Download Measurements of the Geomagnetic Field in the Antarctic Upper

Measurements of the Geomagnetic Field in the
Antarctic Upper Atmosphere Using the Tracer
Molecule 18O16O and Telescope Repair
Senior Honors Thesis
Oberlin College Department of Physics and
Astronomy
Michael J. Brown
[email protected]
April 5, 2007
2
An automated method of reducing spectra of the 2 → 1 rotational transition
of O18 O has been devised. This program expands on the existing framework
to include information about the Zeeman splitting pattern of a molecule with
a permanent magnetic dipole. The bulk of the values obtained for the strength
of the Earth’s magnetic field range from 40-50 µT, which is consistent with
existing measurements. There were anomalously high values for the field on the
last day of observation, which are still not understood.
A plan was also developed to redeploy the telescope in the solar observatory
of Oberlin College. The optical setup was optimized to take advantage of the
existing setup, and the reciever was repaired, both steps toward reviving the
telescope for future use.
Table of Contents
1 Introduction
1.1 Oxygen in the Atmosphere . . . . . . . . . . . . . . . . . . . . .
3
3
2 Theory
2.1 Modeling the Emission Spectra of O18 O . . . . . .
2.1.1 The Rigid Rotor . . . . . . . . . . . . . . .
2.1.2 Sources of Line Broadening . . . . . . . . .
2.1.3 Doppler Broadening . . . . . . . . . . . . .
2.1.4 Pressure Broadening . . . . . . . . . . . . .
2.1.5 Zeeman Splitting of O18 O spectra . . . . .
2.2 O18 O in the Earth’s Atmosphere . . . . . . . . . .
2.2.1 Existing Models of the Atmosphere . . . . .
2.2.2 Brightness Temperature . . . . . . . . . . .
2.2.3 Propagation of partially polarized radiation
2.3 Model Predictions for O18 O emission spectra . . .
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5
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16
3 Data reduction
3.1 Short users guide for Matlab routines .
3.1.1 Importing FITS files to Matlab
3.1.2 Preprocessing . . . . . . . . . . .
3.1.3 Fitting of Spectra . . . . . . . .
3.1.4 Post-Processing . . . . . . . . . .
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4 Results
25
4.1 Fitting of the Spectra . . . . . . . . . . . . . . . . . . . . . . . . 25
5 Reviving the Telescope
5.1 Gaussian Beams and Matrix
5.2 Planning out the telescope .
5.3 Heterodyne Mixing . . . . .
5.4 Repairing The Mixer . . . .
Methods
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29
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31
35
35
6 Conclusions and Future Directions
39
6.1 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1
2TABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTS
Chapter 1
Introduction
Predicting the behavior of the Earth’s atmosphere is a task that is both difficult
and important. The atmosphere is a complex system in constant flux, which
contains tremendous numbers of chemical and photochemical reactions taking
place within several different environments. It is its variable nature which makes
modeling so important, with aims to predict its behavior. While this is highly
important in predictions of short term atmospheric phenomena, such as weather
patterns, it has recently become prevalent as evidence of human-induced long
term climate change mounts.
Mapping long term climate change and predicting short term atmospheric
phenomena both require an extensive understanding of the atmosphere. Both
require models of the atmosphere, which in turn require a tremendous body
of data and testing in order to determine predictive capability or long term
trends. This work presents a body of data and further evidence for models in
place of the magnetic fields above the South Pole in the atmospheric region of
the mesosphere.
Data was taken using the Antarctic Sub-millimeter Telescope/Remote Observatory (AST/RO) during January of 2005 at the Amundsen-Scott South Pole
Station (Stark et al., 2001). Many of the current atmospheric models neglect
the extreme latitudes due to the difficulty of obtaining data at these remote
locations. Data collected at the poles should provide verification and elaboration for the current data taken from satellites and what few ground based
measurements exist.
1.1
Oxygen in the Atmosphere
Molecular oxygen is one of the most important components to humans, and
thus oxygen in the atmosphere has been extensively researched. In addition
to being critical to life near the surface of the Earth, the oxygen molecule is
a key component in a wide variety of different chemical processes that occur
higher in the atmosphere. Oxygen forms approximately 21% of the Earth’s
3
4
CHAPTER 1. INTRODUCTION
lower atmosphere, and is a highly reactive element relative to the other major
constutuents. The formation of the the O zone layer is a photochemical reaction
with oxygen molecules in the upper atmosphere.
Research of oxygen in the upper atmosphere has intensified over recent
decades due to evidence of long term climate change and the possibility that
human waste products have affected the Earth’s climate. There is also a great
need for modeling the behavior of oxygen in the polar regions, as the decay of
the O zone layer is most prevalent in this region. Much of the radiometric work
done investigating oxygen in the upper atmosphere has been done at middle latitudes and involve the primary isotope, 16 O2 (Lenoir, 1967; Liebe, 1981). This
work investigates the emission of O18 O, an isotope which composes about 0.4%
of the oxygen in the atmosphere.
As with most diatomic molecules, the oxygen molecule can effectively be
modeled as a rigid rotor with quantized angular momentum described by the
rotational quantum number J. When the molecule relaxes to a lower rotational
state (J decreases) it will emit a photon to carry away the lost energy. The main
isotope of oxygen is symmetric, which limits the possible rotational quantum
numbers, and thus limits the number of transitions which can be modeled. The
primary isotope of oxygen emits in the 60GHz region, yet this frequency range
is full of interfering absorption lines from water and other interferences, which
pose a problem for modeling. The primary isotope of oxygen is also populous
enough in the lower atmosphere that self-absorption becomes problematic for
resolving the finer structure of emission from higher elevations.
O18 O circumvents several of these problems due to its asymmetry. It emits in
a relatively quiet spectral region for the primary interferences of the atmosphere,
being water and Oxygen. Interactions of this isotope with the Earth’s magnetic
field will split the rotational transitions into a Zeeman splitting pattern, which
can be detected from the ground. This is especially interesting in the Antarctic,
as the upper atmosphere there is relatively poorly documented.
Chapter 2
Theory
The primary motivation of this study is to draw information about the geomagnetic field in the upper atmosphere. In order to extract the desired information
from the data available a model was developed of the radiating molecules and the
surrounding atmosphere, which influenced both the molecules and the emitted
radiation as it travels to the telescope.
2.1
2.1.1
Modeling the Emission Spectra of O18 O
The Rigid Rotor
The transitions observed in this experiment were between two rotational states
of O18 O. The standard model for an asymmetric molecule is that of a rigid
rotor, with two atoms separated by a fixed distance rotating around an axis
perpendicular to the internuclear axis. It is assumed that angular momentum
is quantized in units of h/2π (Townes & Schawlow, 1975, Ed. 1, p. 3). The
classical system described by the rigid rotor can then be described as
Jh
(2.1)
2π
where I is the moment of inertia about axis of rotation, ν is the frequency of
molecular rotation, and J is the angular momentum “quantum number.” The
frequency of the photon emitted during a rotational transition will be the difference between the frequencies associated with the upper and lower quantum
numbers. The primary isotope of oxygen is symmetric, which limits the possible rotational states the molecule can assume. Rotational states associated
with even J values are symmetric under particle exchange, and because two
indistinguishable fermions, such as the two identical oxygen atoms, must be antisymmetric under exchange, 16 O2 cannot assume rotational states with even J
values (Levine, 1975). O18 O is asymmetric and the particles are distinguishable
and thus can assume even values of J. The transition observed in this experiment
is from J = 2 to J = 0.
2πνI =
5
6
2.1.2
CHAPTER 2. THEORY
Sources of Line Broadening
An isolated O18 O molecule would emit a very sharp energy band spread only by
fluctuations required by the uncertainty principle, or so called natural broadening. In the atmosphere, however, there are two primary sources of line broadening which are far greater than natural broadening. A gas at a finite temperature
will have thermal motions which will cause Doppler Broadening, and a gas at
non-zero pressure will experience collisions which will alter the molecule’s energy levels and thus broaden the emitted spectrum, which is called Pressure
Broadening. These two sources of line broadening are discussed in more detail
below.
2.1.3
Doppler Broadening
Any gas at a non-zero temperature will have random thermal motions, which
will lead to a Doppler shift. Any molecule with a component of velocity relative
to the direction of photon propagation will emit a different frequency photon as
measured in the rest frame of the observer. If a molecule at emits a frequency ν0
in its rest frame, and has a component of velocity v parallel to the direction of
propagation, then the photon emitted will have a frequency of νdopp = ν0 (v/c)
in the rest frame of the observer. A neutral gas at temperature T is assumed to
have a distribution of velocities given by the Boltzmann distribution. From this
distribution, the probability of a molecule having a velocity v along the line of
observation is proportional to
"
2 #
−mc2
ǫ
−mv 2
= exp
.
(2.2)
exp
2kT
2kT
ν0
where ǫ is the frequency difference between the observed photon and the photon
emitted in the molecular rest frame. Given a large number of photons, this is
also proportional to the observed intensity at a frequency ν, and will produce a
Doppler lineshape
D(ν, ν0 ) =
(ln 2/π)1/2 e− ln 2((ν−ν0 )/∆νd )
∆νd
2
(2.3)
where ∆νd is the line’s half-width at half maximum. Solving for the half-width
and half-maximum yields
r
r
T
ν0 2kT
−7
∆νd =
ln 2 = 3.581 × 10
ν0 .
(2.4)
c
m
A
where A is the molecular weight (cf Janssen, 1993, p. 339; Townes & Schawlow,
1975, p. 337). The J = 2 → 0 transition of O18 O (M=34amu, ν0 =233GHz)
at T = 300K, has a Doppler width of approximately ∆ν = 2.8 × 104 Hz. The
Doppler linewidth and the linewidth determined form the pressure broadening
are both included in the final model line width.
2.1. MODELING THE EMISSION SPECTRA OF O18 O
2.1.4
7
Pressure Broadening
The model of pressure broadening used in this experiment was developed by
van Vleck & Weisskopf (1945). This theory is applicable to a larger range of
molecules than both the Debye and Lorentz models that preceded it. The distribution arrived at by Van Vlack and Weisskopf is characterized by a line width
parameter ∆ν and a resonant frequency ν0 . It is shown that the amount of pressure broadening is inversely proportional to the mean time between collisions,
and thus directly proportional to the local atmospheric pressure. The absorption
coefficient γ is shown to be
γ=
8π 2 N f
∆ν
.
|µij |2 ν 2
3ckT
(ν − ν0 )2 + (∆ν)2
(2.5)
where N is the molecular number density, µij is the electric dipole moment,
ν0 is emission line center frequency, ∆ν is the line width factor, and f is the
quantum state ratio, determined by Boltzmann statistics (Townes & Schawlow,
1975). For this experiment, the simplification was made such that the absorption
coefficient is modeled as
6.01M Hz
γ=
P T 0.2
(2.6)
mb
as determined for submillimetric lines of the common isotope 16 O2 by Liebe
(1981). This produces a linewidth of approximately 4 × 104 Hz, which is comparable to the Doppler broadening observed, resulting in a linewidth of approximately 105 Hz.
2.1.5
Zeeman Splitting of O 18 O spectra
Molecular oxygen has a permanent magnetic dipole resulting from interaction
of the two unpaired valence electrons. When in a magnetic field this will split
each rotational level into three discreet energy levels based on the relative orientation of the magnetic dipole and the external magnetic field. This splits
each rotational state into three distinct energy levels with quantum number M,
which has integer values between -1 and 1. An energy level schematic of the
2 → 0 transition for O18 O is presented in Figure 2.1. The relevant selection
rules are ∆M = 0, ±1 and 0 → 0 is forbidden. The transitions where ∆M = 0
and ∆M = ±1 are referred to as π transitions, and σ± transitions respectively.
Thus the observed spectra should have six distinct peaks. The energy splitting
of these levels can be determined from
J(J + 1) + S(S + 1) − N (N + 1)
(2.7)
∆E = (−1.0010M H) ×
J(J + 1
where S = 1 for Oxygen, µ0 is the Bohr magneton, H is the strength of the
external magnetic field, and ∆E is relative to the energy of the transition with
no external magnetic field. The quantum factors for the upper and lower states
8
CHAPTER 2. THEORY
of the 2 → 1 transition for O18 O are -1 and 2 respectively, and thus the transition
frequencies should be shifted by
∆ν(Hz) = 1.4015 × 104 × HµT (Mu + 2Ml )
(2.8)
It is known that the Earth’s magnetic field is approximately 20−70µT , and thus
the spectra should be split a fraction of a MHz around the center frequency of
the line. This splitting is of the same order of magnitude as the line broadening
expected, and thus the peaks should be overlap yet be distinguishable. The
angle of observation should also determine the effective magnetic field strength,
as the alignment of the geomagnetic field lines and propagation vectors will
determine what the effective magnetic field splitting the energy states.
2.1. MODELING THE EMISSION SPECTRA OF O18 O
9
Figure 2.1: 2 → 0 rotational transition of O18 O. The sub-levels are split such
that there are six possible transitions (Pardo et al., 1995)
10
CHAPTER 2. THEORY
2.2
O18 O in the Earth’s Atmosphere
In order to model the spectra that have been collected, a model must be used
to determine several important parameters of the atmosphere. A model must
also be developed of how the radiation propagates through the atmosphere. Information such as the neutral pressure and temperature of the atmosphere in
addition to the altitude distribution of O18 O was taken from existing models
of the atmosphere. The emission of O18 O is polarized depending on the specific transition, and thus a model developed by Lenoir (1967) for propagating
differently polarized radiation through the atmosphere was used.
2.2.1
Existing Models of the Atmosphere
The Earth’s atmosphere is usually divided into a set of four layers defined by
trends in the temperature profile. Though these layers are usually bounded by
altitudes, there is no fixed boundary, and thus each layer will vary in extent
based on both time and location. The bottom layer is the troposphere, which
extends from the ground up to about 10km in the Antarctic (Chamberlin, 2001).
The troposphere is characterized by temperature decreasing with altitude until
the tropopause, which separates the troposphere from the stratosphere. The
stratosphere extends from the tropopause to about 40 or 50 km, and is characterized by temperature slowly increasing with altitude. Extending between
the end of the stratosphere and about 80-90 km is the mesosphere, characterized again by a drop in temperature with altitude. The mesosphere has an
upper boundary called the mesopause, which has the lowest temperature in
Earth’s atmosphere, and separates the mesosphere from the top layer of the
atmosphere, the thermosphere. The thermosphere is not shielded from solar
radiation, and thus temperature increases dramatically with altitude. Most of
the radiation produced by O18 O comes from an altitude region between 60 and
75 km (Pardo et al., 1995), which lies in the mesosphere.
The temperature profile of the atmosphere has been known at mid-latitudes
for a long time, and models such as the US Standard Atmosphere (1976) are
reliable for these regions. A detailed temperature profile of the Antarctic latitudes has only been produced recently (Lübken et al. (1999), Hernandez et al.
(1995),Pan & Gardner (2003)). To approximate the atmospheric temperature
profile of the atmosphere over Antarctica, the MSISE Model 1990 model was
used to produce Figures 2.2 and 2.3). This models background temperature,
neutral density, and the concentrations of several atmospheric species as a
function of location, time, and altitude, with predictions at altitudes up to
1000 km. In the lower and middle atmosphere, the model is largely based on
the MAP handbook (Labitzke et al., 1985). Above 72.5 km, the MSISE-90 is
essentially a revision of the 1986 Mass-Spectrometer-Incoherent-Scatter (MSIS)
model(Hedin, 1987). The MSISE-90 and other standard atmospheric models
are obtainable from the National Space Science Data Center website1 .
1 http:// nssdc.gsfc.nasa.gov/ space/ model/ atmos/ atmos
index.html
2.2. O18 O IN THE EARTH’S ATMOSPHERE
2.2.2
11
Brightness Temperature
The data taken in this experiment was collected as a relation of frequency to telescope “temperature,” which is related to the power collected by the telescope,
and distantly related to the standard definition of temperature. “Temperature”
in radiometry typically refer to “radiation temperatures,” or “equivalent temperatures.” These are defined as temperature is related to the emission of a
blackbody. The Planck distribution formula for a blackbody radiator is given
by
ν3
2h
L(ν, T )dν = 2
dν,
(2.9)
c
ehν/kT − 1
where h is Planck’s constant, c is the speed of light, k is Boltzmann’s constant,
T is the temperature of the radiator, and L(ν, T )dν is the radiance (sometimes
called “brightness”), which is the radiated power per unit solid angle. The
spectral radiance density, Lf , is defined as the radiance per unit frequency,
Lf ≡ dL
df . Over small frequency intervals dν, L and Lf are approximately
constant and we can approximate
Z ν+dν
L(ν ′ , T )dν ′ = Lf (ν, T )dν
(2.10)
ν
For low-frequency radiation, this expression can be simplified using the RayleighJeans approximation, hv ≪ kT . In this limit, we make the approximation
ehν/kT ≈ 1 +
hν
+ ...,
kT
(2.11)
which in combination with Equation 2.9 gives the Rayleigh-Jeans law:
Lf (ν, T ) ≈
2ν 2
kT.
c2
(2.12)
This approximation is reasonable for the region of interest, although it breaks
down as hν approaches kT.
Even objects that do not behave like blackbodies can be modeled according to this “temperature” scale. For unpolarized radiation, Tb , known as the
brightness temperature, is the temperature of a blackbody that would emit the
same radiance as over the specified frequency interval. In the Rayleigh-Jeans
limit this is obtained by inverting Equation 2.12
Tb (ν) ≡
c2
2kT ν 2
(2.13)
For polarized radiation the brightness temperature can no longer be represented as a scalar, as it must also account for the different polarizations. This
is done in the method of Lenoir (1967). If the maximum available power per
unit bandwidth is given by W, then the antenna temperature, Ta , is defined
as the temperature of a blackbody that would emit the same power per unit
bandwidth.
W = kTa
(2.14)
12
CHAPTER 2. THEORY
Ta usually varies with frequency. Spectral Flux Density is a method of describing
energy transfer via different polarizations by means of a matrix. A matrix
with a spectral flux density P is a 2x2 matrix with diagonal elements Pxx and
¯
Pyy , describing the power traveling in the z-direction having x and y linear
polarizations respectively. The off diagonal elements of a spectral flux matrix
describe the coherence of the polarizations. If a wave described by a spectral flux
density P0 with a particular polarization, and the antenna having an effective
reception area for that polarization Ae , then the Antenna Temperature is given
by
Ta = P0 Ae /k
(2.15)
Using this to derive a brightness temperature matrix Tb allows radiation with
a finite angular spread to be measured, where Ta assumes that the energy flow
is received from an infinitely small angle. Brightness temperature is often a
more convenient measurement, as it relates the incoming power to radiance
as opposed to spectral flux density. Antenna and Brightness Temperature are
related by
dTa = (Ae /λ2 )TB (θ, φ)dΩ
(2.16)
Combining equations 2.15 and 2.16 provides a relation between an incident
power matrix P0 and the Brightness Temperature matrix TB . The brightness
temperature matrix is Hermetian, and transforms exactly as the Spectral Flux
Density Matrix. Brightness Temperature matrices are discussed in more detail
below.
2.2.3
Propagation of partially polarized radiation
The transitions from J = 2 → 0 emit different polarizations of light dependent
on the change of M. Observations of π transitions result from the emission and
subsequent remission of a photon that is linearly polarized in the direction of
the geomagnetic field vector. Similarly, observations of σ+ and σ− transitions
result from the absorption and re-emission of photons that are right-hand and
left-hand circularly polarized in the plane perpendicular to the geomagnetic
field. In order to model this we need a generalized form of the radiative transfer
equation that can take into account polarization.
From Maxwell’s equations, the propagation of a plane wave traveling in the
+z direction in terms of associated magnetic field H is
(d2 /dz 2 )H(z, ν) − G2 (ν)H(z, ν) = 0
(2.17)
H(z, ν) = exp[−G(ν)]H(0, ν)
(2.18)
which has solutions
where G(ν) is a propagation matrix in the polarization plane. G is dependent
on the medium which the wave is propagating through. The atmosphere is
modeled to have a scalar permittivity ǫ0 and a magnetic permeability tensor
2.2. O18 O IN THE EARTH’S ATMOSPHERE
13
µ = µ0 [I + χ(ν)], where the off diagonal elements arise from the interaction
with the external magnetic field. With these assumptions, G becomes
G(ν) = ik0 [I + χ(ν)/2]
(2.19)
where k0 is defined as k0 = 4π 2 ν 2 ǫ0 ν0
The electric field associated with the wave described above in then modeled
as a false vector(Ep (z, ν) : Epx = −Ey , Epy = Ex
Epx =
G(ν)
H(z, ν)
2iπνǫ0
(2.20)
The results of equations 2.18 and 2.20 can be used to determine the Poynting Vector. The time average of the Poynting Vector can be related to the
energy arriving at the detector due to natural emission. In order to get this
time average Lenoir followed the methods of Born & Wolf (1964), and created
a power-spectrum coherency matrix described by
Sij (ν) =< Epi (ν)H ∗j (ν) >
(2.21)
which is broken into its Hermetian component P (ν) and Antihermetian part
iQ(ν)
S(ν) = P (ν) + iQ(ν)
(2.22)
P (ν) is interesting because the diagonal elements Pxx and Pyy are the power
flowing in the z direction with linear polarization in the x and y direction respectively. Thus the trace of P is the total power flowing along the direction
of propagation. The off diagonal elements represent the coherence of the two
polarizations. The degree of polarization, p, can also be determined from this
matrix by
1/2
4 det P (ν)
p(ν) = 1 −
]
(2.23)
Tr(P (ν)2 )
Maxwell’s equations approximated to first order dictate the relations
(d/dz)Ep (z, ν) = −2iπνµ0 [I + χ(ν)H(z, ν)
(2.24)
(d/dz)H(z, ν) = −2iπνµ0 Ep
(2.25)
These two equations can be manipulated and combined with the earlier equations (Pardo et al., 1995) to produce a differential equation for the time averaged
Poynting vector.
(d/dz)S(z, ν) + G(ν)S(z, ν) + S(z, ν)G∗T (ν) = 0
(2.26)
This equation can be split into two equations, one for the Hermetian part of
S and another for the Antihermetian part. The Hermetian equation can be
rewritten in terms of the brightness temperature matrix Tb as defined in the
14
CHAPTER 2. THEORY
preceding section. The matrix P transforms equivalently to Tb , and thus the
Hermetian part of equation 2.26 can be written as
(d/dz)Tb + G(ν)Tb + Tb G∗T (ν) = 0
(2.27)
In order to include emission by the medium into this equation, the relation must
be set up that the change in brightness temperature plus the loss of brightness
temperature equals the emission for any given length.
(d/dz)Tb + G(ν)Tb + Tb G∗T (ν) = E(ν)
(2.28)
If the medium obeys thermodynamic equilibrium, a simplification can be made.
(d/dz)Tb = 0, in this case, and thus emission and absorption obey the relationship
G(ν)Te + Te G∗T (ν) = E(ν)
(2.29)
where Te is the emission temperature matrix for thermal emission. For the
Earth’s atmosphere, the emission temperature matrix will be diagonal with
elements equal to the kinetic temperature of the layer T(z). Manipulation of
the above equations (Pardo et al., 1995) yields the general solution for the Tb .
The generalized radiative transfer equation is then
Tb (z, ν) = e−G(ν)z Tb eG∗
T
(ν)z
+ T [I − e−2A(ν)x ]
where A is the Hermetian part of the propagation matrix G.
(2.30)
2.2. O18 O IN THE EARTH’S ATMOSPHERE
15
Figure 2.2: Temperature structure and layers of the atmosphere. Atmospheric
temperature profile from the MSISE-90 model for 90◦ S on January 1, 2004 at
0h UT.
Figure 2.3: Vertical mass density profile from the MSISE-90 model for 90◦ S on
January 1, 2004 at 0h UT.
16
2.3
CHAPTER 2. THEORY
Model Predictions for O18O emission spectra
As modeled by Liebe (1981) the z component of the electric field of this wave
can also be modeled as
Az = exp[ikz(I + Nz × 10−6 )]E0
(2.31)
where Nz is defined to be the refractivity matrix in the plane of polarization.
Pardo adapts this to reflect the Zeeman splitting of O18 O, which has a refractivity matrix given by
N0 sin2 (φ) + (N+ + N− ) cos(φ) −i(N+ − N− ) cos(φ)
(2.32)
i(N+ − N− )
N+ + N−
where N0 is the normalized intensity of the π transitions, N+ and N− are the
normalized intensities of the σ+ and σ− transitions respectively, and φ is the
angle between the line of observation and the magnetic field. Thus the predicted
lineshape will vary according to the angle of observation relative to the magnetic
field vector.
The function fitted for this project was a set of voigt lineshapes with a width
dependent on the temperature of the layer they were assumed to have originated
from. This was a variable parameter to an extent, though if the linewidth
of the fit was to narrow or to broad the scans were filtered out. The free
parameters of the fit included the background function, the relative heights of
the σ and π transitions, the intensity of the central σ transitions and the effective
magnetic field. As can be seen in the Refractivity Matrix, scans parallel to the
magnetic field should exhibit no π transitions, and produce a model spectrum
like that shown in Figure 2.4 for a magnetic field of 50 µT and an observation
angle parallel to the magnetic field lines. As the angle between the magnetic
field and the line of observation increases, the relative size of the π transitions
should increase, as shown in Figure 2.5 for the same model magnetic field at an
observation angle 45 degrees away from the magnetic field lines.
The two smaller inverted lineshapes flanking the large central peak are artifacts of the process called frequency switching. This is explained in more detail
in the following chapter. The sinusoidal term most prevelent on the edges of
the scan is a sample background vector, as discussed further in the following
chapter.
2.3. MODEL PREDICTIONS FOR O18 O EMISSION SPECTRA
17
12
10
Antenna Temperature
8
6
4
2
0
−2
−4
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
Frequency from Line Center
1.5
2
2.5
7
x 10
Figure 2.4: Predicted Lineshape with B=50 µT, at altitude of 65 km observing
parallel to the magnetic field. The intensity is arbitrarily set to approximate
the values of the other scans
12
10
Antenna Temperature
8
6
4
2
0
−2
−4
−6
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
Frequency from Line Center
1.5
2
2.5
7
x 10
Figure 2.5: Predicted Lineshape, away from field lines. The same parameters
as Figure 2.4, but now observing at 75◦ away from the magnetic field lines
18
CHAPTER 2. THEORY
Chapter 3
Data reduction
3.1
Short users guide for Matlab routines
This section of the project involved modeling the Zeeman substructure of the
spectra taken earlier by AST/RO, and automating the process of fitting and
characterizing the spectra. This was done by adapting a set of MATLAB
routines developed by Susannah Burrows for fitting a vertical profile of Carbon Monoxide (Burrown, 2005). These routines implemented the LevenburgMarquadt fitting algorithm, and were designed to fit the Zeeman splitting pattern described above.
The information garnered from the data included the geomagnetic field
strength along the axis of observation, taken from the splitting of the pattern,
and the relative intensities of the three different peak types, from which the
angle between the line observation and the geomagnetic field vector should be
able to be determined.
The steps of fitting of the data are outlined below:
1. Importing and selecting spectra (cf 3.1.1): The desired spectra are saved
into the working MATLAB directory. MATLAB then selectively imports
the desired subgroup of files based on data stored in the header of the
FITS file, allowing for selective processing.
2. Preprocessing (cf 3.1.2): The spectra are imported into MATLAB, and
“folded” as described below. The data is trimmed to contain only the
area around the peak and an approximation to the background function
is made.
3. Fitting (cf 3.1.3): The function O18Ofit.m fits the spectra according to the
desired form and returns the resulting fit and other important parameters
into a set of Matlab structures.
4. Post-processing (cf 3.1.4): Files were flagged during the fitting process
if the fit was below a standard residual. These scans were then sorted
through manually to see if an obvious cause of failure could be determined
19
20
3.1.1
CHAPTER 3. DATA REDUCTION
Importing FITS files to Matlab
After initial pre-processing on-site, the spectra taken by AST/RO were saved
in the Flexible Image Transport System (FITS) data format, which can be read
by Matlab. Matlab contains two built-in functions for reading FITS files,
fitsinfo, which reads FITS Headers, and fitsdata, which reads the FITS
data. These were combined into a more efficient function rfits, written by
A. Bolatto and modified for use in this project. rfits combines the header
information and the data into fields of a Matlab structure.
In order to select certain FITS files to process, a list of file names is attained
using the built-in command dir, which returns a struct of information about the
files in a given directory, one field of which will be the file names of every file in
the directory. These files can then be filtered by any desired set of criteria using
the function filter_filevec, which will import files which meet the criteria
input. This allows selection of scans which have a specific property shown in
the header, such as elevation, angle, or time.
3.1.2
Preprocessing
The preprocessing of the data involves three steps. Each spectrum is first
“folded” about the line center, then “trimmed” to include only the area near
the peaks. Finally, an initial guess is made at the background fit. These three
steps are accomplished using the routine trimautofo, which accepts as input
the Matlab cell filevec containing the list of file names.
The “folding” of the spectra reduces the background noise. This process
consists of inverting the spectrum, shifting it by the switching frequency, and
then combining this transformed spectrum and the original. This results in a
large central positive peak flanked by two smaller negative peaks, as illustrated
in Figures 3.1 and 3.2. This “folding” occurs in the function foldspectrum.
This function produces a structure similar to that produced by rfits with three
additional fields. The fields frequency and frequencyfo contain the frequency
vectors (ν) for the unfolded and folded spectra, respectively. The folded data is
stored in the field datafo.
The bandwidth of the telescope greatly exceeded the width of the desired
peeks, resulting in a large part of the spectrum that has no relevant data, yet
could interfere with fitting. The background for the scans had a complex and
inconsistent form which would require a complicated fitting method, and hinder
processing of relevant data. Thus, the spectra were “trimmed” to include only
the region close to the peak. This allowed for a simpler background subtraction
and increased fitting accuracy of the peaks. The spectra were trimmed to only
include a 50MHz region surrounding the peak. The starting and ending data
points included in this scan were given values ni and nf .
A preliminary background function was then fit to a subset of the trimmed
data which excluded the actual peak region. A 20MHz region around the peak
frequency was removed, and the remainder of the trimmed spectrum fit to a
3.1. SHORT USERS GUIDE FOR MATLAB ROUTINES
21
background equation of the form:
B(ν) = b1 + b2 · ν + b3 · sin(b4 · ν + b5 ),
(3.1)
where b1 ...b5 are the fitted parameters contained in the background vector b.
trimautofo outputs three Matlab vectors containing the values of ni , nf , and
b for each spectrum in filevec.
3.1.3
Fitting of Spectra
The routine O18Ofit then outputs important information into Matlab an array
of structures. The format of the structures output by O18Ofit is summarized
in Table 3.1.
O18Ofit uses an implementation of the Levenburg-Marquadt fitting algorithm, which has been implemented in Fortran, which can be linked with
Matlab through a set of gateway functions. The fitting algorithm is implemented in the Fortran file ZeemanO18O.f in order to fit the function by estimating the Jacobian through finite differencing, and then least-squares fitting.
The functional form being fit is returned by the function fvoigtO18O.f. This
function accepts a state vector, and returns a function based on the state vector,
and can easily be modified. The functions were fit to both a voigt lineshape
and a simple Lorentzian with the approximate voigt linewidth, and the better
fit of the two was chosen. Fortran and Matlab can be interfaced using the
functions mvoigtO18O.f and ZeemanO18O.f.
3.1.4
Post-Processing
After fitting, the scans which were flagged as having been poorly fit were reviewed manually to determine the source of the bad fit. This can be done using
the function view, which allows visual inspection of both the scan and the best
fit. Some of the spectra were noisy enough that a precise fit was impossible,
while others has some sort of obvious error that made them useless, and thus
could be quickly discarded. Scans which showed no obvious flaw were re-fit
using the mean of the successfully fitted scans as initial parameters.
After the scans had been either successfully fit or discarded, a simple error
analysis was conducted on the calculated magnetic field. A standard deviation
was calculated for all scans with the same telescope direction in order to determine if the model proposed above satisfactorily describes the mesospheric
magnetic field. Scans in which either the strength of the magnetic field or the
relative intensities of the peeks were more than 3 standard deviations away from
those at the same telescope heading were reviewed manually and then discarded
if the fit was misleading. This produced the final data that is presented in the
results section.
22
CHAPTER 3. DATA REDUCTION
Sample Raw Spectrum
35
Antenna Temperature
30
25
20
15
10
5
2.3391
2.3392
2.3393
2.3394
2.3395 2.3396
Frequency
2.3397
2.3398
2.3399
11
x 10
Figure 3.1: Sample raw spectrum, centered on TA = 0, ν = ν0 .
Figure 3.2: Sample spectrum from Figure 3.1, produced by foldspectrum.
3.1. SHORT USERS GUIDE FOR MATLAB ROUTINES
Field name
coffvec
stda
filevec
time.vec
time.num
resnorm
residual
telaz
telalt
obstime
magfield
std B
23
Description
Matrix containing the fitted “coefficient” state vectors a.
Matrix containing the estimated
standard variances in the state vector a.
Cell containing the names of files
included in the series.
Vector containing the date as a vector of the format [YYYY MM DD hh
mm ss].
Vector containing the time at
which each observation began, as
the number of whole and fractional
days since Jan 1, 2000.
Matrix with rows containing the 2norm 12 |F(a, x) − Tb |22 of the residual vector for each spectrum.
Cell containing the residual vectors
F(a, x) − Tb for each spectrum.
Vector containing the telescope’s
pointing direction in degrees from
geographic north (0◦ ).
Vector containing the telescope’s
radio pointing elevation in degrees
from the surface for each spectrum.
Vector storing the length of observing time for each spectrum.
Vector containing the magnetic
field strength along the axis of
propagation.
Vector containing the estimated
standard variance in the magnetic
field strength.
Table 3.1: Summary of the fields in Matlab structure output by timeseriesfo.
24
CHAPTER 3. DATA REDUCTION
Chapter 4
Results
1083 spectra of O18 O were taken using AST/RO and processed using the method
described above. The results of these scans agree with model predictions fairly
well, though further processing is desirable.
4.1
Fitting of the Spectra
The spectra were fit as described above, and those with exceptionally poor
fits were discarded. The initial processing removed spectra which produced a
calculated magnetic field of more than three standard deviations from the mean,
but upon manual review it was determined that some of these outliers appeared
legitimate fits. A sample ‘normal’ fit spectrum is presented in Figure 4.3.
The spectra that were classified as outliers exhibited a very bizarre spectrum,
which appeared to have two unsplit peaks that were greatly separated, yielding
a magnetic field of up to 130 µT. A sample of one such spectrum is presented
below in Figure4.2.
These bizarre spectra seemed to be relatively isolated in time to the third
day of observation, further lending credibility that they might be a legitimate
phenomenon. The fitting of these spectra is poorer, as they do not seem to fit the
same model as the other spectra due to the sharpness of the peaks, which also
appear to be unsplit. Thus, while the magnetic field is most likely not accurate
for these spectra, they do seem to display some sort of unique behavior.
A plot of the time dependency of the magnetic field shows agreement with
the baseline magnetic field value of about 44 µT as observed earlier by Pardo,
thought the standard deviation is larger, having a value of 15 µT due to non
ideal fits and the bizarre high field scans. If a smaller tolerance on the residuals
of the fit is imposed then the error is significantly reduced, giving the same
magnetic field average but an error of 9 µT. A plot of the calculated magnetic
field as a function of time is shown below.
The observations were all taken at telescope angles between 15◦ and 65◦
and therefor the relative intensities of the polarizations could not be readily
25
26
CHAPTER 4. RESULTS
determined with the fit quality available, though a greater range of angles or
less noisy scans would help this greatly.
27
4.1. FITTING OF THE SPECTRA
05s.3085.2.fits
Antenna Temperature(K)
15
10
5
0
−5
−2.5
−2
−1.5
−1
−0.5
0
0.5
Frequency(Hz)
1
1.5
2
2.5
7
x 10
Figure 4.1: One of the standard fits yielding a magnetic field on 42 µT
05s.1537.2.fits
Antenna Temperature(K)
10
5
0
−5
−2.5
−2
−1.5
−1
−0.5
0
0.5
Frequency(Hz)
1
1.5
2
2.5
7
x 10
Figure 4.2: One of the fits yielding a magnetic field of 132 µT
28
CHAPTER 4. RESULTS
Time Evolution of Magnetic Field
140
120
Magnetic Field uT
100
80
60
40
20
0
7.3233 7.3233 7.3233 7.3233 7.3233 7.3233 7.3233 7.3233 7.3233 7.3233
5
Observation Time
x 10
Figure 4.3: The time series of calculated magnetic field strength. The high field
scans appear only in the third day of observation
Chapter 5
Reviving the Telescope
After the AST/RO project was decommissioned to make way for newer projects,
the telescope was deconstructed. Many of the components were sent to Oberlin
College, where they will be redeployed over the next several years with hopes
of making a working telescope to continue research as described in the previous sections of this paper. The chosen location for the new telescope was a
vacated solar observatory, which already had in place many desirable features.
An outcrop was already set up with a large flat mirror that could be slid out
for observation and focused onto a large spherical mirror, which could begin
focusing the light. The setup of the telescope was planned out, and will be
deployed by others continuing the project in years to come.
In addition to planning out the telescope room, it was noticed that the mixer
in the receiving Dewar was no longer functioning as expected. The mixer block
was removed from the Dewar, repaired, and then put back inside the Dewar. It
now appears to be functioning normally.
5.1
Gaussian Beams and Matrix Methods
A “Gaussian Beam” is defined as a diffraction limited beam of coherent radiation
with energy concentrated at the canter of the axis of propagation and falling
off radially as a Gaussian. Gaussian Beam propagation in free space is readily
derived from the wave equation (G. & R.G., 1965). The amplitude of a Gaussian
beam propagating along the z-axis is cylindrically symmetric about the z-axis,
and near the axis of propagation is given by
ω 2πz
1
π
0
A(r, z) = A0
exp i
]
(5.1)
−
i
+ φ − r2
ω
λ
ω2
λR
where ω is the “spot radius”, defined as the radius where the amplitude of light
is 1/e of its value along the optical axis, and ω0 is the spot radius at the focus
of the beam, called the beam waist. R is the radius of curvature of surfaces
29
30
CHAPTER 5. REVIVING THE TELESCOPE
of constant phase, λ is the wavelength of light, and φ is an adjustment of the
phase which has the equation
λz
tan(φ) =
(5.2)
πω02
From equation 5.1 the coefficient of r2 has both a real and imaginary component. The real component 1/ω 2 indicates the amplitude is related to the function exp[−ir2 /ω 2 ], a Gaussian distribution, while the imaginary part −iπ/λR
describes a quadratic variation of the wave field. Both ω and R are functions
of z, with z = 0 being defined at the beam waist, and vary according to the
equations
2
λz
2
2
ω(z) = ω0 1 +
)
(5.3)
πω02
2 2
πω0
R(z) = z[1 +
]
(5.4)
λz
These equations can be used to determine the propagation of a Gaussian beam
through free space or in the presence of a converging element. In order to make
these calculations simpler, a ‘complex curvature parameter’ q is defined such
that the coefficient of r2 in equation 5.1 is reduced to −πi/λq and q has been
defined as
1
1
1λ
(5.5)
=
+
q
R πω 2
Substituting q into equations 5.3 and 5.4 produces a simple propagation relation
iλ
q = q0 + z, where q0 is the value of q at the beam waist, which is q0 = πω
2.
0
It can be shown (A. Gerrard, 1994) that this complex curvature parameter can
be used to describe a Gaussian beam interacting with a converging element,
according to the ABCD rule
q2 =
Aq1 + B
Cq1 + D
(5.6)
Where A, B, C, and D are elements of a transfer matrix characterizing the
optical system. The propagation from a beam waist in the presence of a focusing
element di away along the line of propagation will produce a second beam waist
a distance df from the focusing element governed by the transfer matrix
A B
A + Cdf B + Ddf + di (A + Cdf )
1 df
1 di
M=
=
0 1
0 1
C D
C
D + Cdi
(5.7)
which can be combined with equation 5.6 and solved for q2 , which yields the
equation
(A + Cdf )q1 + B + Ddf + di (A + Cdf )
q2 =
(5.8)
Cq1 + D + Cdi
Assuming that we are starting from a beam waist, q1 is imaginary at this point
and can be written as q1 = iκ. Using this as an initial condition and solving for
31
5.2. PLANNING OUT THE TELESCOPE
the next beam waist, where the real component of q2 goes to zero, equation 5.8
can be solved for df to produce
df = −
BD + (BC + AD)di + ACd2i + ACκ2
D2 + 2CDdi + (Cdi )2 ) + (Cκ)2
(5.9)
The imaginary component of q2 can also be used to determine the radius of the
beam waist relative to the radius of the initial beam waist. Taking only the
imaginary components of q2 from equation 5.8 yields
Im(q2 ) =
(AD − BC)κ
(D + Cdi )2 + κ2
(5.10)
At the beam waists, both q2 and q1 are imaginary, and thus substituting in
the imaginary part of equation 5.5, and taking advantage of the fact that the
determinant of the ABCD matrix is always equal to one, the radii of curvatures
are described by the equation
2
!
πRi
πRf2
λ
(5.11)
=
λ
CπR2i 2
(D + Cdi )2 +
λ
which can be solved for w2
wf =
wi
(D + Cdi )2 +
5.2
1/2
CπR2i
λ
2
(5.12)
Planning out the telescope
Using the above propagation techniques, a scheme was derived to focus the
incoming light from the sky into the receiver in the solar telescope room. The
existing setup contained a flat mirror on a movable track which could be wheeled
out onto an outcrop of the building to pickup the sunlight. This would reflect the
incoming sunlight onto a large spherical mirror, which would begin to focus the
beam onto a smaller parabolic mirror. This mirror would reflect the incoming
beam at an approximately 90 degree angle, and focus the beam into the Dewar
window, and thus into the feed horn of the receiver. The second parabolic mirror
has not yet been designed, and it desired to use the mobile flat mirror and the
spherical mirror that are already in place. A diagram of the setup is presented
in Figure 5.1.
In Figure5.1, the quantities wx are the diameters of the indicated beam
wastes, dx are the indicated separation distances, and vx represents the diameter
of the beam incident on each respective mirror. As is common practice, the
optical setup is identical in the forward and reverse directions, so to design our
setup we shall imagine that the receiver is acting as a transmitter. Thus, if the
end of the feed horn of the receiver is assumed to be the initial beam waist w1
32
CHAPTER 5. REVIVING THE TELESCOPE
which has a measured diameter of 2 cm, and from equation 5.3, in the absence
of a focusing element, the beam width at v1 is equal to
1/2
d1 λ
v1 = w1 1 +
= 0.02 × (1 + 1.03d1 )1/2
(5.13)
πw12
It is desirable to have the mirror approximately 1 meter away from the receiver,
so that the Dewar can be placed on a table in a convenient space in the room.
This fixes d1 at 1 m, and thus the radius of the beam incident on the curved
mirror will be about 2.6 cm, requiring that the mirror be made with at least a 3
cm diameter. For a single parabolic mirror with an effective focal length ρ the
tracing matrix has the form
1 0
A B
(5.14)
=
− ρ1 1
C D)
Plugging in these values and those obtained put into equation 5.9 above yield
a relation between d2 and the effective focal length of the parabolic mirror
d2 = −
1 − 2.68ρ + 1.05ρ2
2.68 − 2.1ρ + ρ2
(5.15)
and substituting into equation 5.12 yields
w2 =
1+
0.02
−
1.68
ρ2
2.1
ρ
(5.16)
To further constrain the situation, it is desirable to have the parabolic mirror
reflect the light at approximately a right angle, which simplifies wave coherence
and setup. With the spherical mirror already set up and the Dewar in the
desired location on the table, this would constrain the sum of d2 and d3 to
be 2.5 m. The spherical mirror also has a diameter slightly larger than 30cm,
and thus it is desirable to have the beam be slightly smaller than that at the
spherical mirror, so the constraint is made that v2 is equal to 0.3m. d3 can then
be written in terms of ρ as
d3 =
6.62 − 7.87ρ + 3.52ρ2
2.68 − 2.1ρ + ρ2
and the equation for v2 is then
1/2
2
2
)
0.059 (2.68−2.1ρ+ρ )×(1.17−1.87ρ+ρ
4
ρ
v2 =
= 0.3
2.68
1 + ρ2 − 2.1
ρ
(5.17)
(5.18)
Combining equations 5.17 and 5.18 and solving for ρ gives the effective focal
length to be 18cm. The effective focal length of an off-axis parabolic mirror
(Goldsmith, 1998) is given by
2f
=ρ
1 + cos(θ)
(5.19)
5.2. PLANNING OUT THE TELESCOPE
33
For the described system, this mirror would have a focal length of about 16 cm.
When this mirror is created and the system is set up as described above, the
beam coming from the sun should be focused correctly onto the feed horn as
desired.
34
CHAPTER 5. REVIVING THE TELESCOPE
d1
Off−Axis Parabolic Mirror
w1
v1
Feed Horn
d2
w2
d3
To Sun
w3
v2
Spherical
Mirror
d4
Figure 5.1: Developed by Kim et al. (2007)
5.3. HETERODYNE MIXING
5.3
35
Heterodyne Mixing
The radiation entering the receiver has a frequency of over 200 GHz, which
is difficult to transport and amplify using conventional means. Such a signal
cannot readily be transported down conventional wires due to self-inductance
and capacitance effects, and is thus very difficult to amplify. This problem
can be solved by reducing the frequency of radiation to a more manageable
frequency, in this case approximately 1 GHz. This is done in the mixer through
the process of heterodyne mixing.
The radiation from the sky travels down the waveguide and radiation with a
controllable frequency from a local oscillator is superimposed. These waves are
both incident on a device with a non-linear I-V curve, in this case a Josephson
junction. A Josephson junction is composed of two weakly coupled superconductors separated by a thin insulating layer. The quantum mechanics of Josephson
junctions are very complex, and beyond the scope of this work (Josephson,
1974), yet the I-V curve is known to have a component in which the current is
proportional the square of the input voltage. The voltage across the junction
due to the incident light from the sky and from the local oscillator are given by
Vsky = Asky sin(ωsky t + δ)
(5.20)
VLO = ALO sin(ωLO t)
(5.21)
Were the junction a simple resistor, then these two voltages would be simply
additive, but because the junction is a non-linear element, the total voltage
across the junction due to this radiation will be proportional to:
2
2
Vtot = (VLO + Vsky )2 Vtot = VLO
+ Vsky
+ 2 ∗ (Vsky × VLO )
(5.22)
where the term involving Vsky × VLO is critical. Substituting in the values of
these voltages given in equation 5.20 and VtoA2, neglecting the phase offset for
the radiation from the sky, and using the trig identities for products of two sin
functions makes this term into
Vtot = sin((ωsky − ωLO )t) + sin((ωsky + ωLO )t) + sin((ωLO − ωsky )t)... (5.23)
where the important terms are those related to the difference between the two
frequencies. If the local oscillator frequency is chosen such that it is near the
frequency expected from the sky, then these terms will have proportionally small
frequencies. In this experiment the difference in frequency was less than 2GHz.
Several other frequency signals are produced by this process, but these are
immediately filtered out with a bandpass filter, and thus the signal is reduced
to a few GHz, and can be processed further.
5.4
Repairing The Mixer
At the end of observations at the South Pole, this mixer had stopped working,
and thus needed to be repaired. At room temperature the junction functions
36
CHAPTER 5. REVIVING THE TELESCOPE
as a simple resistor. If the Josephson junction were short, then the resistance
between two points on the mixer block would be approximately 20Ω while if
the junction were functioning at room temperature a value of 60Ω would be
expected. Measurements of the mixer led to the idea that the junction had
become a short, and thus was probably broken. The power source was tested
using a dummy load, which matched the mixer circuit, except the junction
was approximated with a 20 Ω resistor. It was found that the power source
was working correctly, furthering the idea that the junction was most likely
destroyed.
The mixer block was taken to the California Institute of Technology and
investigated with the help of Jacob Kooi, its original designer. The mixer block
was disassembled to expose the junction, which was investigated with a high
powered microscope. A picture of the junction using this microscope is presented
below in Figure 5.2, and shows no obvious damage to the junction itself.
It was then noticed that one of the wires connecting to the junction appeared
discontinuous. The mixer must be cooled down to liquid helium temperatures
to function, as the junction must get cold enough to superconduct, and thus it
is likely that with repeated thermal cycling between liquid helium temperatures
and room temperatures the wire broke, as seen in Figure 5.3. This was repaired
by filling the gap with silver paint. After this repair the mixer functioned as
expected at room temperature, and was put back into the Dewar.
5.4. REPAIRING THE MIXER
37
Figure 5.2: The actual junction is the thin strip on the right half of the ‘bow-tie’
structure
Figure 5.3: The wire was repaired using silver paint
38
CHAPTER 5. REVIVING THE TELESCOPE
Chapter 6
Conclusions and Future
Directions
An automated method of reducing data from FITS spectra of the 2 → 1 rotational transition of O18 O has been produced. This spectra expands on the
framework in place for modeling mesospheric carbon monoxide to include information about the Zeeman splitting pattern of a molecule with a permanent
magnetic dipole. The bulk of the values obtained for the strength of the Earth’s
magnetic field range from 40-50 µT, which are consistent with those measurements made previously (Pardo et al., 1995). The anomalously high values for
the field on the last day of observation are still unexplained, though it is thought
to be a combination of the geomagnetic storm and an error in the fitting function.
A weakness of the data reduction method was the approximation that the
signal was coming from a single layer of the atmosphere with a constant temperature and pressure. This approximation allowed a determination of the magnetic
field, yet a more involved fitting process that accounted for the vertical distribution of O18 O, the atmospheric temperature, and background pressure more
rigorously would both yield a better fit, and possibly allow determinations of
vertical trends in magnetic field behavior. The algorithm developed for fitting a
vertical profile of carbon monoxide could be combined with the Zeeman fitting
developed above to produce such a fit, though this will require a significant
amount of further code manipulation.
This data is of particular interest due to the fact that the spectra were
taken in a time overlapping with the beginning of a geomagnetic storm. The
magnetic field behavior is thus somewhat erratic at the end of the data taking
process. These scans will be compared to the scans taken earlier to determine
the short term variability of the magnetic field in the upper atmosphere over
this period. A more thorough understanding of the behavior of a geomagnetic
storm is required to develop a model of the Earth’s magnetic field in the upper
atmosphere during this time, though this data will hopefully provide verification
39
40
CHAPTER 6. CONCLUSIONS AND FUTURE DIRECTIONS
for such a model.
The routines developed for this project are fairly easily to generalize to the
fitting of other molecules, as the form has been generalized to include transitional substructures. A change in the functional form of the file fvoigtO18O
allows fairly straightforward alteration of the fitting method to include any desired molecule and spectral form. Work is continuing on further generalizing
the fitting mechanism to include a wider variety of spectral forms at different
transition frequencies.
In addition to the computational results, a plan has been devised to redeploy
the telescope into the solar observatory at Oberlin. The optical setup has been
optimized theoretically, and will hopefully be realized soon. The mixer of the
telescope has been repaired such that it functions as expected at room temperature, though it remains to be seen if the repairs will hold when the mixer block
is cooled to liquid helium temperatures. The redeployment and calibration of
the telescope before it can be used again will likely take a significant amount of
time, yet the foundations have been laid for future projects to that end.
6.1
Acknowledgments
I’d like to thank Chris Martin for all of the time he’s spent helping me through
this project and his endless patience and understanding. None of this work
would have been possible without his input, and I’m very thankful for the opportunities he has offered.
I’d also like to thank Susannah Burrows and Molly Roberts for their work
developing the MATLAB and FORTRAN routines for CO which were adapted
for use in this project.
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