Lab 2 Working with the X-Band Interferometer

Transcription

1 Lab 2 Working with the X-Band Interferometer Abhimat Krishna Gautam 6 March 2012 ABSTRACT Lab 2 performed experiments with the X-Band Interferometer consisting of two dishes placed along an East-West axis. The interferometer was used to observe the Orion Nebula, the Sun, and the Moon. These observations were used to calculate the baseline length of the interferometer, of m, m, and 9.88 m respectively. They were also used to determine a function for fringe frequency for a given hour angle. This was carried out by a least squares process. For the Sun and the Moon, a nonlinear least squares process was carried out to determine the zero points of their modulating functions. This helped determine their angular radii, and 0.154, respectively. Contents 1 Introduction 2 2 Method 2 3 Data Clean Up Analysis Baseline and Fringe Frequency Least Squares Radius Least Squares, All Over Again Interpretation 20 6 Conclusion 22

2 2 1. Introduction This lab consisted of working with a basic East-West interferometer. The first portion of the lab (Section 4.1.1) calculated the baseline length of the interferometer and a function for the fringe frequency using observations of the Orion Nebula, the Sun, and the Moon. The second portion (Section 4.2.1) determined the angular radii of the Sun and the Moon. The experiments and the calculations revealed the utility of the interferometer, and potential experiments that could be possible with a more complex interferometer. 2. Method The setup for this lab involved the interferometer located on the roof of Wurster Hall. The interferometer consisted of two 0.9 m dishes, spaced approximately 10 m apart along the East to West axis. The signal from each antenna was mixed with the signal from a local oscillator (LO), the HP Synthesizer. This was set a frequency of GHz. The resulting signals from each of the dishes were then sent inside the building separately, and subsequently passed through a MHz filter. These two signals were each mixed with another LO, set at a frequency of 1.55 GHz. The resulting two signals were sent through another set of filters, only allowing MHz. The resulting two signals were then finally mixed together, and the resulting signal was sampled by the HP Voltmeter. The entire process of using LOs was set up to reduce the frequency of the original signal (about 10.7 GHz) down to something that could be much more easily sampled (about 150 MHz). The observed frequency range can be determined by adding the ranges of the first set of filters to the frequency of the first LO. This gives a range of ( GHz+1665 MHz) to ( GHz MHz), or GHz to GHz. This means a center frequency of GHz. Pointing of the telescopes was accomplished through a program written in IDL. Pointing the telescopes required transforming the equatorial coordinates of the object in the sky, the right ascension (RA) and declination, to the altitude and azimuth to which the dishes could point. This was done through the multiplication of rotation matrices. For example, to convert from the celestial equatorial coordinates of an object to the altitude and azimuth of the object, the coordinates were first put into a rectangular coordinate vector, then multiplied by an appropriate rotation matrix to convert to the local equatorial coordinates, and finally multiplied by another rotation matrix to convert to altitude and azimuth. This yielded a vector containing the final rectangular coordinates, from which the appropriate longitude and latitude coordinates could be extracted. The pointing program would then run continuously for the duration of the observation, adjusting the altitude and azimuth of the dishes as the object moved across the sky, by sending the coordinates each time to the POINT2 program. Every half-hour to an hour, the pointing program would also call the HOMER program to home the telescopes, so that the telescope s pointing would remain accurate throughout the observing run. Observing the Sun and the Moon involved slightly

3 3 different techniques. Since their equatorial coordinates continuously change, it was necessary to continually get updated positions of the objects in the sky. This was done through the ISUN and the IMOON program, which returned the current altitude and azimuth of the Sun and the Moon, respectively. Collecting the data from the voltmeter was done through the STARTCHART1 and the NOTSTARTCHART programs. These programs would collect the voltage from the sampler, the local sidereal time, and the equatorial coordinates of the object if the Moon or the Sun was being observed, approximately every 1 second. 3. Data One of our first targets was the point source W3, observed on 12 February - 13 February. However, we did not get very good data for the source (Figure 1). This source was one of the dimmer sources, and I had a bug in the program to track the object that resulted in the homer process being performed only about once every two hours. Attempts at cleaning up the data (using the same process outlined for the Orion Nebula in Section 3.1) were not very successful, so we decided to try a different point source instead, and observing it using an updated tracking program. Our next target was the Orion Nebula, observed on 21 February. We were able to get an observation from an hour angle of to an hour angle of The original voltage data collected from the sampler plotted against the hour angle is shown in Figure 2. The hour angle was calculated by subtracting the Orion Nebula s RA from the Local Sidereal Time (LST) at the time of sampling. Our data for the Sun was collected on 22 February, from an hour angle of to an hour angle of Our original data is shown in the top plot of Figure 3. The voltages collected from the sampler are plotted against the hour angle for each point. The hour angle was calculated by subtracting the Sun s RA at the times of sampling from the LST at the times of sampling. The data for the Moon was collected on 29 February. We attempted to collect the data from hour angles of -6 to 6. This original data from the sampler is shown in the top plot of Figure 4. The hour angle was calculated by subtracting the Moon s RA at the time of sampling from the LST at the time of sampling. As is evident, the entire data does not appear to be showing the Moon, particularly at the edges near hour angles of -6 and 6. This could be due to multiple reasons, such as obstructions, like hills or trees blocking the view to the Moon. In addition, the Moon is much dimmer than the Sun. When comparing the Sun data to the Lunar data, it is easily evident that the voltages collected from the Moon are about an order of magnitude lower than that of the Sun. This would mean that the radiation from the Moon would likely be more easily dimmed by objects like trees, accounting for its data being much noisier than that of the Sun. Discussion about cleaning up the Lunar data is provided below in Section 3.1.

4 4 Voltage (V) W3, Original Data Hour Angle W3, Attempt at Cleaning Up Voltage (V) Hour Angle Fig. 1. The voltages collected during the observation of W3. The top plot shows the original data, and the bottom shows an attempt at cleaning up the data Clean Up All of the data required cleaning up to produce better results during analysis (Section 4). Cleaning up the Orion Nebula data required essentially required two steps. I wrote a function to removehomerpointsbygoingthroughthedata2000pointsatatime. Ineachsubsetofthedata, the function sets any points located more than 4 standard deviations from the mean of the data subset to be equal to the mean of the subset. Operating this function on the Orion Nebula data nearly completely removed the outlying voltages at the points where the homer process was performed. After this, I performed a 4 th degree polynomial fit on the data by using the POLY FIT function in IDL. I then subtracted polynomial fit from the voltages, removing the gradual fluctuation evident in the voltages of the original data. The cleaned up data is shown in the bottom plot of Figure 2.

5 Orion Nebula, Original Data Voltage (V) Hour Angle Orion Nebula, Cleaned Up Power (V) Hour Angle Fig. 2. The voltages collected during the observation of Orion Nebula. The top plot shows the original data, and the bottom plot shows the cleaned up data. This same process was performed for the W3 data, but was not as successful, as evident in Figure 1. This is because there are more fluctuations in the W3 data, likely because of inaccurate pointing resulting from infrequent use of the homer process. For the Sun data, we manually removed most of the points from the data arrays of the voltages and the LSTs at which it was obvious where the telescope performed a homer operation. This modified data is shown in the bottom plot of Figure 3. Cleaning up the Lunar data required much more work than the Solar data. For the Moon, it appeared that the majority of the signal collected near the edges of the observation period was likely not radiation collected from the Moon. This is most clearly evidenced by the lack of a modulating

6 6 Voltage (V) Voltage (V) Sun, Original Data Hour Angle Sun, Homer Removed Hour Angle Fig. 3. The data collected of the sun through the interferometer. The top plot shows the voltages collected. The bottom plot shows the modified voltage data, in which most homer points were removed. function, which we would expect from the Moon since it is not a point source (see below, Section 4.2). Therefore, I focused on working with the data from an hour angle of -4 to an hour angle of 3. This subset of the data most clearly appeared to show an enveloping modulating function as expected. This data still contained spikes resulting from the homer routine for the telescopes, in addition to what appeared to simply be random noise. These were then removed through Fourier filtering. An FFT was performed on the original subset of the data. Then, I removed frequencies from -10 to 10 Hz from the resulting power spectrum. These frequencies would likely cover much of the low frequency noise, like the signals from during the homer positions. Afterwards, I performed an inverse FFT, which yielded a much cleaner and workable set of data. There were still a few

7 7 Voltage (V) Moon, Original Data Hour Angle Moon, Fourier Filtered Data Voltage (V) Hour Angle Fig. 4. The data collected of the Moon through the interferometer. The top plot shows the voltages collected. The bottom plot shows the cleaned up voltage data, the majority of which was done through Fourier filtering. outlying points in the data. These appeared to be due to noise, so I removed all points with a voltage above V and below V from the Fourier filtered data. The resulting, cleanedup, and more manageable, data is shown in the bottom plot of Figure 4. This data yielded much better results when attempting to fit a modulating function in Section

8 8 4. Analysis The analysis of the data mainly consisted of two portions. The first involved determining the length of the baseline between the interferometer, and subsequently the fringe frequency at a given hour angle. This process was carried out with the data from all sources. The second portion involved measuring the zero point of modulating functions of the resolved sources, and subsequently determining their angular radius on the sky Baseline and Fringe Frequency Orion Nebula, FFT Power Frequency (Hz) Orion Nebula, FFT Power Frequency (Hz) Fig. 5. The FFT of the Orion Nebula observation s voltage data. The bottom plot shows a subset of the top plot (which contains the entire power spectrum) to better display the peaks corresponding to the fringe frequencies.

9 9 Power Sun, FFT Frequency (Hz) Sun, FFT Power Frequency (Hz) Fig. 6. The FFT of the Solar observation s voltage data. The bottom plot shows a subset of the top plot (which contains the entire power spectrum) to better display the peaks corresponding to the fringe frequencies. Note that a logarithmic scale for the power axis is used. The first portion of the analysis involved determining the baseline length of the interferometer setup, and using that to subsequently determine the fringe frequency. The fringe frequency is a feature of the voltages collected from the interferometer since the signals from the two dishes are multiplied together with a mixer. The delay of a certain signal captured from one antenna relative to another antenna causes a phase shift. This delay is due to difference in distance the signal has to travel to reach the antenna (τ g ) and differences in the wiring of each antenna before their signals are multiplied [ (τ c ). τ g is dependent solely on the present hour angle (h s ), since due to the geometry τ g (h s ) = c ]sinh cosδ s. The resulting fringe output can be written as:

10 Moon, FFT Power Frequency (Hz) Moon, FFT Power Frequency (Hz) Fig. 7. The FFT of the Lunar observation s voltage data. The bottom plot shows a subset of the top plot (which contains the entire power spectrum) to better display the peaks corresponding to the fringe frequencies. F(t) = cos(2πνt)cos(2πν[t+τ g +τ c ]) 1 = 2 [cos(2πν[τ g +τ c ])+ cos(2πν[2t+τ g +τ c ])] (1) F(h s ) = C(cos(2πν[τ g (h s )+τ c ])) [ ( [ ) ( [ )] = C cos(2πντ c )cos 2πν ]sinh c cosδ s sin(2πντ c )sin 2πν ]sinh c cosδ s ( [ ) ( [ ) = Acos 2π ]sinh λ cosδ s Bsin 2π ]sinh λ cosδ s (2)

11 11 The constants in the front of the steps were eliminated and replaced with arbitrary constants (A and B) since these are what will be determined through the next process, Least Squares (Section 4.1.1), using equation (2). The sum term in equation (1) was removed since the frequency would be very high and that component would average to zero. The fringe frequencies can be seen in the FFT power spectrum of the voltages. The FFT of the Orion Nebula data is shown in Figure 5. The peaks corresponding to the fringe frequencies can be seen in the region between 200 and 300 Hz, and its corresponding lower sideband region. This leads ustoexpect afringefrequenciesinasimilarrangeonceweperformleast squares. Similarly, thefft of the Sun (Figure 6) and the FFT of the Moon (Figure 7) show peaks corresponding to the fringe frequencies as well. A logarithmic scale for the power is required to show the peaks corresponding to the fringe frequencies in the Sun FFT, likely because the low frequencies, corresponding to features like the modulating function, dominate over the fringe frequencies. The opposite is true in the Lunar data, in which the peaks corresponding to the fringe frequencies are easily evident, even without a logarithmic power scale. This is because the low frequencies were removed in an effort to clean up the Lunar data (see Section 3.1) Least Squares [ ] The method of least squares was required to determine the value of λ cosδ from equation (2) for each observation. This was done by cycling through a series of guesses for this value. The guesses were then inputted into equation (2), and the equation was then fitted onto the voltage data for the observation using the method of linear least squares to determine the best values of A and B. At each guess, the sum of the squares of the residuals (s 2 ) is calculated for the least squares fit. Afterwards, [ the lowest value for s 2 corresponded to the guess which is closest to the actual value of λ ]. cosδ [ This] process is illustrated for the Orion Nebula in Figure 8. The minimum s 2 occurs at λ cosδ = In order to calculate the baseline length, B y, we took δ = or δ = radians (which is the declination of the Orion Nebula) and λ = c ν = m (assuming ν = GHz from the central frequency of the observations, as shown in Section 2). [ ] B y = λ cosδ λ/cosδ = m/ = m This is close to what we assumed the baseline length to be, 10 m. In addition, the local fringe frequency is given by the following:

12 Orion Nebula Least Squares s (B y /λ)cosδ Fig. 8. The s 2 values plotted against the guesses for for the Orion Nebula observation. [ λ cosδ ] during the least squares process [ ] f f = λ cosδ cos(h s ) = = Hz, at beginning of obs. = = Hz, at end of obs. This corresponds with the peaks seen in the Orion Nebula FFT in Figure 5. [ This same ] process was carried out for the Sun, as seen in Figure 9. The minimum s 2 occurs at λ cosδ = We calculated the baseline length like before, except this time taking the declination value of the Sun at hour angle of 0 on the observation date.

13 Sun Least Squares s (B y /λ)cosδ Fig. 9. The s 2 values plotted against the guesses for for the Solar observation. [ λ cosδ ] during the least squares process [ ] B y = λ cosδ λ/cosδ = m/ = m This is again close to what we assumed the baseline length to be, 10 m, and very close to the baseline length calculated from the Orion Nebula observation above. In addition, the local fringe frequency for the Sun is given by the following:

14 14 [ ] f f = λ cosδ cos(h s ) = = Hz, at h s = 3.55 The fringe frequency above was calculated at an hour angle of 3.55, since this will be of interest in Section This frequency fits within the peaks seen in the Sun FFT in Figure Moon Least Squares s (B y /λ)cosδ Fig. 10. The s 2 values plotted against the guesses for for the Lunar observation. [ λ cosδ ] during the least squares process The least[ squares] process was done again for the Moon, as seen in Figure 10. The minimum s 2 occurs at λ cosδ = The baseline length could be calculated like before, except this time taking the declination value of the Moon at hour angle of 0 on the observation date.

15 15 [ ] B y = λ cosδ λ/cosδ = m/ = m This value for the baseline length is close to the assumed baseline length of 10 m, but it is much different than the baseline length values calculated through the Orion Nebula and Solar observations. This could be due to the much noisier lunar data, and the Fourier filtering carried out to reduce that noise. The Fourier filtering may have reduced some of the lower frequency components that built up the fringe frequency, resulting in the calculated baseline length to be somewhat lower than the actual value. In addition, the local fringe frequency for the Moon is given by the following: [ ] f f = λ cosδ cos(h s ) = = Hz, at h s = 3.00 Similar to the Sun, the fringe frequency above was calculated at an hour angle of -3.00, since this will be of interest in Section As for the other observations, this frequency also fits within the peaks seen in the Moon FFT in Figure Radius The radius for the non-point sources (i.e. the Sun and the Moon) could be calculated due to the modulating function, evident in the non-constant amplitudes of oscillating voltage data of these sources. This modulating function is a result of these resolved, two-dimensional, objects in the sky, passing through the fringe pattern (which can be thought as projected on to the sky by the interferometer). At every position, the brightness of the object is unevenly added and not added at the fringe maximums and minimums, respectively. The total intensity of the object, therefore, fluctuates as the source moves across the sky, and at any one point in the sky, is given by an integral of all the additions in brightness. The Fourier transform of this distribution on the sky gives the modulating function: MF theory = 1 R which can be approximated as a sum: MF theory R N R R n=+n n= N (R 2 h 2 ) 1/2 cos(2πf f h)d h [ ( n 1 N )] ( ) 1/2cos 2πff Rn N (3)

16 16 where the circle of light is divided into 2N +1 slices. 400 MF theory 300 MF theory f f R Fig. 11. MF theory (equation (3)) is plotted above. For this, N = 200. Since R N is not known, it was set to be 1 for generating the above plot. Since R N 0, the location of the zeroes is not affected. Equation (3) is only a function of f f R, and is plotted in Figure 11. The zeroes of this function correspond to the zeroes of the actual modulating function, so if the fringe frequency at which the zero occurs in the voltage data can be estimated, then the radius, R, of the source can be calculated. To estimate this fringe frequency, the hour angle at which the zero occurs needs to be estimated first. This can be done by fitting the following function around the zero value to get the best estimate for the hour angle at which it occurs: F(h s ) = A(h s H)cos ( 2π [ λ cosδ ) ]sinh s B(h s H)sin ( [ ) 2π ]sinh λ cosδ s (4)

17 17 where H is the hour angle at which the zero occurs. The least squares process can be used again to get a value for H, this time fitting the voltage data around the zero to equation (4) Least Squares, All Over Again The method of least squares [ to determine ] the value of H is similar to the method carried out in Section This time, λ cosδ was left constant at the value determined above in Section for the respective observations. Instead, different values of H were guessed for equation (4) for each observation. The guesses were inputted similarly into equation (4), and the equation was then fitted onto the voltage data for the location around the zero using the method of linear least squares to determine the best values of A and B. As before, at each guess, the sum of the squares of the residuals (s 2 ) is calculated for the least squares fit. Afterwards, the lowest value for s 2 corresponds to the guess which is closest to the actual value of H, the point at which the zero occurs for the data. Figure 12 shows this process for determining the value for H for the sun. The minimum value for s 2 occurs at H = This yielded the fit seen in Figure 13, and it can be seen that the fit does give a modulating function with a zero at an hour angle of As in Section 4.1.1, the fringe frequency at this point can be calculated by setting h s = H: [ ] f f = λ cosδ cos(h s ) [ ] = λ cosδ cos(h) = = Hz, at h s = H Now that the local fringe frequency at the zero is known, the radius can be calculated, using the fact that the first zero of MF theory occurs at f f R = 0.610, as seen in Figure 11: f f R = R = 0.61/f f = 0.61/ = Radians = Repeating this least squares process for the Lunar data, Figure 14 shows the minimum value for s 2 occurs at H = This yielded the fit seen in Figure 15, and the fit does give a modulating function with a zero at an hour angle of , as expected. Like in Section 4.1.1, the fringe frequency for the Moon at this point can be calculated by setting h s = H: [ ] f f = λ cosδ cos(h s ) [ ] = λ cosδ cos(h) = = Hz, at h s = H

18 Sun Zero Least Squares s Hour Angle (rad) Fig. 12. The s 2 values plotted against the guesses for H during the second least squares process for the Solar observation. Now that the local fringe frequency at the zero is known, the radius can be calculated, using the fact that the first zero of MF theory occurs at f f R = 0.610, as seen in Figure 11: f f R = R = 0.61/f f = 0.61/ = Radians = One way to judge the accuracy of the radius calculation is by using the results for the angular radius of the Sun (R) to estimate the actual radius of the Sun (r). This can be found by r = dtanr where d is the distance from the Earth to the Sun. This gives r = m, or 61.78% of the actual Solar radius, ( m. My result is significantly different, but is within the same

19 Sun, Modulating Function Fit Voltage (V) Hour Angle Fig. 13. Plot showing the voltage data of the Sun (solid line) overlaid with the best least squares fit (dotted line) in order to determine the location of the zero. Note that the voltage data from hour angle of 3.54 to an hour angle of 3.57 is set to 0, because a homer process occurred at this time. order of magnitude. The Lunar calculation can be judged similarly. Since the Moon is typically the same angular size as the Sun, using its angular radius in the above calculation should give the actual radius of the Sun. For the Moon, r = m, or 57.38% of the actual Solar radius. This is again off from the actual value, but is within the same order of magnitude. If using the second zero of the Modulating Function instead (f f R = 1.61), for the Sun r = m and for the Moon r = m. These are % and %, respectively, of the actual Solar radius. These are much closer to the actual values. Besides experimental errors, the discrepancy between the size calculated here and the actual size could be due to there being a

20 Moon Zero Least Squares s Hour Angle (rad) Fig. 14. The s 2 values plotted against the guesses for H during the second least squares process for the Lunar observation. difference between the physical Lunar and Solar size, and the Lunar and Solar size as seen at the radio wavelengths the interferometer used. 5. Interpretation This lab helped show the utility of the interferometer in radio astronomy. Using two dishes, placed on an East to West line, [ produced ] fringe patterns. In the first part of the lab, we were able to determine a value for λ cosδ in each observation by fitting the Fringe function onto the voltage data, and were subsequently able to find the baseline length. It is also easy to see how if the baseline length were known very accurately and precisely in future experiments, the declination of

21 Moon, Modulating Function Fit Voltage (V) Hour Angle Fig. 15. Plot showing the (Fourier filtered) voltage data of the Moon (solid line) overlaid with the best least squares fit (dotted line) in order to determine the location of the zero. Note that the phase of the fit and the data match up at the exterior points, but near the center of the fit, the phase of the fit does not match up. This could be due to large amount of noise in the data. the source could be calculated instead. This is likely to be more useful than calculating the baseline length for astronomical purposes. To obtain [ the] fringe frequency at any time, the best fit from the least squares process and the value for λ cosδ was used again. The utility of obtaining the fringe frequency became obvious in the next part of the experiments, where the angular radius of the Sun and the Moon were to be calculated. This time, for the resolved sources, a modulating function was fitted to the voltage data, giving the hour angle where the zero occurred. Comparing this with the theoretical zeroes in the theoretical Modulating Function, helped determine the radius. The fringe frequency at this

22 22 point was calculated using the relationships determined in the first portion of the lab. It is important to note that the value for the angular radius was determined by assuming that the source of the radiation was circular in shape. A different Modulating Function would emerge if the source were some other shape. If the shape of the source is not known, then it can be assumed to be circular. However in this case, this interferometer would only give the length of the object along the East West axis. Interferometers along different axes would be necessary to give the size along other axes. This experiment also leads to the realization of what could happen to resolution as some parameters of the experiment are changed. The resolution of the experiment is proportional to the fringe frequency. As the fringe frequency increases, the spacing of the fringe pattern on the sky decreases. Sources are resolved since they cover up an the fringe pattern unevenly, and as the fringe [ λ cosδ ], the fringe frequency, and frequency grows, more sources can be resolved. Since f f the resolution of the interferometer, can increased by increasing the baseline length and decreasing the wavelength. In other words, spacing the antennas of the interferometer further apart and using higher frequencies gives a greater resolution. This result is similar and analogous to what happens with optical telescopes: a larger diameter and a smaller wavelength yield greater resolution. 6. Conclusion The first part of the lab (Section 4.1.1) determined the baseline length of the interferometer used in this experiment. It also helped determine a function for the fringe frequency at any hour angle. This was useful in the next part of the lab. This part (Section 4.2.1) consisted of finding the radius of the resolved sources in the experiment (i.e. the Sun and the Moon). The lab also provided experience with various methods of improving the usability of the data collected (Section 3.1). It also gave examples of using the least squares process to fit theoretical functions onto data. The practical aspect of using an interferometer and working with the data and theoretical models will definitely be useful in future experiments. Learning about the details of a basic interferometer was very useful, and I found it exciting to see some potential for more elaborate experiments that could result from a more complex interferometer. Onearea I was not very satisfied by was the determination of the radius of the Moon. Thedata seemed to be very noisy, and my Fourier filtering did not completely reduce that noise. Although there appeared to be a zero around an hour angle of -3, the fit showing this was not completely convincing to me. It could have been helpful to try a second attempt at obtaining data for the Moon, but we did not have time to carry this out.