Polarization of the Moon at 90 GHz
Colin Bischoff
Department of Physics
[email protected]
July 14, 2003
Abstract
The CAPMAP experimental apparatus at the Crawford Hill 7-meter antenna is
used to observe the polarization of emission from the moon at 90 GHz. The dielectric constant of the moon’s surface can be deduced from this data and is found
to be 1.6 0.1. Measurement of the dielectric constant is complicated by the presence of instrumental effects, which are studied.
1 Introduction
The goal of the CAPMAP experiment is to make a sensitive detection of polarization
in the cosmic microwave background (CMB). The CMB is a relic of radiation from an
extremely hot, early period in the evolution of the universe. It has a very small (difficult
to detect) polarized component due to Thompson scattering of photons off charged
particles. By measuring this polarization on different angular scales, we can constrain
important cosmological parameters that determine the evolution of the universe up to
the present day [1].
As an offshoot of the CAPMAP project, I made measurements of the moon’s polarization using the CAPMAP detector. Polarization measurements provide a very direct
method of determining the dielectric constant of the moon. Deviations from the expected polarization signal provide interesting information about the response of the
telescope and the detector to a bright, extended source.
2 Theory
An antenna pointed at a point on the moon will measure a temperature, T a , given by
Ta
TCMB R
Tm 1
R
(1)
where R R θ is the reflectivity of the moon, a function of the angle between the
observer and the surface of the moon, TCMB is the temperature of the cosmic microwave
background (3 K) which is the source of most microwave photons reflecting off the
moon, and Tm is the temperature of the moon. For the frequency ranges measured by
1
3 APPARATUS AND OBSERVATIONS
2
the CAPMAP detector (84-100 GHz), the CMB temperature is negligible compared to
emission from the moon [2].
Microwaves emitted by the moon originate in a layer of the moon’s surface with a
depth of several wavelengths. The waves refract at the boundary between the moon’s
surface and the vacuum above. The transmission amplitudes for the polarized components in and out of the plane of incidence are given by the Fresnel equations:
t
2εi cos θi
t
εi cosθt εt cos θi
2εi cos θi
εi cos θi εt cos θt
(2)
where εi is the dielectric constant of the moon (ε moon ), εt is the dielectric constant of
space ( 1), and θ i and θt are related by Snell’s law:
εi sin θi
εt sin θt
(3)
Generally t t , so the transmitted wave is partially polarized in the plane of
incidence. Our detector measures the intensity of the polarized part of the radiation,
which is given by
εt cos θt
(4)
I
t t 2
εi cos θi The first factor in Eq. (4) is due to the difference in size between the incident and
transmitted area elements (ratio of cosines) and the difference in the speed of wave
propagation between the moon and the space around it (ratio of dielectric constants)
[3].
In the approximation that the moon is a smooth sphere, θ t is simply proportional
to the distance of the observed point on the moon from the center of the moon’s disc.
Points at the center of the disc have θ t 0 while points at the edge of the moon have
θt 90 . In this case, I is a function of radial distance on the moon only. Figure 1
plots I r for several values of ε moon . Also, for a smooth ball, the plane of incidence
for all points passes through the center of the moon. This means that polarization from
any point on the moon is in the radial direction.
The effects of surface roughness are not considered in depth in this paper, but the
most obvious effect of a realistically bumpy surface would be that, on small patches,
the polarization vectors would not all be radial. Assuming that bumps were oriented
randomly, this would average out over the size of our beam to produce a radial polarization pattern, but the strength of the polarized signal would be weakened. Other
roughness effects could conceivably skew the θ t values for a beam sized patch, but this
requires delving into particulars of the roughness model.
The expected temperature measured at each point by our polarimeters can be found
by multiplying I r by the temperature of the moon and then convolving the result
with the telescope’s beam pattern. This expression has ε moon as a parameter, which can
be found by comparing the expected curves with measured data.
3 Apparatus and Observations
The CAPMAP experiment is deployed at the Crawford Hill 7-meter antenna, in Crawford Hill, New Jersey. This telescope has offset Cassegrainian optics, with a 7 meter
3 APPARATUS AND OBSERVATIONS
3
Figure 1: The fraction of total intensity that is polarized radially (I ) for a dielectric
sphere with ε equal to 1 5 (red), 1 75 (yellow), 2 0 (green), 2 25 (light blue), and 2 5
(purple).
diameter primary reflector and a 1.2 by 1.8 meter secondary reflector [4]. Combined
with the lenses and feed horns designed for CAPMAP, the Crawford Hill telescope
gives a 0.06 degree beam in our frequency range.
In order to make a sufficiently accurate measurement of the CMB polarization,
CAPMAP uses a cryogenic detector. The electric field is measured along two orthogonal detector axes, each of which feeds into a separate line. Monolithic microwave
integrated circuits (MMICs) are used to provide extremely low noise amplification at
frequencies around 90 GHz. The signals are mixed down to an intermediate frequency
(IF) range of 2-18 GHz and split into three frequency bands, denoted S0, S1, and
S2. These frequency bands correspond to microwaves in the ranges 84-89 GHz, 8993 GHz, and 93-100 GHz respectively. Also, total (unpolarized) power is measured
by two detector diodes, one on each line, denoted D0 and D1. The two total power
channels have to be calibrated separately, mostly because of the variability in MMIC
amplification. Finally, the signals are multiplied together to find the difference between two orthogonal polarizations. For the winter of 2002-2003, CAPMAP had four
radiometers referred to as A, B, C, and D, for a total of 12 polarization channels and 8
4 DATA ANALYSIS
4
total power channels.
Microwaves are strongly scattered by water vapor, so winter is the ideal time to
observe since the colder atmosphere has lower humidity. Cold weather is less important
for observations of the moon, however, since the moon is so bright that it can be seen
even through clouds. The data analyzed for this paper were taken on the 30th of April
(2003) but additional observations were made on February 28th, March 15th, and April
28th. April 30th was one day before a new moon and the data were taken in the morning
(from 0825 until 0847 EST), shortly after the moon had risen. The weather was warm
(14 C), which contributed to high atmospheric temperature readings of 165K in the
total power channels.
Data were taken continuously as the telescope scanned over a 1 1 square, which
followed the moon as it moved through the sky. The moon’s diameter is about half
a degree and each of the four radiometers is pointed about .25 degrees from the telescope’s central ray, so the scan was sufficiently large for all four radiometers to see the
entire disc of the moon. While tracking on the moon, the scan swung back and forth
in its azimuthal (coelevation) coordinate while stepping in elevation by intervals of .01
degrees. Since the data point spacing is considerably smaller than our beam size, this
scan pattern gives full coverage of the moon. The entire April 30th scan took about 22
minutes.
4 Data Analysis
4.1 Telescope Pointing
For each data point, the azimuth and elevation position of the telescope were recorded.
The four radiometers (A, B, C, and D) all point in slightly different directions, so
pointing corrections were applied separately for each radiometer. These corrections
were derived from observations of Jupiter by Denis Barkats [5], which appears as a
point source to our detector (see Figure 2).
The azimuthal position of the telescope is converted to a coelevation coordinate
(which gives real angles on the sky) by multiplying by the cosine of the elevation. Also,
the position of the center of the moon is calculated by ephemeris software (XEphem)
and subtracted. These steps line up the data for all channels and put them into a mooncentered coordinate system where both the x and y coordinates give actual angular
distances on the sky.
4.2 Total Power Data
The data from the radiometer total power channels are useful to determine the overall
temperature of the moon, which multiplies I to determine the polarized intensity. Even
though the moon’s disc is fairly evenly illuminated (very close to a new moon), the
moon’s temperature varies considerably. This is due to the high specific heat of the
moon, which causes a phase lag in the moon’s temperature [6]. The warmer side of the
moon was the smaller coelevation side (left side).
4 DATA ANALYSIS
5
Figure 2: The (inverted) positions of Jupiter as shown in channels A, B, C, and D
(clockwise from top), from D. Barkats (2003).
Unlike the polarization channels, the total power channels are sensitive to the temperature of the atmosphere and the detectors. In order to remove these effects, they
were modeled as three separate contributions. The background temperature (of 165K)
is due to the temperature of the detector and emission from the sky.
Tbackground
Tatmosphere
Tradiometer
(5)
The noise temperature of each radiometer had previously been measured by Matt Hedman [7], using the fact that the atmosphere thickness (and T atmosphere ) is proportional to
sec elevation in order to remove the atmospheric contribution. The radiometer temperature is subtracted from each total power data. The remaining temperature in the
background is due to atmospheric emission. This is subtracted as well. The physical temperature of water in the atmosphere is given by 95% of the temperature on the
ground, independent of the relative humidity [8]. Assuming that water vapor emits and
absorbs microwaves with the same efficiency, then the absorption due to the atmosphere
is given by:
Tatmosphere
Twatervapor 95Tground
Aatmosphere
(6)
Twatervapor
The final correction to the data comes from dividing the data by 1
datacorrected
datauncorrected
1
Tradiometer Tatmosphere
Aatmosphere
A atmosphere .
(7)
The resulting total power data are shown in mapped form in Figure 3. Most of the
total power data are in good agreement with each other and with previously published
results [9]. CD0, DD0, and especially AD1 are all low, however. This suggests that
either their gains or their receiver noise temperatures are overestimated. Both AD1 and
CD0 have noise temperatures 25% higher than the other channels. DD0 doesn’t have
a particularly high noise temperature but has the highest gain and may be saturating.
4 DATA ANALYSIS
6
Figure 3: Maps of the moon in all total power channels. Vertical scale is degrees Kelvin
4 DATA ANALYSIS
7
Figure 4: Maps of the moon in polarization channels AS0, AS1, and AS2. Vertical
scale is degrees Kelvin
4.3 Polarization Data
4.3.1 Instrumental Effects
The data from the polarization channels is calibrated based on measurements made of
Tau A, an astronomical source with a well-defined polarization [10]. Maps of the moon
polarization for channels AD0, AD1, and AD2 are shown in Figure 4. Background
offsets have been subtracted. However, for the polarimeters, these offsets are not due
to the atmosphere (which doesn’t produce polarized radiation) but are an unavoidable
effect of our detector.
The most noticible feature of these plots is their quadrupole pattern. This occurs
because our detector measures polarization by differencing the intensity of microwaves
4 DATA ANALYSIS
8
polarized along two perpendicular axes. When the polarization vector is pointing vertically or horizontally, the measurement axes see equal values for the electric field
intensity and no polarization is detected (see Figure 5). While the polarized intensities
are given in degrees Kelvin, negative values do make sense. The sign of the polarization temperature indicates which of the two orthogonal detector axes the electric field
is more closely aligned with. The sign convention can clearly be seen from Figure 4,
remembering that all polarized radiation from the moon should be oriented radially.
Since the magnitude of the polarized radiation should be the same function of radius
E• X=E• Y=0
no polarization
detected
E• Y=0
(E• X)2-(E• Y)2
is maximally positive
E• X=0
(E• X)2-(E• Y)2
is maximally negative
Y
X
Detector axes
Figure 5: Polarization vectors (green arrows) produce no response if they are oriented
45 degrees from our detector axes. The response changes sign depending on which
axes is aligned with the polarization, creating the quadrapole pattern seen in our data
from the moon’s radial polarization pattern.
(ignoring temperature differences) for any angle, we would expect that all linear cuts
through the center of the moon would be indentical up to a scale factor equal to the sine
of the angle at which the cut was made. Figure 6 plots the maximum value along each
cut as a function of cut angle alongside sin 2θ for cut angles from 0 to 90 degrees.
These values line up well, but indicate that our radiometer is rotated approximately 3
degrees from vertical.
Another feature of the polarization maps are circular notches near the expected
peak in the radial function. These can be seen well in the upper right corners of the
AS1 and AS2 maps in Figure 4. The shape of the notches varies between different
radiometers but is quite consistent between different channels on the same radiome
4 DATA ANALYSIS
9
Figure 6: Linear cuts (such as the ones in Figure 7) are made through the center of the
moon at different angles with respect to the horizontal. The maximum value along the
cut in channel AS0 is marked as a function of cut angle (by an x). The solid curve is
y sin 2θ .
ter (see Figure 7). Additionally, these notches are roughly one beamwidth from the
edge of the moon. These two factors suggest that these notches may be caused by an
intensity quadrapole leakage. This effect occurs when diffraction in the telescope optics rotates the polarization vector of incoming radiation. Since each radiometer looks
through the telescope optics along a different ray, this would naturally affect each radiometer distinctly. Also, the notches are more severe in channels B and C, which look
through grooved lenses that have been shown to exagerrate the intensity quadrapole
leakage term.
An alternative, but less convincing, explanation for the notches is that they are due
to saturation of the detector. This is supported by the location of the notches over what
would be peaks in the polarization signal and also because the moon is a much brighter
source than the CAPMAP experiment was designed to observe. This theory fails to
explain, however, the left peak in Figure 7 which is mostly intact while the right peak
is significantly affected. Also, if a linear cut is made in the channel B polarization
data at only 5 or 10 degrees from the horizontal, where polarization values are greatly
reduced, the same notch is still present.
An additional instrumental effect that can be seen in Figure 7 is that the polarization
value of the center of the moon (0 ) varies for different detectors, while it should be
strictly zero. This is due to a known intensity polarization leakage. The rejection
ratio is the ratio of total power (temperature of the center of the moon) to polarized
response. Rejection ratios calculated from this data can be compared to ones found
during preseason tests of the radiometer (see Table 1) and they don’t appear to be
correlated. Rejection ratios derived from the moon observations which are greater than
2000 or so are not meaningful since they are below the noise level of the polarimeters.
4 DATA ANALYSIS
10
Figure 7: BS0 (light blue), BS1 (green), and BS2 (red) values along linear cuts across
the moon at 45 degrees from the horizontal.
channel
AS0
AS1
AS2
BS0
BS1
BS2
CS0
CS1
CS2
DS0
DS1
DS2
preseason
-229
-280
325
-6750
2140
-714
-1080
1330
-470
-1235
-317
-636
measured
-705
12137
2041
-789
-691
-5303
2267
1495
739
2373
2503
-6682
Table 1: Rejection ratios from preseason measurements and from moon polarization
data.
4.3.2 Calculation of the Dielectric Constant
In order to directly compare the I curves found from equation (4) to polarization measurements, the theoretical curves must be multiplied by the temperature of the moon
(which is not a constant across the disc) and then convolved with the telescope beam
pattern. This process is done here not for an entire map of the moon, but only for linear
cuts made across the moon’s disc at an angle of 45 with respect to the horizontal (this
is where our detector measures the full strength signal).
The moon’s temperature can be found from the total power measurements, but these
have already been convolved with our beam. The actual moon temperature is modeled
4 DATA ANALYSIS
11
simply by assuming zero temperature outside the moon’s disc and a linear temperature
function on the moon, as shown in Figure 8. This model was chosen because it was
a good fit to the measured total power data, but it can be thought of as a first-order
expansion of the function for moon temperature given by Strezhneva and Troitsky [6].
The beam is modeled as a Gaussian function with full-width-half-max equal to .06
Figure 8: Total power channel measurement of the moon (light blue) on a 45 degree
line. A simple model for the moon’s temperature is drawn in red.
degrees.
The comparison of polarization measurements to theoretical expectations is shown
in Figure 9. The previously mentioned notches prevent any comparisons between peak
values, which is unfortunate, since height of the peak in I is much more sensitive to
variations in the dielectric constant than its behavior away from the peak. However, the
data show that the dielectric constant of the moon’s surface (ε moon ) is a value between
1.5 and 1.7.
This value for εmoon , while not particularly exact, is reasonable considering that
the moon’s surface is rock and dust and is comprised largely of silicon (i.e. it is not
metallic). Table 2 compares these results with previous results that have been obtained
at longer wavelengths.
wavelength (cm)
0.33
2.1
3.2
6.3
6
11
21
21
ε moon
1.6 0.1
1.8
1.55
1.9
2.00 0.05
2.05 0.05
2.3 0.15
2.1 0.3
observers
Present work
Baars et al. (1965)
Soboleva (1962)
Golnev and Soboleva (1964)
Davies and Gardner (1966)
Davies and Gardner (1966)
Davies and Gardner (1966)
Heiles and Drake (1963)
Table 2: Past measurements of ε moon [2]
5 CONCLUSION
12
Figure 9: Expected measurements for ε moon equal to 1.5 (light blue), 1.6 (green), and
1.7 (red). Polarization measurements are shown for AS0 (plus signs), AS1 (x’s), and
AS2 (triangles).
By subtracting the expected measurement and the (mostly constant) rejection ratio
term from a polarization map of the moon, an actual map of the notches caused by
beam effects can be found. Figure 10 shows these patterns for radiometers A, B, C,
and D. The strong correlation between the location of these notches and the peaks
of the polarization signal reinforce the theory that saturation of the polarimeters is
responsible.
5 Conclusion
Instrumental effects greatly reduced the precision with which ε moon could be measured.
The effects themselves are quite interesting though, and may be useful in understanding
other data taken for the CAPMAP experiment. Despite the large uncertainty, finding
that the measured value ε moon is roughly correct provides useful confirmation that polarimeter gains (which are still being refined) are certainly close to the right values.
Since these measurements were taken, the detector has been moved to a different
location in the telescope’s focal plane. Additional measurements of the moon taken in
this configuration could confirm or disprove the theory that the notches are due to a
beam effect.
5 CONCLUSION
13
Figure 10: Maps of the instrumental “notch” effect. Vertical scale is degrees Kelvin.
REFERENCES
14
References
[1] Hedman, M.M., 2002. The Princeton IQU Experiment and Constraints on the Polarization of the Cosmic Microwave Background at 90 GHz. Ph.D. thesis, Princeton University.
[2] Davies, R.D. and F.F. Gardner, 1966. Linear Polarization of the Moon at 6, 11,
and 21 cm Wavelengths. Australian Journal of Physics, 19:823-836.
[3] Hecht, E., 1998. Optics (3rd Ed.). Addison Wesley.
[4] Chu, T.S., R.W. Wilson, R.W. England, D.A. Gray, and W.E. Legg, 1978. The
Crawford Hill 7-Meter Millimeter Wave Antenna. The Bell System Technical
Journal, 57(5):1257-1288.
[5] Barkats, D., 2003. Jupiter Pointing Memo. CAPMAP group internal memo.
[6] Strezhneva, K.M. and V.S. Troitsky, 1962. The Phase Dependence of Radio
Emission of the Moon on 3.2 cm. International Astronomical Union Symposium,
14:501-510.
[7] Hedman, M.M., 2003. February Sky-Dip Results. CAPMAP group internal
memo.
[8] Grossman, E., 1989. AT - Atmospheric Transmission Software: User’s Manual,
36-37.
[9] Gary, B., J. Stacey, and F.D. Drake, 1965. Radiometric Mapping of the Moon at 3
Millimeters Wavelength. Astrophysical Journal Supplement Series, 12:239-262.
[10] Hedman, M.M., 2003. Calibration Status. CAPMAP group internal memo.