of model resolution, ST^ust/7dust
a = 2.0 model t h a t satisfies ST^ust/Tdust
Log(2 5e + 00)
« 2 5%
Log(12 8)
and <5T/T of the best-fit one-component
< 2.5%, in Galactic coordinate Mollweide projection with
the Galactic center at the center and longitude increasing t o the left.
145
Appendix C
More Complex Dust Models for
Galactic-Plane Spectra
146
C.l
One-component Free-a Model
FIRAS pix # 0372, glon = 132.08, glat =
-1.76
10000.
a = 1.47 ± 0.03
T = 20.31 ± 0.27
T = 2.70e-04 ± 8.39e-06
1000.
100.
A
one-comp ~"
[CM] intensity = 0.29 ± 1.60e-02
[Nil] intensity = 0.01 ± 5.90e-03
* *-*
10.
1.
0.
0.
100
1000
frequency (GHz)
FIRAS pix # 0377, g on = 130.59 glat = 3.05
1 0000.
1000.
100.
=
=;
I
[Cll] intensity = 0.24 ± 1.42e-02
[Nil] intensity = 0.01 ± 5.47e-03
a = 1.59 ± 0.03
T = 18.86 ± 0.22
T
= 3.29e - 0 4 ± 9.38e-06
~ 1.22
:
i
-
x1one-comp
% '.
10.
ra T = -0.96
U r = 0.94
r T l = -0.99
1.
0.
1
r
0.
100
1000
frequency (GHz)
FIRAS pix # 2750, g on = 139.40, glat =
1.23
10000. 00
a = 1.57 ± 0.03
1000. 00 r T = 19.43 ± 0.21
T = 4.26e-04 ± .11e-05
100. 00
x 2 one-comp i' • lr\o
-'0
[CM] intensity = 0.38 ± 1.91e-02
[Nil] intensity = 0.03 ± 7.09e-03
10. 0 0 r
:
l
1|
jj&ff^
1. 00 r
y
r„, T = -0.95
r„ T = 0.94
rT T = -0.99
b~J^"^
0. 10
L
i
0. 01
1000
100
frequency (GHz)
Figure C.l: Sample 7° spectra and their best-fit one-component free-a models. In each plot, red
curve represents model predictions; orange x 's are dust spectra deduced from DIRBE, FIRAS and
WMAP measurements. Best-fit parameters and Xdof of the fits are printed on each plot. This set of
plots represents examples of good fits between model and data for spectra along the Galactic plane.
147
FIRAS pix # 4 3 4 9 , glon = 2.34, glat = - 0 . 7 9
10000.00 1
1000.00
100.00
=;
a = 2.10 ± 0.01
T = 21.03 ± 0.08
T = 2 . 2 4 e - 0 3 ± 1.89e-- 0 5
v2
= 1 7 89
E A one-comp
[Cll] intensity = 1.81 ± 4 . 6 .a e - 0 2 I
[Nil] intensity =
5 e - 0 2 -J
:
1
' •"-'•^
10.00
ra> T = - 0 . 9 8
ra T = 0.97
rT T = - 0 . 9 9
1.00 r
0.10
0.01
:
- K - ' l
-i
I
1
100
1000
frequency (GHz)
FIRAS pix # 4350, glon = 3.30, glat = 2.76
10000.
a = 2.16 ± 0.01
= 20.02 ± 0.10
: T = 1.45e - 0 3 ± 1 .56e-- 0 5
r x° on -comp ~= 7.31
:
1000.
r T
100.
[Cll] intensity = 1.11 ± 3 . 0 5 e - 0 2 ;
[Nil] intensity = r1 1 ' T I .nftnHfrnmWii^
0^"^
1
10.
1.
r
-|
ra. T = - 0 . 9 8
ra T = 0.97
rT T = - 0 . 9 9
/ - "
0.
!
i-"
0.
100
1000
frequency (GHz)
FIRAS pix § 5164, glon = 310.59, glat = - 3 . 0 5
10000.00
1
1000.00 r
100.00
:
=
a = 1.99 ± 0.01
T = 20.02 ± 0 . 1 1
T = 8 . 8 3 e - 0 4 ± 1 .05e- - 0 5
v*
= 4 50
A
one-comp
[Cll] intensity = 1.04 ± 2 . 2 2 e - 0 2
[Nil] intensity = 0.12 ± 6 . 2 3 e - 0 3
^g00l^^^^
:
^
'^^
^.*J*J
:
10.00
J&^
1.00 r
r„, T = - 0 . 9 8
ra T = 0.97
rT T = - 0 . 9 9
L^^
i
0.10
0.01
>
i
I
1
100
1000
frequency (GHz)
Figure C.2: Sample 7° spectra and their best-fit one-component free-a models. In each plot, red
curve represents model predictions; orange x 's are dust spectra deduced from DIRBE, FIRAS and
WMAP measurements. Best-fit parameters and x\0f ° f t n e f^s
are
P r m ted on each plot. This set
of plots represents examples of poor fits between model and data. These plots show that models
have difficulties in fitting FIRAS low-band measurements.
148
C.2
T w o - c o m p o n e n t 0:1,2 = 2.0 M o d e l
FIRAS ptx § 0 3 7 2 , glon = 132 0 8 , glat =
- 1 76
1000 00 =
frequency (GHz)
FIRAS pix # 0 3 7 7 , glon = 130 59
glat =
3 05
10000 oo
1000 00
100 00
5 37 ± 2 9 5 e - 0 1
T, = 2 96e-03 ± 6 53e-04
10 00
16 78 ± 8 4 8 e - 0 2
"l~~2«e-04 ± 7 8 0 e - 0 6
1 00
[CM] m t e n s i t y X ^ O 23 ± 1 3 4 e - 0 2
[Nil] ntensity = OS^I ± 5 47e 03
0 10
X2 „.„-„„,„ = 0 9 6
0 01
1000
frequency (GHz)
FIRAS pix § 2 7 5 0 , glon = 139 4 0 , glat =
1 23
10000 00
000
frequency (GHz)
Figure C.3: Examples of best-fit two-component a = 2.0 models and their corresponding spectra.
This group represents fits that have a Xdof ~ 1- In
eacn
plot, light and dark blue curves represent
model predictions for the two dust components; the red curve represents combined model of dust
and emission lines.
149
FIRAS pix § 5 2 1 8 , glon =
2 9 4 2 0 , glat
-1 5 6
10 00 r
frequency (GHz)
FIRAS pix ft 5 2 3 2 , glon =
10000 00
2 8 4 4 2 , glat
=
1 25
1
-
^
~=
r~
--
2 00
19 36 ± 5 5 2 e - 0 2
T, = 7 16e-04 ± 6 45e-06
T 2 = 6 03 ± l . 4 4 e - 0 1
T
2 = 4 81e-03 ± 4 39e-04
^ -^^
r
-"-^^J
:
-:
:
-=
68e-02 :
-^^ ^ p ^ '
[CM] intensity = 1 01 ± 1
[Nil] mtensity = 0 07 ± 5 26e-- 0 3
=".,< _, ^^^
0 10
1
x't™ . „ ,
= 2 00
\
1
0 01
frequency (GHz)
FIRAS p i x
§
5 2 3 5 , glon
= 2 8 2 2 3 , glat =
0000 00
-1
62
. .,
X
1 000 00 r
-j
100 00 r
-1
2 00
-^"^
10 00
^<p^-^
1 00
0 10
0 01
r
s^
T, = 18 46 ± 4 2 6 e - 0 2
X 4 ^ ^8 6 4 e - 0 4 ± 6 2 8 e - 0 6
T2 = 5 5 8 - < 1
T
2
33e-01
= 6 O9e-03x± 5 78e-04
[Cll] intensity = 0^83 ± 1 61 e--02
[Nil] intensity = 0 0 6 \ t 5 38e-03
^
X\w> - „ m p = 1 46
. ,
1
1
\
\
frequency (GHz)
Figure C.4: Examples of best-fit two-component a = 2.0 models and their corresponding spectra.
This group represents fits that have a Xdoi ~ 2. In each plot, light and dark blue curves represent
model predictions for the two dust components; the red curve represents combined model of dust
and emission lines.
150
FIRAS pix § 4 3 4 9 , glon =
2 3 4 , glat
-0 79
frequency (GHz)
FIRAS pix § 4 3 5 0 , glon
10000 00
=
3 3 0 , glat
~
=
2 76
,
E
1000 00 r
100 00
0-
^
—
:
^^
a =
0 10
^
><;
I
T, - 21 45 ± 2 3 1 e - 0 2
1 25e-03 ± 3 19e-06
10 00 r
1 00
2 00
^
S
T2 = 1 81 ± 8 9 0 e - 0 1
T
2 = 8 48e-02 ± 1 13e-01
^
~~^^^
[CM] intensity = 1 1 1 ± 2 87e-- 0 2
[Nil] ntensity = 0 15 ± 9 71 e - 0 3
X'two
-1
-=
;
:
=omp = 7 01
0 01
frequency (GHz)
FIRAS pix § 5 1 6 4 , glon -
3 1 0 5 9 , glat
-3 05
1 000 00
100 00 E-
requency (GHz)
Figure C.5: Examples of best-fit two-component a — 2.0 models and their corresponding spectra.
This group represents fits that have a x^ of > > 2. In each plot, light and dark blue curves represent
model predictions for the two dust components; the red curve represents combined model of dust
and emission lines.
151
C.3
T h r e e - c o m p o n e n t 0^2,3 = 2.0 M o d e l
FIRAS pix # 4 1 7 0 , glon = 3 2 8 7 8 , glot =
- 0 60
1000 00 =-
frequency (GHz)
FIRAS pix § 4 3 4 8 , glon -
1 0 5 , glat =
1 46
ty = 1 90
ty = 0 34
1 35e-02
frequency (GHz)
10000 00
requency (GHz)
Figure C.6: Examples of best-fit three-component 0:1,2,3 = 2.0 models and their corresponding
spectra. In each plot, both blue curves and the light green curve represent model predictions for
the three dust components; the red curve represents combined model of dust and emission lines.
The pixels shown here have a x^ of > 2 with their two-component a i j 2 = 2.0 fits.
152
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Vita
Zhuohan Liang immigrated to the United States and enrolled to study at New
Utrecht High School in Brooklyn, New York in 1997. In 2000, she graduated Valedictorian of her class and went on to receive a B.S. in Physics, a B.S. in Mathematics
(both with Highest Distinction), and a Minor in Philosophy from the University of
Rochester in 2005. At U of R, she was named a Barry M. Goldwater Scholar, given
the Excellence in Undergraduate Teaching Award, and awarded a tuition-free fifth
year to pursue a self-designed program of study on Human Thought and the Relation between Human Beings and Nature. She conducted research under the guidance
of Dr. Steve Valone on viscosity under shock loading conditions, and her presentation of the results was chosen for the Outstanding Undergraduate Oral Presentation
Award in Material Science at Los Alamos National Laboratory Symposium in the
summer of 2003. She enrolled in the Ph.D. program in the Department of Physics
and Astronomy at the Johns Hopkins University in 2005, and was supported by the
Clare Boothe Luce Graduate Fellowship for the first two years. Her thesis work was
conducted under the guidance of Professor Charles Bennett, on modeling interstellar
dust thermal emission at millimeter and microwave wavelengths.
166
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