The Practicality of a Seemingly Simple Project to Learn the Secrets of the Universe. Or How Physics is Like Making a Cake.

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1 Utah State University From the SelectedWorks of David Smith April, 2014 The Practicality of a Seemingly Simple Project to Learn the Secrets of the Universe. Or How Physics is Like Making a Cake. David Alan Smith, Utah State University This work is licensed under a Creative Commons CC_BY International License. Available at:

2 The Practicality of a Seemingly Simple Project to Learn the Secrets of the Universe. Or How Physics is Like Making a Cake April 14, 2014 A senior seminar presented by David A. Smith

3 Outline Introduction/Motivation Radio Telescope Design Data Collection Data Analysis Final Thoughts

4 Introduction/Motivation Purpose of Senior Seminar?

5 Introduction/Motivation Demonstrate Learning

6 Introduction/Motivation Present Research

7 But Why The Title?

8 I ve built lots of antennas But Why The Title?

10 But Why The Title? I ve built lots of antennas Didn t seem too difficult

11 But Why The Title? I ve built lots of antennas Didn t seem too difficult Parabolic dish antenna

12 But Why The Title? I ve built lots of antennas Didn t seem too difficult Parabolic dish antenna Radio telescope just an antenna

13 But Why The Title? I ve built lots of antennas Didn t seem too difficult Parabolic dish antenna Radio telescope just an antenna ARRL Antenna Book

14 But Why The Title? F c = d(inches) The circular feed can be made of copper, brass, aluminum, or even tin in the form of a coffee or juice can The circular feed must be within a proper size (diameter) range for the frequency being used. This feed operates in the dominant circular waveguide mode knows as the TE 11 mode. The guide must be large enough to pass the TE 11 mode with no attenuation, but smaller than the diameter that permits the next higher TM 01 mode to propagate. ARRL Antenna Book, Page Published by the American Radio Relay League, Newington, CT 1991, 16 th Edition

15 But Why The Title? F c = d(inches) The circular feed can be made of copper, brass, aluminum, or even tin in the form of a coffee or juice can The circular feed must be within a proper size (diameter) range for the frequency being used. This feed operates in the dominant circular waveguide mode knows as the TE 11 mode. The guide must be large enough to pass the TE 11 mode with no attenuation, but smaller than the diameter that permits the next higher TM 01 mode to propagate. 20 years ago these words meant nothing to me

16 But Why The Title? F c = d(inches) The circular feed can be made of copper, brass, aluminum, or even tin in the form of a coffee or juice can The circular feed must be within a proper size (diameter) range for the frequency being used. This feed operates in the dominant circular waveguide mode knows as the TE 11 mode. The guide must be large enough to pass the TE 11 mode with no attenuation, but smaller than the diameter that permits the next higher TM 01 mode to propagate. 20 years ago these words meant nothing to me Now I understand what they mean!

17 But Why The Title? F c = d(inches) Where did this equation come from?

18 But Why The Title? F c = d(inches) Where did this equation come from? Would it be possible to derive it?

19 But Why The Title? F c = d(inches) Where did this equation come from? Would it be possible to derive it? Became important part of project

20 So how is physics like making a cake?

21 So how is physics like making a cake? Two ways to make a cake

22 Buy a box So how is physics like making a cake? Two ways to make a cake

23 So how is physics like making a cake? Two ways to make a cake Buy a box Add some stuff

24 So how is physics like making a cake? Two ways to make a cake Buy a box Add some stuff Eat cake!

25 So how is physics like making a cake? Two ways to make a cake Buy a box Add some stuff Eat cake! Or

26 So how is physics like making a cake? Two ways to make a cake Buy a box Add some stuff Eat cake! Or Learn about the ingredients

27 So how is physics like making a cake? Two ways to make a cake Buy a box Add some stuff Eat cake! Or Learn about the ingredients Understand how they work together

28 So how is physics like making a cake? Two ways to make a cake Buy a box Add some stuff Eat cake! Or Learn about the ingredients Understand how they work together Master the art of baking a cake

29 So how is physics like making a cake? Two ways to make a cake Buy a box Add some stuff Eat cake! Or Learn about the ingredients Understand how they work together Master the art of baking a cake From scratch!

30 Use the box KE = 1 2 mv2 F c = d(inches)

31 Use the tools E γmc 2 E kin = mc u2 c 2 1???

32 Radio Telescope Design A telescope is an instrument that aids in the observation of remote objects by collecting electromagnetic radiation Reflector/Dobsonian Mount Electromagnetic radiation between nanometers

Radio Telescope Design Parabolic Reflector 6-foot diameter Focal length 17.

33 Radio Telescope Design Parabolic Reflector 6-foot diameter Focal length inches Dish gain at 4GHz 35dB LNB Gain 66 db System Gain 101 db 3 db double signal; 10dB increases by 10 1 watt watts! Electromagnetic radiation between 7-8 centimeters

34 Radio Telescope Design y-axis x-axis Image:

Radio Telescope Design Detail of feed system Diamond LNB Low Noise Block Down Converter Input 3.7-4.

35 Radio Telescope Design Detail of feed system Diamond LNB Low Noise Block Down Converter Input GHz Output MHz Wave Guide TE 11 mode Inner diameter 2.10 (5.334 cm) According to the ARRL formula: TE 11 F c = TM 01 F c = = 3.3 GHz = 4.3 GHz

36 Radio Telescope Design Let s try to derive the cutoff frequency Start with the following assumptions: Wave guide is perfect conductor Inside the material E = 0 and B = 0 At boundary E = 0 and B = 0 k is real The area inside the wave guide approximate free space Then generally, i kz ωt E x, y, z, t) = E 0 x, y e B x, y, z, t) = B 0 x, y ei kz ωt The electric/magnetic fields must satisfy Maxwell s equations in the interior of the wave guide: E = 0 B = 0 E = B t B = 1 E c 2 t The goal is to find functions for E 0 and B 0 so the fields obey Maxwell s Equations subject to the boundary conditions. Confined waves are not generally transvers. To fit the BC longitudinal components E z and B z are included such that E 0 = E 0x x + E 0y y + E 0z z B 0 = B 0x x + B 0y y + B 0z z Please see Introduction to Electrodynamics, Fourth Edition, by David J. Griffiths, pp

37 Radio Telescope Design Each of those components are a functions of x and y. Next the curl of E and B are evaluated. According to Griffiths, these equations can all be solved for E x, E y, B x, B y. Griffiths states, It suffices, then, to determine the longitudinal components. If we knew those, we could quickly calculate all the others, just by differentiating. Inserting equations for E x, E y, B x, B y into the divergence equations yields uncoupled equations for E z and B z leaving us with, 2 x y 2 + ω c 2 k 2 E z = 0 2 x y 2 + ω c 2 k 2 B z = 0 TE Waves when E z = 0 TM Waves when B z = 0 We want the TE mode, so E z = 0 The proof of the above statement is given in Griffiths, page 427

38 Radio Telescope Design Looking back for a moment to Maxwell s Equations then taking the curl of both sides, B = μ 0 ε 0 E t 2 B = μ 0 ε 0 2 B t 2 B z = B 0z ei kz ωt 2 B z = μ 0 ε 0 2 B z t 2 In cylindrical coords B z is a function of r and φ so the Laplacian is recast. Making the full substitution for B z we have 2 B z = 1 r r r B 0ze i kz ωt r + 1 r 2 i kz ωt B 0z e φ B 0z e i kz ωt z 2 = μ 0 ε 0 2 B 0z e i kz ωt t 2 Then doing derivatives, cancelling exponentials, and rearranging we see that 1 r r r B 0z r + 1 r 2 B 0z φ 2 = B 0z k 2 μ 0 ε 0 ω 2 We can now use separation of variables

39 Radio Telescope Design B 0z = R r Φ φ B 0z r 2 B 0z φ 2 = R r Φ = 2 Φ φ 2 R Subbing in for B 0z, multipling by r 2, dividing by R r Φ φ, then moving things around we get r R r R r r r2 k 2 μ 0 ε 0 ω 2 = 1 2 Φ Φ φ 2 Depends on R Depends on Φ. Since equal, constant. Call the constant m 2 Φ = A cos m φ φ + B sin m φ φ

40 Radio Telescope Design r R r r R r r2 k 2 μ 0 ε 0 ω 2 = m r 2 Two new variables: h = k 2 μ 0 ε 0 ω 2 x = r k 2 μ 0 ε 0 ω 2 x hr r = x h dr = dx h 1 r = h x Multiply through by R r2, inserting the new variables, and using the product rule give us 2 R x R x x R 1 m 2 r x 2 = 0 This is a Bessel Equation of Order n with a general solution given as R x = A m J m x + B m Y m (x) J m x is a Bessel Function of the first kind, Y m (x) is a Bessel Function of the second kind. Apparently, Y m 0 which would cause the solution to be unnatural, leaving us with R = A m J m hr

41 Radio Telescope Design R x = A m J m x + B m Y m (x) Plot of Bessel Function of the First Kind (J m x )

42 Radio Telescope Design R x = A m J m x + B m Y m (x) Plot of Bessel Function of the Second Kind (Y m x ) blows up at x=0

43 Radio Telescope Design Plot of Derivative of Bessel Function of the First Kind (J m x ) MatLab syntax: numeric::fsolve(diff(besselj(1, x), x) = 0, x= 1..2) x

44 Radio Telescope Design We finally have our solution for B 0z. Putting everything together we have, B z = A m J m hr A cos m φ φ + B sin m φ φ ei kz ωt The decoupled second-order equations for E and B are identical in form. Hence, i kz ωt E z = A m J m hr A cos m φ φ + B sin m φ φ e E z is a function of r and φ. Plugging E z into Maxwell s Equations in cylindrical form allows us to solve for E φ. Hence, E = 1 r E z φ E z z r + E z z E z r φ + 1 r re z r E z φ z = B z t E φ = h r A mn J m h r r A cos m φ φ + B sin m φ φ e ihzz = 0 Since J m h r r is the only term that depends on r, and since at the boundary r=r it follows that E φ must equal zero when r=r, then it follows that J m h r R = 0. Then we can say that h r R χ mn h r = χ mn R χ mn is the nth root of the derivative of the Bessel function of the first kind, of the mth order. So the TE 11 mode means the 1 st root of the Bessel function of the first kind of the 1 st order. See Advanced Engineering Electromagnetics, by Constantine Balanis, Wiley, New York, 1989, Sect. 9.2.

45 Radio Telescope Design h 2 = h r 2 + h z 2 h z = h 2 h r 2 h z = h 2 χ mn R 2 The cutoff frequency is when h z = 0 h = χ mn h R c = ω c μ 0 με 0 = χ mn R χ mn R = ω c μ 0 με 0 χ mn R = 2πf c μ 0 με 0 f c = χ mn μ 0 με 0 2πR f c = χ mn 2πR c

46 Radio Telescope Design c 186, miles second = 11,802,852,688 inches second χ f c = π D 2 = D Hz = D MHz

47 Radio Telescope Design c 186, miles second = 11,802,852,688 inches second χ f c = π D 2 = D Hz = D MHz

48 Radio Telescope Design

49 Radio Telescope Design Front view Freq: Modified for voltage output Mic cable to laptop Radio SkyPipe Back view Detail of modification Thanks Tyler Hooker!

50 Data Collection Drift Scan Technique Allow Sun to pass through beam of radio telescope Difficult to point accurately Path of Sun due to rotation of Earth Radio Telescope Beam Width Sun

51 Data Collection?? AOS LOS Background signal (quiet sky) x-axis=time, y-axis=relative signal power Screen shot of typical Radio SkyPipe chart.

52 Data Collection Interesting bumps. Telescope lobes?

53 Data Collection Main Lobe Side Lobes

54 Large dip right in center. Due to feed system? Data Collection

55 Analysis Procedure Smooth the chart (averaging) Get max, min, half-power Get calibration signal 300 K building (house) Plug numbers into Excel Solar Brightness Temp Data Analysis Smoothed Chart Data Points Sun Max Quiet Sky Calibration Half-power Calculations True Sun True Calibration HP elapsed time HP beam width Antenna Sun Temp Solar Temp Antenna Temp Smoothed Calibration

56 Data Analysis Calculations True Calibration=Calibration Quiet Sky True Sun=Sun Max Quiet Sky Solar Antenna Temp= True Sun True Calib Temp Half-power= Sun Max + Quiet Sky 1 2 Half-power elapsed time=hpt 1 HPt 2 Half power elapsed time Half-power Beam Width= 0.25 Solar Temp=Solar Ant Temp HPBW Solar Observations with ESA-Haystack, Joachim Köppen, Strasbourg, 2010

57 Data Analysis The Solar Atmosphere Solar Temp at 4.0 GHz 54,111 Wilson and Thursby, ,000 Kundu, ,000 Koglin, 1994

58 10: : : : : : : : : : : : : : : : : : : : : : : : : : : : :47.7 Data Analysis Solar Maximum 6442 Quiet Sky 2985 Calibration Signal 5106 Half-Power Figure Half-Power Time 1 13 Half-Power Time Elapsed Time (mins) 18.2 HPBW (Degrees) 4.6 True Calibration 2121 True Solar Signal 3457 Signal to Calibration 1.6 Calibration Temp (F) 70 Calibration Temp (K) Antenna Temp (K) Solar Temp (K) Series1 Antenna Temp (K) Screen shot from Excel

59 Data Analysis Solar Max Quiet Sky Calibration True Calibration True Solar Signal HPBW Solar Max/Quiet Sky Ratio Calibration/Quiet Sky Ratio Solar Max/Calibration Ratio True Solar/True Calibration Ratio Estimated Solar Temp Screen shot from Excel

60 Final Thoughts Secrets of the Universe Bessel Functions Limb Darkening Limb Brightening Solar Atmosphere Brightness Temperature HPBW Radio SkyPipe

61 Final Thoughts Thank You Dr. C. D. Hoyle Advisor/Professor/Mentor Dr. Ken Owens Professor/Mentor Dr. Ryan Campbell Professor/Mentor Physics Department and Staff

62 Final Thoughts Thank You Shannon!

63 Final Thoughts List of references ARRL Antenna Book, Page Published by the American Radio Relay League, Newington, CT 1991, 16 th Edition Introduction to Electrodynamics, Fourth Edition, by David J. Griffiths, pp Advanced Engineering Electromagnetics, by Constantine Balanis, Wiley, New York, 1989, Sect Solar Observations with ESA-Haystack, Joachim Köppen, Strasbourg, 2010 Several other excellent papers Handbook of Geophysics and the Space Environment, Air Force Geophysics Laboratory, 1985 A Brief Theoretical View On Cylindrical Waveguides And Optical Fibers, Vardiambasis Statistical Analysis of Solar Radio Observations at 4 GHz, Wilson and Thursby, 1999 Microwave Observations of the Quiet Sun, Shibasaki Black Body Temperature of the Sun at 4 GHz, Koglin, 1994 Radio Astronomy Experiments at 4 GHz, Tapping

Radio Observation of Milky Way at MHz. Amateur Radio Astronomy Observation of the Milky Way at MHz from the Northern Hemisphere

Radio Observation of Milky Way at MHz. Amateur Radio Astronomy Observation of the Milky Way at MHz from the Northern Hemisphere Amateur Radio Astronomy Observation of the Milky Way at 1453.5MHz from the Northern Hemisphere Dr David Morgan February 2011 Introduction The measurements reported here were made in March and April 2007