UBC Undergraduate Research

Lunar Brightness Temperature Measurement with CHIME Zhang, Yuze 2020-04

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Lunar Brightness Temperature Measurement withCHIMEbyYuze ZhangA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFBachelor of Combined Physics and AstronomyinTHE FACULTY OF SCIENCE(Physics and Astronomy)The University of British Columbia(Vancouver)April 2020c© Yuze Zhang, 2020AbstractHalf a century after measurement of the lunar brightness temperature for prepa-rations of the moon landing, Canadian Hydrogen Intensity Mapping Experiment(CHIME) provides improved instrumentation and a more sophisticated methodfor the same measurement. Such measurement also helps our understanding ofCHIME itself and extends its limit, such as better calibration of antenna beam pat-terns and understanding of the artifacts within CHIME data. Previous research in1950s has provided multiple data points with lunar brightness temperatures from230 to 240 Kelvin between 100MHz and 1000MHz. However, a recent measure-ment performed by Murchison Widefield Array (MWA) indicates a much lowervalue of the lunar brightness temperature around 180 Kelvin at 150 MHz. Giventhat the CHIME band is also in low frequencies, a series of measurements can beperformed between 400MHz and 800MHz to test the result from MWA which canpotentially reject 1950s’ results and enhance our knowledge about lunar brightnesstemperature in low frequencies. This research is aimed to measure the lunar bright-ness temperature within the CHIME band from 400MHz to 800MHz. To measurethe lunar brightness temperature from CHIME’s visibility data, Fringestop wasperformed. Subsequently, an averaged background intensity was measured andsubtracted from the intensity obtained after Fringestop. Rayleigh Jeans Law wasused to finalize the calculation of the lunar brightness temperature. There is in-consistency in my result. Between the two measured lunar brightness temperaturevalues at different times and frequencies, one agrees with the 1950s measurementwhile the other agrees with the data measured by MWA in 2017. Nevertheless, thisstudy helps to pave the foundation of lunar brightness temperature measurementwith CHIME. More future measurements can be performed and averaged with im-iiproved methods.Key words: Canadian Hydrogen Intensity Mapping Experiment(CHIME), Lu-nar brightness temperature measurement, Fringestop, Visibility to skyiiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 CHIME, MWA, and Hydrogen Intensity Mapping . . . . . . . . . 11.2 History of Lunar Brightness Temperature Measurement . . . . . . 31.3 Aim of the Thesis and Future Prospective . . . . . . . . . . . . . 62 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Basic Radio Interferometry . . . . . . . . . . . . . . . . . . . . . 72.2 Generalization and Antenna Beam Pattern . . . . . . . . . . . . . 102.3 Visibility to Sky . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Intensity to Brightness Temperature . . . . . . . . . . . . . . . . 142.5 Background Subtraction and Uncertainty . . . . . . . . . . . . . . 153 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1 Finding Lunar Transit . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Visibility Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Fringestop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20iv3.4 Dividing Antenna Beam Ratio . . . . . . . . . . . . . . . . . . . 223.5 Background Subtraction and Uncertainty Calculation . . . . . . . 244 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.1 Background Measurement with Method 1 . . . . . . . . . . . . . 334.2 Background Measurement with Method 2 . . . . . . . . . . . . . 404.3 A Different Temperature at 565.625MHz? . . . . . . . . . . . . . 405 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . 485.1 Measured Lunar Brightness Temperature . . . . . . . . . . . . . . 485.2 Explanation of Inconsistency and Implication . . . . . . . . . . . 495.3 Limitations and Potential Improvements . . . . . . . . . . . . . . 50Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52vList of TablesTable 4.1 Values of background intensities taken on different dates in twopolarization for a frequency of 638.28MHz. The intensities arein units of Jansky. All the values have been rounded up to 2decimal places. . . . . . . . . . . . . . . . . . . . . . . . . . . 37Table 5.1 Values of lunar brightness temperature with uncertainty mea-sured in two polarization using two different methods. . . . . . 48viList of FiguresFigure 1.1 A picture of CHIME. As can be seen, the main structure ofCHIME is four parabolic cylinders with focal lines at their foci. 2Figure 1.2 Scattered data of lunar brightness temperature as a function offrequency measured by Russian radio astronomers in 1963(inblue) and Mckinley et al(in red). The attempted fit of the 1963data shows a brightness temperature from 220 to 240K. In con-trast, the temperature measured in 2017 is around 180K. Thisis roughly 40 to 60K less than the Russian measurements. Thegreen band on the plot is the frequency range of CHIME from400 to 800MHz. KRJ is the temperature in Kelvin in Rayleigh-Jeans limit. Data is from V. D. Krotikov & V. S. Troitski andalso from Mckinley et al. [6] [7] . . . . . . . . . . . . . . . . 4Figure 2.1 A simplified radio interferometer baseline. The signals fromdirection sˆ received by two antennas with time varying ampli-tude have a time delay induced by incoming angle. Inducedsinusoidal voltage signals are multiplied at the multiplier andaveraged over time to obtain visibility. Figure from A. R.Thompson et al. [12] . . . . . . . . . . . . . . . . . . . . . . 8viiFigure 2.2 A generalized sky to baseline relation. The baseline and thedirection of the source are considered vectors within certaincoordinates. Infinitesimal intensity are subtended by solid an-gle dΩ. The interferometer outputs visibility which are Fouriercomponents of sky intensities. Figure from A. R. Thompson etal. [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Figure 2.3 The CHIME first light measured in 2017 which is the signalfrom source Cyg. A. As can be seen, the data in real andimaginary parts of the complex visibility is shown in thin blueand orange lines. The intensity of the source Cyg. A, on theother hand, is shown in thick green line. The synthesized beamwidth of the Cyg. A during the transit is around 1000 seconds.The data is taken during the transit of Cyg. A. . . . . . . . . . 12Figure 3.1 The declination of the moon during transits of the moon through-out one year and two months period. As can be seen, thedeclination of the moon varies from around -23 degrees to 23degrees sinusoidally over roughly 14 periods during approxi-mately 14 months. The yellow and blue dots on the plot indi-cate the two transits of the moon which have the closest dec-lination to 22.01446 degrees. The green dashed line indicatesthe declination of 22.01446 degrees. . . . . . . . . . . . . . . 18Figure 3.2 is a plot of signal after Fringestop containing antenna angu-lar response (beam ratio) in north-south polarization in Jansky.The x axis here is time during the lunar transit on September23rd, 2019. As can be seen, the signal over time is a curve witha peak value around 210 Jansky. . . . . . . . . . . . . . . . . 21Figure 3.3 is a plot of signal after Fringestop containing antenna angularresponse in east-west polarization in Jansky. The x axis hereis time during the lunar transit on September 23rd, 2019. Ascan be seen, the signal over time is a curve with a peak valuearound 220 Jansky. . . . . . . . . . . . . . . . . . . . . . . . 22viiiFigure 3.4 (a) and (b) show the antenna beam in two different polariza-tion. The antenna angular response in two polarization in theimages is part of a 3D antenna angular response. As can beseen, the response at different angle is different. The magni-tude of the response here is in units of dB. . . . . . . . . . . . 23Figure 3.5 shows the measured intensity at the location of the moon innorth-south polarization during the lunar transit on September23rd. As can be seen, the peak of the curve is between 15:11and 15:20. The beam width is about 1000 seconds. The peakof the curve is around 230 Jansky. . . . . . . . . . . . . . . . 25Figure 3.6 shows the measured intensity at the location of the moon ineast-west polarization during the lunar transit on September23rd. As can be seen, the peak of the curve is between 15:11and 15:20. The beam width is about 1000 seconds. The peakintensity of the curve is around 260 Jansky. . . . . . . . . . . 26Figure 3.7 shows the measured intensity of the background on Septem-ber 30th in north-south polarization at the same location of themoon at transit on September 23rd. As can be seen, the peakof the curve is around 8000 seconds after 1.56985e9 in unixtime (13:26 Sep. 30th, 2019). The peak intensity of the curveis around 25 Jansky. . . . . . . . . . . . . . . . . . . . . . . 27Figure 3.8 shows the measured intensity of the background on September30th in east-west polarization at the same location of the moonat transit on September 23rd. As can be seen, the peak of thecurve is around 8000 seconds after 1.56985e9 in unix time.The peak intensity of the curve is around 25 Jansky. . . . . . . 28Figure 3.9 shows the measured intensity of the background on September23rd in north-south polarization at a point with a smaller rightascension than the right ascension of the moon. As can be seen,the peak of the curve is around 50000 seconds after 1.5692e9in unix time. The peak intensity of the curve is around 40 Jansky. 29ixFigure 3.10 shows the measured intensity of the background on September23rd in north-south polarization at a point with a larger rightascension than the right ascension of the moon. As can be seen,the peak of the curve is around 53000 seconds after 1.5692e9in unix time. The peak intensity of the curve is around 20Jansky. Also, the feature of the moon is still visible in this plotbetween 15:03 and 15:20. . . . . . . . . . . . . . . . . . . . 30Figure 4.1 shows the measured intensity of the background on Septem-ber 21st in north-south polarization at the same location of themoon at its transit on September 23rd. As can be seen, thepeak of the curve is around 76000 seconds after 1.569e9 inunix time. The peak intensity of the curve is around 40 Jansky. 34Figure 4.2 shows the measured intensity of the background on September21st in east-west polarization at the same location of the moonat its transit on September 23rd. As can be seen, the peak ofthe curve is around 14:26 (76000 seconds after 1.569e9 in unixtime). The peak intensity of the curve is around 40 Jansky. . . 35Figure 4.3 shows the measured intensity of the background on Septem-ber 28th in north-south polarization at the same location of themoon at lunar transit on September 23rd. As can be seen, thepeak of the curve is around 15:40 (5250 seconds after 1.56968e9in unix time). The peak intensity of the curve is around 20 Jansky. 36Figure 4.4 shows the measured intensity of the background on September28th in east-west polarization at the same location of the moonat lunar transit on September 23rd. As can be seen, the peakof the curve is around 15:40 (5250 seconds after 1.56968e9 inunix time). The peak intensity of the curve is around 30 Jansky. 37xFigure 4.5 shows the measured intensity of the background on Septem-ber 29th in north-south polarization at the same location of themoon at lunar transit on September 23rd. As can be seen, thepeak of the curve is roughly between 15:30 and 15:45 (71500and 72000 seconds after 1.56968e9 in unix time). The peakvalue of the curve is around 20 Jansky. . . . . . . . . . . . . . 38Figure 4.6 shows the measured intensity of the background on September29th in east-west polarization at the same location of the moonat lunar transit on September 23rd. As can be seen, the peakof the curve is between 15:38 and 15:46 (71500 and 72000seconds after 1.5697e9 in unix time). The peak value of thecurve is around 25 Jansky. . . . . . . . . . . . . . . . . . . . 39Figure 4.7 shows the declination of the moon at lunar transits in past 100days before April 19th. The green dashed line indicates thedeclination of 22.01446 degrees. The orange and blue dotsare two transit times that the moon declination is closest to22.01446 degrees. . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 4.8 is a plot of signal after Fringestop in north-south polarization inJansky which contains antenna angular response (beam ratio).The x axis here is time during the lunar transit on March 6th,2020. As can be seen, the signal over time is a curve with apeak value around 125 Jansky. . . . . . . . . . . . . . . . . . 42Figure 4.9 is a plot of signal after Fringestop in east-west polarization inJansky which contains antenna angular response (beam ratio).The x axis here is time during the lunar transit on March 6th,2020. As can be seen, the signal over time is a curve with apeak value around 130 Jansky. . . . . . . . . . . . . . . . . . 43Figure 4.10 shows the measured intensity at the location of the moon innorth-south polarization during the lunar transit on March 6th.As can be seen, the peak of the curve is close to 5:03 (71000after 1.5834e9 unix time). The beam width is about 1000 sec-onds. The peak intensity of the curve is around 190 Jansky. . . 44xiFigure 4.11 shows the measured intensity at the location of the moon ineast-west polarization during the lunar transit on March 6th.As can be seen, the peak of the curve is close to 5:03 (71000seconds after 1.5834e9 in unix time). The beam width is about1000 seconds. The peak intensity of the curve is around 220Jansky. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 4.12 shows the measured background intensity in north-south polar-ization on March 6th before the lunar transit. As can be seen,the peak of the curve is close to 3:56 on Mar. 6th, 2020 (7000seconds after 1.58346e9 in unix time). The peak intensity isaround 30 Jansky. . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 4.13 shows the measured background intensity in east-west polar-ization on March 6th before lunar transit. As can be seen, thepeak of the curve is about 3:48 (6500 seconds after 1.58346e9in unix time). The peak intensity is around 65 Jansky. . . . . . 47Figure 5.1 Scattered data of lunar brightness temperature as a function offrequency measured by Russian radio astronomers in 1963(inblue) and Mckinley et al(in red). The result from this projectis also marked on the plot. The orange square dot with uncer-tainty is the lunar brightness temperature measured at 638.28MHzwith uncertainty. The brown square dot is the lunar bright-ness temperature estimated at 565.625MHz. The attempted fitof the 1963 data shows a brightness temperature from 220 to240K. The temperature measured in 2017 is around 180K. Thegreen band on the plot is the frequency range of CHIME from400 to 800MHz. KRJ is the temperature in Kelvin in Rayleigh-Jeans limit. Previous data is from V. D. Krotikov & V. S. Troit-ski and also from Mckinley et al. [6] [7] . . . . . . . . . . . . 50xiiAcknowledgementI would like to express my special thanks of gratitude to my supervisor professorMark Halpern as well as our course instructor Robert Kiefl who gave me the oppor-tunity to do this wonderful project on the Lunar Brightness Temperature Measure-ment with CHIME, which also helped me in doing a lot of research, and I come togain so much experience that I am really thankful to them.Furthermore, my completion of this project could not have been accomplishedwithout the support of postdoctoral fellows and PhD students at CHIME, TristanPinsonneault Marotte, Dallas Wulf, and Carolin Hofer; and the entire CHIME teamat CNRC-DRAO(CHIME site) for the data collection and previous work.xiiiChapter 1Introduction1.1 CHIME, MWA, and Hydrogen Intensity MappingCanadian Hydrogen Intensity Mapping Experiment (CHIME) is an interferometricradio telescope designed to measure the large scale neutral hydrogen 21cm emis-sion power spectrum across the redshift range 0.8 to 2.5. This power spectrumis primarily used to measure the baryon acoustic oscillation (BAO) scale closeto the transition from matter dominated to cosmological constant dominated Uni-verse. BAOs are primordial acoustic waves caused by slight over and under densi-ties in the early Universe before epoch of photon decoupling. The acoustic wavepropagates radially outwards until the Universe expansion rate exceeds the rate ofphoton scattering. The baryon matter without photon lost momentum and formedoverdensity with dark matter in an overall matter distribution of the Universe witha consistent scale of about 500 million light years. The characteristics of BAOwas previously measured by Cosmic Microwave Background experiments (suchas Planck) as a ”standard yardstick” which can be used to map out the expansionhistory of the Universe.CHIME maps over 3 percent of the total observable volume of the Universeand makes 3D maps of the BAOs. The neutral hydrogen measurement provides apromising avenue for testing the ΛCDM model of the Universe. However, funda-mentally, it is difficult to observe neutral hydrogen for two reasons. First, 21cmemission signal is weak and covered by bright foreground celestial objects [7].1Figure 1.1: A picture of CHIME. As can be seen, the main structure ofCHIME is four parabolic cylinders with focal lines at their foci.Second, systematic error caused by instrumental effects contaminates the signal,and thus, such measurements require a high calibration precision [7].To collect the 21cm signals, CHIME uses cloverleaf antennas with dual po-larization aligned in directions of north-south and east-west. These antennas aremounted on the focal lines of 4 parabolic cylinders with surfaces covered by metalmesh to focus signals. The signals from antennas, after passing through low noiseamplifiers (LNA), are sent to supercomputers to be averaged over time and multi-plied to form visibility data which can be used to analyze different source withinthe CHIME band.Similar to CHIME, Murchison Widefield Array (MWA) is also designed tomeasure the 21cm emission. Instead of focusing on the period of redshift range 0.8to 2.5, MWA is designed to detect the signal from the Epoch of Reionization (EoR)and the cosmic dark ages [7]. During the cosmic dark ages, there is no formedgalaxy scale structures that can emit light. However, baryons, mostly consist ofneutral hydrogen, have formed large scale structures within the universe during2this time and can be observed in radio band. Unlike CHIME, MWA uses dualpolarization dipole elements on a 4m by 4m steel mesh setting flat on the groundto detect the 21cm signal.1.2 History of Lunar Brightness TemperatureMeasurementIn early 20th century, the only information about the lunar physical condition wasthe sunlight reflected from its surface. Naturally, this would only contain informa-tion on the state of substances on the lunar surface and its microrelief (throughness,graininess, etc.) [6]. By now, however, the properties of the lunar surface sub-stances from lunar emission have been well studied (based on observation throughtelescopes and samples from moon landings). Unlike any of the Earth rocks, thematerial on the moon surface is rough and finely crushed based on their reflectingability and polarization properties [6].In 1949, the measurement of the moon’s own thermal radio emission at 1.25cm by Piddington and Minnet reached a conclusion that the upper cover of themoon has a double-layer dust structure which also has an estimated effective elec-tric conductivity [6]. This research indicates that the radio emission of the moon isdetermined by a thick layer of matter and contains information of the entire layer’sphysical properties [6]. A series of later measurements following the Piddingtonand Minnet’s measurement in radio band indicates various physical conditions onits surface including the temperature, thermal and electric properties of the sub-stances [6].In fact, some intensive follow up researches were conducted by Russians inthe 1960’s before the moon landing using multiple standard disk radio telescopeswith diameters under 10 meters [6]. In 1963, these Russian researches were aimedto provide a brightness temperature of the moon averaged over time for futurelunar missions which never took place. In these Russian studies, the structureof the outer cover of the moon, thermal properties of the material of the moon’scrust, density/dielectric constant of the rocks in the moon outer crust, nature of thematerial of the outer cover of the moon, and thermal state of the lunar interior wereall investigated via the data taken in radio frequencies [6].3The data leads to brightness temperature they measured which is shown inFigure 1.2. As can be seen, nevertheless, there is a huge room for improvement onthe data measured in 1963. The data taken between 100 and 4000 MHz is limited.The temperature is measured at only a few frequencies with fairly large uncertainty.This data is only considered sufficient in the 1960’s.Figure 1.2: Scattered data of lunar brightness temperature as a function offrequency measured by Russian radio astronomers in 1963(in blue) andMckinley et al(in red). The attempted fit of the 1963 data shows abrightness temperature from 220 to 240K. In contrast, the temperaturemeasured in 2017 is around 180K. This is roughly 40 to 60K less thanthe Russian measurements. The green band on the plot is the frequencyrange of CHIME from 400 to 800MHz. KRJ is the temperature in Kelvinin Rayleigh-Jeans limit. Data is from V. D. Krotikov & V. S. Troitskiand also from Mckinley et al. [6] [7]By contrast, new data taken by MWA in 2017 indicates a much different num-ber of the lunar brightness temperature at low frequencies. The temperature mea-sured by MWA has a value around 180K which is more than 50 degrees lower thanthe fitted temperature from the 1963 Russian results [7]. This is a huge differenceconsidering the brightness temperature of the moon is only 4 times of the differ-ence. The MWA measurement used lunar occultation method developed in 20144to measure the occulted sky temperature behind the moon which is the main ob-jective of their project [7]. This technique allows a measurement of the relationbetween the flux density of the moon, Sm(ν), and the frequency. The condition ofsuch measurement is an adequate sampling of the lunar disk’s angular scales [7].The flux density of the moon was later converted to the brightness temperature, ∆Tin Kelvin, by the equation∆T (ν) =10−26c2Sm(ν)2kΩν2(1.1)where ν is the frequncy in Hz, c is the speed of light in SI unit, k is the Boltzmannconstant in SI unit, Sm(ν) is the measured flux density of the moon in Jansky,and Ω is the solid angle of the moon [7]. Using the lunar occultation method,the measurement can achieve high precision despite the variation of measurementsensitivity with respect to the baseline length of a radio interferometer [7].In the MWA research, the actual temperature of the moon was considered asthe difference of the temperature measured (i.e. the total lunar brightness temper-ature, Tlunar(ν)) and the sky temperature, the averaged brightness temperature ofthe occulted patch of sky (i.e. the background temperature, Tsky) [7]. Here, the skytemperature is considered as the combined temperature of the galaxy, TGal , cosmicmicrowave background (CMB), TCMB, and the temperature from the era of reion-ization (EoR), TEoR [7]. The temperature of the CMB used here is TCMB = 2.725K.As a side note, the ultimate goal of the MWA project is an accurate measurementof EoR temperature [7]. The measured temperature, on the other hand, includes theintrinsic thermal temperature of the moon, Tmoon, the temperature of the reflectedEarth shine, Tre f l−Earth, and the temperature of the reflected Galactic shine, Tre f lGal .The reflected Earth shine is contributed by the radio emission from Earth reflectedoff on the moon surface back to the antenna. Similarly, the reflected galactic shineis contributed by the radio emission reflected off the moon surface from the MilkyWay. The equation of the lunar brightness temperature is, therefore,∆T (ν) = Tlunar(ν)−Tsky(ν)= [Tmoon+Tre f l−Earth(ν)+Tre f l−Gal(ν)]− [TGal(ν)+TCMB+TEoR(ν)].(1.2)5However, in MWA’s original measurement of the lunar brightness temperature, thesignal from the EoR was not considered for the measurement of the moon andgalactic synchrotron emission because the EoR temperature is more than 2 ordersof magnitude weaker than galactic temperature from synchrotron emission, TGal[7].1.3 Aim of the Thesis and Future ProspectiveIn this paper, several new measurements of the lunar brightness temperature withCHIME will be presented. These measurements are set to potentially test the newresult of the lunar brightness temperature measured by MWA in 2017, and the re-sult of this study might approve or disapprove Russian measurements in the 1950’swhile providing new insight into the radio emission of the moon in low frequencies.Furthermore, the study will provide the method necessary for future measure-ments of the lunar brightness temperature with CHIME. The measurement of themoon can offer useful information about the beam pattern of the CHIME feeds andthe artifacts within the measurements [2]. By using the lunar brightness tempera-ture data, it is expected to further improve the 0.1 percent calibration of CHIMEfeeds done previously with other sources [4]. The artifacts in most of the CHIMEmeasurements, the ripple in the intensity with respect to frequency, are interfer-ence within the CHIME cylinders. This interference is, in theory, caused by thestanding wave formed between the focal line of the telescope and the surface of thecylinder [8]. The measured lunar brightness temperature can help to further under-stand this extra feature in the signal and improve the quality of CHIME visibilityfor measuring other sources in the future.6Chapter 2TheoryThe measurement of the lunar temperature involves understanding of the radio in-terferometry and synthesis imaging. The basic theory will be introduced with asimplified model followed by its generalization and discussion including introduc-ing antenna beam pattern. The theory in transforming CHIME visibility data tosky intensity will also be shown afterwards. In this section, we also introduce theconcept of subtracting background temperature and uncertainty measurement.2.1 Basic Radio InterferometryRadio telescopes are designed to detect electromagnetic waves, within the fre-quency band of detection. For CHIME, the band frequency is from 400 to 800MHz. Incoming EM waves from sky are considered as sinusoidal waves with cer-tain time changing amplitude due to earth spin. The signal picked up by cloverleafantennas at CHIME will induce current in the circuit that results in a sinusoidalvoltage [3].For a single source far away, the incoming radiations are considered parallelfor different antennas on the ground as shown in Figure 2.1. The source locationgives an angle with respect to the baseline of the interferometer. Thus, it results ina time delay of the signals between different antennas:τg =~b·sˆc=bcosθc(2.1)7Figure 2.1: A simplified radio interferometer baseline. The signals from di-rection sˆ received by two antennas with time varying amplitude have atime delay induced by incoming angle. Induced sinusoidal voltage sig-nals are multiplied at the multiplier and averaged over time to obtainvisibility. Figure from A. R. Thompson et al. [12]At the antenna further away from the source, the phase of the sinusoidal voltage isτg smaller than the antenna closer to the source.Signals V1 and V2 are designed to be multiplied by a gain of the amplifier and gothrough a band. Frequencies of signals from multiple antennas after going throughthe amplifier and the band before reaching the multiplier is the central frequencyof the band, and this frequency is not changed [12]. Subsequently, signals from2 antennas are multiplied together at the multiplier and averaged over a time in-8terval T which is large enough to filter out the more rapidly varying terms in timeafterwards. The output of the multiplier is proportional toV1V2 =V (t)V (t− τg) ∝ F (2.2), whereF = 2sin(2piνt)sin(2piν(t− τg))= cos(2piντg)− cos(4piνt)cos(2piντg)− sin(4piνt)sin(2piντg)(2.3)[12]. During the time averaging process, suppose the time interval for the averagingis 2T, to average the multiplied signal, we have:〈V1V2〉= 12T∫ T−TV (t)V (t− τg)dt (2.4)Accordingly, we can get rid of the rapidly varying terms, and the last 2 terms of thesignal will be gone. Now F is left with [12]:Favg = cos(2piντg)= cos(2pibcosθλ)(2.5)This gives the real part of the visibility detected by the radio interferometer:Rr =V 22cos(2piντg)=V 22cos(ωτg)=V 22cos(2pibcos(θ)λ)(2.6)Normally, there is an additional 90◦ phase shift on V1 which directly contributes tothe complex part of the visibility to avoid null result from a integral of an odd dis-tribution of sources when mapping a region of sky [12]. The result of the complex9part, after the same process used to get the real part, is:Rc = iV 22sin(2piντg)= iV 22sin(ωτg)= iV 22sin(2pibcosθλ)(2.7)Thus, we can form a complex visibility from these two signals at a particular fre-quency and a particular time of a source at a particular angle:R =V 22eiωτg (2.8)2.2 Generalization and Antenna Beam PatternFor CHIME, however, the beam pattern covers a large angular area of the sky, sothe calculation has to be generalized to higher dimension [2]. The location of thesource on the sky is subtended by a solid angle, but if the source is small, we canconsider it as a point source in certain angular direction on the sky with intensityI(sˆ) where sˆ denotes the source direction. Aside from the intensity of the source,antenna beam pattern A(sˆ) in the direction of the source is also required for themeasurement [12]. Between a pair of antennas, we denote the beam pattern asAi(sˆ) and A j(sˆ).In reality, the antenna is receiving signals from multiple sources over the entiresky. As shown in Figure 2.2, the total visibility obtained, therefore, is the inverseFourier transform of the sky intensity, namely the sum of all point sources’ inten-sities along with the beam pattern within the mapping regionR =∫ΩAi(sˆ)A∗j(sˆ)I(sˆ)e2piiφdΩ (2.9)whereφ =ν~bc· sˆ =~bλ· sˆ. (2.10)That is, the visibility obtained is the Fourier component of the sky intensities [10].10Figure 2.2: A generalized sky to baseline relation. The baseline and the di-rection of the source are considered vectors within certain coordinates.Infinitesimal intensity are subtended by solid angle dΩ. The interferom-eter outputs visibility which are Fourier components of sky intensities.Figure from A. R. Thompson et al. [12]2.3 Visibility to SkyGiven the visibility as Fourier components of the sky intensities, it is intuitive toconsider the sky signal as the Fourier transform of the visibility data. However,in practice, it is easier for one to depend on previous CHIME measurements ofother sources to understand the visibility to sky signal transformation (or so calledFringestop).Figure 2.3 shows the visibility signal and the intensity of the source Cyg. Ameasured by CHIME in 2017 (CHIME first light). The imaginary and real partsof the complex visibility are plotted in thin blue and yellow lines. The intensity ofCyg. A is plotted in thick green line. The x axis gives the time of transit (a timewhen the sources pass the meridian) in unix time after the start of the transit (thetime when the data starts). Obviously, there is a phase in the visibility data which11Figure 2.3: The CHIME first light measured in 2017 which is the signal fromsource Cyg. A. As can be seen, the data in real and imaginary partsof the complex visibility is shown in thin blue and orange lines. Theintensity of the source Cyg. A, on the other hand, is shown in thickgreen line. The synthesized beam width of the Cyg. A during the transitis around 1000 seconds. The data is taken during the transit of Cyg. A.results in the sinusoidal pattern contained in the wave packet centered at the timeof transit. From the plot, the intensity of the source Cyg. A is simply the visibilityof Cyg. A without the phase.As mentioned before in the basic interferometry and its generalization, thephase of the visibility of a point source such as Cyg. A isφ =ν~bc· sˆ =~bλ· sˆ = ντg. (2.11)If one considers the moon as a point source, the visibility of the moon will be(Rm)i j =V 2m2e2pii(φm)i j (2.12)where (Rm)i j is the visibilty of the moon for a single baseline (i and j denotes a pairof different antennas). Vm is the voltage induced by the signal from the moon. Thisis proportion to the intensity of the moon. (φm)i j is the phase of the moon whichis from Equation 2.11. The phase of the moon depends on the time delay betweentwo antennas i and j from one baseline. To get rid of the phase of the moon, thevisibility of the moon from a single baseline is multiplied by the exponential of the12opposite phase.(Rm)i je−2pii(φm)i j =V 2m2e2pii(φm)i j e−2pii(φm)i j (2.13)However, the visibility of a single baseline is a Fourier component of the entireregion mapped by CHIME. The subtraction of the phase from a single baseline’svisibility is thenRi je−2pii(φm)i j (2.14)where Ri j is the total visibility (Fourier component of the sky) of a single baselinewith antennas i and j.After the subtraction of the moon phase, the phase in the total visibility ofa single baseline is from all the other sources on the sky except the moon. Fordifferent baselines, the phases are different. For a single source at a particularfrequency, the phase of the source depends on the length of the baseline. The longerthe baseline, the longer the time delay, τg, and, therefore, the larger the phase.Since different baselines are different Fourier components of the sky intensity witha random phase distribution, obviously, one can average the visibility with randomphases from different baselines to approximately zero [12].|A|2I(t) = 1N∑i, jRi je−2pii(φm)i j (2.15)The summation in Equation 2.15 allows visibility of other sources to be averagedto zero while the lunar intensity (the moon visibility without phase) adds up tomultiple of the number of baselines. In this equation, N is the number of baselinesbeing averaged, and |A|2 is the antenna beam ratio (antenna beam patter/angularresponse) where |A|2 = Ai(sˆ)A∗j(sˆ). This is equivalent to a discrete Fourier trans-form. The summation can be divided by the number of baselines to obtain theintensity of the moon at a particular time with the antenna beam pattern (or angularresponse/beam ratio). To obtain the true intensity of the moon, the antenna beampattern can be divided from the averaged signal.132.4 Intensity to Brightness TemperatureThe moon, similar to many terrestrial objects in the solar system, emits radiationfrom it’s surface, which can be approximated as black body [5]. The side of themoon facing the sun with higher temperature reflects visible sunlight. However, theentire moon also emits within a different frequency range. Since the moon is not ahigh energy source, the peak of emission from the dark side needs to be below thevisible frequency [9]. The Russian data in Figure 1.2 clearly indicates the presenceof spectrum within radio frequencies which can be related to the lunar brightnesstemperature through the CHIME band [6].From the intensity of the moon, one can obtain the brightness temperature ofthe moon via Planck’s Law [11]B(ν ,T ) =2hν3c21e−hνkT −1. (2.16)In radio frequencies, one can approximate the black body curve with Rayleigh-Jeans Limit [11]B(ν ,T )≈ 2ν2kBTc2. (2.17)With the intensity of the moon I(ν ,T ), the brightness temperature is, therefore,TRJ =c2I2ν2kB. (2.18)This equation can also take on another form as shown below. If the intensity of asource I is constant within the beam of CHIME, or the solid angle Ω over which aradio telescope is sensitive, its flux density S is related to intensity as S = IΩ, andits flux density and brightness temperature are related in turn byTRJ =λ 22kBΩS (2.19)where λ is the wavelength and Ω is the solid angle of the moon.142.5 Background Subtraction and UncertaintyThe brightness temperature from the last section, nevertheless, is not the actualbrightness temperature of the moon. This measured temperature also contains abackground temperature. Similar to the MWA measurement, According to Equa-tion 1.2, the actual temperature of the moon is the temperature from the differenceof the measured intensity, Imeasured , and the background intensity, Ibg (here Tmoon isnot the same as the one used in introduction, and this notation will be kept for therest of the thesis) [7].Tmoon =λ 22kBΩ(Imeasured− Ibg) = λ22kBΩ∆I (2.20)The background intensity can be measured at the same location of the moon on thesky on a different day, since the moon transit location is changing over a monthlyperiod. Multiple background intensities can be averaged over one month to obtainthe averaged background intensity. This background intensity can later be sub-tracted from the measured intensity at the moon position during the lunar transit(transit on a particular day). The difference in intensity can be used for gettingthe lunar brightness temperature. However, the intensity of the background wouldbe difficult to measure on a date too far away from the date of the lunar measure-ment (more than one month away) because the background sky is changing over aroughly annual period due to the earth’s motion around the sun. That is, the loca-tion of a fixed right ascension and declination on the sky is changing in the azimuthand elevation coordinate. An example is the location of the galactic center in ob-server’s frame (azimuth and elevation coordinate) changing with respect to timeduring a year. If too many dates have elapsed between the measurement of themoon and the measurement of the background, the transit time of the backgroundhas to be recalculated (since the background transit time is not close to the lunartransit time measured previously).Multiple measurements from the background temperature also provide an un-certainty of the measurement. The uncertainty in intensity can be calculated viathe equationδ∆I =√∑(Ibg− Ibg)2n(2.21)15where Ibg is the averaged background intensity, and n is the number of backgroundintensity measured. The propagation of uncertainty can be used to obtain the un-certainty in temperature. The equation is shown belowδTmoon =√(∂Tmoon∂ Ibg)2δ∆I2 =∂Tmoon∂ Ibgδ∆I =λ 22kBΩδ∆I (2.22)where δTmoon is the uncertainty of lunar temperature.16Chapter 3MethodThe lunar brightness temperature measurement with CHIME involves data pro-cessing and coding. The first step of the data analysis is locating the moon on thesky during lunar transits. This is followed by locating the data file in the CHIMEonline database. The final step is Fringestop and obtaining lunar brightness tem-perature with uncertainty. In this section, some examples of the intensity resultsand uncertainty measurements will be presented as an overview of the softwareused, data analysis, and experiment procedure. A more complete set of results willbe shown in the following result section.3.1 Finding Lunar TransitThe first step of obtaining the lunar brightness temperature is to find the locationof the moon on the sky and to choose when to take the data. This is related tothe antenna angular response, and certain location of the moon on the sky couldpotentially reduce the workload on finding the antenna angular response (or beamratio) in the direction of the moon afterwards. A python code was used to find thelunar transit on different days throughout from March 2019 to March 2020, andthe transit with the lunar declination closest to 22.01446 degrees (declination ofsource TAU-A) was obtained to match the TAU-A beam ratio. This beam ratio wascalculated before and was used as a reference for all sources in Tau-A direction onthe sky.17Figure 3.1: The declination of the moon during transits of the moon through-out one year and two months period. As can be seen, the declinationof the moon varies from around -23 degrees to 23 degrees sinusoidallyover roughly 14 periods during approximately 14 months. The yellowand blue dots on the plot indicate the two transits of the moon whichhave the closest declination to 22.01446 degrees. The green dashed lineindicates the declination of 22.01446 degrees.Figure 3.1. shows the declination of the moon at lunar transits each day through-out last year (a bit more than a year). As revealed in the figure, the lunar transitsfrom September 23rd to September 24th, and the the lunar transit from August 26thto August 27th (blue and yellow dots) have the declination closest to the 22.01446degrees (green dashed line), declination of TAU-A, with a minimal difference. Infact, almost all the other declination values of the moon during transits on otherdays are smaller than the values obtained on September 23rd and August 26th.The plot also revealed the rotation of the moon around the earth over a monthlyperiod as known.The data of the visibility during the transit between September 23rd and septem-ber 24th was chosen from the CHIME archive data stored online to be further18analyzed. Additionally, from the code, more precise time of the transits duringthese two days were obtained. On September 23rd, the time of the lunar tran-sit is at 15:14:11.042273 UTC time which corresponds to a unix time of around1569251651 (all time mentioned in this thesis except unix time are in UTC time).On August 26th, the time of the lunar transit is at 16:28:31.759174 UTC time whichcorresponds to a unix time of around 1566836912. The epoch of the unix time isset to be January 1st, 1970, 00:00:00 UTC.3.2 Visibility DataAfter the unix time was obtained for the tranist of the moon at declination of22.01446 degrees, the corresponding HDF5 data file was located in the CHIMEserver, and by choosing information about frequency, a list of visibility as a func-tion of time and baselines (pairs of antennas) was obtained. The HDF5 data filewith time during the September 23rd lunar transit was used for analysis because theAugust 26th lunar transit data is missing in the Coconut, the computer at CHIMEsite, database. The August 26th lunar transit data can still be on Cedar, the su-percomputer in Computecanada, database. However, for convenience, the data onSeptember 23rd was used. As a side note, sometimes, data are missing in thedatabase due to maintenance or excessive noise, majorly human activities such ascellphone signals. This list of visibility was used later to plot the intensity of themoon during transit. A python code was used to obtain the information of fre-quency and antenna locations. Here, data at several different frequencies were ob-tained for further analysis. More data can be used for more results in later researchvia the same code by only changing several parameters. One of the frequencies ofinterest, for example, is at 638.28MHz with a bandwidth of 0.390625MHz.Both a product map and an input map were used for specifying the baselinearray of CHIME. The input map used here is an array of objects including CHIMEantennas, noise source, holography antennas on the 25.6 meter John A. Galt tele-scope, and blank objects. Apparently, except the CHIME antennas, all the otherobjects were ignored for the purpose of this research. The ignored objects wasidentified from the input map array. The product map is also an array but has itselements as pairs of objects within the input map. That is, the pairs of objects can19be any combinations of two elements mentioned above in the input map. Onlycombinations of 2 CHIME antennas were used in this measurement, and all theother pairs were deleted from the product map. In the other word, only the CHIMEbaselines were used in the measurement. In the product map array, index numbersof elements in the input map array were used as identities of these objects. Afterdeleting the ignored objects in the two arrays, input map and product map, theseobject indices can be considered as antenna indices. An antenna index correspondsto the position of the antenna in meters in the antenna array of the interferometricradio telescope. Given there are total 1024 antennas with 2 polarization, the num-ber assigned to antenna indices is from 0 to 2047. To form a baseline, two CHIMEantenna objects need to have the same polarization. Also, the elevation of all theCHIME antennas is set to be 0.00m as default.3.3 FringestopThe product of the signal of the moon and antenna angular response was obtainedafter Fringestop. Fringestop is a technique to eliminate the lunar phase withinvisibility due to the motion of the moon across the sky (or essentially the earthspin throughout a day). To Fringestop a source, the source’s right ascension anddeclination were included in the code along with the time array of the data file,the chosen frequency, and the information of the baseline array (i.e. product mapand input map). For the moon on September 23rd at transit, the right ascensionand declination are 111.102196 degrees and 22.021439 degrees respectively. Thefrequency in this example, as mentioned above, was chosen to be 638.28MHz. Thetime during transit was obtained from the HDF5 file.Figure 3.2 shows the real part of the signal from north-south polarization afterFringestop. As can be seen, there is a peak between 15:11 and 15:20 on Sep. 23,2019, between 51000 and 52000 seconds after 1.5692e9 unix time. This matchesthe time of lunar transit at 15:14. As mentioned in the theory, without Fringstop, thevisibility data versus time would have phase variation. That is, the visibility datais a wave packet under the blue curve in the figure. The relatively smooth curveof the signal after Fringestop indicates that the Fringestop was successful. The thepeak value of the curve in Janksy is 209.144. This value should be the product of20Figure 3.2: is a plot of signal after Fringestop containing antenna angularresponse (beam ratio) in north-south polarization in Jansky. The x axishere is time during the lunar transit on September 23rd, 2019. As canbe seen, the signal over time is a curve with a peak value around 210Jansky.the intensity of the moon and the beam ratio. Figure 3.3 shows the real part of thesignal from east-west polarization. As can be seen, there is a peak between 15:11and 15:20 on Sep. 23rd. This matches the time of the lunar transit time and thetime of the peak in north-south polarization as mentioned above. Similarly, therelatively smooth curve of the signal after Fringestop indicates that the Fringestopwas successful. The real part of the peak value of the curve in Jansky is 224.751.This is the product of the intensity of the moon and the beam ratio. The differencebetween peak values in two different polarization is small compared to the totalvalue of the peak signal.21Figure 3.3: is a plot of signal after Fringestop containing antenna angularresponse in east-west polarization in Jansky. The x axis here is timeduring the lunar transit on September 23rd, 2019. As can be seen, thesignal over time is a curve with a peak value around 220 Jansky.3.4 Dividing Antenna Beam RatioThe beam ratio is equivalent to the antenna angular response. The CHIME an-tenna is designed that the response of the antenna to a signal from different direc-tions are different. The angular response of an older version of CHIME antenna isshown below in Figure 3.4 The actual beam ratio used in the measurement fromthe latest version of CHIME antenna is the beam ratio calibrated with respect tothe source Cyg. A. That is, the 0 degree reference is pointing at the direction ofCyg. A with a maximum response. As mentioned in section 3.1, the time of thelunar transit chosen is the time when the declination of the moon at transit matchesthe declination of the source TAU-A. The TAU-A declination, as mentioned be-fore, is 22.01446 degrees. The beam ratio in both polarization was obtained fromthe previously measured TAU-A beam ratio. The beam ratio between TAU-A andCyg. A at 638.28MHz in north-south polarization is 0.89308992+0.03206855i and22Figure 3.4: (a) and (b) show the antenna beam in two different polarization.The antenna angular response in two polarization in the images is part ofa 3D antenna angular response. As can be seen, the response at differentangle is different. The magnitude of the response here is in units of dB.23in east-west polarization is 0.86+0.036i. These two beam ratios in two differentpolarization was divided from two signals after Fringestop obtained in the last sec-tion. The result gives the measured intensity at the location of the moon duringlunar transit.Figure 3.5 shows the measured intensity at the location of the moon in north-south polarization during the transit on September 23rd. As can be seen, the transittime on plot matches the transit time of the moon and the transit time shown in thelast section for signals after Fringestop. The peak of the intensity after dividing thebeam ratio in north-south polarization is 233.879 in Jansky. This is the sum of theintensity of the moon and the intensity of the background. The beam width of thelunar transit here is similar to the beam width of the transit of Cyg. A shown in thetheory section. This further validates the Fringestop since the beam width dependson the angular speed of the source across the sky during the transit. This angularspeed, results from the earth rotation, is approximately the same for both Cyg. Aand the moon. Therefore, both transits have approximately the same beam width.Figure 3.6 shows the measured intensity at the location of the moon in east-westpolarization during the transit on September 23rd. As can be seen, the transit timehere matches the transit time of the moon and the transit time shown in previousthree plots. The peak of the intensity after dividing the beam ratio in east-westpolarization is 260.901 in Jansky. The beam width of the lunar transit here is thesame as the beam width in north-south polarization which is around 1000 seconds.3.5 Background Subtraction and UncertaintyCalculationAs mentioned in the theory, the intensity from the last section is not the lunarintensity. Instead, it is the sum of the intensity of the moon and the intensity ofthe background. To obtain the intensity of the moon, the background intensity wassubtracted from the total measured intensity. Since the moon is having differentdeclination and right ascension over a monthly period, the background intensitywas measured at the same location of the moon on September 23rd at differenttimes during the month. This intensity was then subtracted from the total intensity24Figure 3.5: shows the measured intensity at the location of the moon in north-south polarization during the lunar transit on September 23rd. As can beseen, the peak of the curve is between 15:11 and 15:20. The beam widthis about 1000 seconds. The peak of the curve is around 230 Jansky.to get the intensity of the moon.The example data chosen here for the background analysis has a time arrayon the day of September 30th, and the data was taken for the lunar transit hap-pened on September 23rd. That is, the data used here has time around 15:14:11 onSeptember 30th instead of September 23rd. The location of the sky background isat the same location of the moon on September 23rd at its transit time (right ascen-sion and declination are 111.102196 degrees and 22.021439 degrees respectively).That is, this is the same background as the background occulted by the moon onSeptember 23rd at lunar transit. There might be minor difference in the transit timeat 111.102196 degrees right ascension and 22.021439 degrees declination on the30th compared to the 23rd. As mentioned in the theory, this is majorly due to theearth’s motion around the sun throughout a year, and a fixed location in right as-cension and declination changes in the azimuth and elevation coordinate during an25Figure 3.6: shows the measured intensity at the location of the moon in east-west polarization during the lunar transit on September 23rd. As canbe seen, the peak of the curve is between 15:11 and 15:20. The beamwidth is about 1000 seconds. The peak intensity of the curve is around260 Jansky.annual period.Figure 3.7 shows the intensity of the background taken on September 30th innorth-south polarization at the same right ascension and declination as the moon onSeptember 23rd at lunar transit. As can be seen, there is a peak around 15:40 (8000seconds after 1.56985e9 unix time). This time, however, is about 25 minutes laterthan 15:14:11 which is the same as the lunar transit time on September 23rd. Asmentioned above, this is due to the motion of the earth around the sun, so the sky ischanging during an annually cycle. The peak intensity is 25.454 in Jansky whichis the intensity of the background in north-south polarization. Similarly, Figure 3.8shows the intensity of the background on September 30th in east-west polarizationat the same right ascension and declination as the moon on September 23rd at lunartransit. The peak is also around 8000 seconds after 1.56985e9 unix time. This time26Figure 3.7: shows the measured intensity of the background on September30th in north-south polarization at the same location of the moon attransit on September 23rd. As can be seen, the peak of the curve isaround 8000 seconds after 1.56985e9 in unix time (13:26 Sep. 30th,2019). The peak intensity of the curve is around 25 Jansky.matches the time in the north-south polarization. The peak intensity is 26.380 inJansky which is the intensity of the background in east-west polarization.Another way to take the background measurement is to measure the sky aroundthe moon right before and after the time of transit. That is, different points on thesky with right ascension and declination close to the moon can also be used forbackground intensity measurement. The advantage of this method is that the fea-ture of the moon will still be visible in the plots, and the moon feature can be usedto confirm the result. To simplify the calculation of the background intensity, pointson the sky around the moon with the declination same as the lunar declination dur-ing the transit were used for the background measurements (i.e. the points on thesky with only different right ascension were used for the background measure-ments.) Figure 3.9 and Figure ?? are plots of two such examples both measured27Figure 3.8: shows the measured intensity of the background on September30th in east-west polarization at the same location of the moon at transiton September 23rd. As can be seen, the peak of the curve is around8000 seconds after 1.56985e9 in unix time. The peak intensity of thecurve is around 25 Jansky.in north-south polarization. One point on the sky has right ascension larger thanthe moon, and this point transited at an earlier time before lunar transit. The otherpoint on the sky has right ascension smaller than the moon, and this point transitedat a later time after the lunar transit.Figure 3.9 shows the intensity of the point on the sky with a larger right ascen-sion than the moon on September 23rd. The peak is around 50000 seconds after theunix time of 1.5692e9. This corresponds to 14:46 UTC time on September 23rdwhich is about 30 minutes before the lunar transit. The peak intensity of the curveis 37.691 in Jansky.Figure 3.10 shows the intensity of the point on the sky with a right ascensionsmaller than the right ascension of the moon at lunar transit, and this point on thesky transited at a later time after the lunar transit on September 23rd. As can be28Figure 3.9: shows the measured intensity of the background on September23rd in north-south polarization at a point with a smaller right ascensionthan the right ascension of the moon. As can be seen, the peak of thecurve is around 50000 seconds after 1.5692e9 in unix time. The peakintensity of the curve is around 40 Jansky.seen, there is still remaining feature of the moon with fringes between 15:11 and15:20. The peak of the curve, however, is at a later time which is approximately15:36 (53000 seconds after the 1.5692e9 unix time). The peak intensity of thecurve is 22.351 in Jansky.In this research, for the first method, multiple background measurements weretaken on different days in two polarization with the same right ascension and dec-lination of the moon on September 23rd at transit (111.102196 degrees in rightascension and 22.021439 degrees in declination). These measurements in each po-larization were averaged to a single averaged background intensity. To obtain theintensity of the moon, averaged background intensity was then subtracted from thetotal intensity measured in the same polarization which was taken on September23rd at lunar transit. The intensity of the moon at the frequency of 638.28MHz was29Figure 3.10: shows the measured intensity of the background on September23rd in north-south polarization at a point with a larger right ascensionthan the right ascension of the moon. As can be seen, the peak of thecurve is around 53000 seconds after 1.5692e9 in unix time. The peakintensity of the curve is around 20 Jansky. Also, the feature of themoon is still visible in this plot between 15:03 and 15:20.then plugged into the Rayleigh Jeans law from Equation 2.19 to get the brightnesstemperature of the moon. Similar analysis was also done in the second methodwhere 9 different sky intensities around the moon were taken and averaged to ob-tain a background intensity which was used in the calculation of a different lunarbrightness temperature in north-south polarization.Besides, the uncertainty of the brightness temperature was also calculated frommultiple background measurements. Equation 2.21 and 2.22 was used to calculatethe uncertainty in temperature. The calculated temperature and its uncertainty fromthe measurements of the background and the measurement of the moon will bepresented in the result section with greater details. There are measurements attwo different frequencies in two polarization with each having their corresponding30background measurements. All the plots of the measurements, numbers obtainedfrom the measurements, and uncertainties will be presented in the result section.31Chapter 4ResultThe result presented in this section leads to the value of lunar brightness temper-ature with uncertainty at 638.28MHz and 565.625MHz within the CHIME band.The temperature results from this research can potentially reject the result of tem-perature measurement from either the Russian research in 1963 or the MWA mea-surement in 2017. Of the data files used for brightness temperature measurementat the position of the moon during September 23rd and March 6th lunar transit,two frequencies of data were analyzed. For these frequencies, temperatures werecalculated from the intensities measured in two polarization. Two methods wereused in background measurements. For the first method (Method 1), for each polar-ization, multiple different background intensities were measured on different datesduring the same month at the same location at lunar transit happened on September23rd. For the second method (Method 2), multiple points around the moon duringthe September 23rd lunar transit were measured instead. Result related to thesemeasurements will be presented in this section.As mentioned in the method section, the visibility data contains informationof the frequency that the data was taken in. Given the CHIME frequency is from400MHz to 800MHz, and the frequency array has a length of 1024, there are, inprinciple, 1024 different frequencies that the moon brightness temperature can bemeasured at. However, due to maintenance and man made issues, data in someof the frequencies across the CHIME band might not be available. Only two fre-quencies were chosen for this project. The frequencies used are 638.28MHz and32565.625MHz.4.1 Background Measurement with Method 1As mentioned in the last section, at 638.28MHz, the total intensity measured at thelocation at lunar transit on September 23rd gives a value of 233.879 in Jansky innorth-south polarization and 260.901032317045 in Jansky in east-west polariza-tion. The corresponding plots of the lunar transit at the frequency of 638.28MHzin two polarization were shown in the method section. This intensity, however, isnot the intensity of the moon itself. The background subtraction was also doneby measuring background intensities on different days during September. An ex-ample was shown in the method section, and the background measurement in theexample was done on September 30th. Here, more background measurements areintroduced.The second background measurement is chosen to be on September 21st. Themeasurement was done at the same location of the moon at transit on September23rd. The measurement was also done in two polarization. Figure 4.1 shows thatthe peak is around 1569076000 unix time. This corresponds to a time of 14:26UTC time on the 21st. As mentioned in the method section, the lunar transit hap-pened at 15:14:11 UTC time on the 23rd. Compare to the result of the measuredbackground on the 30th, the transit of the location with right ascension 111.102196degrees and declination 22.021439 degrees (the moon location at lunar transit onthe 23rd) happened earlier than 15:14:11 rather than later (the transit on the 30this later). This indicates that the measurement is successful because the location ofobjects on the sky (fixed right ascension and declination) moved in one directionduring that time of the year. The peak intensity of the measurement is 41.952 inJanksy. Figure 4.2 shows the background intensity measured on September 21st ineast-west polarization at the same location of the moon at lunar transit on Septem-ber 23rd. The peak is around 14:26 (1569076000 unix time). This matches themeasurement done in north-south polarization. The peak intensity value is 57.793in Jansky.The next measurement is chosen to be on September 28th, two days beforethe measurement on the 30th. The measurement was done at the same location of33Figure 4.1: shows the measured intensity of the background on September21st in north-south polarization at the same location of the moon at itstransit on September 23rd. As can be seen, the peak of the curve isaround 76000 seconds after 1.569e9 in unix time. The peak intensity ofthe curve is around 40 Jansky.the moon at transit on September 23rd. The measurement was also done in twopolarization. Figure 4.3 shows that the peak of the curve is about 5250 secondsafter unix time of 1.56968e9. This corresponds to a UTC time of 15:40 which isabout 25 minutes after 15:14:11, the time of lunar transit on September 23rd. Thisis also approximately the same time as the peak measured on the 30th shown in themethod section. Therefore, the measurement is successful given the 28th is only2 days away from the 30th. The peak intensity of the measurement is 22.571 inJansky. Figure 4.4, on the other hand, shows the background intensity measuredon September 28th in east-west polarization at the same location of the moon atlunar transit on September 23rd. The peak of the curve is about 15:40, and itroughly matches the time from the measurement in north-south polarization. Thepeak intensity of this measurement is 27.163 in Jansky.34Figure 4.2: shows the measured intensity of the background on September21st in east-west polarization at the same location of the moon at itstransit on September 23rd. As can be seen, the peak of the curve isaround 14:26 (76000 seconds after 1.569e9 in unix time). The peakintensity of the curve is around 40 Jansky.The last background measurement done for the frequency of 638.28MHz ison September 29th, one day before the 30th. The measurement was done at thesame location of the moon at transit on September 23rd and was also done in twopolarization. Figure 4.5 shows that the peak of the curve is between 71500 and72000 seconds after 1.5697e9 unix time. This corresponds to a UTC time of 15:30to 15:45 which is about 15 to 30 minutes after 15:14:11, the time of lunar tran-sit on September 23rd. This is close to the same time from the measurement onthe 30th shown in the method section. Therefore, the measurement is successfulgiven the 30th is only 1 day later than the 29th. The peak intensity of the measure-ment is 17.838 in Jansky. Figure 4.6 shows the background intensity measured onSeptember 29th in east-west polarization at the same location of the moon at lunartransit on September 23rd. The peak of the curve is between 15:30 and 15:45. This35Figure 4.3: shows the measured intensity of the background on September28th in north-south polarization at the same location of the moon atlunar transit on September 23rd. As can be seen, the peak of the curveis around 15:40 (5250 seconds after 1.56968e9 in unix time). The peakintensity of the curve is around 20 Jansky.matches the measurement in north-south polarization. The peak intensity of themeasurement is 26.929 in Jansky.The values of the background intensity measurements are shown in Table 4.1.Values for each polarization were averaged to obtain an averaged background in-tensity. The averaged background intensity in north-south polarization is approxi-mately Ibg,Spol=26.95 Jansky. The averaged background intensity in east-west po-larization is approximately Ibg,E pol=34.57 Jansky. The lunar intensity in north-south polarization at the frequency of 638.28MHz is ISpol=233.88-26.95=206.93Jansky. The lunar intensity in east-west polarization is IE pol=260.9-34.57=226.33Jansky. After the Rayleigh Jeans Law in Equation 2.19 was used, the lunar bright-ness temperature with the solid angle of the moon Ωm = Amd2m =piR2md2mwas obtained.The lunar brightness temperature in north south polarization at the frequency of36Figure 4.4: shows the measured intensity of the background on September28th in east-west polarization at the same location of the moon at lunartransit on September 23rd. As can be seen, the peak of the curve isaround 15:40 (5250 seconds after 1.56968e9 in unix time). The peakintensity of the curve is around 30 Jansky.Date North-South Pol. East-West Pol.Sep. 21st 41.95 Jy 57.79 JySep. 28th 22.57 Jy 27.16 JySep. 29th 17.84 Jy 26.93 JySep. 30th 25.45 Jy 26.38 JyTable 4.1: Values of background intensities taken on different dates in twopolarization for a frequency of 638.28MHz. The intensities are in unitsof Jansky. All the values have been rounded up to 2 decimal places.638.28MHz is Tmoon,Spol=258.17K. The brightness temperature of the moon in east-west polarization is Tmoon,E pol=282.37K.Besides, the uncertainties of the background was also calculated via Equation2.21. The uncertainty of background in north-south polarization is approximately37Figure 4.5: shows the measured intensity of the background on September29th in north-south polarization at the same location of the moon atlunar transit on September 23rd. As can be seen, the peak of the curveis roughly between 15:30 and 15:45 (71500 and 72000 seconds after1.56968e9 in unix time). The peak value of the curve is around 20Jansky.9.08 Jansky. The uncertainty of background in east-west polarization is approx-imately 13.41 Jansky. Using propagation of uncertainty in Equation 2.22, theuncertainty in temperature in north-south polarization is δTSpol=11.33K, and theuncertainty in temperature in east-west polarization is δTE pol=16.73K.The averaged temperature at the frequency of 638.28MHz between two polar-ization is, therefore,Tmoon =Tmoon,Spol +Tmoon,E pol2= 270.27K (4.1)38Figure 4.6: shows the measured intensity of the background on September29th in east-west polarization at the same location of the moon at lunartransit on September 23rd. As can be seen, the peak of the curve isbetween 15:38 and 15:46 (71500 and 72000 seconds after 1.5697e9 inunix time). The peak value of the curve is around 25 Jansky.. The uncertainty of the lunar temperature is thenδT =√(∂Tmoon∂Tmoon,Spol)2δTmoon,Spol2+(∂Tmoon∂Tmoon,E pol)2δTmoon,E pol2=√(12)211.332+(12)216.732 = 10.1K(4.2)To sum up, the averaged brightness temperature of the moon at a frequency of638.28MHz has a value of 270.27K with uncertainty of 10.1K.394.2 Background Measurement with Method 2Based on method 2, 9 different background measurements in north-south polar-ization were taken place on the same day of the lunar transit, September 23rd,before and after the lunar transit time, 15:14:11 UTC time. Two examples of themeasurements were shown in the method section before. The 9 measurementsgive 9 different intensity values: 30.523, 30.434, 35.147, 46.328, 47.164, 37.691,35.742, 29.155 which are all in unit of Jansky. The averaged intensity is thenIbg=36.52Jy. The measured brightness temperature of the moon in north-south po-larization, Imeasured=233.88Jy, was used to calculate the actual intensity of the moonImoon=233.88-36.52=197.36Jy. Then, from the Rayleigh Jeans Law in Equation2.19, the brightness temperature of the moon is Tmoon=246.23K. The uncertainty ofthe background intensity was calculated from Equation 2.21, and it has a numberof δ I=6.04Jy. From Equation 2.22, the uncertainty in temperature is δT =7.54K.The temperature measured using Method 2 in north-south polarization is 11.94Klower than the temperature measured using Method 2. This temperature is 24.04Klower than the averaged temperature between two polarization obtained by usingMethod 1. Moreover, the temperature in north-south polarization from Method 2is 36.14K lower than the temperature obtained in east-west polarization by usingMethod 1.4.3 A Different Temperature at 565.625MHz?To form a more general idea on the temperature of the moon across the CHIMEband from 400MHz to 800MHz, a different date and a different frequency werechosen for a new measurement to compare with the measurement at 638.28MHz onSeptember 23rd. The way to find a different day when the declination of the moonis close to 22.01446 degrees is similar to the one mentioned in the method section.However, the time period used for searching this date is only 100 days instead of365 days. The two dates which have declination closest to 22.01446 degrees areJanuary 9th and March 6th with transit time 06:40:21.509183 and 05:00:19.225533in UTC time respectively. A plot of the lunar transit is shown in Figure 4.7. Ascan be seen, the orange and the blue dots correspond to two transit times that thelunar declination is closest to 22.01446 degrees. The lunar transit on March 6th40was chosen for the analysis. At the lunar transit on March 6th, the right ascensionand the declination of the moon are 119.847015 degrees and 22.06322 degreesrespectively.Figure 4.7: shows the declination of the moon at lunar transits in past 100days before April 19th. The green dashed line indicates the declinationof 22.01446 degrees. The orange and blue dots are two transit times thatthe moon declination is closest to 22.01446 degrees.The same procedure including Fringestop and dividing beam ratio were usedduring this analysis at a frequency of 565.625MHz. Figure 4.8 shows the signal af-ter Fringestop in north-south polarization. As can be seen, the peak of the curve isclose to 71000 seconds after unix time of 1.5834e9. This matches the lunar transittime on March 6th which is 05:00:19.225533 in UTC time. This time correspondsto a unix time of 1583470819. The same time indicates that the Fringestop wassuccessful at 565.625MHz. The beam width is around 1000 seconds which also41matches the width for Cyg. A transit. The real part of the peak value of the curvein Jansky is 124.460. Figure 4.9 shows the signal after Fringestop in east-west po-Figure 4.8: is a plot of signal after Fringestop in north-south polarization inJansky which contains antenna angular response (beam ratio). The xaxis here is time during the lunar transit on March 6th, 2020. As canbe seen, the signal over time is a curve with a peak value around 125Jansky.larization. The peak of the curve is also close to 5:03 on Mar. 6th, 2020 (71000after the unix time of 1.5834e9). This matches all the result in north-south polar-ization. The same transit time indicates that the Fringestop was successful. Thepeak value in east-west polarization in Jansky is 131.432.Similarly, beam ratio was divided from the Fringestopped signal to get theactual intensity measured at the location of the moon during the lunar transit onSeptember 23rd. The beam ratio at the frequency of 565.625MHz, different fromthe one at 638.28MHz, is 0.663-0.01i in north-south polarization, and 0.606-0.008i42Figure 4.9: is a plot of signal after Fringestop in east-west polarization inJansky which contains antenna angular response (beam ratio). The xaxis here is time during the lunar transit on March 6th, 2020. As canbe seen, the signal over time is a curve with a peak value around 130Jansky.in east-west polarization.After dividing the beam ratio from the signal, the intensity measured in twopolarization at the location of the moon during lunar transit on March 6th is shownbelow in two plots. Figure 4.10 shows the intensity in north-south polarization.The peak of the curve is close to 71000 seconds after unix time of 1.5834e9. Thismatches the plot of the Fringestopped signal before dividing the beam ratio. Thepeak value of the intensity in north-south polarization in Jansky is 187.745. Fig-ure 4.11 shows the intensity in east-west polarization at the location of the moonduring March 6th lunar transit. The peak of the curve is also close to 5:03 (71000seconds after unix time of 1.5834e9). This is the same for the measurement in43Figure 4.10: shows the measured intensity at the location of the moon innorth-south polarization during the lunar transit on March 6th. As canbe seen, the peak of the curve is close to 5:03 (71000 after 1.5834e9unix time). The beam width is about 1000 seconds. The peak intensityof the curve is around 190 Jansky.north-south polarization and the measurement of Fringestopped signals before di-viding the beam ratio. The peak value of the measured intensity in east-west polar-ization in Jansky is 217.01.To obtain the actual lunar intensity, background intensity was also measured at565,625MHz at a different point on the sky close to the location of the moon duringits transit. The measurement was done at a right ascension 5 degrees larger thanthe right ascension of the moon on the same day. This sky location (the locationthat the measurement was taken) also has 22.06322 degrees in declination whichis the same as the moon declination on March 6th. As shown in Figure 4.12, innorth-south polarization, the peak occurs a bit before the unix time of 1583467000which corresponds to 3:48 in UTC time. The peak of the background intensity after44Figure 4.11: shows the measured intensity at the location of the moon in east-west polarization during the lunar transit on March 6th. As can be seen,the peak of the curve is close to 5:03 (71000 seconds after 1.5834e9 inunix time). The beam width is about 1000 seconds. The peak intensityof the curve is around 220 Jansky.Fringestop and dividing the beam ratio is 31.422 in Jansky. Figure 4.13 shows theintensity of the background in east-west polarization on March 6th before the lunartransit. The peak is close to 1583466500 in unix time. This corresponds to 3:48 inUTC time which is before the time of lunar transit on March 6th (05:00:19). Thepeak of the background intensity after Fringestop and dividing the beam ratio is66.44 in Jansky.By using these measurements, one can estimate the intensity of the moon inboth north-south and east-west polarization at 565.625MHz. The estimated in-tensity in north-south polarization is Imoon,Spol=166.58Jy. The intensity in east-west polarization is Imoon,E pol=150.57Jy. Using the Rayleigh Jeans Law in Equa-tion 2.19, the estimated lunar brightness temperature in north-south polarization45Figure 4.12: shows the measured background intensity in north-south polar-ization on March 6th before the lunar transit. As can be seen, the peakof the curve is close to 3:56 on Mar. 6th, 2020 (7000 seconds after1.58346e9 in unix time). The peak intensity is around 30 Jansky.is Tmoon,Spol=207.83K, and in east-west polarization is Tmoon,E pol=187.85K. Theaveraged estimated brightness temperature is Tmoon=197.84K.46Figure 4.13: shows the measured background intensity in east-west polariza-tion on March 6th before lunar transit. As can be seen, the peak of thecurve is about 3:48 (6500 seconds after 1.58346e9 in unix time). Thepeak intensity is around 65 Jansky.47Chapter 5Conclusion and Discussion5.1 Measured Lunar Brightness TemperatureThe measurement at 638.28MHz provides four different temperatures with uncer-tainties. Their uncertainties were measured in two different methods, and the tem-peratures were measured in 2 different polarization. The results are shown in Table5.1. These temperature values are not too high compared to the Russian resultsPolarization Method 1 Method 2north-south pol. 258.17±11.33K 246.23±7.54Keast-west pol. 282.37±16.73K N/Aaveraged 270.27±10.1K N/ATable 5.1: Values of lunar brightness temperature with uncertainty measuredin two polarization using two different methods.[6]. The result that is closest to the Russian result in this project is the lunarbrightness temperature measured in north-south polarization using Method 2 ofthe background calculation [6]. The largest value is the lunar brightness tempera-ture measured in east-west polarization using Method 1. The value that is closestto the Russian value with uncertainty is shown in Figure 5.1 as the orange squaredot with uncertainty. This indicates that the result at 638.28MHz measured in thisproject matches previous Russian results.On the other hand, the estimated measurement taken on March 6th at 565.625MHz48is drastically different from the measurement done on September 23rd at 638.28MHz.The estimated lunar brightness temperature at 565.625MHz is 197.84K. Com-pare to both the Russian result, around 240K, and the result in this research at638.28MHz, at least 246.23K, 197.84K is much closer to the result measured byMWA in 2017 [6][7]. As can be observed from the plot, the estimated result fromthis project at 565.625MHz is marked by a brown square dot.The result presented above shows successful measurements of the lunar bright-ness temperature with CHIME. This research paved the foundation for future mea-surements of lunar brightness temperature. In addition, the study can potentiallyhelp future research improve the CHIME instrumentation such as better antennabeam calibration and better understanding of artifacts within CHIME signals.5.2 Explanation of Inconsistency and ImplicationAs mentioned above, in contrast to the result measured at 638.28MHz, the resultat 565.625MHz is marked by a brown square dot in the plot which matches theMWA result instead of the Russian result. However, there is one Russian data pointwhich is close to the result measured by MWA in 2017 around the same frequency[6]. Thus, the scatter plot probably indicates a temperature rise of more than 40Karound the frequency of 600MHz, around the middle of the CHIME band. If themeasurement in 2017 is correct, the combined results of three projects might implya feature in the spectrum of the moon in low frequencies. That is, temperature ofthe moon between 100 and 600MHz is much smaller than lunar temperature acrossthe rest of the radio frequencies.Based on the result of this project, the result from MWA, and the result mea-sured in 1963, more lunar brightness temperature measurements are encouragedacross the CHIME band. Further measurements can provide better understandingof the spectrum of lunar emissions in low frequencies. Furthermore, given there islittle atmospheric absorption in radio frequency range, the analysis of the spectrumin low frequencies might provide an understanding of the chemical composition ofsubstances on the lunar surface and the properties of the moon itself which wererarely investigated before.49Figure 5.1: Scattered data of lunar brightness temperature as a function offrequency measured by Russian radio astronomers in 1963(in blue) andMckinley et al(in red). The result from this project is also marked on theplot. The orange square dot with uncertainty is the lunar brightness tem-perature measured at 638.28MHz with uncertainty. The brown squaredot is the lunar brightness temperature estimated at 565.625MHz. Theattempted fit of the 1963 data shows a brightness temperature from 220to 240K. The temperature measured in 2017 is around 180K. The greenband on the plot is the frequency range of CHIME from 400 to 800MHz.KRJ is the temperature in Kelvin in Rayleigh-Jeans limit. Previous datais from V. D. Krotikov & V. S. Troitski and also from Mckinley et al.[6] [7]5.3 Limitations and Potential ImprovementsDuring this project, there are several factors that were not carefully considered. Thefirst one is that, instead of being a point source, the moon should be considered asan extended source which covers an area of a disk on the sky. Therefore, duringthe measurement, the solid angle in the Rayleigh Jeans Law in Equation 2.19 needsto be changed into the beam solid angle if the size of the moon is not containedwithin the beam. Unfortunately, integrating over the beam to obtain the beam solid50angle could be difficult. Nevertheless, a potential way to solve the beam solidangle problem is using the temperature measurement from other sources online,such as Haslam at 408MHz [1]. Via the temperature measured by Haslam and thebackground intensity at 408MHz which can be measured by CHIME, the beamsolid angle can be obtained using the Rayleigh Jeans Law [1]. Subsequently, thisbeam solid angle can be applied to the calculation of lunar brightness temperatureacross the CHIME band.Another factor that was ignored is the reflected Earth shine and the reflectedgalactic shine [7]. The calculation of the reflected Earth shine isTre f l,Earth =TEarthΩEarthΩre f l' 300K 22602' 3.3K(5.1)where ωEarth is the angular size of the earth and Ωre f l is the angular size of the re-flection (here, 2 and 60 are in degrees). Therefore, the contribution of the reflectedEarth shine is small compared to the lunar brightness temperature. The contribu-tion from the galactic shine is expected to be even smaller. However, consideringthese values in the lunar brightness temperature measurement might provide moreaccurate results.Finally, as can be seen in the plots in two different polarizations in the resultsection for both 638.28MHz and 565.625MHz, the signal in east west polarizationis slightly higher than the signal in north south polarization. This might indicatethat the radiation from the moon might be polarized. Therefore, further analysis onthe signal and its polarization related property might reveal certain composition ofthe lunar surface.51Bibliography[1] → pages 51[2] P. Berger, L. B. Newburgh, M. Amiri, K. Bandura, J.-F. Cliche, L. Connor,M. Deng, N. Denman, M. Dobbs, M. Fandino, A. J. Gilbert, D. Good,M. Halpern, D. Hanna, A. D. Hincks, G. Hinshaw, C. Ho¨fer, A. M. Johnson,T. L. Landecker, K. W. Masui, J. Mena Parra, N. Oppermann, U.-L. Pen,J. B. Peterson, A. Recnik, T. Robishaw, J. R. Shaw, S. Siegel, K. Sigurdson,K. Smith, E. Storer, I. Tretyakov, K. Van Gassen, K. Vanderlinde, andD. Wiebe. Holographic beam mapping of the chime pathfinder array.volume 9906, pages 99060D–99060D–16. SPIE, 2016. ISBN 0277-786X.→ pages 6, 10[3] M. Deng. The cloverleaf antenna : a compact wide-bandwidthdual-polarization feed for CHIME. PhD thesis, 2014. → pages 7[4] D. C. Good. Redundant baseline calibration in CHIME : a firstimplementation application as beam probe. PhD thesis, 2017. → pages 6[5] G.-P. Hu, Y.-C. Zheng, A.-A. Xu, and Z.-S. Tang. Microwave brightnesstemperature of the moon: The possibility of setting a calibration source ofthe lunar surface. IEEE Geoscience and Remote Sensing Letters, 13(2):182–186, 2016. → pages 14[6] V. D. Krotikov and V. S. Troitskiı˘. Radio emission and nature of the moon.Soviet Physics Uspekhi, 6(6):841–871, 1964. → pages vii, xii, 3, 4, 14, 48,49, 50[7] B. McKinley, G. Bernardi, C. M. Trott, J. L. B. Line, R. B. Wayth, A. R.Offringa, B. Pindor, C. H. Jordan, M. Sokolowski, S. J. Tingay, E. Lenc,N. Hurley-Walker, J. D. Bowman, F. Briggs, and R. L. Webster. Measuringthe global 21-cm signal with the mwa-i: improved measurements of thegalactic synchrotron background using lunar occultation. Monthly Notices of52the Royal Astronomical Society, 481(4):5034–5045, 2018. → pages vii, xii,1, 2, 4, 5, 6, 15, 49, 50, 51[8] A. Popping and R. Braun. The standing wave phenomenon in radiotelescopes - frequency modulation of the wsrt primary beam. Astronomyastrophysics, 479(3):903–913, 2008;2007;. → pages 6[9] P. Ryan and L. Pesek. Solar system. A. Lane, London, 1978. ISBN9780670656363;0670656364;0713910437;9780713910438;. → pages 14[10] D. Smith, A. Young, and D. Davidson. Reconsidering the advantages of thethree-dimensional representation of the interferometric transform forimaging with non-coplanar baselines and wide fields of view. ASTRONOMYASTROPHYSICS, 603:A40, 2017. → pages 10[11] S. M. Stewart, R. B. Johnson, and T. . F. eBooks A-Z. Blackbody radiation:a history of thermal radiation computational aids and numerical methods.CRC Press, Taylor Francis Group, Boca Raton, 1 edition, 2017;2016;. ISBN1315372088;9781315372082;9781482263145;1482263149;1482263122;9781482263121;.→ pages 14[12] A. R. Thompson, J. M. Moran, G. W. Swenson, and W. O. Library.Interferometry and synthesis in radio astronomy. Wiley, New York, 2nd;2.aufl.; edition, 2001;2008;. ISBN 9780471254928;0471254924;. → pagesvii, viii, 8, 9, 10, 11, 1353

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