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A measurement of the degree scale anisotropy in the cosmic microwave background radiation

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A MEASUREMENT OF THE DEGREE SCALE
ANISOTROPY IN THE COSMIC MICROWAVE
BACKGROUND RADIATION
CALVIN BARTH NETTERFIELD
A DISSERTATION
PRESENTED TO THE FACULTY
OF PRINCETON UNIVERSITY
IN CANDIDACY FOR THE DEGREE
OF DOCTOR OF PHILOSOPHY
RECOMMENDED FOR ACCEPTANCE
BY THE DEPARTMENT OF PHYSICS
JANUARY, 1995
UMI Number: 9519119
OMI Microform 9519119
Copyright 1995, by UMI Company. All rights reserved.
This microform edition is protected against unauthorized
copying under Title 17, United States Code.
UMI
300 North Zeeb Road
Ann Arbor, MI 48103
Copyright 1994 by Calvin Barth Netterfield. All rights reserved.
ii
ABSTRACT
A MEASUREMENT OF THE DEGREE SCALE ANISOTROPY IN
THE COSMIC MICROWAVE BACKGROUND RADIATION
Calvin Barth Netterfield
A measurement of the degree scale anisotropy in the Cosmic Microwave Background
Radiation (CMBR) is described. Two High Electron Mobility Transistor (HEMT)
amplifier-based radiometers provide frequency coverage from 26 GHz to 46 GHz in 6
frequency bands and two polarizations. The observations were made from Saskatoon,
Saskatchewan in January and February of 1994. There are two primary results:
1) A previous detection of the anisotropy (Wollack et al., 1993) is confirmed. The
spectral index of the anisotropy is consistent with that of the CMBR. Both free-free
emission and dust emission are ruled out as the sole source of the anisotropy. The
root-mean-squared amplitude of the fluctuations for the combined data set is 44^2 piK,
including a 14% calibration uncertainty. Expressed as the square root of the angular
power spectrum in a band of multipoles between te - 69!^, this is
ST, = ^(2^ + l)<|<f >/4;r = 42+ _\ 2 /jK.
2) The chopping scheme provides spatial sensitivity from 0. = 50 to (' = 150 in 5
multipole bands, allowing a determination of the shape of the spatial power spectrum.
With the fluctuations expressed as a power law in multipole moment, ST t = ST e e (fl/£ e ) m ,
a maximum likelihood analysis finds 5rte = A5+J6juK (not including the 14% calibration
uncertainty) and m - -0.4^, with f.e = 82. Thus, a mildly falling spectrum is favored
at intermediate angular scales.
iii
He is before all things, and in him, all things hold together. (Col 1:17)
iv
ACKNOWLEDGMENTS
First I would like to thank the BigPlate team. (Lord) Lyman Page has been an ideal
advisor, who has made learning and hard work both profitable and fun. It would be
difficult to overstate the effort and contribution made to the project by my fellow
graduate student Ed Wollack. The experimental skills of Norm Jarosik saved the day
on more than one occasion (even if he did make us eat at Rob Roy's). And of course,
the experience and keen insight provided by Dave Wilkinson kept us asking the right
questions.
The HEMT amplifiers, developed at NRAO by Marian Pospieszalski made this
experiment possible. The field sight and services in Saskatoon provided by George
Sofko and Mike McKibbin of the University of Saskatchewan and by Larry Snodgrass
of the SRC are greatly appreciated.
Much effort was applied to this project by many others, including Chris Barns (reaction
bar servo), Carrie Brown (weather station), Weihsueh Chiu (heater cards), Randi
Cohen (chopper), Peter Csatorday (heater card), Cathy Cukras (weather station), Peter
Kalmus (heater card), John Kulvicki (chopper), Wendy Lane (analysis), Young Lee (VF's), Naser Quershi (preamps and buckers), and Peter Wolanin (cables). Thanks to
Brian Crone for measuring the mirror.
The physics department staff contributes to any experiment done here. In particular,
the expertise of Laszlo Varga, Bill Dix, and Glenn Atkinson in the Princeton machine
shop have been invaluable. The oversight and advice from Ted Griffiths in the student
v
shop is greatly appreciated. Rosalie Staloff, Helen Porto, Helga Murray, and Kathy
Warren in the purchasing office always got the order through, even if I needed it
yesterday. Marion Fugill kept the Grav Group organized, and made sure that the FAXs
always got through. The efforts of John Washington in receiving, John Kalagen, in the
metal stockroom, and Claude Champaign in the stock room are appreciated.
There are many others who have contributed to my learning here at Princeton. My
General's Project advisor, Jeff Peterson first got me excited about the CMBR, and Greg
Tucker was instrumental in improving the quality of'bart spec'. Mark Dragovan, Ed
Groth, Jim Peebles, Bharat Ratra, and Dave Spergel have each provided me with
valuable ideas and insights which have made their mark on the project. Insight from
fellow (past and current) graduate students, Don Alvarez, Ken Ganga, Charles
Ferenbaugh, John Ruhl, Huges Sicotte, Chris Smith, and Suzanne Staggs has also
enriched my experience here.
I also give heart felt thanks to my family. Thanks to Joshua for sleeping through the
night at 1 month and being such a good baby. Most of all, I would like to thank my
wife, Sue. Thanks for patience with all of the late nights and extended absences. And
for lunch in the lab. And for breakfast on the roof. And for good advice and ideas.
And for your continuous encouragement.
vi
TABLE OF CONTENTS
Page
ABSTRACT
iii
ACKNOWLEDGMENTS
v
TABLE OF CONTENTS
vii
LIST OF ILLUSTRATIONS
x
LIST OF TABLES
xiii
Chapter 1 Introduction
1
Chapter 2 The Receiver
5
Receivers
5
The Q Band Receiver Design
9
Determination of the Radiometer Band-pass
14
Receiver Noise
22
Future Considerations
29
Summary
29
Chapter 3 The Telescope
31
Introduction
31
Overview
32
vii
Optics
33
The Horn
34
The Main Beam
39
Sidelobes
44
The Chopper
46
Electronics and data acquisition
50
Summary
53
Chapter 4
The Data
55
Introduction
55
Calibration
56
Chopping
65
Data Selection and Preparation
72
Data Quality
77
Summary
92
Chapter 5
The Analysis....
94
Introduction
94
Likelihood Analysis
95
Frequency and Amplitude Analysis
97
The Spatial Dependence of the Fluctuations
102
Summary
116
Chapter 6 Conclusions
118
viii
Appendix A The Synthesis Vectors
122
Appendix B
126
The Correlations
BIBLIOGRAPHY
128
ix
LIST OF ILLUSTRATIONS
Figure
Page
Figure 1. The CMBR Angular Spectrum as of February, 1994
3
Figure 2: The Q-band radiometer
9
Figure 3. Diode linearity
13
Figure 4. Setup for band-pass measurement
17
Figure 5. Gain compression in the band pass determination
18
Figure 6. The full-band band-pass for the Q radiometer B chain
20
Figure 7. The band-passes for the Q radiometer B chain
20
Figure 8. The B2 triplexor band-pass
21
Figure 9. The cold load used for lab calibration
23
Figure 10. A lab calibration of the Q radiometer
24
Figure 11. The NET as a function of load temperature
25
Figure 12. The noise spectrum of the Q radiometer
27
Figure 13. The telescope
32
Figure 14. Calculated illumination of the primary
34
Figure 15. The Q-band horn and mandrel
35
Figure 16. The horn, baffles, and vacuum window
38
Figure 17. Q-band horn beam pattern
39
x
Figure 18. The Q-band Channel B E-plane beam map
40
Figure 19. The beam determination for channel Q-Bl using Cas-A
41
Figure 20. Q Band Channel B H-plane beam map
42
Figure 21. The Ka beam patterns
44
Figure 22. The chopper drive arrangement
47
Figure 23. The chopper PID control
48
Figure 24. The chopper wave form
49
Figure 25. A block diagram of the telescope electronics
51
Figure 26. Unit step response of data system
52
Figure 27. Determination of the Cas-A flux density scale
58
Figure 28: Cas-A measured by channel QBI
60
Figure 29: Evening and morning chop paths around Cas-A
61
Figure 30. Angular dependence of sky noise
67
Figure 31. The Q and Ka synthesized antenna patterns
69
Figure 32. Contour plots ofQ synthesized antenna patterns
70
Figure 33. Distribution of spatial temperature gradient fluctuations
75
Figure 34. The Ka 3 point data at two cut levels
77
Figure 35. Noise spectra of demodulated data
78
Figure 36. Low frequency spectra of demodulated data
78
Figure 37. Ka band offsets vs. time
80
Figure 38. Q band offsets vs. time
81
xi
Figure 39. Ka 3 point distributions at different time scales
83
Figure 40. The Ka East and West data compared
87
Figure 41. The Q East and West data compared
88
Figure 42. The Ka and Q data compared
89
Figure 43. SK93 Ka and SK94 Q and Ka data
90
Figure 44. The rms and frequency spectral index analysis
99
Figure 45. The SK93 and SK930L window functions
104
Figure 46. The window functions
105
Figure 47. The amplitude and spatial spectral index likelihood analysis
109
Figure 48. Fluctuation amplitude for each synthesized chop
113
Figure 49. The CMBR anisotropy spectrum
120
xii
LIST OF TABLES
Table
Page
Table 1. Amplification and attenuation of receiver components
10
Table 2. Detector diode resistor and operating output values
13
Table 3. The error budget for centroid determination
19
Table 4. Effective bandwidths and centroids for the Q radiometer
21
Table 5. The NET of the Q radiometer with a 19K cold load
27
Table 6. Q Receiver NET at 7.8 Hz Obtained Looking at the Sky
28
Table 7. Summary of radiometer specifications
30
Table 8. Summary of Q-beam widths
43
Table 9. Telescope offsets for 3-point chop
50
Table 10. BigPlate telescope specifications
54
Table 11: The predicted antenna temperature of Cas-A
59
Table 12. Measured amplitudes of Cas-A
62
Table 13: Calibration coefficients determined from Cas-A
63
Table 14: Determination of atmospheric attenuation
65
Table 15. v values for the SK94 synthesis vectors
72
Table 16. Amount of data at different cut levels
74
xiii
Table 17. Offset drift and reduced x2 in a bin
81
Table 18. Reduced x2 for quadrature phase tests
83
Table 19. Signal reduced x2 and naive sky rms
91
Table 20. Offsets and sensitivities on the sky
92
Table 21. Summary of the spectral index and rms analysis
101
Table 22. Summary of the amplitude and spatial spectral index analysis
110
Table 23. Amplitude of the fluctuations for each synthesized chop
Ill
Table 24. The polarization of the CMBR
116
Table 25. The Ka chopper position and synthesis vectors
123
Table 26. The Q chopper position and synthesis vectors
125
Table 27. Correlation matrix for the Ka synthesized beams
125
Table 28. Ka 3 point temporal correlation matrix
126
Table 29. Ka 7 point temporal correlation matrix
126
Table 30. Q 3 point temporal correlation matrix
127
Table 31. Q 7 point temporal correlation matrix
127
xiv
Chapter 1
Introduction
One of the most powerful probes into the history of large scale structure formation in
the universe is the anisotropy in the Cosmic Microwave Background Radiation
(CMBR). A host of theories of structure formation have been put forth with differing
predictions for the angular power spectrum of the anisotropy.1 With precise
measurements of of the anisotropy, combined with other measurements the large scale
structure in the universe (the best of which is the distribution of galaxies2), one might
hope to understand the formation of large scale structure.
The characterization of the anisotropy in the CMBR has proved to be a very
challenging task. The signals are extremely small, on the order of 30 fxK, requiring
extremely sensitive detectors, very long integration times, or both. The extreme
faintness of the signals also makes the experiments susceptible to a host of possible
contaminants, including (but not limited to) telescope offset variations, objects in the
telescope sidelobes, atmospheric signals (if the experiment is done from the ground)
and foreground astrophysical sources. Consequently, very careful attention to all of
•M. White, D. Scott, and J. Silk, "Anisotropics in the Cosmic Microwave
Background", Annual Review of Astronomy and Astrophysics, 1994 32: 319.
2J.R.
Bond, "Cosmic Structure Formation & The Background Radiation", Proceedings
of the IUCAA Dedication Ceremonies held in Prune, India, 28-30 Dec. 1992.
1
Chapter 1. Introduction
2
these details must be paid in the design of a CMBR anisotropy experiment.
Specifically, the telescope must have low and stable offsets, low sidelobes, and either
operate outside the atmosphere, or be able to discriminate against atmospheric induced
signals. And the experiment must be able to measure the temperature spectral index of
the fluctuations to verify that they do indeed have the spectrum of the CMBR, and not
one of the possible foreground contaminants.
Figure 1 was presented by P.J.E. Peebles at the February 1994 Lake Louise Winter
Institute and summarizes the state of CMBR anisotropy measurements. Two theories
are also included as examples. The theory that peaks at f, » 200 is the 'standard'
adiabatic, Q.b = 0.05, CDM theory. The other is an isocurvature theory. The data
points are, starting with low i (large angular scale) COBE Quadrupole, COBE3,
FIRS4, Tenerife, SP91 (ACME-HEMT)5, SK936, PYTHON7, ARGO8, MAX9 (two
3G.
Smoot, et al., "Structure in the COBE Differential Microwave Radiometer First
Year Maps", The Astrophysical Journal , 396 (1992): L I .
4K.
Ganga, L. Page, E. Cheng, and S. Meyer, "The Amplitude and Spectral Index of
Large Scale Anisotropics in the Cosmic Microwave Background Radiation", The
Astrophysical Journal (1994) accepted.
5J.
Schuster et al., "A Degree Scale Measurement of Anisotropy of the Cosmic
Microwave Background Radiation", The Astrophysical Journal, 412 (1993): L47.
6E.J.
Wollack, et al. "A Measurement of the Anisotropy in the Cosmic Microwave
Background Radiation at Degree Angular Scales", The Astrophysical Journal, 419
(1993): L49.
7M.
Dragovan, et al. "Anisotropy in the Microwave Sky at Intermediate Angular
Scales", The Astrophysical Journal, All (1994): L I .
8P.
de Bernardis, et al. "Degree-Scale Observations of Cosmic Microwave Background
Anisotropics", The Astrophysical Journal, 422 (1994): L33.
Chapter 1. Introduction
3
points), MSAM 2pt (full, and with 'sources' removed), and MSAM 3pt10. From this
plot, it is clear that CMBR measurements should be capable of distinguishing between
theories. It is also apparent that at the time the plot was made, they had not.
4
3
o
0
1
100
10
1000
Multipole moment I
Figure 1. The CMBR Angular Spectrum as of February. 1994
Another important consideration for any experiment, especially ones as susceptible to
contamination as a CMBR anisotropy measurement, is that it be confirmed. At the
very least, this means that the measurements should be repeated by the same group.
9J.O.
Gundersen, et al. "A Degree Scale Anisotropy Measurement of the Cosmic
Microwave Background Near the Star Gamma Ursa Minor", The Astrophysical
Journal , 413 (1993): L I .
10E.S.
Cheng, et al. "A Measurement of the Medium-Scale Anisotropy in the Cosmic
Microwave Background Radiation", The Astrophysical Journal, 422 (1994): L37.
Chapter 1. Introduction
4
Wherever possible, they also need to be repeated and verified by another independent
group. At the time this plot was made, only the bottom four points (COBE, FIRS, and
Tenerife) had been confirmed.
Given this state of affairs, there were several requirements that needed to be met by the
SK94 experiment. The most important was that it be able to confirm the results of the
SK93 experiment. Because SK93 was only able to rule out free-free emission as the
sole source of the signal at the 90% confidence level, increased frequency coverage
was desired. Added to this was the requirement that the experiment be able to probe a
variety of angular scales, since it is not just the amplitude of the fluctuations that hold
information, but also their spectral variation. In fact, at around ('«= 70, where the SK93
experiment is sensitive, the amplitudes for many theories (including the ones presented
in Figure 1) are very similar. It is the spatial spectral index that differs.
The SK94 experiment, like SK93, used the BigPlate telescope, and observed from on
the ground in Saskatoon, Saskatchewan. The Ka radiometer from SK93 was again
used. For extended frequency coverage, a second radiometer that is sensitive to higher
frequencies was added. The chopper throw was modified to allow a greater range of
angular scales to be probed, and some deficiencies in the ground screen noted during
the SK93 run were remedied.
The experiment was successful in confirming the results of SK93 and tightening the
constraints on the spectral index of the fluctuations. It was also able to probe a variety
of angular scales, placing limits on the shape of the spectrum at intermediate angular
scales, and demonstrating the effectiveness of the observing strategy.
Chapter 2
The Receiver
Receivers
One of the largest challenges in measuring the anisotropy in the CMBR has been the
development of receivers of adequately low noise. It is the meeting of this difficult
challenge which has finally allowed the detection of the primordial fluctuations after
more than 20 years of effort.
The minimum signal detectable by a coherent receiver is given by the radiometer
equation1,
(1)
Here, ATmjn is the smallest signal detectable with a signal to noise of 1 (or,
equivalently, the noise of the radiometer), T
is the system temperature, At^is the RF
bandwidth, and ris the effective integration time. The sensitivity of a detector may
also be expressed in terms of its spectral noise density, S(v), which is what is measured
by a spectrum analyzer. The rms noise coming from a receiver is A 7^ = S{ v)yjAvaud,
]M.E.
Tiuri and A.V. Raisanen, "Radio Telescope Receivers" in Radio Astronomy by
Kraus, J.D. (Powell: Cygnus-Quasar Books, 1986), 7-8.
5
Chapter 2. The Receiver
6
where Aoaud is the audio bandwidth. Since Avaud = 1/(2r), ATmm = S(v)/<J2t. Thus,
(1) can be rewritten as
Typical units for S(v) are mK/VHz. Another common measure of a receiver's noise is
its Noise Equivalent Temperature {NET), which is 42 smaller than S(v), since
ATmls = NET/Vr• Typical units for this are mk • s'2. The radiometer equation in terms
of NET is
<2>
mT=T»-\y^
The NET is a more common measure of the noise of a receiver than the spectral noise
density, since integration time is normally the parameter of interest rather than the
audio bandwidth.
To make a sensitive detector, then, one wishes to have a low system temperature, and a
large //bandwidth. Recently, several different techniques have been utilized to achieve
this. At higher frequencies, (80 GHz and above) bolometric systems have been
successfully used, and at lower frequencies (46 GHz and lower), receivers employing
High Electron Mobility Transistor (HEMT) amplifiers are the state of the art.
The White Dish2 and PYTHON3 experiments have utilized bolometers which are
cooled to 70 mK and which are coupled to the sky using single mode W band
2G.S.
Tucker et al., "Cryogenic Bolometric Radiometer and Telescope", Review of
Scientific Instruments, 65,2 (February, 1994): 301.
3J.
Ruhl, "A Search for Anisotropy in the Cosmic Microwave Background Radiation"
(Ph.D. thesis, Princeton University, 1993)
Chapter 2. The Receiver
7
(~90 GHz) optics. While these devices have an intrinsic NET of less than 50 ^K*s1/2,
poor optical efficiency in the filter chain, and long bolometer time constants result in a
NET of 1.8 mK-s"2 at 5 Hz when looking at a 20 K sky4. Multimode optics have been
used in the FIRS5 and MAX6 experiments. For both of these, the primary CMBR data
channel is at 170 GHz. In the FIRS experiment, a sensitivity of 320 |aK-s1/2 to the
CMBR was achieved.
At frequencies below 50 GHz, receivers utilizing HEMT amplifiers have become the
standard. Ka band* amplifiers developed by Marian Pospieszalski at NRAO have been
used by the ACME-HEMT7 and SK938 experiments. These devices exhibit noise
temperatures of 30-50 K between 26 GHz and 36 GHz. The NET of these
radiometers is -1.1 mK-s1/2 when looking at a 15 K sky.
4Ruhl,
16.
5L.A.
Page, E.S. Cheng, and S. Meyer, "A Large-Scale Microwave Background
Anisotropy Measurement at Millimeter and Submillimeter Wavelengths", The
Astrophysical Journal, 3 5 5 (May 20, 1990): L I .
6M.J.
Devlin et al, "Measurements of the Anisotropy in the Cosmic Microwave
Background Radiation at 0.5° Angular Scales Near the Star Gamma Ursae Minoris",
The Astrophysical Journal, 430 (July 20, 1994): L I .
t Ka band is used here to represent the single mode bandwith of WR-28 waveguide,
which is 26 to 40 GHz
7T.
Gaier, et al, "A Degree-Scale Measurement of Anisotropy of the Cosmic
Background Radiation", The Astrophysical Journal, 398 (October 10, 1992): L I .
8E.J.
Wollack, "A Measurement of the Degree Scale Cosmic Background Radiation
Anisotropy at 27.5, 30.5, and 33.5 GHz" (Ph.D. thesis, Princeton University, 1994)
Chapter 2. The Receiver
8
To date, bolometer systems have achieved the lowest noise while observing the sky,
but these systems require sophisticated cryogenics to cool the detectors well below
1 K. With the HEMT systems, on the other hand, the amplifiers need only be cooled to
20 K or below to perform well, which greatly simplifies the design of the system. With
the ACME-HEMT experiment, the amplifiers were cooled using a simple 4He dewar,
while with the SK93-Ka experiment, the devices were cooled using a closed cycle
mechanical refrigerator. It is this simplicity which is one of the greatest advantages of
HEMT amplifier based radiometers.
As mentioned in the Introduction, it is important when making measurements of the
CMBR to cover a range of frequencies in order to determine whether the signals that
are observed have the spectrum of the CMBR. While the SK93-Ka experiment did
divide its band-pass into three bins, it was only able rule out free-free emission as the
sole source of the signal at a 90% confidence level9. The primary reason for the
development of a new receiver is to improve this limit. Since the contamination from
background sources decreases with increased frequency, and since HEMT amplifiers
were not yet available at frequencies above the 60 GHz oxygen line, it was decided to
build the receiver in Q band (36-46 GHz). Thus, for the 1994 Saskatoon observing
trip, two receivers are used, one in Ka band, and one in Q band. The Ka band
radiometer is the same as was used in the SK93 observing trip, and is discussed
elsewhere.10
9Wollack,
et al.
10Wollack,
Chapter 2.
9
Chapter 2. The Receiver
Corrugated Feecj
15K stage
B Channel
(Horizontal)
HEMT amp!
~31 dB
OO
linn 111
Preamps
OMT
A Channel
1(Vertical)
Vacuum Port
f
Diode
Detectors
Bandpass
Triplexor
Level set
Attenuator
(~6dB)
36-46 GHz
Amplifier
(~52 dB)
isolator
(Huges)
B
A Channel
Channel
301K receiver
OO
IIIIII
M
Figure 2: The Q-band radiometer.
The Q Band Receiver Design
The Q band radiometer used in the SK94 experiment (shown in Figure 2) is a HEMT
based receiver with three frequency bands between 36 GHz and 46 GHz and two
Chapter 2. The Receiver
10
polarizations (The electric field for A is vertical and for B is horizontal) for a total of six
channels. The front end, comprised of a custom built corrugated feed, an orthomode
transducer (OMT) and a pair of amplifiers, is held at 15 K by a mechanical refrigerator.
The signal then enters the room temperature receiver where it is further amplified, split
into frequency bands, and square law detected with diodes. Table 1 summarizes the
amplification and attenuation levels of the various components.
Device
Gain or level
Front end system noiset
-86 dBm
1st stage Amplifiers
+31 dB
Waveguide and Isolators
~-l dB
2nd stage Amplifiers
+52 dB
Level set Attenuator
-6.5 dB
Triplexor
-3.5 dB
Level into diode detectors
-17 dBm
Table 1. Amplification and attenuation of receiver components.
At the heart of the radiometer is a pair of HEMT amplifiers developed by the National
Radio Astronomy Observatory (NRAO) in Charlottesville, Virginia. They provide
-30 dB of gain, and operate well between 36 GHz and 46 GHz. Two different types
are used. The A channel HEMT amplifier (NRAO SN-B33) utilizes Gallium Arsenide
transistors in five stages and has a noise temperature of 30-40 K. In the B channel
amplifier (NRAO SN-B28), the first stage transistor is replaced with an Indium
Phosphide transistor and provides a very impressive noise temperature of less than
10 K. While improved noise temperature may be gained by cooling the devices further
t Assuming 7^=5OK and a 3.5GHz Bandwidth. PdBm= 101og(f/1.0 mW)
Chapter 2. The Receiver
11
using liquid helium11, the extreme convenience of using a mechanical refrigerator at
15 K more than compensates for the increased noise.
To maintain the cold stage at 15 K, a CTI model 250 mechanical refrigerator is used.
It has two cooling stages, one capable of sinking 5 W at 77 K and the other rated to
sink 1 W at 16.5 K. The 15 K stage resides within a vacuum dewar and the corrugated
feed horn looks out through a 0.015 inch thick polypropylene window. The 77 K stage
cools a shield that surrounds the cold stage to reduce radiative cooling. The signal
leaves the dewar through 0.01 inch wall stainless waveguide that has been gold plated
to reduce loss, and through a 0.001 inch thick kapton window.
The OMT was acquired from Gamma-F corporation in Torrance, California. It is
specified to operate from 36 to 43 GHz, with a VSWR of less than 1.2:1 and less than
-30 dB of cross-polar coupling. It continues to have reasonable band-pass
characteristics up to at least 46 GHz.
The second stage temperature amplifiers were acquired from DBS Microwave Inc., in
El Dorado Hills, California. The noise of a second stage amplifier is normally much
less important than that of the first, since the signal (and first stage system noise) has
already been amplified by the first stage gain. Thus, room temperature amplifiers are
used in the second stage. They are specified to have greater than 50 dB gain and less
than ± 4 dB gain variation with frequency between 36 and 46 GHz. However, to
preserve the effective band width of each of the frequency channels, less than ± 2 dB of
nM.W.
Pospieszalski, "Modeling of Noise Parameters of MOSFETS's and MODFET's
and Their Frequency and Temperature Dependence", IEEE Transactions on
Microwave Theory and Techniques, 42,9 (September, 1989): 1340.
Chapter 2. The Receiver
12
variation is permitted in each of the sub-bands 36 - 39.5 GHz, 39.5 - 43 GHz and 43 46 GHz.
The three sub-bands are defined using a channel dropping multiplexor from Pacific
Millimeter Products (Model number 3647, SNOOl and SN002). These devices are built
in suspended strip line and offer a much more compact package with less microphonic
sensitivity than is possible using a waveguide filter chain. However, they do have
substantial insertion loss (-3 dB in the 43-46 GHz band), and exhibit overlap between
channels (see below).
The signal is detected after the triplexor using diode detectors obtained from Millitech
Corporation and Pacific Millimeter. These devices are specified to be linear in power
up to -20 dBm, or 0.01 mW12, with a sensitivity of 1000 mV/mW. Above this power
they tend to become increasingly non-linear. At high power (typically above 0 dBm)
their behavior is well described as being linear in E-field, or as varying as the square
root of the power. However, by placing a resistor across the output of the diode,
linearity can often be improved up to much higher power levels, although at the cost of
decreased sensitivity. It has been reported13 that with sufficiently low video resistance,
the output of the diode may increase with a power law index greater than 1, before
falling off, but this effect is not observed with these devices. Figure 3 shows the
response of the diode used in Bl with two different resistor values. This effect is useful
12Millitech
13R.G.
Catalog (1991-92), Millitech Corporation, South Deerfield, MA, p39.
Harrison, "Non-squarelaw Behavior of Diode Detectors Analyzed by the RitzGalerkin Method", IEEE Transactions on Microwave Theory and Techniques, 42,5
(May, 1994): 840.
Chapter 2. The Receiver
13
in leveling the outputs from the three bands. Table 2 summarizes the video resistances
and resulting output voltages with a 18 K cold load replacing the horn.
Channel
^video
^diode
S1A
560
31 mV
S2A
2.2K
13.6 mV
S3A
20K
6.0 mV
SIB
560
17.6 mV
S2B
2.2K
27.8 mV
S3B
20K
10.3 mV
Table 2. Detector diode resistor and operating output values.
60
Ru=20 kn
40
R„=2.2
kn
20
0
0
X
J.
0.02
I
J.
0.04
Incident Power (mW)
X
0.06
0.08
Figure 3. Diode linearity with two different resistive loads placed across the diodes.
To avoid standing waves in the signal chain, isolation is required between the first and
second stage amplifiers and between the second stage amplifier and the triplexor, due
to these devices generally poor VSWR. For example, the room temperature amplifiers
Chapter 2. The Receiver
14
have a VSWR of 2:1. A Hughes isolator is used between the amplifiers, and a variable
attenuator is used between the second stage amplifier and the triplexor. The attenuator
is set to -6.5 dB, which provides adequate isolation, and served as a level set to define
the power levels at the diodes.
The sensitivity of the diode detectors and the gains of the room temperature amplifiers
change with temperature. It is therefore necessary to regulate the temperature of room
temperature receiver. The receiver needs to be able to operate with the ambient
temperature ranging from -40°C to 0°C. This is achieved by double regulation. The
outside of the receiver is regulated to 10°C, while the inside box is regulated to 28°C.
This way, a change in ambient temperature will not greatly effect the loading on the
inner receiver box. The temperature servo used on the inside of the box regulates the
temperature of its sensor to 10 mK.
The radiometer was in general found to be quite reliable, operating for two months
with only 18 hours of down time. However, a crack developed in a soft solder
waveguide joint during shipment in the A chain, causing it to become extremely
microphonic and ruining the data acquired from it. Soft solder joints should never be
used in a microwave system where reliability is required.
Determination of the Radiometer Band-pass
As was mentioned, each of the polarization chains in the receiver is broken up into
three sub-bands. The band-pass, G(v) tells how the gain of the system varies with
frequency. Ideally, the response of each of the bands would be zero for frequencies
outside the pass band and uniform inside the pass band. This is not the case for real
radiometers.
Chapter 2. The Receiver
15
It is important to know the band-pass of each of the radiometer channels for several
reasons. The centroid, or effective center, of each of the bands is needed for calibrating
on power-law radiating sources, such as the supernova remnant Cas-A, and for
determining the spectral index of the CMBR fluctuations. The effective bandwidth of
each of the bands is needed in the analysis of the system noise.
A useful definition of the centroid frequency is
=
J uG( v)dv
jG(v)dv '
which is just the mean frequency of the band-pass, weighed by the gain. For a flat gain
response, this will just be the center of the band. This definition is used in determining
the spectral index of the sky signal, which will be frequency independent if it is due to
the CMBR (see Chapter 5).
For finding the effective centroid for Cas-A, which has a spectral index in temperature
of a = -2.7,
Va
n ..an,.^„\
uaG(o)dv
J
J G( v)dv
is a useful definition. This value of vc may be placed directly into (5) in Chapter 4 for
determining the effective temperature of a power-law source, such as Cas-A. In
practice, these two definitions of vc lead to very nearly the same centroid. The
effective bandwidth is given by
\\G{v)di)t
\G\D)dv'
Chapter 2. The Receiver
16
While band-passes for most of the components going into the radiometer were
provided by the manufacturers, they are not all of sufficient quality to accurately
deduce the overall band-pass parameters. Furthermore, simply combining the data
from the various manufacturers will not take into account the effects of standing waves
and other interactions that might develop in the completed radiometer. Consequently,
the band-passes of the receiver channels are measured.
Figure 4 is shows the waveguide pluming used in making the measurement. A Q-band
sweeper (HP model 8690B with a 8697A-H50 Q-band module) produces a narrow
band output that is swept continuously from 34 GHz to 49 GHz. This signal is injected
into the front end of the receiver, and the output of the receiver is measured.
Approximately 80 dB of attenuation is needed to make up for the 80 dB of gain from
the amplifiers. To determine how the power produced by the sweeper varies with its
output frequency, the signal is split off with a -20 dB directional coupler and sampled
with a power meter (HP model 432A). This signal is used to divide out the sweeper
fluctuations and is very repeatable (~0.2 dB). A sweep of the sweeper is set to take
about 5 seconds because of the time constant of the power meter. For each of the
measurements, 100 sweeps are averaged to reduce the noise.
The 300 K emission from the warm attenuators following the sweeper results in about
+3 dBm at the output of the DBS room temperature amplifiers, and the power from the
sweeper at its peak (38 GHz) is set to about half of that. To avoid compression in the
diodes, the last stage level-set attenuator in the room temperature receiver is set to
keep the diodes in a linear regime. However, the room temperature amplifier before
the level set attenuator shows signs of non-linearity.
Chapter 2. The Receiver
17
Powek\
Meter ^v
Variable Attenuators
Dewar
20 dB coupler
HP Sweeper
Rectangular to Rounc
Transition
Round to Round
Transition
OMT
Load
HEMT B-28
To room temperature
Receiver (B Chain)
Figure 4. Setup for band-pass measurement.
Fortunately it is possible to measure and remove the effects of the compression in the
amplifiers. To do this, it is assumed that when the amplifier compresses, its gain drops
an equal amount at all frequencies. Thus, if the sweeper is transmitting power in BJ,
and is compressing the amplifier, then the output from B2 and B3 due to the 300 K
background will decrease. The factor by which the B2 and B3 are reduced should be
the same as that by which B1 is reduced. Figure 5 shows compression vs. sweeper
output frequency. Channel B3 is sampled when the sweeper produces power below
41 GHz, and Channel BJ is used when the sweeper outputs above 41 GHz. The
compression is about 10% at the largest. This curve is used to correct the effects of
the compression in the band-pass measurement. It is not needed in the analysis of the
sky data.
Chapter 2. The Receiver
18
0.4-
0.2- -
ie ' ' ' ie ' ' ' io' ' ' '
' ' ' U ' ' ' 4e' ' ' ' is
Frequency (GHz)
Figure 5. The measured gain compression used in the band pass determination.
The band-pass of the -20db coupler and in the variable attenuators are also measured
and taken out. The band-pass of the transitions from rectangular waveguide to the
OMT is not measured, and neither is the band-pass of the horn. A measurement of the
Ka band horn (of similar design) shows that it has a VSWR of less than 1.2:1, which
implies that the effect of the horn on the band-pass should be small. The largest quoted
uncertainty comes from the power head itself The data sheet that came with it has
calibrations at 5 frequencies from 33 GHz to 50 GHz, which show a variation of about
0.6 dB. The frequency resolution of these numbers is not enough to correct the bandpasses.
Table 3 summarizes the error budget for the centroid determination. It is assumed that
the corrections due to the attenuators and the gain compression are good to 50%
(which is fairly conservative), and that the error due to the power head not being flat is
Chapter 2. The Receiver
19
about the same size as the other corrections, since the amplitudes of the variations are
comparable.
Term
Correction (GHz)
Uncertainty (GHz)
Gain Compression
0.1
0.05
Attenuators/Coupler
0.1
0.05
Power Head
0.0
0.1
0.12
Total Uncertainty
Table 3. The error budget for centroid determination.
The band-pass of receiver chain B before the triplexor is shown in Figure 6, corrected
for the attenuators and gain compression. The band-passes of Bl, B2, and B3 are
shown in Figure 7. They are reasonably well separated except for a rather large
overlap between Bl and B2. In an otherwise ideal radiometer, this overlap would
introduce correlations in the noise between Bl and B2. The effective bandwidths and
centroids for the three B channels are shown in Table 4. The synchrotron centroid is
used to find the expected brightness of Cas-A for calibration.
Because this receiver uses single mode coherent components, the gain outside of band
falls very quickly. The combination of the gain rolloff of the amplifiers and the band
pass filters result in relative gains of less than -40 dB by 34 GHz at the low frequency
end and 48 GHz at the high frequency end. The WR-22 waveguide cutoff below 33
GHz further eliminates leakage at the low frequency end of the band.
Chapter 2. The Receiver
0.&-
0.4-
0.2--
Frequency (GHz)
Figure 6. The full-band band-pass for the Q radiometer B chain.
B2
B3
OS-
0.6-
0.4-
0.2--
Frequency (GHz)
Figure 7. The band-passes for the Q radiometer B chain.
Chapter 2. The Receiver
Channel
21
Effective Bandwidth (GHz)
|JG(«)duf
jG2(u)du
Centroid (GHz)
j uG( u)du
\G(u)du
Synchrotron Centroid (GHz)
tes?r.«=-
2.8
B1
2.5
38.2
38.2
B2
4.1
40.7
40.6
B3
3.3
44.1
44.1
Table 4. Effective bandwidths and centroids for the Q radiometer.
The full band channel gives an effective bandwidth of 4.6 GHz, with a centroid at
39.1 GHz. This is only slightly larger than the bandwidth of B2 by itself. If the
receiver is ever to be used broad band, without the triplexor, then it will become useful
to flatten the band-pass to increase the effective band width. There is significant gain
over more than 10 GHz of bandwith.
o.a.
0. 6 -
J
0.4-
0.2--
Frequency (GHz)
Figure 8. The B2 triplexor band-pass.
Chapter 2. The Receiver
22
In Figure 7, there is an undesirable bump in Channel B2 at around 39 GHz. The data
from the manufacturer, Pacific Millimeter, shows this feature, but at about -7db, which
is smaller than what we see here. Therefore the Triplexor is measured in the middle
band. To do this, the band-pass of B2 (from Figure 7) is divided by the band-pass of
the receiver without the triplexor (from Figure 6). Figure 8 shows the band-pass of the
B2 channel of the triplexor. The feature is visible here, at about the -7 dB level, in
agreement with the data from Pacific Millimeter. It appears larger in the full system
band-pass because the amplifier gain is rising quickly there.
Receiver Noise
The primary amplifier (B) in Q-band is NRAO amplifier SN-B-28. This amplifier uses
an Indium-Phosphide device in its first stage and has been tuned for good performance
between 36 and 46 GHz. It boasts a very impressive noise temperature of less than
10K, but suffers rather severely from low frequency fluctuations. Similar fluctuations
were observed in the Ka-band HEMTs ,4'15 at a lower level.
To calibrate the radiometer and to measure its noise temperature, the horn is replaced
with a temperature regulated and adjustable cold load (shown in Figure 9). An
absorbing load (Eccosorb MF-116, from Emerson Cuming in Canton, MA) in round
stainless waveguide is thermally linked to a copper block, which is temperature
regulated. The temperature is measured with a #10 diode from LakeShore
14N.
Jarosik, et al., 'Measurements of the Low Frequency Noise Properties of a 30
GHz High-Electron-Mobility-Transistor Amplifier,' June 1993, JPL contract #959556.
15Wollack,
111.
23
Chapter 2. The Receiver
Cryotronics, in Westerville, OH. A cold load of similar design in Ka band has been
shown to have less than -30dB reflection16 (< -15dB in E-field). Consequently,
replacing the horn with the cold load will have at most a few percent effect on the
behavior of the radiometer. The #10 diode used has only been calibrated to IK, and no
measurement has been made as to the quality of the thermal link between the load and
the temperature sensor.
Copper Coldstrap
u
S t a i n l e s s Tube
7 n i l w a l l , 2 5 6 nil I D
\l
Ferrisorb Load
r
- _ i - . _. l — ,
Figure 9. The cold load used for lab calibration.
16Wollack,
49.
Chapter 2. The Receiver
24
Diode voltage
(mV)
3o
io ' ' ' ' ^0 ' ' '
do
Cold Load Temperature (K)
Figure 10. Diode voltage vs. cold load temperature for a lab calibration of the Q radiometer.
To measure the system temperature, and to roughly calibrate the receiver in the lab, the
cold load is servoed to a temperature, and after the load has come to thermal
equilibrium, the diode voltage is measured. Approximately 15 minutes is allowed for
the load to equilibrate. Because of uncertainties in thermometry, this calibration is not
used in the analysis of the astrophysical data. The final calibration of the radiometer is
discussed in Chapter 4.
Figure 10 shows the results of a lab based calibration and noise temperature
measurement made for the 36-39.5 GHz (BJ) band. It gives a noise temperature of less
than 8 K, which is consistent with the measurements of this device made at NRAO by
Marian Pospieszalski. An offset in the preamp or diode will produce a systematic error
in this measurement, but this has been measured to be at most 50 fiV, or 0.16 K.
Although the signal into the diodes have been attenuated 3 .7 dB over the normal
Chapter 2. The Receiver
25
operating level, there is some indication of compression beginning to be a factor at the
highest cold load temperature. The slope of the line, which will be used later to
calibrate the detector NET, is 30.1 ± 1
, including uncertainty in the calibration of
the LakeShore diode. This does not include uncertainty in the surface temperature of
the cold load.
If the radiometer equation, (2), held with this device, a noise level on the sky of only
540 fiK-s1/2 (without chopping), would have been achieved in the B1 channel alone. In
addition, averaging together the three sub-bands would have reduced the total noise
even further, to as low as 350 |iK-s1/2. The system noise would have been dominated
by the temperature of the sky. Unfortunately, this turned out not to be the case.
Rather, low frequency fluctuations in the gain of the amplifier prevent this noise level
from being achieved unless one operates at an audio frequency of over 1 kHz.
NET (mKs1
-£o
io ' ' '
4o ' ' '
^0 ' ' '
Cold Load Temperature (K)
Figure 11. The NET as a function of load temperature
for the Q radiometer at three audio frequencies.
do
Chapter 2. The Receiver
26
To measure how the NET varies with audio frequency, power spectra of the output of
the diode are taken at each of the cold load temperatures plotted in Figure 10. Figure
11 shows how the NET varies with cold load temperature at each of three audio
frequencies.
Two things can be seen from Figure 11. The NET does increase with cold load
temperature, as expected from (2). But unfortunately, the spectrum is not white, as
predicted by (2), but rather increases dramatically at lower audio frequencies. This is
consistent with the excess fluctuations at low frequencies being due to gain
fluctuations17.
(3)
Here, AG =AG(o) represents the gain fluctuations, and Tsys=Tload+Trec•
Under this
model, the X intercept of each of the three lines should be -Trec, which they are. There
is once again some indication of nonlinearity (caused by diode compression or perhaps
cold load thermometry) when the cold load temperature is 70 K.
The NET spectrum with the cold load at 19K is tabulated in Table 5 and plotted in
Figure 12. The data have been converted from the output of the spectrum analyzer to
temperature units using NET = Sy (u)/(J2 • 30.1 ± 1 mf). This calibration is from the
line in Figure 10. The data are fitted to NET = yja +bu . The fit gives a = 0.394,
b = 13.9, and c = -0.757, or, in the form of (3), with T v s = 21K,
17Tiuri
and Raisanen, 7-12
Chapter 2. The Receiver
Frequency (Hz)
NET (mK'S1/2)
20,000
0.609
1000
0.655
250
0.771
200
0.820
150
0.870
100
0.912
50
1.05
16
1.43
8
1.82
2.30
4
Table 5. The NET of the Q radiometer with a 19K cold load.
15-
E
i—
Z
LJ
0.5-
200
400
600
Frequency(Hz)
800
Figure 12. The noise spectrum of the Q radiometer with a I9K load.
The noise is at twice the high frequency limit at 32 Hz.
28
Chapter 2. The Receiver
This curve is plotted in Figure 12. The bandwidth predicted by the fit (2.2 GHz) is
narrower than what is measured (2.5 GHz). This discrepancy is consistent with a 6%
error in either the DC signal or the measurement of the spectral noise density.
Equation (3) appears to be a good model. At 8 Hz, the noise is more than 3 times
larger than would be predicted by (2).
Since the excess noise appears to be due to gain fluctuations, one might expect it to be
correlated from channel to channel within one filter chain. As with the Ka receiver, this
is the case. The correlations are measured from data acquired looking at the sky in
Saskatoon. At 7.8Hz, the correlation coefficient between B1 and B2,
Pn
= a
n A^i^)
~0-7. About 0.1 of this can be attributed to atmospheric
fluctuations, as determined by looking at the correlation coefficient between BJ and
A J, which should otherwise be zero. The unfortunate result of this correlated noise is
that the error bars will not be reduced by ~V3 (the case if the channels were
uncorrelated) if all three channels are combined.
B1
B2
B3
All B
2.4 mK-s"2
2.5 mK-s"2
4.1 mK-s"2
2.1 mK-s"2
Table 6. Q Receiver NET at 7.8 Hz Obtained Looking at the Sky
Table 6 reports the NET at 7.8 Hz obtained with the receiver looking at the sky. The
noise for B1 (2.4 mK-s1/2) is 30% greater than measured in the lab, in Table 5
(1.8 mK-s1/2). The calibration for the sky noise measurements is based on the
supernova remnant Cas-A and differs from the lab calibration by -20% (see
Chapter 4). The remaining 10% discrepancy is attributable to atmospheric fluctuations
adding to the noise.
Chapter 2. The Receiver
29
B3 has the greatest noise since the temperature of the atmosphere is highest there (39K
for B3 rather than 18K for Bl) due to the oxygen line. When all three B channels are
combined, a reduction in the NET of 1.14 is realized relative to Bl alone, rather than
the factor of 1.6 consistent with the unequal distributions being uncorrelated.
Future Considerations
While this radiometer design is simple and reliable, it by no means utilizes the HEMT
amplifiers to their full potential. The radiometer suffered heavily from the correlated
low frequency fluctuations, resulting in a noise level of 2.1 mK-s1/2 at 7.8 Hz. If the
modulation could be performed at a much higher rate, or if the fluctuations could be
monitored and taken out, then ultimately a noise level from the ground of 350 |iK-s1/2
could be achieved. From a balloon, with Tsys= ~10K, and a 10 GHz bandwidth, a noise
level of 100 fxK-s1/2 should be ultimately possible.
Summary
Two receivers were used in the 1994 Saskatoon observing trip, one in Ka band and one
in Q band. Both receivers are based on HEMT amplifiers provided by NRAO, and
have six channels each, in three frequency bins and two polarizations. Table 7
summarizes their performance. The Ka data are taken from E. Wollack's thesis (1994).
Due to a failure of a waveguide joint in shipping, the A radiometer chain in Q band is
not used.
Chapter 2. The Receiver
Channel
Centroid
Effective
NET at 7.8 Hz
NET at 7.8 Hz
Frequency
Bandwidth
(Lab)
(Sky)
KaAl
27.5 ±0.15 GHz
3 GHz
1.4 mK'Sl/J
1.1 mK-s"2
Ka A2
30.5 ±0.15 GHz
3 GHz
1.3 mK-s"2
1.5 mK-s"2
Ka A3
33.5 ±0.15 GHz
3 GHz
1.6 mK-s"2
1.3 mK-s"2
KaBl
27.5 ±0.15 GHz
3 GHz
1.1 mK-s1'2
1.1 mK-s"2
Ka B2
30.5 ±0.15 GHz
3 GHz
1.5 mK-s"2
1.3 mK-s"2
Ka B3
34.0 ±0.15 GHz
4 GHz
1.4 mK-s"2
1.4 mK-s"2
Q-Bl
38.2 ±0.12 GHz
2.5 GHz
1.8 mK-s"2
2.4 mK-s"2
Q-B2
40.7 ±0.12 GHz
4.1 GHz
Q-B3
2.5 mK-s"2
44.1 ±0.12 GHz
4.1 mK-s"2
3.3 GHz
Table 7. Summary of radiometer specifications.
The NET entries are in antenna temperature, referenced to the horn.
Chapter 3
The Telescope
Introduction
In the design of a telescope used to measure the anisotropy of the CMBR from the
ground, three challenges must be met. The telescope must have very low sidelobes, the
telescope must be able to reject atmospheric signal, and the telescope must have small
and stable offsets. These criteria stem from the extreme faintness of the signals being
observed. If one wishes to keep the level of spurious telescope induced signals below
10% of the signal being observed, then, assuming a 30|iK signal, the telescope must
introduce no more than 3(iK of contamination to the final data set. While one might
hope to identify and at least partially remove telescope effects in analysis, the chances
of success at this endeavor are greatly enhanced if the contamination is small to begin
with.
The BigPlate telescope used in the SK93 and SK94 observing seasons meets these
challenges by using a clean optical design with extensive shielding, by employing a
flexible chopping scheme on the sky for atmospheric noise suppression, and by
minimizing the number of optical elements effected by the chopping.
31
Chapter 3. The Telescope
32
Fixed Ground Screen
lear Sun Screen
Parabolic Prinar;
Ground Screen-,
/-Detector
| f
| pEi. >ctronlcs
HuSbiE
n
,
Figure 13. The telescope.
Overview
The BigPlate telescope optics is comprised of a corrugated feed illuminating a
parabolic primary, followed by a chopping flat with a vertical chopping axis. These
elements, as well as the receiver, some electronics, and near ground screens reside on a
rotating frame mounted on a 2' 8" diameter crane bearing (Avon Bearing Co.,
Columbus, OH). The telescope is steerable in azimuth, but is fixed in elevation to
52.15°, which is the elevation of the NCP from the observing sight in Saskatoon,
Saskatchewan. Surrounding the telescope is a fixed ground screen, made from
aluminum covered Styrofoam housing insulation, which is 9.4 feet tall to the north and
Chapter 3. The Telescope
33
12.3 feet tall to the south, with an extra 4'x8' extension due south. The layout of the
telescope is summarized in Figure 13. Data acquisition and telescope control is
provided by a '386 PC connected by ethernet to a pair of UNIX workstations, which
provide mass storage and analysis capabilities.
Optics
The telescope beam is defined by a diffraction limited corrugated feed underilluminating a parabolic primary mirror. This results in an essentially frequency
independent beam across each of the two receivers. The beam solid angle produced by
a n o p t i c a l e l e m e n t ( i n t h e d i f f r a c t i o n l i m i t , a s i n t h i s c a s e ) i s g i v e n b y 1 Q = X 2 / Ae ,
where X is the wavelength of the radiation, and Ae is the effective area under
illumination. In this limit, the beam size from the horn decreases with increased
frequency, reducing the illumination on the primary by the amount required for a
frequency independent beam. A numerical treatment2 shows that the Q-band beam
varies as 0O = 0395GW_(o.926 + 0.0731(39.5 GHz/ l>)2). Thus, the beam width is
expected to vary by 3.3% between 36 and 46 GHz. The Ka and Q receivers both
illuminate the primary at the same level, which causes the Q beam to be smaller than
the Ka beam by ~30GHz / 40GHz.
The primary mirror is an off-axis parabola (0Feed= 55°) with a 50 cm focal length. It
was custom made by Alpha Machine Shop out of a single piece of QC7 aluminum. The
!Kraus,
2The
6-5.
numerical calculations pertaining to the telescope optics were performed using
par_f eed, a program adapted from Sletton {Reflector and Lens Antennas, 1988,
Artech House, Boston) by L. Page and E. Wollack.
Chapter 3. The Telescope
34
rms surface error was measured to be Sjum. To minimize sidelobe levels, the primary
mirror is substantially under-illuminated, as shown in Figure 14. The illumination at the
edge of the primary is -24dB.
40 -
20 -
E
0
v
-20 -
-40 _J
-40
I
_l
L-20
I
L_
_1
0
I
L_
-J
20
I
L.
40
X (cm)
Figure 14. Calculated illumination of the primary.
The contours are in 3dB intervals. The first contour is -3dB.
The Horn
A cylindrical corrugated feed illuminates the primary. Lower sidelobes and greater
symmetry between polarizations may be achieved with corrugated horns than with less
costly smooth walled horns. To manufacture the horn, a mandrel was cut from 6061
35
Chapter 3. The Telescope
aluminum by the Princeton machine shop and then sent out to Custom Microwave Inc.
in Longmont, Colorado for electro-forming. The mandrel is electroplated (first with a
thin layer of gold, and then with copper) to sufficient thickness for strength (0.08"
minimum) and the aluminum mandrel is then etched out. The mandrel and resulting
horn are shown in Figure 15.
co oO
Figure 15. The Q-band horn and mandrel.
Chapter 3. The Telescope
36
The design of the Q-band horn is drawn from the Ka design described in E. Wollack's
thesis, with reference to a book by Clarricoats and Olver3, and papers by James4 and
Thomas5. The fundamental design of a corrugated feed is quite simple. It helps to
imagine waves leaving the horn rather than them entering, although the process is
completely reversible. The intention is to couple the TE11 mode propagating in round
waveguide to free space without generating any other modes and without causing
reflections. The first corrugated element is the throat. It couples the TE11 mode in the
round waveguide, where the E-field is attached to the wall to the HE11 mode in the
flare where the field is detached. This is accomplished by smoothly varying the depth
of the corrugations from X/2 at the first corrugation in the throat to X/4 in the flare.
The idea here is that X/2 slots have the same electrical behavior as a smooth wall. Slots
of depth X/4 in the flare of the horn decouple the E-fields from the wall of the horn,
which minimizes perturbations in the transition from the horn to free space: if the E
field is zero at the wall of the horn, then the absence of the wall after the wave leaves
the horn should produce a minimal effect. The only complication is to make the horn
work over a broad frequency band without generating unwanted modes, which would
result in higher sidelobes.
3P.J.B.
Clarricoats and A.D. Olver, Corrugated Horns for Microwave Antennas,
(London, UK: Peter Peregrinus Ltd., 1984)
4G.L.
James, "TE11 to HE11 Mode Converters for Small Angle Corrugated Horns",
IEEE Transactions on Antennas and Propagation, AP-30:6 (November, 1982): 1057.
5B.M.
Thomas, "Antenna Design Notes", IEEE Transactions on Antennas and
Propagation, AP-26:2 (March, 1978), 367.
Chapter 3. The Telescope
37
The aperture of the horn has a radius of 0.5833" to provide the appropriate illumination
pattern on the primary, and has a 6° flare angle. The pattern from the horn is calculated
by integrating over the HE11 mode from a corrugated waveguide. For a small flare
angle, diffraction limited horn, such as this one, this calculation is a good
approximation. In the flare, for good operation, the slot period needs to be small
compared to the wavelength. In practice, A/4 is adequate. This translates to a slot
period of 0.0618" for a maximum operating frequency of 47.8 GHz. In the interest of
machinability and keeping the slot depth to width ratio below 4:1 for electro-forming
reasons, the ridges in the horn (slots in the mandrel) were set to 0.030". The slot depth
in the flare is set A./4 at 38 GHz, which places the balanced HE11 frequency (where
there is minimum cross-polar mixing) at 39.7 GHz, which is near the center of the
useful band-pass of the receiver. The guide wavelength asymptotically approaches the
free space wavelength as the horn opens, and thus the slot depth changes gradually
from 0.081" just after the throat to 0.077" at the last corrugation.
For lowest reflections, the radius of the smooth wall waveguide entering the throat
should be greater than X/2 at the lowest frequency. This would lead to a 0.164"
opening radius. But in the interest of not generating the HE 12 mode at the top of the
band, this is reduced to 0.1545", or 0.471 X at 36 GHz. As discussed earlier, the first
slot depth should ideally be X/2. But at what wavelength? According to Claricoats, for
best VSWR, it should be near the bottom of the band. But Thomas says that to avoid
generating the EH11 mode, X/2 > d > XI4 at all frequencies. Thus the first slot depth is
set to X/2 (0.123") at the top of the band, 48 GHz. The depth of the groves must then
vary smoothly (as -Jx) to the slot depth in the flare in 10 groves.
Chapter 3. The Telescope
38
The horn is held at 15K inside the dewar to reduce emission from the horn, as shown in
Figure 16, and looks out through a 0.015" polypropylene window. To avoid ice
buildup on the window, a space is defined over the window using a Styrofoam ring and
Glad Wrap window where warmed air is blown. The dielectric constants of the
windows and the Styrofoam ring are very small, and they do not have any effect on the
beam.
Heater Window
Styrofoam Ring
Warned Airspace
Vacuum Window
Baffles
77K Shield
Horn and Mount
15K Stage
\\
Figure 16. The horn, baffles, and vacuum window.
A map of the horn is shown in Figure 17. The map was made at 39.5 GHz, with the
horn in the dewar as in Figure 16, to include effects from the vacuum window and the
baffles. The calculation is for an infinitely long horn (essentially corrugated
waveguide), and is only expected to reproduce the main beam, and not the sidelobes.
The more worrisome feature is that the E-plane (the plane of the map and the E-fields
Chapter 3. The Telescope
39
parallel) and H-plane maps are not equal. The assumption made in the calculation is
that the pattern from the horn can be completely described by the scalar value of the
field, which is a very good assumption for large diameter horns, but falls apart as the
horn diameter approaches the wavelength of the radiation, as apparently happens here.
A wider E-plane horn pattern results a larger area being illuminated on the primary, and
thus a narrower E-plane beam on the sky.
0
CD
I"
• E Plane
* H Plane
Calculated
—10 -
"O
oo
o>
*\ B
\
w
V-
o
\
\
<D
^ -20
0
«•a .•
I •
o/
//
-30 -40
a
\
a
\
*V
«
V
-20
0
20
40
6 a (degrees)
Figure 17. Q-band horn beam pattern.
The Main Beam
The telescope is calculated to have a FWHM beam width of 1.06° at 39.5 GHz. The
beam pattern was measured on the roof of Jadwin Hall to determine the actual beam
width in both the E and H planes. Data from observations of Cas-A are also used.
Chapter 3. The Telescope
40
For the roof measurements, a 39.5GHz Gunn Oscillator is placed on top of Fine
Tower, which is at an angle of -24° from where the telescope is placed on the roof of
Jadwin. Since the telescope looks at a fixed angle of 52.15°, the entire telescope is
tipped by -28° to look directly at the source.
To measure the E-plane map, two different methods are employed to steer the beam.
One is to rotate the telescope base, and measure its position with the base encoder that
is accurate to 1 part in 216, or 0.0055°. This process is done in two steps, with the
source brightness increased by 30dB at 1.3°. Foil backed eccosorb is placed in front of
the horn at the beginning and end of every scan to accurately determine zero. The
results of this map are shown in Figure 18.
o
Measured
Calculated
-20
CD
"O
c
Oo
o
O
-5
0
5
Q (degrees)
Figure 18. The Q-band Channel B E-plane beam map, measured and calculated.
Chapter 3. The Telescope
41
In the second method, the beam is swept over the source using the chopper rather than
the base. This method has the disadvantage of not providing a simple way of finding
zero. Nevertheless, it gives a map very similar to that of Figure 18. The most useful
result of using both techniques is that it allowed us to discover and correct a 2.97%
calibration error in the chopper encoder.
1
C
'o
o
> °-5
•
o
Relative Azimuth (dea)
Figure 19. The E-plane beam determination for channel Q-Bl using Cas-A.
The beam width is also measured in the field using the bright super-nova remnant
Cas-A. The beam is chopped over it with the chopper and is binned in elevation of
Cas-A relative to the beam into 0.1° bins, shifted in azimuth, coadded, and normalized
to unity amplitude. A plot of all of the data from morning Cas-A for channel B1 is
given in Figure 19. A Gaussian is then fitted to determine the beam width and azimuth
pointing error. The pointing error was 0° in the morning and 0.12° in the evening. The
Chapter 3. The Telescope
42
difference is due to an elevation pointing error of 0.06°. The beam widths from Cas-A
are corrected for the calibration error in the chopper encoder.
Mapping the H plane of B requires moving the moving the beam in elevation and is
thus more difficult than mapping the E plane, since the telescope is not steerable in
elevation. To change the elevation, the entire telescope, including the base, is rotated
by raising and lowering the jack, and the elevation is read out with a WWII gunner's
level that is precise to 0.01° The resulting map is shown in Figure 20. It agrees well
with the calculated pattern for the main lobe, but seems to indicate the existence of
excess power at negative angles. This may be due to a mispointing of the horn, or may
be due to the somewhat shaky technique. The existence of an elevation pointing
system with a well-defined rotation axis will be useful to better measure the vertical
map.
1.2
1
o.a
U 0.6
0.4
0.2
0
-2
-1
0
1
Q (degrees)
Figure 20. Q Band Channel B H-plane beam map.
2
Chapter 3. The Telescope
43
The Q-band maps done on the roof are done at 39.5 GHz. However, the telescope
beam width is calculated to vary as 0O - 0395GH,(o. 926 + 0.0731(39.5 GHz/ vf \ The
beam widths at the centroids of each of the three frequency channels are therefore
extrapolated using this relation. Table 8 summarizes the beam width determinations.
The entries for Cas-A have been corrected for the calibration error in the chopper
encoder.
39.5GHz
38.5GHz (Bl)
40.8 GHz (B2)
44.1 GHz (B3)
Calculated
1.06
1.06
1.05
1.04
H-Plane
1.084±0.02
1.09±0.02
1.08±0.02
1.07±0.02
1.007±0.005
0.999±0.006
0.983±0.007
1.012±0.005
1.004*0.005
0.993±0.005
1.050±0.02
1.042±0.02
1.031±0.02
E-Plane (Cas-A)
E-Plane (Base)
1.009±0.005
+ f%)/2
Table 8. Summary of Q-beam widths.
The effect of the E-plane of the horn being wider than the H-plane can be seen in these
data: the E-plane map of the main beam is narrower than the H-plane map. The
quadratic mean of the E-plane and H-plane beam widths is therefore given for
determination of the beam solid angle.
The mapping of the Ka beam is done using the same procedure as for the Q beam. The
Ka optics are discussed in E.J. Wollack's thesis, but updated beam maps are presented
here, in Figure 21. The Ka beam is more symmetric than the Q beam, but still exhibits
slightly higher than calculated sidelobes. The beam parameters are summarized in
Table 10.
Chapter 3. The Telescope
o
44
• A E-plane
- - A H-plane
— B E-plane
B H-plane
-20
Calculated
H-plane P
Calculated
E-plane
-60
10
-5
o
5
10
6 (degrees)
Figure 21. The Ka beam patterns
Sidelobes
One of the most severe constraints on the telescope is that it have low sidelobes. The
most dangerous source of contamination is the sun. At 8200K, it is the brightest object
in the sky, and, since it moves through the sky at nearly the same rate as the celestial
sphere, it has the best chance of having a signature similar to that of the CMBR. If one
demands that it produce a signal of less than 3(iK, then the telescope gain at the angle
of the sun must be less than -85dB relative to the main lobe. This is achieved by
building a telescope with clean optics, as described above, and by the extensive use of
shielding. In the SK93 observing trip, the sun shield was not tall enough to
geometrically block the sun at all times. This was corrected for the SK94 trip by the
addition of several more 4'x8' panels of shielding to the ground screen.
45
Chapter 3. The Telescope
To measure the attenuation provided by the ground screens, a 39.5 GHz Gunn
Oscillator is placed 25m in front of the telescope and the signal from it measured with
the fixed ground screens in place. The shielding is then removed and the measurement
repeated. It is found that the shielding provides 36dB of attenuation. If we
conservatively estimate the telescope gain at the angle of the source (-52°) to be less
than -65 dB (the calculated response of the telescope here is around -83 dB) then the
gain directly in front of the telescope is less than -101 dB. When the source is placed
behind the telescope at approximately the angle of the sun early in the observing
season, the signal is a further 15 dB lower than in front of the telescope. Thus, the
sensitivity of the telescope to the sun is estimated at less than -115 dB relative to the
main beam.
The sidelobe level can also be estimated by estimating the effective temperature of the
source, and comparing that to the level of the signal measured from it. The effective
temperature of the source is given by
$ A
T =-
C „2
where Sv is the flux at the telescope due to the source, u is the frequency of the source,
and Q.Tel is the telescope beam solid angle. If we assume that the flux is constant over
the illumination pattern of the transmitting horn, then we can estimate the flux from the
transmitter as
'source
where P is the power of the source, r is the distance between the source and the
telescope, and Au is the //bandwidth of the receiving channel. The source is at 39.5
GHz, has 140mW of power and has a beam solid angle of 0.04 str. This yields an
Chapter 3. The Telescope
46
effective source temperature of 1010K. With the source on the ground 25 m in front
of the telescope, a signal level of 1.5K from the source is observed, which gives a
sidelobe level of -98 dB. From behind the telescope at the angle of the sun, the level is
50 mK, or -113 dB. These numbers are consistent with the other estimation technique.
Given a -113 dB sidelobe level, the contribution from the sun is estimated to be less
than 5pK. However, due to site constraints, it is not possible to raise the source to the
full height that the sun reached at the end of the season. Nevertheless, since the entire
telescope is entirely shaded geometrically by the ground screen at all times, it is not
expected that the sidelobe level at the sun ever approached the critical level of-85dB.
The Chopper
The final element in the optics is a 3'x5' honeycomb chopper which steers the beam on
the sky. It has a maximum throw of ±3.25°, which moves the beam 8° on the sky, and
can make a 2° transition in -32 ms. The edge illumination on the chopper is measured
to be less than -45dB. Essentially the same chopper was used in the SK93 experiment,
although the plate was replaced by one with lower camber and the throw was
increased.
The chopper is driven magnetically. Two 7.9 cm diameter coils (with 570 turns of 24
gauge wire each) are mounted on the back surface of the plate 11.4" from the rotation
axis, and move in a 0.5T magnetic field produced by permanent magnets mounted on a
hinged reaction bar, as shown in Figure 22. The plate has a mass of 4.1 kg, and a
moment of inertia of 0.31 kg-m2. Current is provided to the coils (wired in parallel) by
an Apex PA-04 Op-Amp, capable of sourcing 10A at 90V. There are 570 turns in each
of the coils. From this we can estimate, at full power, a 21ms minimum transition time
Chapter 3. The Telescope
47
for a 2° step, assuming 50% of the coil is in the field. In reality a transition time (5% to
95%) of 31 ms is actually achieved for a 2° step. Air is blown over the coils during
operation with a pair of 5" muffin fans to keep them from over-heating, and to
minimize the warming of the chopper. In operation, the front surface of the mirror in
front of the coils comes to 3.5°C above the temperature in the center of the plate.
Chopper noves +/- 3.25 dee
—Chopper
Coil
Pernanent Magnet
Pivot
10 cm
Reaction Bar
Figure 22. The chopper drive arrangement.
The control of the chopper is by a Proportional Integral Derivative (PID) loop
(Figure 23). The chopper position is read out from a Lucas-Shavitz RVDT (model
RSYN-8/30), and the position request comes from a DAC on the PC. The values of
the PID gains are determined empirically: a request signal is injected and the actual
wave form monitored. Increasing the proportional gain decreases the response time of
the loop, but can cause overshoot. Increasing the derivative gain introduces damping
and kills the overshoot. The gain (and the damping) are increased to the maximum
value possible without causing voltage or current clipping with the desired request
wave form. If the servo loop does go to the rail, the vibration from the chopper
48
Chapter 3. The Telescope
increases noticeably, with little or no increase in performance. The integral term
guarantees that the average position of the chopper is zero, and thus compensates for
changing wind loading on the chopper. It can be thought of as increased gain at low
frequencies. The plate has a resonance at 180 Hz, so the request signal has a notch
filter to cut any gain there and avoid oscillation.
Proportional
3XE
Position Request In
Chopper Angle In
1V/degree
Error
Integral
59XiEdt
Derivative
-0.14 X dE/dt
Current Request Out
1V/Amp
180 Hz
Notch Filter
Figure 23. The chopper PID control.
The permanent magnets which the coils react against are mounted on a hinged reaction
bar which weighs (with the magnets) 34 kg. The intention here is to absorb the angular
momentum of a chop into the reaction bar rather than shaking the frame. The reaction
bar, however, must be somewhat coupled to the frame to keep it from swinging out of
range during a base move. This is done by the use of another magnetic servo very
similar to the one driving the chopper.
While the chopper is capable of chopping in a step pattern, (as in a two or three point
pattern) it is not used in that fashion. Rather, in the interest of reducing vibration, it is
Chapter 3. The Telescope
49
driven in a ±3.0° linear pattern with rounded endpoints. The measured chop pattern is
shown in Figure 24. The chop frequency is 3.90625 Hz. The rms variation of
individual chops from this pattern is 0.03 degrees, and is ignored. This continuous
chop has the added benefit of greater flexibility in analysis (see Chapter 4).
- 2
J
0
I
I
0.05
L
0.1
0.15
I
0.2
0.25
Time (s)
Figure 24. The chopper wave form.
Emission from the plate does produce a small offset. Moving the chopper changes the
incident angle off of the chopper, and thus changes the emission. This results in a
symmetric offset of a few hundred microkelvin. A theoretical discussion of the
expected magnitude of these offsets is given in Wollack. The experimental magnitudes
of the offsets for all of the data are given in Chapter 4, but Table 9 reports the offsets
obtained if the data is analyzed as a three point chop (see Chapter 4 for an explanation
of how this is done). The offsets from the anti-symmetric chops (2 pt, 4 pt, 6 pt, etc.)
50
Chapter 3. The Telescope
are consistent with 0|iK. The temperature scale here is referenced to the telescope, and
not to above the atmosphere.
Channel
Offset
K-,^7 (27.5 GHz)
114 ±7 uK
K i > A2 (30.5 GHz)
103 ± 7 nK
K 9 A3 (33.5 GHz)
105 ± 7 uK
Ka B1 (27.5 GHz)
-423 ± 7 uK
Ka B2 (30.5 GHz)
-492 ± 7 uK
K„ B3 (33.5 GHz)
-626 ± 7 uK
QB1 (38.5 GHz)
-615 ± 10 uK
Q B2 (40.8 GHz)
-661 ± 10 uK
-831 ± 15 uK
Q 55(44.1 GHz)
Table 9. Telescope offsets for 3-point chop.
Electronics and data acquisition
Telescope control and data acquisition is provided by a '386 PC mounted outside near
the telescope. The six receiver channels are read with V-F converters on the telescope,
and thirty auxiliary data channels are read using a 12 bit A-D card internal to the PC.
The base position is read by a 16 bit absolute encoder. The base control and chopper
position request come from two 12 bit DACs. A block diagram of the data acquisition
is presented in Figure 25.
The primary data from the six receiver channels comes from diode detectors, whose
output is ~20mV during operation. An initial gain of 100 is provided by preamp cards
connected to the detectors. The signal is then passed to another box (still on the
telescope) where a DC offset is removed another factor of-10 gain is provided, and
the signal is filtered. Except for the fixed offset which is removed, the system is DC
51
Chapter 3. The Telescope
coupled. The signals are digitized using V-F converters (AD562AQ). They are
clocked at 4.096 MHz, which gives a maximum frequency out of 2.048 MHz. With a
250 Hz gate on the counter (in the PC), which is the sample rate, this corresponds to
8192 counts full scale. The gain is set to be approximately ImK/count according to the
lab calibrations, which gives the system a range of 8.2 K, which is adequate for all but
the most cloudy (and thus useless) days. For data acquired when the weather was
acceptable, the total sky temperature variation was 3 K. The chopper position and the
base position were recorded at the same rate as the primary data.
I
1
Diode Detectors
Preamps
On Telescope
DC Bucking
Filters
Base
V-Fs
Base Cont
Digital BOB
Aux
Sensors
Chopper
Weather
Chop Cont
Analog BOB
In Shed
Black Rack
•386 PC
PC Console
Near Telescope
Workstations
Storage
Ethernet
1
I
Figure 25. A block diagram of the telescope electronics.
Eight pole Bessel (phase linear) filters with a 120 Hz 3dB roll-off are used. At low
frequencies, a Bessel filter can be treated as just introducing a time delay to the signal
which is constant at all frequencies. Since the filter rolls off much higher than the
highest frequency of interest (24 Hz), this approximation is very good. Figure 26
Chapter 3. The Telescope
52
shows the measured step response of the data system, including the V-F's. The time
delay introduced by the filters is taken into account by using the chopper position
measured through the data system (Figure 24) in all of the analysis.
1
0.8
03U)
c
oCL
<flV
or
0.6
0.4
0.2
0
J
I
I
-0.04
I
1
I
-0.02
.
I
I
I
0
Time (s)
I
I
I
0.02
I
I
I
I
L
0.04
Figure 26. Unit step response of data system.
Thirty channels of auxiliary data are acquired at 0.5 Hz using a commercial 12 bit A-D
board in the PC (Analog Devices RTI815). These signals included a weather station,
hardware temperatures, chopper control diagnostic signals and temperatures of the
surface of the chopper.
The pointing is also controlled by the PC. The base position is read out by a 16 bit
absolute encoder (Teledyne Gurley model 25/043-16). Based on the pointing error, the
PC outputs a base velocity request to the servo amplifier. The rms pointing error due
to the base control is generally 0.005° (one count from the encoder) but there were
Chapter 3. The Telescope
53
conditions when this error could grow to ten times that (0.05°): if the workstation
which served as a file server was backing up to tape, its response latency for ethernet
requests increased dramatically, which upset the dynamics of the software control loop.
This has been corrected.
Timing for the data acquisition is based on an external 4.096 MHz clock. This serves
as a reference for the V-F converters, and, divided down, as the gate for the counters
and the interrupt line to the PC for timing chopper position updates and data reads.
It is imperative in a system like this that there be access to the data while the
experiment is running. This is achieved at two levels. First of all, on its display, the PC
plots the data as it is acquired, and a running average (coherent with the chops) of each
of the channels. A summary of the engineering data is also displayed. This allows for
rapid detection of major system failures and provides an indication of data quality in
real time. The monitor and keyboard are located inside, 30 m from the PC. At a
second level, the PC is connected by ethernet to a pair of UNIX workstations where
the data are recorded. 16MB of data is acquired per hour. Thus, access to the data for
more elaborate analysis is provided immediately. The results of the experiment were
largely known while still acquiring data in the field.
Summary
The BigPlate telescope which is used in the SK93 and SK94 observing seasons is
comprised of a corrugated feed illuminating a parabolic primary. The beam is steered
on the sky using a 3'x5' chopper. The telescope pointing is fully adjustable in azimuth,
but fixed in elevation. A summary of the specifications of the BigPlate telescope is
given in Table 10. For a more complete list of offsets, see Chapter 4.
Chapter 3. The Telescope
54
Beam Size:
(Kn receiver)
(Q Receiver)
Frequency
FWHM
27.5 GHz
1.44
30.5 GHz
1.42
33.5 GHz
1.41
38.5 GHz
1.05
40.8 GHz
1.04
44.1 GHz
1.03
Offset Magnitudes:
0 - 830 nK (analysis dependent)
Sidelobe Levels:
-98dB on Ground, -113 dB from Sun
Chopper Throw:
± 3.0° (7.4° on sky) Triangle at 3.91 Hz
Elevation Pointing:
52.15° ±0.06° (Fixed)
Azimuth Pointing Error
0.05°
rms Repeatability:
Table 10.
Big] }late
0.005° typ., 0.05° max.
telescope specifications.
Chapter 4
The Data
Introduction
In this chapter, the calibration and observing strategy for the SK94 experiment are
discussed. Also described are the selection of clean data, and the verification of the
signals.
The BigPlate telescope is calibrated using the supernova remnant Cassiopeia-A. The
use of a celestial calibrator has the advantage of inherently including all telescope and
atmospheric efficiencies, and of being readily usable by other groups, providing a
standard temperature scale. As CMBR experiments become increasingly sensitive, the
calibration is becoming the primary source of uncertainty in determining the amplitude
of the anisotropy.
In choice of an observing scheme for the SK94 experiment, there are several things
which are taken into account. First, the experiment is capable of preferentially rejecting
contaminating atmospheric signals over celestial signals. This is achieved through
differencing on the sky. Second, SK94 observes the same sky as SK93 in order to
confirm it. Third, since many theories predict a dependence on angular scale at around
t« 100, the observing strategy is intentionally chosen to be able to probe a variety of
angular scales.
55
Chapter 4. The Data
56
While adding greatly to the convenience (and economy) of making the observation,
performing the experiment from the ground (as opposed to a balloon or satellite)
results in undesirable contamination by the atmosphere. From a cold, dry location such
as Saskatoon, Saskatchewan, where the SK experiments are run, there are certainly
times where the seeing is very good. But just as certainly there are times when it is not.
It is important to able to select the times of good seeing in a way that will not bias the
answer. By selecting data on the basis of the stability of spatial thermal gradients in the
atmosphere (which are not used in the analysis) this goal is achieved.
The observing scheme used in the SK experiments has a number of symmetries which
are probed to verify that the signal is truly on the sky. Data are acquired with the
telescope in two different orientations in such a way that the signal seen in one is seen
in the other ~ 11 hours later. Data are acquired at several different frequencies to show
that the signal observed has the spectrum of the CMBR. Data are acquired with two
different radiometers (which would presumably have different susceptibilities to
electrical, microphonic or sidelobe contamination) over a two month period. With the
Ka radiometer, the signals seen in with vertical and horizontal polarization are
compared. And the data is compared to that acquired in the SK93 experiment. Also,
since the data are acquired over a two month period, a signal locked to solar time
rather than sidereal time would be shifted by several hours over the period of the
experiment. The consistency of the symmetries add considerably to the believability of
the result.
Calibration
The primary calibrator used for both the Ka and Q radiometers is Cas-A. This
supernova remnant is one of the brightest objects in the sky in these bands, and has
57
Chapter 4. The Data
been measured extensively for use as a celestial calibrator. In general, it is much more
convenient and reliable to use a celestial calibrator than to use ground based lab
calibration, because the effects of telescope efficiency and atmospheric attenuation are
intrinsically included. Of equal importance, the use of a celestial temperature scale
allows the calibrations of various experiments to be tied together.
The flux density scale for Cas-A was acquired by fitting a power law to data compiled
by Baars et al.1 (from 8.2 GHz to 31.4 GHz) and a measurement by Mezger et al.2 at
250 GHz1\ The Baars et al. paper gives a flux density scale based on data from
300 MHz to 31.4 GHz, but the 31.4 GHz measurement has 20% uncertainty. The next
highest data point is at 22.3 GHz. For this flux scale, only the high frequency data
from Baars, and the 250 GHz measurement from Mezger are used.
The Baars paper reports a time dependent decrease in flux density of the form
7] = (0.9903 + 0.0031og(u))'
'J.W.M. Baars, et al., "The Absolute Spectrum of Cas-A; An Accurate Flux Density
Scale and a Set of Secondary Calibrators", Astronomy and Astrophysics, 61 (1977),
99-106.
2P.G.
Mezger, et al., "Maps of Cassiopeia A and the Crab nebula at A, 1.2 mm",
Astronomy and Astrophysics, 167 (1986), 145-150.
t There are several other measurements referred to in the Mezger et al. paper, but they
are not used here. One is a reference to a paper which does not appear to exist. One is
a personal communication. One (Kenny and Dent), the authors of the Mezger et al.
paper claim, has some severe systematic errors, and another is re-analyzed by the
Mezger et al. authors to a value approximately a factor of two different from reported
in the original paper.
Chapter 4. The Data
58
where 77 is the factor by which Cas-A has dimmed, o is frequency in GHz, and t is the
number of years since the measurement. The values of the measurements have been
corrected by this factor to epoch 1994.
A plot of the fit data is given in Figure 27. Fitting to a power law of the form
log(S( u))=a + b\og( u)
(4)
with S(o) in Jy and v in GHz gives a = 3.316 ± 0.034 and b = -0.695 ± 0.029 (epoch
1994). The reduced %2 of the fit is 0.74.
2.4iog(S)
[Jy]
2.2--
2--
2.2
log(u) [GHz]
Figure 27. Determination of the Cas-A flux density scale.
The antenna temperature of Cas-A is obtained from the flux density by
(5)
T(u) = -§^xl0-16/C
2kbv Q
Chapter 4. The Data
59
where Q is the beam solid angle (see Chapter 3). Table 11 gives the antenna
temperature from Cas-A for each of the telescope bands. For the frequency of the
band, the centroid for a synchrotron spectrum was chosen, rather than the centroid for
a black body (see Chapter 2). In the final column, the first error reported is due to the
uncertainty in the temperature scale of Cas-A, and the second is due to uncertainty in
the beam solid angle and the frequency centroid.
Channel
u(GHz)
S(v) (Jy)
Q (10"4s/r)
T (mK)
Ka Al/Bl
27.5 ±0.15
206.6 ± 27
7.16±0.11
12.4 ± 1.6 ±0.21
Kq A2/B2
30.5 ±0.15
192.2 ±25
6.96±0.10
9.68 ± 1.2 ±0.16
Ka A3
33.5 ± 0.15
180.1 ±23
6.82±0.10
7.67 ± 1.0 ±0.13
KaB3
34.0 ±0.15
178.2 ±23
6.82±0.10
7.37 ± 1.0 ±0.13
QB1
38.3 ±0.12
164.1 ±21
3.81±0.08
9.57 ± 1.2 ±0.20
QB1
40.6 ±0.12
157.5 ±20
3.75±0.08
8.30 ± 1.1 ± 0.17
QB3
3.67±0.07
6.79 ±0.9 ±0.14
44.1 ±0.12
148.7 ± 19
Table 11: The predicted antenna temperature of Cas-A. The first error for T is due to uncertainty in
the temperature scale of Cas-A. The second is due to uncertainties in u and Q.
To determine the amplitude of Cas-A in A-D units (ADU), as measured by the
telescope, the beam was swept over Cas-A with a rounded linear chop, as in Figure 24
in Chapter 3, but with a smaller amplitude (1.54° for Q and 2.00° for Ka). The
brightness of Cas-A at a given time was determined by demodulating the data with a
vector generated from a Gaussian of appropriate beam width and position (determined
in Chapter 3 using both measurements on the roof and Cas-A). Essentially, this is just
finding the amplitude a = ^ttgt, where t, represents the data and g, has the form of the
signal expected from a point source (Cas-A), normalized here so that ^ gf = 1. The
demodulated data were binned in time into bins of 0.05° and plotted vs. position
60
Chapter 4. The Data
(Figure 28 for Q-Bl). The amplitude of a best fit Gaussian beam is taken as the
measured amplitude of Cas-A.
QI
-1
I
I
•
•
I
-0.5
•
.
.
•
I
I
I
I
0
Relative Altitude (deg)
i
I
0.5
i
i
1
1„ .
1
Figure 28: Cas-A measured by channel Q Bl
Cas-A passes through an elevation of 52.15° (the pointing elevation of the telescope)
twice a day, at an azimuth of +52.04° (in the morning in February), and -52.04° (in the
evening). There is a discrepancy (~7% for Q and -12% for Ka) between the results of
measurements taken in the morning and those taken in the evening. This effect is
explained by a -25 Jy HII region (2311+611)3-4-5 which is intercepted by the beam
during the evening runs, but not during the morning runs.
3E.
Kallas and W. Reich, "A 21cm Radio Continuum Survey of the Galactic Plane
Between / = 93° and / = 162°", Astronomy and Astrophysics Supplement Series, 42
(November 1980), 227-243.
Chapter 4. The Data
61
62
2311+611 HII Region
PM Chop path
AM Chop path
K0 beam (-3dB)
60
T3
58
56
Cas-A
346
348
350
RA (deg)
352
354
356
Figure 29: Evening and morning chop paths around Cas-A.
Figure 29 shows the relative positions of Cas-A and the offending HII region, along
with the Ka beam width and chop pattern. Because the telescope is steerable only in
azimuth, and not in elevation, a source can only be observed when it passes through the
elevation of the telescope, which normally happens twice per day. Since the chopper
4M.
Fich, "A Complete VLA Survey in the Outer Galaxy", The Astronomical Journal,
92:4 (October, 1986), 787.
5R.H.
Becker, R.L. White, and A.L. Edwards, "A New Catalog of 53,522 4.85 GHz
Sources", The Astrophysical Journal Supplement Series, 75 (January, 1991), 1-229.
62
Chapter 4. The Data
only can steer the beam in azimuth, the path taken by the chopped beam is different
between the morning and evening runs, as the sky has rotated relative to the horizon.
Table 12 lists the results of four measurements of Cas-A in Q band. The column
marked <r gives the standard deviation of the four fits. This is taken to be the
uncertainty in the fit. If the trials were assumed to be Normally distributed, a/*Jn
could be used instead, so this is a conservative estimate of the statistical uncertainties.
The column marked all reports the measured amplitude if all of the trials are combined,
and then fit. This is taken as the measured amplitude of Cas-A.
CT
All
8.2
0.46
7.65
5.55
0.34
5.41
Trial 1
Trial 2
Trial 3
Trial 4
Q-Bl
7.90
7.12
7.67
Q-B2
5.68
4.92
5.53
5.08
3.82
4.76
Q-B3
4.8
0.55
4.54
Table 12. Measured amplitudes of Cas-A in A-D units (ADU).
The temperature scale for Cas-A assumes that the brightness of the source is
proportional to its antenna temperature, according to the Rayleigh-Jeans law,6
B = ^T„.
c
(6)
In reality, however, the brightness of a source is actually given by the Plank law,
m
(
R
)
6Kraus,
3-27
2hv
c2
'
1
ehu,kT-Y
63
Chapter 4. The Data
In cases where hv « kT, the Plank law reduces to the Raleigh-Jeans law. To relate the
antenna temperature, Tant to thermodynamic temperature, equate (6) and (7) and solve
for Tant to find
(8)
Tant
anl =
hv
ghutkT
j'
A small change in antenna temperature is related to a small change in thermodynamic
temperature by differentiating (8) to find
Table 13 gives the determination of the calibration coefficients, and their final
uncertainties. The raw data in ADU are multiplied by
CCas-A
= (TfU /TCas-a)^,/^
to convert them to mK. Where two errors are reported, the first is due to uncertainty
in the temperature scale for Cas-A, and the second is due to measurement uncertainties
in the fit, the beam solid angle, and the channel centroid.
(mK)
TV
(ADU)
m.
cCas-A
^
(-Ek-\
K.A1
12.4 ± 1.6 ±0.21
15.00 ±0.3
0.98
0.844 ±0.11 ±0.03
K„A2
9.68 ± 1.2 ±0.16
11.63 ±0.3
0.98
0.849 ±0.11 ±0.03
Kn A3
7.67 ± 1.0 ±0.13
8.40 ± 0.2
0.98
0.931 ±0.12 ±0.03
KaBl
12.4 ± 1.6 ±0.21
14.00 ± 0.4
0.98
0.904 ±0.12 ±0.03
Kq B2
9.68 ± 1.2 ±0.16
10.66 ± 0.2
0.98
0.926 ±0.12 ±0.03
Kn B3
7.37 ± 1.0 ± 0.13
7.41 ±0.2
0.98
1.015 ±0.13 ±0.03
QB1
9.57 ± 1.2 ±0.20
7.65 ±0.5
0.96
1.30 ±0.17 ±0.08
QB2
8.30 ± 1.1 ± 0.17
5.41 ±0.3
0.96
1.59 ±0.21 ±0.09
Channel
'TCas-A
VADUZ
QB3
0.95
6.79 ±0.9 ±0.14
4.54 ± 0.6
1.58 ± 0.21 ± 0.13
Table 13: Calibration coefficients determined from Cas-A. The first column is taken from Table 11.
The first error is due to uncertainty in the temperature scale of Cas-A. The second is due to
uncertainties in v, O., and the fit to Cas-A.
64
Chapter 4. The Data
The uncertainty in the overall calibration (which contributes to the uncertainty in the
measurement of the amplitude of the CMBR fluctuations) is dominated by the
uncertainty in the temperature scale of Cas-A. Adding the temperature scale and
measurement uncertainties in quadrature, yields an overall calibration uncertainty of
14%.
The uncertainty in the temperature scale for Cas-A effects all of the channels nearly
equally, which means that the contribution to the uncertainty in spectral index
determination is just the uncertainty in b in the fit to Equation (4), which is 0.029. The
uncertainty in spectral index due to error in the measurement of Cas-A taken from the
center band of Ka to the center band of Q is 0.2. However, within a given radiometer,
the error is much less. An error in the beam width will effect the calibration of an
entire radiometer equally. And the atmospheric fluctuations which are the dominant
source of uncertainty in the fits are common to all channels of a radiometer. A
reasonable estimate of the uncertainty in spectral index within a radiometer is 0.1.
The radiometers are also calibrated in the lab using cold loads (see Chapter 2). To
relate the calibration from Cas-A to this, the expected transmission of the atmosphere
needs to be determined. This transmission is given by
n
'aim
=
>
where Tatm is the measured temperature of the sky at the observation angle, and Tphy is
the physical temperature of the sky, which is around 265K. This assumes that the sky
temperature is constant across the band, which it is not, so this will only be
approximate. Sky dips were performed to measure Tatm. The resulting determination
Chapter 4. The Data
65
of rjAtm is given in Table 14. The sky temperatures for Ka band7 are included for
reference. Over the course of the observing season, Tatm varied by 3K in Q-Bl for
observations made during acceptable weather. The resultant calibration uncertainty
(0.5% for Bl) is negligible compared to the other calibration uncertainties.
The calibration coefficients derived from the cold load temperature scale corrected for
the atmosphere and referenced to the CMBR are also given in Table 14. They are
different than the coefficients derived from Cas-A (see Table 13). This may be due to a
problem in the thermometry of the cold loads, or (more likely) because the room
temperature receivers are at a different temperature in the lab than in the field.
Because of the uncertainties inherent in the lab based cold load measurement, they are
not used.
Channel
^zenith
Tatm (K)
'Tatm
fCL (-£&-)
VADUZ
Ka Al/Bl
8.9
11.3
0.96
1.06
Ka A2/B2
7.7
9.8
0.96
1.06
Kq A3/B3
9.5
12.0
0.95
1.07
Q-Bl
14.1
17.8
0.935
1.09
Q-B2
17.6
22.3
0.919
1.18
Q-B3
0.864
30.5
38.6
Table 14: Determination of atmosp leric transmission.
1.24
Chopping
In any CMBR anisotropy experiment done from the ground, it is important to chop, or
difference the beam on the sky. If one were to look at a single spot for a long period,
7Wollack,
55
Chapter 4. The Data
66
fluctuations in the atmosphere could be expected to completely wash out any signal
which one might hope to see. Chopping the beam between two points on the sky and
then taking the difference (a 'two point chop'), as in the ACME south pole experiment8
removes any sensitivity to changes in the average temperature of the atmosphere.
Sensitivity to spatial temperature gradients in the atmosphere still remains, however.
The gradients can be quite large, and need to be removed. A common solution to this
is to do a double difference, or three point chop, as in the OVRO9, Tenerife10, or
(effectively) SK9311 experiments. In these cases, sensitivity to atmospheric gradients is
eliminated. Now the sensitivity is to spatial curvature. In cases where even more
atmospheric suppression is required, a four point chop can been used, as in the
PYTHON12 experiment.
These techniques are successful because the fluctuations from the atmosphere fall
rapidly with decreasing angular scale. A two point chop is sensitive to larger angular
scales than a three point chop of the same beam size and total throw angle. Each of
these chopping strategies is sensitive to range of angular scales, which is limited at
large scales by the nature of the chop, and at small scales by the beam size.
8Gaier,
et al.
9S.T.
Meyers, A.C.S. Readhead and C.R. Lawrence, "Limits on the Anisotropy of the
Microwave Background Radiation on arcminute Scales. II. The RING Experiment",
The Astrophysical Journal, 405 (March, 1993): 8-29.
10R.D.
Davies, et al. "Observations of the Microwave Background on a Scale of 8° -1.
The Observing System", Monthly Notices of the Royal Astronomical Society, 258
(1992), 605-615.
nWollack
12M.
et al.
Dragovan, et al.
Chapter 4. The Data
67
Figure 3013 shows a plot of <(TrT2)2> vs. 0n-072, which is how the mean square sky
temperature difference varies with the displacement of two beams on the sky. The
variance of the atmospheric noise over these scales grows with the square of the
displacement of the beams on the sky. Equivalently, the rms grows linearly with
displacement. This is consistent with the fluctuations on the sky being, over the angles
observed here, just spatial gradients. Thus, one expects much greater sensitivity to the
atmosphere with a two point chop than with a chop insensitive to spatial gradients, like
a three point chop.
A 20
5
10
Point Separation (degrees)
Figure 30. Angular dependence of sky noise.
13Wendy
Lane, private communication.
68
Chapter 4. The Data
One of the major goals of the SK94 experiment is to probe a variety of angular scales.
As shown in Chapter 3 in Figure 24, the beam is moved on the sky in a triangular
pattern. The radiometer is sampled 64 times per chop. The data are later demodulated
with a synthesis vector, S, as in
US,T,).
By choice of synthesis vector, S,, the relative weighting of each spatial point in the chop
can be set, allowing the synthesis of arbitrary effective antenna sensitivity patterns,
H(\), given by
where a - (l/ yjS\n(2))FWHM is the beam width of the telescope, and X, is the
position on the sky corresponding to synthesis vector element 5,. The synthesized
antenna patterns (or, equivalently, the synthesis vectors) are normalized such that
(9)
J|//(i)|A = 2,
or so that the integral of the area under the positive lobes of the effective antenna
pattern is equal to 1.
Figure 31 shows the family of synthesized beams (for a single chop) used for Q (solid)
and Ka (dashed). The synthesis vectors for the 3 point through 7 point chops used to
generate these curves are orthogonal in time, and have sensitivity to different angular
scales.
While some effort was placed at generating optimal arbitrary synthesis vectors using
non-linear optimization, it was found that using sinusoidal harmonics of the chop as
synthesis vectors provided nearly maximal sensitivity, good suppression of atmospheric
Chapter 4. The Data
69
attenuation, and superior orthogonality than was obtained otherwise. Pure sinusoidal
synthesis vectors are therefore used in this analysis. The three point chop is generated
from a cosine at twice the frequency of the chop (7.81 Hz). The 7 point chop is
generated from a cosine at 6 times the chop frequency (27.34 Hz). While the synthesis
vectors are orthogonal in time, they are not necessarily orthogonal in space.
In addition to this family, the antenna pattern from the SK93 experiment is synthesized
as shown. Figure 31 shows the effective antenna patterns from a single chop. In this
representation, the Ka curves have a smaller amplitude than the Q curves, because of
the normalization in (9): the Ka beam is wider than the Q beam.
The synthesis vectors are given in Appendix A, as are the spatial correlations between
the synthesized beams.
_l l r , i i > , . i i i j u i l | l l l | i . l_ _
0.4 f Synthesized
0.2 r SK93
/"
/'
1 '
1
, 1 , ' ' 1 , ' ' ! ' ' ' 1 ' , ,_
~
y\
0
-0.2
-0.4
0.4
-4->
0.2
t
0
-0.2
-0.4
0.4
0.2
0
-0.2
-0.4
. 1 . . . 1 . , .
-4
-2
0
2
4
-4
Position (degrees)
-2
0
Figure 31. The Q and Ka synthesized antenna patterns.
2
4
70
Chapter 4. The Data
Data are acquired for 20 seconds with the chop centered to the East of the North
Celestial Pole (NCP) by 7.2° in azimuth (4.4° on the sky), and then for 20 seconds
centered to the West (including 4 seconds of base motion). Since the axis of the
chopper is vertical, the beam is chopped in azimuth, and not in declination along lines
of constant RA. The data are averaged into 48 bins (7.5° each) in RA for the 3-9
point synthesis, and into 24 bins (15° each) for comparison with the SK93 experiment
(hereafter, SK930L). For the symmetry tests in the remainder of this chapter, all of the
data are averaged into 24 bins.
_"1' ' V T 1 | 1 -1 1 | 1 1 1 | 1 1 1 -J I 1 —I—r—i——i—r—r |—r—i—i—|- i > i | i i
Synthesized SKjLV *>• f/-"Ox
-I i 03
\
^N ^
L 04
1 Q6
j
-
•
^ j
05
"i
07
-~\<~\ -
i i i . i i i i i i i . i i i i i i i
8
0
2
4
6
8
0, (deg)
-
0
2
4
-
6
Figure 32. Contour plots of Q synthesized antenna patterns.
Binning in RA is included. The NCP is at (0,0).
Since the data are divided into 15° bins for the SK930L analysis, the effect of the Q
and Ka beams being of different widths is greatly reduced, since the width of the
synthesized pattern is dominated by the width of the bin. Figure 32 shows contour
Chapter 4. The Data
71
plots of the Q synthesized antenna patterns with the base pointed to the east, including
binning in RA. The NCP is at (0,0) and lines of constant RA are straight lines from the
NCP. The mirror image of this (0X —» -6X) is seen when the base is pointed to the
West. Thus, close to (but not exactly) the same signal should be seen in the West as is
seen the East ~11 hours earlier (inverted for the antisymmetric chops). The SK930L
antenna pattern is wider (in 0) than the others because it is binned into a 15° bin in RA
rather than a 7.5° bin.
The rms noise of the data after application of a synthesis vector St will depend on the
synthesis vector used. A measure of this is k; defined here as the ratio of the NET after
application of the synthesis vector to the NET of the raw data. Thus,
where N is the number of samples. Table 15 gives vfor each of the synthesis vectors
used in the SK94 experiment. The overall sensitivity of the telescope decreases as the
spot to spot spacing of the telescope approaches the beam width of the telescope. The
effect of this is to increase /r for synthesis vectors with many lobes. Furthermore, it will
be seen in the next chapter, that under the normalization of the synthesized beams (9),
increasing the number of lobes in the synthesized beam further reduces its sensitivity.
Competing with this is the fact that the correlated noise in the receivers (see Chapter 2)
decreases with increased demodulation frequency. The net effect* is to render the Ka
t The figure of merit used here is N
E
T
j
. The NET is given in Table 20 and
's g*ven 'n Table 22 in Chapter 5. This gives the NET referenced to STt (see
Chapter 5).
72
Chapter 4. The Data
7pt chop 2.6 times less sensitive to the anisotropy than the Ka 3 point chop, and the Q
7pt chop 3 .3 times less sensitive.
In principle, the analysis could be continued to beyond the 7 point chop. However,
these are increasingly less sensitive (4 times for 8pt and 6 times for 9pt) to true sky
signals. Consequently, they would have little statistical weight relative to the rest of
the data set. A more important reason for not including them in the analysis is that
given the error bars, it is very difficult to verify that the data are not contaminated at
the <30fiK level necessary, as will be seen.
Radiometer
3 Point
4 Point
5 Point
6 Point
7 Point
SK930L
K,
2.485
3.014
3.940
4.994
7.687
3.361
Q
2.328
2.608
3.015
2.404
4.338
2.953
Table 15. /rvalues for the SK94 synthesis vectors. For a single difference experiment, K= 1.41.
Data Selection and Preparation
While Saskatoon was chosen as the observing sight because of its reputation as a city
with many cold clear days, even there, the skies are not always blue. In fact, January of
1994 had the least amount of sunshine ever recorded in Saskatoon14. While the
telescope was tarped when it was snowing, there was still plenty of time when the
weather was clearly poor, but the telescope was run anyway. During the 6 weeks of
observing, 363 hours of data with the Q radiometer and 234 hours of data with the Ka
radiometer were acquired. By no means all of these data are clean enough to use.
l4The
Star Phoenix (Saskatoon), 2 February, 1994.
Chapter 4. The Data
73
The first level of cutting is designed to remove periods of large atmospheric noise.
This is done by evaluating, for each 15 min segment of data, the mean deviation of 8
second averages demodulated with a two point synthesis vector, which is sensitive to
the horizontal component of spatial thermal gradients. This number is large when the
spatial slope is changing and small when it is stable, and has been found to be a good
indicator of the amount of atmospheric noise. The mean deviation is used because it is
less sensitive than the standard deviation to outliers, which are more likely to be due to
birds or airplanes in the beam than to atmospheric noise. The mean deviation can be
related to the standard deviation by Ax = yf^ax for normal distributions. A more
general expression for the cut levels (used here) is given by
(io)
where a is the standard deviation of the demodulated data, r is the integration time,
and 6eff -J6H(0)dO is the effective throw in degrees. The units of C,are mk • s12/deg.
This measurement of the stability of the atmosphere may be compared from experiment
to experiment. The equivalent detector NET of the cut is given by
NET = £6eff / k .
For Ka94, 6eff /k = 2.25° and for Q94, 6eff /k = 2.37°. The temperature scale for ^is
in antenna temperature, referenced to the horn.
A small amount of data is accepted by the two point cut in which the atmosphere is still
poor. A cut based on the mean deviations of three point through five point synthesized
beams removes this (~2 hours total). We furthermore require that a 15 min segment of
data be accepted only if the 15 min segments before and after it are of good quality.
Thus, the sky must be stable for at least 45 min before any data are accepted at all. The
Chapter 4. The Data
74
amount of data that is kept is presented in Table 16. The cut levels and hours of data
accepted from the SK93 experiment are also included for reference.
Cut Level (C)
Radiometer
SK93-Ka
SK94-Ka
<4.5
130
< 3.0
79
No Cut
SK94-Q
Hours of Data
< 2.5
%
< 1.7
%
No Cut
< 2.5
243
140
107
363
158
< 1-7
98
Table 16. Amount of data at different cut levels.
Figure 33 shows the distribution of the spatial gradient fluctuations at the three
different cut levels. The periods which are removed below the cut level are removed
because of the requirement that the 15 min periods before and after also be of good
quality. The cut levels were selected by looking at the raw data (and looking out the
window) for evidence of poor atmosphere, and then determining at what gradient
fluctuation level this corresponded to. Two cut levels are analyzed to insure that the
placement of this cut level is not critical. Qualitatively speaking, the data that were
kept corresponded to clear skies, and the data that were rejected corresponded to
cloudy skies. There were a few periods when it was very cold with a high thin overcast
that were accepted.
The distributions from Ka and Q have some qualitative differences. The lower cut off
on both distributions is due to the system noise, which is higher in Q than in Ka. The
second hump at around 200
mk-sI%
!g
in the Q distribution is due to data acquired during
a two week period of very poor weather in January. The Ka distribution has more data
75
Chapter 4. The Data
between 5 mk s"(%.g and 10
mk s'%.
g.
It is expected that this is due to a number of warmer
but mainly clear days in the end of February when the Ka data were acquired. The best
data, as expected, were acquired when it was very cold (below -25°C) and clear.
i
100
i
T r,T
,"l"Tj
i
i
i
i
- Q
i i "TT'j
i
i1 t •"!"!
r i|
t
i
i
i
i i-rr
No Cut
C< 2.5
1-7
1
m
;
L
50
J
V_
-O
I
0 -H
50
.
1
; Ka
40
30
r
w
-J
I I 1• + + i — — I
J
20
ii
ti
10
0
i i i
I I I 11
1
1—I
I
I I I 11
1
1—I
I
I
I I I-
-
ir
r-iW]
J
:
L
i
I
J]
r
1
l
.i
1—I
:
L
J
.
^
:
,. i
1
10
100
Gradient Fluctuations (mK s 1/2 /°)
1000
Figure 33. Distribution of spatial temperature gradient fluctuations.
The data are then blanked for base moves (4 seconds every 20 seconds) and for spikes
which may be due to birds, planes, or occasional glitches in the data system. An entire
chop of data are blanked if any of the 64 samples in it vary by more than 3 .5 sigma
from the mean. The amount of data removed in this step (2.6%) is consistent with the
data being Normally distributed. True glitches and spikes are quite rare (a few per
day).
The data are then multiplied by the synthesis vector, and binned according to the Right
Ascension of the center of the chop. Data with the base pointed in the East are binned
76
Chapter 4. The Data
up separately from data with the base pointed in the West. Any 16 sec average which
does not lie within 3a of the mean for the bin is rejected. This removes a further -1%
of the data. Error bars and the channel to channel correlation matrixes are generated
from the distribution of 16 second averages of demodulated data. The correlation
matrixes are listed in Appendix B.
Channel to channel averages are made before the application of the synthesis vector
using weights derived from the channel to channel covariance matrix. For example, to
combine the three Q channels into one, data from the three channels are combined as
Xd'w-
4* = -^—:w = KT'.
2><
1=1
d is a data point, and w, are the weights derived from the covariance matrix, C. The
error bars are generated after the data are combined, which ensures accurate error bars
for combined channel data, even in the presence of large channel to channel
correlations. For the 3pt to 7pt synthesis vector analysis in Chapter 5, the combined
data are used. For SK930L, where frequency information will be important, the
frequency and polarization channels are not combined. For the tests in the remainder of
this chapter the data are combined.
The atmospheric cut has little effect on the signals. As an example, Figure 34 shows
the combined Ka 3 point data at two different cut levels.
77
Chapter 4. The Data
0.2
0.1
0
-0.1
-0.2
0
0
1
20
25
Bin RA (hours)
Figure 34. The Ka 3 point data at two cut levels.
Data Quality
Once the data have been selected and cleaned, their quality is evaluated. Because of
the extremely small signals which are being measured, contamination by a host of
sources, including the sun, ground pickup, electronic pickup, or drifting offsets. The
observing strategy employed provides a variety of symmetries which are probed to
ensure that the data are not being contaminated.
Chapter 4. The Data
15
10
5
I?
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o
15
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10
cl
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15
3 i i ii | i i M | i i i i | i i i i|( i i r ] i i i i | i i i i | i i i i |m
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i | m
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m
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5
„ ] i i i i | i i i i | i i ii | i i i i | i i i r] i i i i | i i i i | i i i i | i i i i | i ii ii
0
K„5
05
15
10
5
0
i ....i .... i .... i
. i .... i .... i i ... i
0.01
0.02
0.03
Frequency (Hz)
0.04
0
0.01
0.02
0.03
Frequency (Hz)
0.04
Figure 35. Noise spectra of demodulated data.
40
30
20
10
I?
o
>
40
| 30
w
20
f
I
0)
t/)
10
0 ^ i it i | i 1 i i | i i i i | i i i i i i I 7 i
o
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40
ri i i
i i t i
i i i i
i i i i
JL
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30
i«
-8
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V)
i i i [ i ii i | i i t i ] i i t i ] i i ii: j r i i i pTT i f i i i i " t iTT | i [ i i:
05
40
JL
K„5
30
20
10
i : * . • 'i
12
3
Frequency (mHz)
4
12
3
Frequency (mHz)
Figure 36. Low frequency spectra of demodulated data.
4
J
Chapter 4. The Data
79
Of primary importance is whether the telescope output remains stable over extended
periods of time. One probe of the instrument's stability is to take a power spectrum of
the demodulated data stream, and look for power at very low frequencies. Figures 35
and 3615 show power spectra of data after multiplication by the synthesis vectors on
two different time scales. These data are for the longest stretches of good data (£ < 2.5
H^^Xeg) and are not blanked for base moves. While the three, four, and five point data
streams are stable to time scales of several hours, the two point data are unstable within
a few minutes. This is almost certainly due to the large amount of power at large
angular scales in the atmosphere, and renders the two point data unusable.
While the offsets (described in Chapter 3 and listed in Table 20) in the BigPlate
telescope are relatively small (being in all cases less than lmK, and for many of the
synthesis vectors, less than 100 (iK) they are still generally large compared to the sky
signals of interest. It is therefore important to verify their stability. The variation of
the demodulated data with time is shown in figures 37 and 38 for 3pt-5pt chops and for
the SK94 overlap data. The data are averaged into 1 hour bins and plotted vs. time. If
there were no significant variation of the offsets with time, and no signal on the sky,
these plots should be consistent with a horizontal line at the level of the mean offset.
Drifts in the telescope offsets, excess noise at low frequencies, or a true sky signal can
cause significant variations.
15Wendy
Lane, private communication
80
Chapter 4. The Data
1
1
1
1
1
•'
1
1
»
'
'
'
|
1
'
I
.
,
0.5
0
1
\
f
h*
l
-0.5
-1
1
0.5 -
4pt
0
-0.5
-1
E,
~
1
5
0.5 —
01
0
,
5pt
i. Sffe & n
-0.5
-1
1
T
0.5 L
0
r fcfji
SK93 Overlap
T
i
1
Ift
-0.5
-1
i
t
1
45
i
•
i
>
50
Day of Year
1
55
Figure 37. Ka band offsets vs. time.
To distinguish between a true sky signal and excess noise or drifts, the data are binned
into 48 RA bins, and the reduced y} of the data is calculated, with the mean of each bin
removed. For the Ka data, the removal of the means left 250 degrees of freedom, and
for the Q data, there remained 280. A line was also fit to the data to measure the level
of a linear drift. Table 17 summarizes the results of these fits.
Chapter 4. The Data
81
1—1—1—1—1—r
1
3pt
0.5
0
-0.5
H—I—I—I——I—I—I—I——I—I—I—I——I—I—I—I——h
- 1
1
4pt
0.5
0
5" -0.5
E
«w
-1
O 1
H—I—I—I——I—I—I—h
5pt
0.5
0
-0.5
-1
H—I—I—I——I—I—I—I——I—I—I—I——I—I—I—h
1
SK93 Overlap
0.5
0
-0.5
1
- 1
15
10
20
•
I
I
I
I
1
25
Day of Year
1
L_
It , 1
JO
35
40
Figure 38. Q band offsets vs. time.
3 pt
4 pt
5 Pt
6 pt
7 Pt
SK930L
0.3 ±0.7
-0.6 ±0.5
0.1 ±0.5
0.1 ±0.6
0.7 ±0.8
0.6 ± 1
nK/day
nK/day
nK/day
uK/day
UK/day
uK/day
X2
1.24
1.79
1.07
0.98
1.05
1.05
Q Slope
-2.8 ±0.4
-2.8 ± 0.4
-1.4 ±0.4
-0.7 ±0.4
0.4 ±0.5
1.7 ± 1
nK/day
nK/day
uK/day
uK/day
(iK/day
uK/day
1.19
1.16
Ka Slope
y2
1.54
1.53
1.08
1.03
Table 17. Offset drift and reduced y} in a bin.
The drifts are all very small, and may be ignored. However, several of the y} entries
indicate some troubles. With -250 degrees of freedom, the reduced y} distribution may
Chapter 4. The Data
82
be approximated as being normally distributed with mean = 1 and a = 0.063. There are
many entries in Table 17 more than 2a from 1.0, so not all of the fluctuations in the
30 min bins can be explained by the standard error derived from the wobble (16 sec)
averages. The worst channel is the Ka-4pt, where there appears to be a 34% noise
excess. A similar effect was seen in the OVRO experiment16 and was attributed there
to atmospheric noise. An obvious solution to this is to make the error bars based on
the distribution of 30 min averages going into each bin. Unfortunately, this would
leave less than 10 samples in each bin from which to generate the errors. The
uncertainly in the resulting error bars would be about the same size as the discrepancy
seen here. Another solution is to simply inflate the error bars generated from the 16
second wobble averages by the amount needed to make the reduced x2 entries in
Table 17 equal to 1. In essence, the relative error bars are then generated from the 16
second averages, and the overall normalization from the 30 minute averages. It is this
approach which is used.
Figure 39 shows the distribution of the 16 second, 30 minute and 60 minute averages
for the Ka 3 point data in the East, with the offset removed. When base moves and
time spent looking in the West are taken into account, only 44 16 second integrations
go into a 30 minute average of the East data. The sky signal has not been removed
from these data, and not all of the 30 and 60 minute averages have a full 44 or 88
wobbles included in them, due to breaks in the data. Nevertheless, it is encouraging to
see that the distributions are still somewhat Gaussian, and are still integrating down.
16A.C.S.
Readhead et al. "A Limit on the Anisotropy of the Microwave Background
Radiation on Arc Minute Scales", The Astrophysical Journal, 346 (1989). 566-587.
83
Chapter 4. The Data
150
100
:
East Wobble averages
L ff= 1.33842 mK
: n = 8602
50
-4
0
-2
4
2
Signal (mK)
- i
•
'
•
i
i
•
•
20 r 30 min averages
15 r- (~44 East Wobbles/bin)
10 L a— 0.280885 mK
E_ n = 357
5
•
•
i
i
i
i
-
i
i
i
I.
-E
JU
n
2
i
P
1"jjbui
Pi 1
...
—i-Jl
-
l
[U
J1
1
0
p
J
i ~
1
2
Signal (mK)
i
r
:
:
^
i
i
i
|
i
i
i
'
60 min averages
(~88 East Wobbles/bin)
a= 0.231054 mK
n = 200
^ '
'
'
-
'
2
'
'
-
1
'
'
1 ;
'P
rW
r* 1 " . ,
^
1
;
-
r
^
|I
, .
i
0
1
,1
I
2
Signal (mK)
Figure 39. Ka 3 point distributions at different time scales.
A number of other tests are performed to insure that the data were not corrupted.
Though the BigPlate chopper is designed to not shake the detector, there is still some
coupling. And while the 1.2 Hz vibration from the mechanical refrigerator which cools
the detector (see Chapter 2) is asynchronous to the chop, it also could couple to the
radiometer. Therefore it is of interest to determine whether any microphonics
contaminate the data. In general, the phase of any microphonic vibration of the
telescope will not necessarily be of the same phase as a sky signal.
Ka
3 pt
4 pt
5pt
6 pt
7pt
1.2
0.7
1.0
1.2
1.3
0.7
1.0
Q
1.1
1.4
1.1
Table 18. Reduced y} with 23 degrees of freedom for quadrature phase tests
Chapter 4. The Data
84
The data are demodulated with pure harmonics from twice the chop frequency to 6
times the chop frequency whose phase was adjusted to minimize sensitivity on the sky.
The data are binned on the sky in the same way as the sky sensitive data, but in 24 bins
(rather than 48) to increase sensitivity. Table 18 summarizes the reduced %2 with 23
degrees of freedom of the binned data. All of these are consistent with no signal.
There is an offset in the telescope which is due to the emission of the chopping plate
(see Chapter 3). Additionally, the chopper near where coils are mounted is warmed by
about ~3K. A change in either the absolute chopper temperature, or the temperature in
front of the coils relative to the chopper average can cause a change in the offset.
Because of these concerns, the chopper temperature is measured at several locations as
data were acquired.
The chopper temperature and the center to coil difference are both regressed out of the
data to determine their effect. When data are selected at a higher cut level than are
actually used in the analysis (£<5.0
a small but statistically significant
correlation is found between the plate temperature and the signal, but when only the
data which are actually used are fit, (£<2.5
mk"sI%.
g)
no significant correlation is seen.
In neither case does regressing out the plate temperature have a significant effect on the
final form of the binned data. No correlation is found between the data and the
chopper center to coil temperature difference.
A misalignment in the vertical axis of the chopper can also cause an offset, which
changes as the sky temperature changes. Since the data system is DC coupled, the
absolute sky temperature is known and was regressed out of the data. There is no
significant correlation between the sky temperature and the demodulated data, and its
removal has no significant effect on the binned data. Since no dependence on the
Chapter 4. The Data
85
binned data are seen with either the chopper or sky temperatures, they are not
regressed out in the final analysis.
Once the demodulated data have been cleaned and binned in RA, there are a number of
symmetries which are probed to verify that any signal seen is actually on the sky.
Since data are acquired alternatively with the telescope looking to the East of the NCP
and to the West of the NCP, the celestial signal seen in the East and the celestial signal
seen in the West -11 hours later are very nearly equal (a small difference can be
expected since the chopper moves in azimuth, and not declination, as discussed earlier).
Any celestial signals seen by the Ka radiometer and by the Q radiometer will also be
nearly equal (a small difference can be expected since the Ka and Q beams are not
equal). Figures 40 and 41 show the data acquired in the East and in the West for the
Ka and Q band data. They are plotted vs. the RA of the center of the chop. Thus, as
plotted, the East and West curves ought to line up. To increase signal to noise for
these symmetry tests, the data have been averaged into 24 bins of 15° each.
For the synthesized patterns of odd symmetry (4 and 6 point chops), the negative of the
signal seen in the East should be seen in the West. This has been taken into account
here. Figure 42 shows the combined Q East and West data compared to the combined
Ka East and West data.
In Figures 40 through 42, Pearson's r is shown as a function of lag between the two
signals. If the two signals were normally distributed, uncorrelated fields, then r would
Chapter 4. The Data
86
have a mean of zero and standard deviation of 0.217. This does not hold if the
distributions are not normal however, which may be the case in the presence of true sky
signals. Thus, many of the plots of r appear rather bumpy. It is therefore unwise to
make statistical statements from Pearson's r given here. Still, large values of r mean
large correlations between the signals. For data with %2 significantly greater than 1
(i.e., greater than ~1.6. See Table 19) Pearson's r favors zero lag. For the E vs. W
plots, this indicates that the same thing is seen with the telescope pointed East, as is
seen 11 hours later with the telescope pointed in the West, consistent with the signal
being on the sky. For the Ka vs. Q plots, a peak at zero lag indicates that the same
thing is seen with both radiometers. Since the Q and Ka data were acquired
consecutively over a two month period, if the signal were locked to solar time (rather
than sidereal time), it would be shifted by several bins. It is not, consistent with the
signal being on the sky.
17W.H.
Press, et al., Numerical Recipes in C, The Art of Scientific Computing,
(Cambridge: The Cambridge University Press, 1988), 503.
87
The Data
Radiometer Signal
Pearson's r vs Lag
I i i i i I i i i i I i i i i I i i i i I i i i i I i I i i i i I i i i i I i i i i I i i i i I i i
200
1
0.5
0
0
-0.5
•200
111111111111111111 i i 11111 ;11 i i 1111111111111 i i 11111
200
1
0.5
0
0
-0.5
•200
11111ii11111111111111111 ;1111111111111111111111 ii
200
1
0.5
0
0
-0.5
200
1111111111111111111111111 ;111111111111111111111111
200
1
0.5
0
0
-0.5
•200
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I I 11111 ;1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
200
1
0.5
0
0
-0.5
200
1111111111111111111111111 ;111111111111111111111111
200
1
0.5
0
0
-0.5
200
11' • • i ' 11' i • ' ' ' 11 • • • i • ' • •
111111111111111111111111
0
-10
5
10
15
RA (Hours)
20
-5
0
5
Lag (hours of RA)
Figure 40. The Ka East and West data compared.
10
88
Chapter 4. The Data
Radiometer Signal
I i i I.I I i i i i I i i i i I i i i i I
Pearson's r vs Lag
i
i i
L i
[
<
i
i
1
i |
i
1
I | I
i
i
i | i
i
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0.5
0
a.
ro
-0.5
-200
I I I i [ I I I I 1 I I I I|I i I I|I I I I
11111111111111 M 11111111
200
1
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a.
-0.5
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I I I | I I I I | I I II | I I I I | I I I I
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200
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0
ina.
-0.5
-200
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1
0.5
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-200
I I I I|I I I I|I I I I|I I I I|I I I I
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200
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i i i i i I i i i i I i i i i i i i i i iI i i i i iI i i
10
15
RA (Hours)
20
-10
-5
0
5
Lag (hours of RA)
Figure 41. The Q East and West data compared.
10
The Data
89
Radiometer Signal
Pearson's r vs Lag
1
200
0.5
0
0
-0.5
•200
| I I I I | I I I I | I I I I | I I I I | II I I j I | I I I I | I I I I | I I I I | I I I I | I I
200
1
0.5
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0
-0.5
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|I I I I|I I I I|I I I I j I I I I|I I I I j I|I I I I|I I I I|I I I I|I I I I|I
200
1
0.5
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200
1
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|I I I I|I I I I|I I I I |I I I I| I I I I j I |I I I I |I I I I|I I I I| I I I I|I
200
1
0.5
0
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200
1
0.5
0
0
-0.5
-200
0
5
10
15
RA (Houre)
20
-10
-5
0
5
Lag (hours of RA)
Figure 42. The Ka and Q data compared.
10
90
Chapter 4. The Data
The same region of the sky is observed in the SK94 experiment as was observed in the
SK93 experiment, so with proper choice of synthesis vector, the data from the two
year's experiment are compared. Figure 43 shows the SK93 Ka along with the SK94 Q
and Ka data demodulated with the SK930L synthesis vectors. The three curves look
very similar, once again giving reassurances that the signal is real and on the sky.
200
100
D
C
gi
in
100
K o 93
K 0 94
-200
Q 94
-300
20
25
Bin RA (hours)
Figure 43. SK93 Ka and SK94 Q and Ka data.
Table 19 reports the reduced x2 obtained from the sum and difference of each of the
pairs presented in Figures 40 to 43. Also included is A - B for the Ka data. If the noise
is Gaussian, then the %2 entries for the channel differences (e.g., E - W, Ka - Q, or A -
B) should be distributed around 1 ± 0.3, as they are for all but some entries in the 8pt
and 9 pt chop data, indicating that the telescope sees the same signal for each of these
pairs. The y} entries for the Ka + Q combinations are large, indicating highly significant
signals on the sky. Spatial correlations have not been taken into account in the entries
91
Chapter 4. The Data
involving averages of channels (i.e., Ka + Q or Ka East + West), so it is not
appropriate to draw statistical conclusions from these numbers, except to say that the
signals are inconsistent with zero signal and normally distributed noise. These
symmetries serve as compelling evidence that the signals being observed are truly
locked to the celestial sphere.
Also listed in Table 19 is the rms amplitude of the sky signals naively predicted from
the data by simply subtracting in quadrature the error bars from the rms of the data. A
more elaborate analysis including correlations on the sky is presented in the next
chapter.
Ka
Ka
Ka
Ka
Q
Q
Q
Ka+Q
Ka-Q
Ka+Q
E+W
E-W
A-B
rms
E+W
E-W
rms
r
X2
rms
y-
Y.2
y2
(pK)
X2
y2
(UK)
3pt
4.6
0.9
1.0
53
2.3
0.7
55
6.2
0.8
54
4 pt
2.1
0.8
l.i
27
2.0
l.i
42
3.0
1.1
31
5 pt
2.9
0.8
0.8
30
1.1
l.i
12
3.2
0.7
28
6 Pt
1.6
1.2
0.9
17
1.1
0.7
3
1.8
0.8
18
7pt
1.2
1.2
1.3
15
1.9
0.8
46
2.0
1.1
27
8 Pt
1.4
1.6
27
1.2
1.0
21
1.8
0.8
30
9 pt
1.1
0.9
0
2.0
1.5
62
1.6
1.1
34
SK93
5.2
0.8
52
3.2
0.9
66
7.0
1.3
52
Chop
(MK)
OL
Table 19. Signal reduced %2 with 23 degrees of freedom and naive sky rms
For reference, the 8pt and 9pt data are included in Table 19. For the Ka 8pt data, the
E-W entry has a larger reduced x2 than the E+W entry. This may be a completely
benign 1 in 20 occurrence, or may be an indication that the data are contaminated. An
additional worry is that a plot of Pearson's r vs. lag for these data does not show a peak
at zero lag as one might expect. The Q 9pt E+W data shows a large y} of 2.0,
92
Chapter 4. The Data
indicating a significant detection. However, the E-W entry here is also large, and the
plot of Pearson's r vs. lag shows no maximum at zero lag, where it should if the signal
were locked to the sky. Since these data have very large uncertainties, a small error in
the error bars will result in a large change in the predicted amplitude of the sky signal.
Because of these concerns, the 8pt and 9pt data are not used in the final analysis in the
next chapter.
Table 20 reports the offsets and sensitivities on the sky for each of the chops. The
offsets and sensitivities here are reported referenced to above the atmosphere and in the
temperature scale of the CMBR.
Sensitivity (mK*s1/2)
Offsets (|iK)
Ka
Ka
Ka
Q
Ka
Ka
Ka
Q
A
B
All
B
A
B
All
B
3 pt
-109
556
116
685
3.9
4.3
3.6
5.6
4 pt
12
-47
13
56
3.3
3.6
2.7
5.1
5 pt
57
223
129
278
3.5
3.8
2.8
5.1
6 pt
4
7
7
12
4.4
4.6
3.4
5.5
7 pt
15
66
37
77
5.9
5.8
4.3
6.2
3.9
3.3
4.4
100
-320
-80 -360
Table 20. Offsets and sensitivities on the sky.
5.7
SK93
Summary
The SK94 experiment ran for six weeks in January and February of 1994 and acquired
298 hours of usable data. Two radiometers were used, giving frequency coverage
between 26 GHz and 46 GHz. Data are selected on the basis of the stability of the
horizontal component of the atmospheric spatial gradient (see Table 16).
Chapter 4. The Data
93
The telescope was calibrated using the super-nova remnant, Cas-A. A 14% absolute
calibration uncertainty is arrived at, which is dominated by uncertainty in temperature
scale of Cas-A. The channel to channel uncertainties result in a spectral index
uncertainty of 0.2 between radiometers and 0.1 within a single radiometer.
By appropriate choice of synthesis vectors which are orthogonal in time, 3 point
through 7 point effective antenna patterns are synthesized. Chopping is in azimuth, or
approximately along lines of constant RA. Data are acquired with the chop centered
4.4° to the East of the NCP, and then 4.4° to the West of the NCP. For the 3 point
through 7 point chop analysis, the data are binned into 48 RA bins. A synthesis vector
designed to mimic the effective antenna pattern from the SK93 experiment is also
applied to the data.
A variety of symmetries are available to verify that the signals observed are fixed to the
sky. Signals acquired with the base pointed to the West are repeated 11 hours later
when the telescope is pointed in the East, as required. The signals seen in the Q
radiometer are consistent with those seen by the Ka radiometer. The signals seen with
vertical polarization in the Ka radiometer are consistent with those seen with the
horizontal polarization. And the signals acquired by use of the SK93 overlap synthesis
vector are consistent with those acquired in the SK93 experiment.
Chapter 5
The Analysis
Introduction
In the previous chapter, the calibration, acquisition, selection, and verification of the
data were discussed. Having found confidence that the observed signals are truly on
the sky, it is now appropriate to see what the data can say about the anisotropy in the
CMBR.
The first issue is to verify that the anisotropy is due to the CMBR, and not to
foreground contamination. To do this, the extended frequency coverage of the
telescope will be used to verify that the fluctuations do have the spectrum of the
CMBR. Since the sky coverage of the SK94 experiment overlaps the coverage of the
SK93 experiment, both years' data are used in this analysis. This has the added benefit
permitting verification of the SK93 experiment.
Once it has been shown that the fluctuations are due to the CMBR, the spatial
spectrum predicted by the data can be studied. The shape of the spectrum, as well as
its normalization, can be used to differentiate theories of early structure formation. By
probing a large range of angular scales with one experiment, the effect of calibration
uncertainty may be minimized. Since the Ka radiometer observes with two orthogonal
polarizations, it is also possible to place limits on the polarization of the CMBR.
94
Chapter 5. The Analysis
95
Likelihood Analysis
In this analysis, the determination of parameters, (such as the amplitude of the
fluctuations, and their frequency and spatial spectral indexes) will be made using
Maximum Likelihood tests. The likelihood is defined as
_ exp(- + t r M-'t)
(2^ /2 |M| 1/2
where t is a data set and M is a corresponding covariance matrix. L is the probability
density that a system described by M would produce t. In a Maximum Likelihood
analysis, M is considered unknown, and is written as a function of one or more
variables. The variables are adjusted to find the maximum value of L. This is taken as
the most likely model, and the corresponding variables are taken as being at their most
likely values. Confidence intervals for the parameters can be estimated from the
distribution of the Likelihood.
For the Saskatoon analysis, the elements of the data vector t, correspond to data taken
looking at a specified place in the sky with a specified antenna pattern with a specified
frequency and polarization. The covariance matrix M is expressed as
(12)
Here,
describes the noise, and is derived from the data, while (s,sj) describes the
covariance due to the sky, including both spatial and frequency effects, s, is the
convolution of the effective antenna pattern H : (x) with the true signal on the sky, T(x),
(s,sj) = (J i/x1Ji/x2//,(x1)7,(x1)//J(x2)7,(x2))
= jAj«fr2tfi(x1)//J.(x2)(7'(x1)7Xx2)).
Chapter 5. The Analysis
96
The effective antenna patterns //,(*) in (13) and following are the same as those
described in Chapter 4 and shown in Figure 32. Note that there are effective antenna
patterns centered in 48 positions (24 for SK930L data) around the NCP, for data
acquired in the East and West, for two radiometers, and for 5 different synthesis
vectors. An alternate notation might be to write the effective antenna patterns as
H*p(Xt ,x), where X, reports the position on the sky of the center of the chop, b refers
to the base position (either East or West), r refers to the radiometer (either Ka or Q),
and p refers to the chop (3pt to 7pt or SK930L). Rather than carrying around all of
these parameters, just remember that the / denotes some combination of them.
The correlation function (r(x,)7"(x2)) describes the true fluctuations on the sky. As
mentioned earlier, one would like to know the amplitude of the fluctuations on the sky,
their frequency spectrum, and their spatial spectrum. The procedure that will be
followed in this analysis is to express (T{x] )T(x2)) as a function of the parameters of
interest, and then vary the parameters to find the maximum value of L.
Ideally, one would express (7\x,)7Xx2)) as a function of all three parameters
(amplitude, frequency spectral index, and spatial spectrum), and then search the three
(or more) dimensional space for the maximum likelihood, but given the computational
complexity of evaluating the likelihood, this is not tractable. A less ambitious (but
equally valid) approach is to first verify that the fluctuations on the sky have the
frequency spectrum of the CMBR, by assuming a fixed spatial spectrum and then
varying the frequency spectrum and the amplitude of the fluctuations to find the best
values. Once this has been done, the frequency dependence can be fixed and the spatial
spectrum allowed to vary.
Chapter 5. The Analysis
97
Frequency and Amplitude Analysis
The analysis involving the determination of the frequency spectrum and amplitude of
the fluctuations used in the analysis of the SK94 data is essentially identical to the
analysis used for the SK93 data set.1 The SK94 data were first demodulated with a
synthesis vector that produced a synthesized antenna pattern very close to that of the
SK93 data (see Chapter 4). This allows the data from both years to be compared and to
be analyzed together, increasing the statistical weight of the data set.
The procedure used is to express the true signal on the sky, (T(x ] )T{\ 2 )), as a
function of the root mean squared (rms) amplitude of the fluctuations and of frequency,
but as having no spatial correlations. In other words,
(14)
<r(X,)7-(*2)) = A2(££)'<5(I, -X ).
2
The two free parameters are A, which is a measure of the amplitude of the spatial
fluctuations, and ft, which is the spectral index of the fluctuations. If the fluctuations
are entirely due to the CMBR, then ft- 0.
and f2 are the frequencies of the two
samples, and f0 is a normalization frequency. For this analysis, it is taken to be the
lowest frequency used. It is 38.2 GHz if only the Q band data is being used, and is
27.5 GHz if any of the Ka data is used. It is important to normalize the frequencies as
shown to de-couple the dependence of L on A from its dependence on /?. If the spatial
covariance matrix (-V^) is renormalized to be the correlation matrix, as
(•s,^ = {slSj)/^M^ with /?=0 and A=l, then A will be the rms of the data due
to fluctuations on the sky.
Pollack, et al.
98
Chapter 5. The Analysis
A few computational notes are in order. First, in the calculation of
) (equation
(13)) using (14) for (r(xt)T(x2)), the terms involving A and >5can be factored out of
the integral in (13), so that the spatial correlation matrix need only be evaluated once.
Second, the matrixes involved are very large. Since there are 15 frequency/polarization
channels (6 from each of Ka93 and Ka94, and 3 from Q94) each observing 48 spots on
the sky (24 in the East and 24 in the West), t, is 720 elements long, and M is a 720x720
matrix. Simply evaluating (11) by naively inverting M, and finding its determinant
would be both computationally very costly and subject to roundoff. Fortunately there
is a better approach that uses Cholesky Factorization.
The key point is that M _1 is not needed, but only t T M-'t and |M|. Since M is
symmetric, it may be factored using Cholesky factorization2 into M = NTN, where N is
triangular. Thus |M|1/2 = trace(N). The argument in the exponent of (11) becomes
t r M- 1 t = t 7 "(N r N)- , t
= t r N- 1 (N r )- , t.
Now solve for y = t r N _1 , by finding y in yN = t. This may be done quite quickly, since
N is triangular. Thus, trM"'t = yyr. This technique for solving (11) is much faster and
much more stable than directly finding the inverse of M.
A contour plot of the Likelihood as a function of A and P for the combined Ka93, Ka94
and Q94 data sets with the high atmospheric cut level is shown in Figure 44. The
Likelihood has been normalized so that the peak value is 1, since the only thing of
interest is relative likelihood. The contours are at 0.8, 0.6, 0.4, and 0.2.
2W.H.
Press et al., Numerical Recipes in C, 2d ed. (Cambridge: Cambridge University
Press, 1993)
99
Chapter 5. The Analysis
Dust Emission
CMBR
Free-free Emission
20
40
60
80
100
Figure 44. Likelihood contours for rms and frequency spectral index analysis.
Contours are at 0.8, 0.6, 0.4 and 0.2. The Ka93, Ka94 and Q94 data sets
at the upper cut levels are used.
There are two approaches for obtaining confidence limits for A and p. One is to reduce
the distribution from having two independent variables to having only one by simply
fixing one of the variables at its most likely value. The other is, for each value of one
variable, to integrate over all possibilities of the second, again reducing the distribution
to being the function of only one variable. In both cases, the confidence limits for the
remaining variable are obtained in the usual way. Because of the symmetry of the
likelihood distributions presented here, both techniques give essentially the same
confidence limits.
Chapter 5. The Analysis
100
Table 213 summarizes the results of the analysis, with each of Ka93, Ka94 and Q94
analyzed separately, and then together. These data show a significant anisotropy. The
difference between the Ka93 and Ka94 data is also analyzed and is consistent with zero
as expected if the signals were the same between the two years. This adds to the
confidence that the signal observed is truly on the sky. The analysis is also done with
two different atmospheric cut levels as discussed in Chapter 4. The results are
consistent with each other.
The spectral index of the fluctuations is consistent with that of the CMBR (J3 = 0), and
not consistent with that of the most likely sources of foreground contamination, freefree emission (/? = -2) and dust (J3 = 1.7). The rms fluctuations due to foreground
sources are estimated to be ~9 juK for this sky coverage.4 Additionally, since most
sources have a flat or falling spectrum in flux, and would thus have a spectral index of
P <« -2, the fluctuations are also not consistent with being due to sources. The most
compelling explanation for the anisotropy is that it is due to anisotropy in the CMBR.
This assumes that all of the anisotropy has a common spectral index. However, rough
limits may also be placed on a simple two component (free-free and CMBR) model.
Assume that the two components add in quadrature, and that
&CMBR/&FF -
iPff ~P)IifiCMBR ~P) Then with
-0.9, which is the more pessimistic
la limit on p from the low atmospheric cut {C,< 1.7
3Taken
mK s'^
eg)
analysis of all three data
from Netterfield et al., "The Anisotropy in the Cosmic Microwave Background
At Degree Angular Scales", Submitted to The Astrophysical Journal, Letters (October,
1994).
4Wollack
et al.
Chapter 5. The Analysis
101
sets in Table 21, one finds ACMBR ~ 35 fiK and A^= 27 ^K. With the limit on /?from
the higher cut data, the maximum contribution due to free-free radiation is negligible.
Data Set
Ka93
Ka94
Q94
Ka93 + Ka94
(UK)
3.0
44!JO
-0.69
4.5
37-
-0-28!?S
+13
42
^-10
36-
1.7
52-
-0.77:^
52-
2.5
49-
-0 .70:2s
47-
1.7
52:;:
-5.97;!f
48—
2.5
6i-
+1.35l!4V4
57:|;
3.0/1.7
49-12
-0.62^
47-
44 +11
+048
-0
43-0
v nj
54
42-
1.7/1.7
51-
+056
U'54
Jn-0
-0
69
47-
2.5/2.5
48-
0 1io+044
-0.56
44-
3.0/1.7/1.7
47:y
44:n
-0 40+044
-0.52
0
]
7+0
42
Tyj. 1 / _045
44—
4.5/2.5
Ka94 + Q94
Ka93 + Ka94 + Q94
AT;
Cut Level, C,
(-='%)
4.5/2.5/2.5
^ r m s
P
(UK)
42-
Tab e 21. Summary o ' spectral index and rms analysis
The combined Ka93, Ka94 and Q94 data set, analyzed with the upper cut levels, is
taken as the most reliable measure of the rms since it contains the most data and thus
has the lowest error bars. Including the calibration uncertainty of 14% results in a rms
of 44— iiK.
While the full SK93-SK94 data set has more than three times the data as the SK93 data
set alone does, the error bars on the full data set are not V3 smaller than the SK93 data
set. This is because the errors are not dominated by instrument noise. Rather there are
three approximately equal terms. The 14% calibration uncertainty results in a 6.2
error. The sample variance given the number of independent fields that were observed
is 6.5 (iK. Instrument noise contributes the remaining -5.5 ^K. Reducing the
Chapter 5. The Analysis
102
uncertainty in the rms in future experiments will require the improvement of all three of
these terms.
The last column in Table 21 reports the amplitude of the fluctuations as
This is a more experiment-independent parameter than the rms, and is described in the
next section.
The Spatial Dependence of the Fluctuations
Having obtained confidence that the fluctuations seen in the data have the spectrum of
the CMBR, and do not have the spectrum of any known astrophysical foreground
sources, it is appropriate to attempt to glean information regarding the spatial power
spectrum of the fluctuations. It is the spatial spectrum that will prove to have relevance
in distinguishing between cosmological models.
Since the sky is a sphere, it is appropriate to expand the temperature fluctuations on the
sky, T(x), in spherical harmonics, as
e,m
Assuming rotational invariance, and invoking properties of the spherical harmonics, one
finds that the correlation function of the sky can be expressed as5
5P.J.E.
Peebles, Principles of Physical Cosmology, (Princeton: Princeton University
Press, 1992), 517.
Chapter 5. The Analysis
103
(is)
i
where 0 n is the angle between x, and x2, P f (cos(0)) represents the Legendre
polynomials, and ct = ^|a"'| ^ The c/s are a representation of the spatial spectrum of
the sky fluctuations, with low C's representing large angular scales, and high P. 's
representing small angular scales.
Inserting (15) into (13), one finds for the spatial part of the covariance matrix,
{S,Sj) =
=
4 ^J
(
*
+
1)
H
J
(X2 )Z (2f' +
I A 2 H > ^X1
" *2 )
i (*2
(*r *2)
Introducing the definition
(16)
WjJ = Ji/xJi/x2//1(x1)//J(x2)/>,(x1 x2)
one arrives at
(I?)
for the spatial correlation matrix. This is a natural form for parameterizing the spatial
power spectrum cf, and applying a Maximum Likelihood test, but before doing so, it
will be illuminating to first examine some properties of the window function, (16) and
the accompanying form of the correlation matrix, (17).
Note first that given a theoretical spatial power spectrum, ce, and the appropriate
window function, Wt', one can glean from (17) the predicted rms one would expect
from a given data set (as in the analysis in the last section), since
(18)
rms = J(s~s) = ^£(2* +
•
104
Chapter 5. The Analysis
This has been the traditional use of the window function, and often when the window
function is presented, it is presented as the special case W('. Here also, unless
otherwise noted, 'window function' will refer to Wt'.
5
0
_l
0
100
X
j.
200
300
Multipole Moment L
Figure 45. The SK93 and SK.930L window functions.
Note also that Wt' is a measure of the sensitivity of the effective antenna pattern H(x)
to fluctuations at different angular scales, and is thus a good indication of the angular
scales at which a chop pattern is sensitive. Figure 45 shows the window functions from
the synthesized beams analyzed in the previous section. While the three window
functions are not identical, they are very similar. Since the rms in (18) is proportional
to -yjwf, this is only expected to have a 4% effect between the three data sets, and is
ignored.
Chapter 5. The Analysis
105
Figure 46 shows the window functions for the 3 point to 7 point chops for Q94 (as
described in Chapter 4). It is seen here that the SK94 experiment is sensitive to a wide
range of angular scales, from i « 30 to t » 170.
3pt
— Q Window Functions
K„ Window Functions
4pt
5pt
6pt
7pt
0.5
0
50
100
150
I
200
250
Figure 46. The window functions
It is also illuminating to see how the 'delta function' correlation matrix used in (14)
relates to the correlation matrix in (17). Equating the spatial parts of (14) and (17)
(and neglecting any constant terms since (14) is normalized anyway) gives
(19)
£(2* + l)c,P,acflcos(012)).
106
Chapter 5. The Analysis
Pf is an orthogonal set of functions, with ||/^||2 = 2/(2^ +1), and Pe (1) = 1,6 Thus, (19)
is just expressing <5(cos(012)) as a generalized Fourier expansion in P(. Solving for the
generalized Fourier coefficients, ce yields
c(
oc J Pf (cos( 0))S(cos( 0))d(cos( 0))
oc 1
So the 'delta function' correlation matrix in (14) is equivalent to using (17) with
constant ce.
It is now appropriate to use (17) to build a covariance matrix for use in a Maximum
Likelihood test. Since the spatial spectrum of the sky is directly available as c(, one
need merely choose a parameterization for ct and proceed as with the rms and spectral
index analysis in the last section. An obvious parameterization is to express ct as a
power-law, as c( = A
with A and n as the independent parameters. However, it has
been recently suggested that expressing the spectrum as
(20)
<57; - ^f\2i + X)ctjAn
is more natural than using ce explicitly. Many theories (and the measurements) expect
that ST( should be essentially flat over a wide range of f,7, and since the rms in (18)
also is proportional to
the square root makes the mapping of the error bars from
rms to STt much more natural8.
6E.
Kreyszig, Advanced Engineering Mathematics, 6th ed.(New York. John Wiley &
Sons,1988), 231.
7J.R.
Bond, "Cosmic Structure Formation & The Background Radiation", Proceedings
of the IUCAA Dedication Ceremonies held in Prune, India, 28-30 Dec. 1992.
8P.J.E.
Peebles, "Semiemperical Cosmogonies", Lake Louise Winter Institute
proceedings preprint, February, 1994.
107
Chapter 5. The Analysis
Combining (18) and (20) the rms can be related to ST( as
rms
frSTWIt
Making the approximation that STe is constant over the spatial band-pass defined
by Wl', one arrives at
(21)
rms = ST,J^'/e
The appropriate value for te is the mean value of t, weighted by Wt' /C, or
(22)
2X
t
e. =
In the last section, in column 5 of Table 21, (21) was used to find <57^ from the rms. In
this section, STt will be found by parameterizing it and then maximizing the likelihood.
It is natural to express <57^ as a power law:
(23)
(tV
STe=STti j-J
The i dependence is normalized to t e , which serves to decouple the dependence of the
Likelihood on ST(f from its dependence on m. When only one chop pattern is being
analyzed, te is evaluated using (22) directly. When more than one chop pattern is
used, the sum is extended over the window function for each synthesis vector,
weighted using the noise of the data channel (from Table 20 in Chapter 4) as
i
i
With this parameterization, then, we have
108
Chapter 5. The Analysis
^sr,y(m.r
ce
£(2i+\)
and
(24)
M-2#(£)V
One small caveat to this is that (23) diverges in an unphysical way at small f. when m is
negative. This problem is dealt with by cutting off the power law at f, = 10, so that for
f, <10, ST( = ST, (10/P-J". This has very little effect on the results of the analysis, since
the window functions have very little power below £ =10.
The approach that is followed, then, is to use (24) to build the covariance matrix M t j in
(12), and evaluate the likelihood (11), varying 5T(t and m to find the maximum of the
likelihood, as in the previous section. The temporal correlation matrix, atJ is diagonal,
since the synthesis vectors are orthogonal in time. In this case, it has been assumed
that the signal is all CMBR, so the frequency channels have been combined into one
signal for each radiometer before analysis. This was done before the application of the
synthesis vectors or error bar generation, as described in Chapter 4, so that the effects
of channel to channel noise correlations are intrinsically accounted for.
Thus, the data vector t, for the full analysis, contains data from 2 radiometers, with 5
synthesized beams, each with 48 positions in the East and 48 in the West, so t, is 960
elements long, and the covariance matrix M is 960x960.
The window functions W? are calculated beforehand using (16), so that they need only
be evaluated once, and not once for each ST(e, m combination investigated. Also, given
the symmetries of the problem (M is symmetric which gives about a factor of 2,
and W'f' depends only on the relative spacing of Ht and Hp which gives about a factor
Chapter 5. The Analysis
109
of 48), only -10,000 window functions need be evaluated, rather than the full 9602 one
might expect. Nevertheless, it is the evaluation of W? which consumes the most time
(-48 hours on a DEC Alpha 3000-300, which has a spec_fp floating point index of 82).
0 and K0 beams (3 pt to 7 pt) la = 82
CM
,
CM (
1
0
1
,
I
,
20
,
,
|
|
40
,
,
J
|
60
1
,
I
80
100
6Tt (mK)
Figure 47. The amplitude and spatial spectral index likelihood contours.
Contours are at 0.8, 0.6, 0.4, and 0.2.
Figure 47 shows plots the likelihood contours for this analysis, using the Ka and Q
data, and 3 pt through 7 pt beams, with the upper cut level. The data here tends to
favor a mildly falling spectrum, although a flat spectrum is also consistent with the data.
Table 22 summarizes the results of several combinations of data sets and cut levels.
They are all consistent with each other. The results of the last row (both radiometers,
3 pt - 7 pt chop, high cut level, and using the 30 min based error bars) are taken as the
best measurement of the parameters. The uncertainties do not include the 14%
calibration error.
Chapter 5. The Analysis
Ke
(UK)
Spectral Index
74
45:5
-0.22:^
3 pt - 6 pt
79
42:'
3 pt - 7 pt
86
4i:67
-0 5"03
76
47
Synthesis
Atmospheric Cut
Vectors
(0
3 pt - 5 pt
Q
3 pt - 5 pt
2.5
2-5
mK
%
g
81.4
3 pt - 6 pt
3 pt - 5 pt
J 7
- l l
0
-06
—Q 3+06
v
*
J
-0.5
86
74
48:;°
-o r04
48:^
-0 3+04
48::
-0 7+0-
1-7
43^
-0 3+0,4
2-5
43:1.
—0 5+04
1-7
2-5
3 pt - 6 pt
39-lfJ
-04
2+06
45+s
3 pt - 7 pt
Ka&Q
-9
^ '-14
Spatial
1
K,
*.
1
0
O O+
Radiometer
110
2-5
79
-13
0
w
r05
- *-06
y J
-
-06
^-05
'-04
16s Error bars
3 pt - 7 pt
2-5
82
4 9 :
s
-05
J
-0.4
-0 5+03
-04
16s Error bars
1-7
44::
0
1
0 0
f
1
2-5
45:^
-0
4+03
-04
Table 22. Results of the amplitude and spatial spectral index analysis.
The errors do not include the 14% calibration uncertainty.
Also included in Table 22 is the analysis done with the error bars based on the scatter
of 16 sec averages, rather than on the scatter of 30 min averages, as discussed in
Chapter 4. The results using the two error estimation techniques are consistent with
each other, although the results with the (smaller) 16 sec based error bars tend to favor
a slightly larger ST(e, and a slightly more negative spatial spectral index.
An easier way to visualize the spatial spectrum predicted by the data set is to analyze
each of the synthesis vector outputs separately. As shown in Figure 1, each synthesis
vector probes a different range of t s. Table 23 lists ST(e and the range of t s probed
for the signals from each of the synthesis vectors, including the combined SK93 overlap
Chapter 5. The Analysis
111
analysis from Table 21. The limits on t delineate the region where the Window
Function is greater than e~°5 of the maximum. For cases when the likelihood for STf c is
not clearly distinguished from zero, the maximum of the likelihood, and the 95%
confidence upper limit are given. The 14% calibration uncertainty, which effects all of
the entries equally, is not included.
Also listed in Table 23 is
Wt' ji for each of the chop patterns. This is useful for
converting between rms and ST(e using (21). The rms is also given. The results here
are very similar to the naive estimates in Table 19 in Chapter 4.
Synthesis Vector
3T(e (nK)
j?
Ka3pt
43^'
Q3pt
sr;;
56:?;
60%
SK930L
42_+r
Ka4pt
33:;0
0 4 pt
Ka5pt
47+18
-19
41:;4
0 5 pt
(18, < 5 1 )
Kg6pt
33-n
0 6 pt
Ka7pt
rms ^ K )
1.15
49
1.22
62
69-22
*yj+20
' -20
1.06
45
0.88
29
sr'46
0.94
44
96+21
-19
0.73
30
101^25
0.78
14
0.63
21
(0, < 56)
+21
1 1J-19
115
120:^
0.68
0
(31, < 6 8 )
134^
0.55
17
+21
75
139!^
0.60
45
Q7pt
' -35
Table 23. Amplitude of the luctuations for each synthesized chop.
The Q 7pt data finds a most likely amplitude of the fluctuations of 751^ |aK, which is
the largest fluctuations found by any of the chops, and goes against the falling trend
demonstrated by the rest of the data. It is possible that this could be the result of a true
signal on the sky (either CMBR or sources, which one expects to be an increasing
Chapter 5. The Analysis
112
problem at higher €). But since the Q-7 pt data also have the lowest sensitivity (and
the lowest statistical weight), this just as likely to be a completely innocuous 1.5 sigma
point. Greater signal to noise from future experiments will resolve this.
The data from Table 23 are plotted in Figure 48. The tendency for a falling spectrum is
apparent here. It is tempting to attempt to fit a curve to these data as a measure of the
spatial spectrum. However, this ignores the finite widths of the window functions, the
spatial correlations between each of the data sets, and neglects the bin to bin
correlations within the data sets which have much to say about the shape of the
spectrum. The correct thing to do is to apply the full correlation analysis presented in
the previous section. Table 23 and Figure 48 should only be used for visualization.
The best power law fit from the full likelihood analysis is shown in Figure 48, and two
representative theories with very different predictions for the spatial spectrum. The
spectrum that rises at large t is COBE normalized standard CDM (O = 1, Qb= 0.05,
H = 50 km / s / Mpc, A = 0, no reionization).9 The falling spectrum is one of several
CDM seeded Isocurvature models presented recently in a paper by Peebles.10
While both theories are reasonable fits to the data, the falling spectrum does appear to
be favored. To quantify this, the spectra for each of the two theories are placed
directly into (17) and the likelihood for each theory was evaluated. Relative to the
power law fit, the Isocurvature theory has a likelihood of 0.32 while CDM has a
likelihood of 0.05.
9P.J
Steinhardt, private communication.
10P.J.E.
Peebles, "Semiempirical Seeded Isocurvature Cosmogonies", The
Astrophysical Journal, 432 (September 1, 1994): LI. This is theory #1.
Chapter 5. The Analysis
113
100
80
60
40
20
0
0
50
100
150
Multipoie moment I
Figure 48. Fluctuation amplitude for each synthesized chop
If the normalization of the theories are permitted to vary, the relative likelihoods
become 0.37 and 0.06 for the Isocurvature and CDM spectra respectively, with both
theories favoring a normalization -10% larger than what were originally given. It is
significant that using the use of spectral information allows discrimination between
theories, independent of the normalization of the theory, or, equivalently, the
calibration of the telescope. This will be a helpful technique in a field where 25%
calibration uncertainties are not uncommon, and doing much better than 10% is very
difficult.
There are several possible sources of systematic errors besides calibration error that
need to be addressed.
One might worry that the tendency for a falling spectrum is a result of atmospheric
contamination, which is known to have more power on large angular scales, as was
Chapter 5. The Analysis
114
seen in the previous chapter, especially since the 30 min error bars were larger than the
16 second error bars in several of the synthesized beams. Another possible worry is
that the error bars which have been attributed to the data are too small, and that there is
excess atmospheric noise on even longer time scales than 30 minutes.
It is reassuring, however, that this large correction to the error bars (34% in the Ka 4
point) has a rather small effect on the result. Additionally, the quality of the y} tests
presented in Table 19 in Chapter 4 implies that the new error bars are close to correct.
In order to flatten the spectrum, one would need to decrease the amount of true signal
in the largest angular scale synthesized beams, but these signals are the most robust.
The combined 3 point data has a reduced x2 of 6.2. It is also reassuring that the results
from the two atmospheric cut levels are so consistent.
Beam uncertainty also effects the results. Re-running the entire analysis with the beam
width reduced by 0.1° (five times the beam uncertainty estimated in Chapter 3) results
in a decrease in the spatial spectral index m of 0.12 and a decrease in the amplitude of
4 /uK (not counting calibration effects which do not effect m). Interpolating linearly to
a beam uncertainty of 0.02° yields an uncertainty in m of 0.02.
In Chapter 3 it was noted that the rms variation of the chopper pattern was 0.02°. This
has the effect of effectively increasing the beam width by that amount. From the above
argument, to correct for the effect, m should be increased by 0.02. This shift is small
compared to the other uncertainties, but is one sided.
It is known that the phase of the chopper did not vary by more than 2 .8°. For the three
point synthesis vector, which is at the second harmonic of the chop, this causes a 5.6°
Chapter 5. The Analysis
115
phase shift. For the 7 point chop (6th harmonic) this is a 16.8° phase shift. The
resulting shift in m can be estimating by solving
fM"
ST X
ST2
cosW
COS(0 2 )
for m. A phase error of this magnitude results in an error in the spectral index m of at
most 0.04.
Adding the systematic errors in m from the chopper phase and the rms chopper
variation (which both tend to increase m) yields a one sided systematic uncertainty in m
of +0.06. The random error from the beam size uncertainty is insignificant.
Another possible systematic error is contribution from foreground point sources. The
spectral index found in the first section of this chapter is consistent with that of the
CMBR, giving assurances that for the synthesized beams peaking at lower t, the
signals are not dominated by foreground contamination. However, no such test has
been made for the higher (. data, where the signal to noise is too low. And at smaller
scales the contribution from point sources is expected to grow. Contamination by
point sources, which would be seen at a larger level with the high /? synthesized beams,
would tend to increase m.
Contamination by diffuse free-free emission may also be a problem. The limits on the
frequency spectral index of the fluctuations placed using the entire data set disfavor all
but a small contribution to the signal from free-free emission. However, the limits
placed with data at the lower atmospheric cut, while still completely consistent with the
CMBR being the sole source of the signal, also allow a fairly significant contribution
from free-free emission. Without compelling evidence that there is not a systematic
bias to the frequency spectral index introduced by residual atmospheric signals in the
Chapter 5. The Analysis
116
data, it is wise to consider that possibility. Contamination of the data sensitive to large
angular scales could tend to decrease the spatial spectral index, m. Nevertheless, it is
still likely that the CMBR is the sole source of the anisotropy at low C .
These data prefer a falling spectrum, and disfavor (but do not rule out) 'standard' CDM.
By finding the spatial spectral index of the fluctuations, this analysis allows comparison
of the data with theories, independent of their normalization, or equivalently, the
calibration of the telescope.
The level of polarization of the fluctuations may also be tested using the above
procedures. Rather than analyzing the A+B data from the Ka radiometer, A-B is
tested. The differencing is performed before application of the synthesis vector, so all
correlations are once again included. For the spatial correlations, (24) is used with
m = 0. (i.e., ST, is constant). These data are consistent with the anisotropy being
unpolarized. Table 24 summarizes the results.
Chop
3 point
4 point
5 point
6 point
7 point
All
Bound
24 uK
40 uK
60 uK
113 uK
19 uK
27 uK
Table 24. Upper limits (95%) on the anisotropy of the polarization of the CMBR.
Summary
The analysis of the SK94 experiment is performed in two stages. The first stage
confirms the results of the SK93 experiment and places tighter limits on the frequency
spectral index of the fluctuations, greatly increasing confidence that the fluctuations are
indeed due to the CMBR. The second stage probes the spatial spectral index of the
fluctuations, with the intent of discriminating between theories.
117
Chapter 5. The Analysis
In the first stage, analysis, the SK93 beam is synthesized and data from both years are
combined. The SK93 and SK94 results are found to be consistent with each other.
The amplitude of the fluctuations for the combined data set is
6T( = yl((2d +1) <|a™|2 >/4tt = 42^3
at ie - 69^2- The error bars include nearly
equal contributions from calibration uncertainty, sample variance given the sky
coverage, and instrument noise. Free-free emission and dust emission are ruled out as
the sole sources of the anisotropy at ~ 3 a.
The second stage analysis probes the spatial spectrum of the fluctuations. The
spectrum is parameterized as 5T( = ST(e(C/Ce)m. A full likelihood analysis of all the data
(3 pt through 7 pt chop, both radiometers, upper cut level) finds ST(e - 45*J6
(not
including the 14% calibration uncertainty) and m - -0.4!^, at Ce = 82. Small
uncertainties in the chopper pattern contribute an additional systematic uncertainty of
0.06. Thus, a mildly falling spectrum is favored. The likelihood of standard CDM
relative to the best fit power law spectrum is 0.06, independent of theory normalization
or telescope calibration.
A 95% upper limit on the polarization of the anisotropy in the CMBR of 19 ^iK is also
set.
Chapter 6
Conclusions
The SK94 experiment had two major goals. The first, and most important, was to
verify the results of the SK93, and to put tighter limits on possible foreground
contamination. The second was to probe a range of angular scales, with the intent of
measuring the spatial spectrum of the anisotropy. The experiment was successful on
both counts.
In covering the same sky as was covered in SK93, SK94 was able to repeat the SK93
measurement. The form of the resulting signals are consistent between the two years.
Additionally, the SK94 experiment was able to tighten the limits on the temperature
spectral index, ruling out free-free emission and dust emission as being the sole source
of the anisotropy. Rather, the spectral index is consistent with that of the CMBR. The
combined analysis finds an rms amplitude of the fluctuations of 44^3, or, equivalently,
5Tt =
+1) <K|2 >/Ak = 42+J?/uK at P. e = 69^2- The errors are improved only
slightly from SK93 alone. The final uncertainty has approximately equal contributions
from the 14% calibration uncertainty, the sample variance, and receiver noise. In order
to improve the determination of amplitude of the anisotropy, all three error terms need
to be reduced.
By employing a large amplitude continuous chop, the SK94 experiment allowed the
synthesis of effective antenna patterns which are sensitive to a large range of angular
118
Chapter 6. Conculsions
119
scales. A Maximum Likelihood analysis of the data is used to find a best fit power law
to the spatial spectrum. With ST( = ST(e(?/1X >
=
45^/^C (not including the 14%
calibration uncertainty) and m = -0.4^, at t - 82. Favoring a falling spectrum at
intermediate angular scales, SK94 disfavors (but does not rule out) 'standard' CDM
independent of theory normalization.
Figure 49 summarizes the state of the determination of the CMBR spectrum as of the
writing of this thesis. The skeletal star data points represent the amplitude of the
fluctuations seen by each of the SK94 synthesized beams independently. The open star
represents the SK93/SK94 combined data set analysis. The remaining data points
(square) are, starting with lowest i, COBE Quadrupole, COBE, FIRS, Tenerife, SP91
(ACME-HEMT), SK93, PYTHON, ARGO, MAX (6 fields), MS AM 2pt (full, and
with 'sources' removed), and MSAM 3ptt. Also included are two representative
theories with very different predictions for the spectrum The theory peaking at t « 60
is a CDM seeded isocurvature theory.1 The theory peaking at I » 220 is standard
CDM.2
When taken alone, SK94 has some definite predictions about the spatial spectrum of
the CMBR. However, when taken in conjunction with the other experiments which
have been done to date, the picture is not at all clear. The falling spectrum predicted by
+ These data are from L. Page, private communication, except the MAX points, which
are from A C. Clapp et al. Conversion from GACF amplitude to ST( for the MAX
points was adapted from M. White and D. Scott "Quoting Experimental Information",
Preprint, 1994.
'P.J.E. Peebles, "Isocurvature Cosmogonies"
2P.J.
Steinhardt, Private communication.
Chapter 6. Conculsions
120
SK94 is not favored by either MSAM (taking the whole data set) or MAX (except for
the disfavored mu-pegasus point). This possible discrepancy may be due to systematic
errors contaminating some (or all!) of the experiments. Or the spectrum may not be
smoothly varrying over these angular scales. Or the fluctuations may not be Gaussian,
and therefore not well described by an angular power spectrum. Clearly, there is more
work to be done before the spatial spectrum is understood.
100
80
60
i—*
•o
40
20
100
200
300
Multipole moment I
Figure 49. The CMBR anisotropy spectrum.
In order to begin to sort out the situation, it will be important to repeat the
measurements and to verify them. Currently, the BigPlate telescope is being upgraded
and readied for another observing season. A new Q radiometer is being built to
attempt to remove the gain fluctuation which plagued the radiometer used here, and the
telescope beam size is being reduced to 0.5°, to allow a larger region of the spectrum to
be probed. Additionally, the same fields will be observed with SK95, allowing
Chapter 6. Conculsions
121
verification of SK94. Ultimately, in order to adequately understand the spatial
spectrum of the CMBR, a map of an extended region of sky will have to be produced.
Appendix A
The Synthesis Vectors
Chopper
SK930L
SK930L
3 pt
4 pt
5 pt
6 pt
7 pt
East
West
-3.00035
0.0547538
0.0650077
0.0811737
0.105
0.140624
0
-0.0104424
-2.94658
0.0526496
0.0594093
0.0688158
0.0802038
0.0932252
0
-0.0104424
-2.84159
0.0485221
0.0486946
0.0459812
0.0364665
0.0144038
0
-0.0341752
-2.69691
0.04253
0.0337863
0.0161464
-0.0158827
-0.0692727
0
-0.0341752
Angle
(Deg)
-2.52151
0.0349035
0.0159684
-0.0161464
-0.0644811
-0.1296
0
-0.0569586
-2.32177
0.0259356
-0.00322468
-0.0459812
-0.0978518
-0.146244
0
-0.0569586
-2.10284
0.0159711
-0.0221401
-0.0688158
-0.108114
-0.113595
-0.019424
-0.0569586
-1.87878
0.00539277
-0.0391487
-0.0811737
-0.0928444
-0.0426579
-0.0252511
-0.0569586
-1.64704
-0.00539277
-0.052786
-0.0811737
-0.0556488
0.0426579
-0.0553582
0
-1.41658
-0.0159711
-0.0618773
-0.0688158
-0.00531131
0.113595
-0.0660415
0
-1.1874
-0.0259356
-0.0656398
-0.0459812
0.0462805
0.146244
-0.0796383
0.0740462
-0.953099
-0.0349035
-0.0637495
-0.0161464
0.0869428
0.1296
-0.0728399
0.0749955
-0.721359
-0.04253
-0.0563691
0.0161464
0.107073
0.0692727
0
0.079742
0.0844886
-0.49474
-0.0485221
-0.0441342
0.0459812
0.101917
-0.0144038
0
-0.266841
-0.0526496
-0.0280985
0.0688158
0.0726925
-0.0932252
0.0738111
0.0854378
-0.0415018
-0.0547538
-0.00964295
0.0811737
0.0263012
-0.140624
0.0738111
0.0854378
0.183837
-0.0547538
0.00964295
0.0811737
-0.0263012
-0.140624
0.0874078
0.0721475
0.406615
-0.0526496
0.0280985
0.0688158
-0.0726925
-0.0932252
0.0874078
0.0721475
0.631954
-0.0485221
0.0441342
0.0459812
-0.101917
-0.0144038
0.0864367
0
0.849611
-0.04253
0.0563691
0.0161464
-0.107073
0.0692727
0.0815806
0
1.07111
-0.0349035
0.0637495
-0.0161464
-0.0869428
0.1296
0.0767245
-0.0711982
1.28748
-0.0259356
0.0656398
-0.0459812
-0.0462805
0.146244
0.0757535
-0.0778434
1.50642
-0.0159711
0.0618773
-0.0688158
0.00531131
0.113595
0
-0.0645531
1.72152
-0.00539277
0.052786
-0.0811737
0.0556488
0.0426579
0
-0.0541106
1.93405
0.00539277
0.0391487
-0.0811737
0.0928444
-0.0426579
-0.058272
-0.0246821
2.14659
0.0159711
0.0221401
-0.0688158
0.108114
-0.113595
-0.058272
-0.0189861
2.35656
0.0259356
0.00322468
-0.0459812
0.0978518
-0.146244
-0.058272
0
2.54862
0.0349035
-0.0159684
-0.0161464
0.0644811
-0.1296
-0.058272
0
(Ka synthesis vectors cont.)
122
Appendix A. The Synthesis Vectors
123
2.72018
0.04253
-0.0337863
0.0161464
0.0158827
-0.0692727
-0.0349632
0
2.85461
0.0485221
-0.0486946
0.0459812
-0.0364665
0.0144038
-0.0349632
0
2.95064
0.0526496
-0.0594093
0.0688158
-0.0802038
0.0932252
-0.0106832
0
2.99929
0.0547538
-0.0650077
0.0811737
-0.105
0.140624
-0.0106832
0
2.99801
0.0547538
-0.0650077
0.0811737
-0.105
0.140624
-0.0106832
0
2.94295
0.0526496
-0.0594093
0.0688158
-0.0802038
0.0932252
-0.0106832
0
2.84053
0.0485221
-0.0486946
0.0459812
-0.0364665
0.0144038
-0.0349632
0
0.0161464
0.0158827
-0.0692727
-0.0349632
0
2.69457
0.04253
-0.0337863
2.51532
0.0349035
-0.0159684
-0.0161464
0.0644811
-0.1296
-0.058272
0
2.31303
0.0259356
0.00322468
-0.0459812
0.0978518
-0.146244
-0.058272
0
2.09793
0.0159711
0.0221401
-0.0688158
0.108114
-0.113595
-0.058272
-0.0189861
1.87516
0.00539277
0.0391487
-0.0811737
0.0928444
-0.0426579
-0.058272
-0.0246821
1.64469
-0.00539277
0.052786
-0.0811737
0.0556488
0.0426579
0
-0.0541106
1.41424
-0.0159711
0.0618773
-0.0688158
0.00531131
0.113595
0
-0.0645531
1.1825
-0.0259356
0.0656398
-0.0459812
-0.0462805
0.146244
0.0757535
-0.0778434
0.950757
-0.0349035
0.0637495
-0.0161464
-0.0869428
0.1296
0.0767245
-0.0711982
0.722857
-0.04253
0.0563691
0.0161464
-0.107073
0.0692727
0.0815806
0
0.0864367
0
0.492397
-0.0485221
0.0441342
0.0459812
-0.101917
-0.0144038
0.264498
-0.0526496
0.0280985
0.0688158
-0.0726925
-0.0932252
0.0874078
0.0721475
0.0404395
-0.0547538
0.00964295
0.0811737
-0.0263012
-0.140624
0.0874078
0.0721475
-0.184899
-0.0547538
-0.00964295
0.0811737
0.0263012
-0.140624
0.0738111
0.0854378
-0.407677
-0.0526496
-0.0280985
0.0688158
0.0726925
-0.0932252
0.0738111
0.0854378
-0.630455
-0.0485221
-0.0441342
0.0459812
0.101917
-0.0144038
0
0.0844886
-0.850673
-0.04253
-0.0563691
0.0161464
0.107073
0.0692727
0
0.079742
-1.06961
-0.0349035
-0.0637495
-0.0161464
0.0869428
0.1296
-0.0728399
0.0749955
-1.28855
-0.0259356
-0.0656398
-0.0459812
0.0462805
0.146244
-0.0796383
0.0740462
-1.50364
-0.0159711
-0.0618773
-0.0688158
-0.00531131
0.113595
-0.0660415
0
-1.72002
-0.00539277
-0.052786
-0.0811737
-0.0556488
0.0426579
-0.0553582
0
-1.93512
0.00539277
-0.0391487
-0.0811737
-0.0928444
-0.0426579
-0.0252511
-0.0569586
-2.14637
0.0159711
-0.0221401
-0.0688158
-0.108114
-0.113595
-0.019424
-0.0569586
-2.35506
0.0259356
-0.00322468
-0.0459812
-0.0978518
-0.146244
0
-0.0569586
-2.54711
0.0349035
0.0159684
-0.0161464
-0.0644811
-0.1296
0
-0.0569586
-2.71996
0.04253
0.0337863
0.0161464
-0.0158827
-0.0692727
0
-0.0341752
-2.85696
0.0485221
0.0486946
0.0459812
0.0364665
0.0144038
0
-0.0341752
-2.95042
0.0526496
0.0594093
0.0688158
0.0802038
0.0932252
0
-0.0104424
-3.00035
0.0547538
0.0650077
0.0811737
0.105
0.140624
0
-0.0104424
~able 25. The Ka chopper position am synthesis vectors.
The beam on the sky is at 0sky = 20ccos(52.15) = 1.230c.
This Appendix lists lists the chopper positions and synthesis vectors (see Chapter 4).
To convert the chopper position to the position on the sky, use 0sky = 20ccos(52.15) =
1.230c The East and West SK930L synthesis vectors are different because for SK93,
the chop was centered 8° from the pole, and for SK94, it was centered 7.2° from the
pole. Thus, the synthesisized SK93 beam needs to be offset relative to the chop.
124
Appendix A. The Synthesis Vectors
Chopper
Angle
3 pt
4 pt
5 pt
6 pt
7 pt
SK930L
East
SK930L
West
-3.00035
0.0515311
0.0561324
0.0636696
0.0736666
0.0864249
0
-2.94658
0.0495508
0.0512983
0.0539765
0.0562697
0.0572945
0
-0.00938362
-0.00938362
-2.84159
0.0456663
0.0420465
0.0360659
0.0255843
0.0088523
0
-0.0307101
-2.69691
0.0400268
0.0291736
0.0126646
-0.011143
-0.0425737
0
-0.0307101
-2.52151
0.0328492
0.0137883
-0.0126646
-0.0452389
-0.0796497
0
-0.0511834
-2.32177
0.0244092
-0.00278443
-0.0360659
-0.0686512
-0.089879
0
-0.0511834
-2.10284
0.0150311
-0.0191174
-0.0539765
-0.075851
-0.0698135
-0.0170621
-0.0511834
-1.87878
0.00507537
-0.0338039
-0.0636696
-0.0651381
-0.0262168
-0.0221808
-0.0511834
-1.64704
-0.00507537
-0.0455793
-0.0636696
-0.0390423
0.0262168
-0.048627
0
-1.41658
-0.0150311
-0.0534294
-0.0539765
-0.00372633
0.0698135
-0.0580114
0
-1.1874
-0.0244092
-0.0566782
-0.0360659
0.0324696
0.089879
-0.069955
0.0665384
-0.953099
-0.0328492
-0.055046
-0.0126646
0.0609977
0.0796497
-0.0639831
0.0673914
0.0425737
0
0.0716567
0.075922
-0.721359
-0.0400268
-0.0486732
0.0126646
0.0751206
-0.49474
-0.0456663
-0.0381087
0.0360659
0.0715032
-0.0088523
0
-0.266841
-0.0495508
-0.0242623
0.0539765
0.0509999
-0.0572945
0.0648363
0.076775
-0.0415018
-0.0515311
-0.00832643
0.0636696
0.0184525
-0.0864249
0.0648363
0.076775
0.0648322
0.183837
-0.0515311
0.00832643
0.0636696
-0.0184525
-0.0864249
0.0767798
0.406615
-0.0495508
0.0242623
0.0539765
-0.0509999
-0.0572945
0.0767798
0.0648322
0.631954
-0.0456663
0.0381087
0.0360659
-0.0715032
-0.0088523
0.0759267
0
0.849611
-0.0400268
0.0486732
0.0126646
-0.0751206
0.0425737
0.0716612
0
1.07111
-0.0328492
0.055046
-0.0126646
-0.0609977
0.0796497
0.0673956
-0.0639792
1.28748
-0.0244092
0.0566782
-0.0360659
-0.0324696
0.089879
0.0665426
-0.0699506
1.50642
-0.0150311
0.0534294
-0.0539765
0.00372633
0.0698135
0
-0.0580078
1.72152
-0.00507537
0.0455793
-0.0636696
0.0390423
0.0262168
0
-0.0486242
1.93405
0.00507537
0.0338039
-0.0636696
0.0651381
-0.0262168
-0.0511865
-0.0221795
2.14659
0.0150311
0.0191174
-0.0539765
0.075851
-0.0698135
-0.0511865
-0.017061
2.35656
0.0244092
0.00278443
-0.0360659
0.0686512
-0.089879
-0.0511865
0
2.54862
0.0328492
-0.0137883
-0.0126646
0.0452389
-0.0796497
-0.0511865
0
2.72018
0.0400268
-0.0291736
0.0126646
0.011143
-0.0425737
-0.0307119
0
2.85461
0.0456663
-0.0420465
0.0360659
-0.0255843
0.0088523
-0.0307119
0
2.95064
0.0495508
-0.0512983
0.0539765
-0.0562697
0.0572945
-0.00938419
0
2.99929
0.0515311
-0.0561324
0.0636696
-0.0736666
0.0864249
-0.00938419
0
2.99801
0.0515311
-0.0561324
0.0636696
-0.0736666
0.0864249
-0.00938419
0
2.94295
0.0495508
-0.0512983
0.0539765
-0.0562697
0.0572945
-0.00938419
0
2.84053
0.0456663
-0.0420465
0.0360659
-0.0255843
0.0088523
-0.0307119
0
2.69457
0.0400268
-0.0291736
0.0126646
0.011143
-0.0425737
-0.0307119
0
2.51532
0.0328492
-0.0137883
-0.0126646
0.0452389
-0.0796497
-0.0511865
0
2.31303
0.0244092
0.00278443
-0.0360659
0.0686512
-0.089879
-0.0511865
0
2.09793
0.0150311
0.0191174
-0.0539765
0.075851
-0.0698135
-0.0511865
-0.017061
1.87516
0.00507537
0.0338039
-0.0636696
0.0651381
-0.0262168
-0.0511865
-0.0221795
1.64469
-0.00507537
0.0455793
-0.0636696
0.0390423
0.0262168
0
-0.0486242
1.41424
-0.0150311
0.0534294
-0.0539765
0.00372633
0.0698135
0
-0.0580078
-0.0699506
1.1825
-0.0244092
0.0566782
-0.0360659
-0.0324696
0.089879
0.0665426
0.950757
-0.0328492
0.055046
-0.0126646
-0.0609977
0.0796497
0.0673956
-0.0639792
0.722857
-0.0400268
0.0486732
0.0126646
-0.0751206
0.0425737
0.0716612
0
0.492397
-0.0456663
0.0381087
0.0360659
-0.0715032
-0.0088523
0.0759267
0
Appendix A. The Synthesis Vectors
0.264498
-0.0495508
0.0404395
-0.184899
125
0.0242623
0.0539765
-0.0509999
-0.0572945
0.0767798
0.0648322
-0.0515311
0.00832643
0.0636696
-0.0184525
-0.0864249
0.0767798
0.0648322
-0.0515311
-0.00832643
0.0636696
0.0184525
-0.0864249
0.0648363
0.076775
-0.407677
-0.0495508
-0.0242623
0.0539765
0.0509999
-0.0572945
0.0648363
0.076775
-0.630455
-0.0456663
-0.0381087
0.0360659
0.0715032
-0.0088523
0
0.075922
-0.850673
-0.0400268
-0.0486732
0.0126646
0.0751206
0.0425737
0
0.0716567
-1.06961
-0.0328492
-0.055046
-0.0126646
0.0609977
0.0796497
-0.0639831
0.0673914
-1.28855
-0.0244092
-0.0566782
-0.0360659
0.0324696
0.089879
-0.069955
0.0665384
-1.50364
-0.0150311
-0.0534294
-0.0539765
-0.00372633
0.0698135
-0.0580114
0
-1.72002
-0.00507537
-0.0455793
-0.0636696
-0.0390423
0.0262168
-0.048627
0
-1.93512
0.00507537
-0.0338039
-0.0636696
-0.0651381
-0.0262168
-0.0221808
-0.0511834
-2.14637
0.0150311
-0.0191174
-0.0539765
-0.075851
-0.0698135
-0.0170621
-0.0511834
-2.35506
0.0244092
-0.00278443
-0.0360659
-0.0686512
-0.089879
0
-0.0511834
-2.54711
0.0328492
0.0137883
-0.0126646
-0.0452389
-0.0796497
0
-0.0511834
-2.71996
0.0400268
0.0291736
0.0126646
-0.011143
-0.0425737
0
-0.0307101
-2.85696
0.0456663
0.0420465
0.0360659
0.0255843
0.0088523
0
-0.0307101
-2.95042
0.0495508
0.0512983
0.0539765
0.0562697
0.0572945
0
-0.00938362
-3.00035
0.0515311
0.0561324
0.0636696
0.0736666
0.0864249
0
-0.00938362
Table 26. The Q chopper position and synthesis vectors.
Table 27 lists the zero lag spatial correlations between the synthesized beams. The sign
of the correlations reflects the arbitrary sign of the synthesis vectors. The correlations
between symetric and antisymetric synthesized beams (eg, 3pt to 4 pt) are small at
zero lag, as listed here. However, the corellation when the beams are shifted by a 7.5°
bin are larger (-0.1) and should be considered. The full correlation matrix is 960x960.
3 pt
4 pt
5 pt
6 pt
7 Pt
3 pt
1
0.01
-0.21
-0.01
-0.16
4 pt
0.01
1
0.01
-0.25
-0.01
5 pt
-0.21
0.01
1
0.01
-0.28
6 Pt
-0.01
-0.25
0.01
1
0.01
-0.01
-0.28
0.01
-0.16
1
7 Pt
Table 27. The Spatial correlation matrix for the Ka synthesized beams.
These are all for zero lag. The matrix is symmetric.
Appendix B
The Correlations
This appendix gives the temporal channel to channel temporal correlation matrixes as
determined from the sky data. The Ka data lists correlations between 6 channels, and
the Q data lists correlations between 3 channels. Entries within a radiometer chain (i.e
A1-A2) include both radiometer induced correlations and atmospheric correlations.
Entries between radiometer chains (i.e. Al-Bl) only include atmospheric correlations.
Tables are given for 3 point (which is produced by demodulating at 7.8 Hz) and for 7
point (23 .4 Hz) data. The correlations in the Q radiometer are much larger than those
in the Ka radiometer, and fall off more slowly. The Ka table indicates that the
atmospheric correlations are insignificant in the 7pt chop.
A1
A2
A3
B1
B2
B3
A1
1.00
0.56
0.51
0.33
0.26
0.25
A2
0.56
1.00
0.48
0.29
0.22
0.21
A3
0.51
0.48
1.00
0.25
0.20
0.20
B1
0.33
0.29
0.25
1.00
0.69
0.69
B2
0.26
0.22
0.20
0.69
1.00
0.68
0.69
0.68
1.00
0.25
0.21
0.20
B3
Table 28. Ka 3 point temporal correlation matrix.
A1
A2
A3
B1
B2
B3
A1
1.00
0.19
0.14
0.01
-0.01
0.00
A2
0.19
1.00
0.17
-0.02
-0.00
-0.01
A3
0.14
0.17
1.00
0.01
-0.00
0.02
B1
0.01
-0.02
0.01
1.00
0.24
0.22
B2
-0.01
-0.00
-0.00
0.24
1.00
0.24
0.00
0.02
1.00
-0.01
0.22
0.24
B3
Table 29. Ka 7 point temporal correlation matrix.
126
Appendix B. The Correlations
B1
B2
B3
B1
1.00
0.77
0.64
B2
0.77
1.00
0.85
0.85
1.00
0.64
B3
Table 30. Q 3 point temporal correlation matrix.
B1
B2
B3
B1
1.00
0.58
0.45
B2
0.58
1.00
0.71
1.00
0.71
0.45
B3
Table 31. Q 7 point temporal correlation matrix.
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