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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 5 6 6 7 7 10 10 11 12 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 19 20 20 21 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 29 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. Bibliography A. Gerrard, J. 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