by Angelica Vialpando Jennifer Miyashiro Zenaida Ahumada
Overview Part I: “Observing Andromeda Using the SRT” Part II: Lesson Plan “The Parabola and the Dish”
Part I “Observing Andromeda Using the SRT”
Andromeda in Greek Mythology
Andromeda: the Constellation
Andromeda: the Constellation
Andromeda: the Galaxy
Hypothesis
Procedure
Procedural Adjustment #1 Because the intensities appeared higher at the edges of our observed frequencies, we increased the frequency range by increasing the number of bins to 50. Second set of data collected from 9:17 – 10:19 a.m. on 19 July 2004.
Analysis Determined the equation of the line to be: y=2.25x + 382.57 Subtracted the line from the averaged data.
Analysis Narrowed the range of frequencies (1421.0 to 1423.4). Determined the slope of this range of frequencies: y = 0.019x – 2.766 Removed slope of this range of frequencies from the data.
Analysis Converted the frequencies to velocities using the formula: v = -(υ – υ0)/ υ0 *c - Where v = velocity
- υ0 = 1420.52 MHz
- υ = observed frequencies
- c = 3*105 km/s
Preliminary Conclusion It was unclear if the radio signals detected by the SRT were from M31. - Peaks were not clearly defined.
Procedural Adjustment #2 We decided to see if a longer observation of M31 would yield a stronger (and more noticeable) signal. Data collected from 11:30 pm ~ 11:30 a.m. beginning on 19 July 2004. Offset - azimuth:12 degrees, elevation: 3 degrees. Central frequency: 1422.2 MHz Number of bins: 30 Spacing: 0.08 MHz
Analysis Determined the equation of the line to be: y=1.92x + 329.55 Subtracted the line from the averaged data.
Analysis Determined the equation of a “good fit” parabola to be: y= 0.0029x2 – 0.0867x – 0.067 Subtracted the parabola from the averaged data.
Conclusion The data did not definitively show a pattern that would indicate M31. Possible reasons a signal was not observed: SRT not properly aimed at M31
Looking Ahead Future investigations could explore: Calibration of SRT to insure correct offsets Frequency patterns observed when the SRT is aimed at “nothing” as compared to when aimed at the M31 The lower-limits of signal intensity detectable by the SRT.
Part II Lesson Plan “The Parabola and the Dish”
The Parabola and the Dish Math Modeling on Excel In this problem we will review and apply: Properties of Parabola Calculating Distance on Coordinate Plane Calculating an Angle of a Triangle (with 3 lengths) Properties of Slope
Resources NCTM Standards Connections and Geometry Student Worksheet The Parabola and the Dish
The Problem You have found a mangled Small Radio Telescope. All that could be determined is that the focal length is 1.04 m and the angle from the focal point to the edge of the dish is 66 degrees.
Fix it Up Our job is to repair the Width (x value) and Height (y value) by finding: x and y values the distances E and V the angle α using the Law of Cosines Determine each angle at any given point by finding: the slope with respect to the horizon the angle θ using the arctangent function
Width (x value) and Height (y value): the equation of the parabola y = (1/(4F))*(x-h)2+k, where F is the focal length and (h,k) is the vertex. To simplify this equation let the vertex be (0,0) and substitute the focal length, F= 1.04. What is your new equation?
Width (x value) and Height (y value): x and y values The new equation of the dish is You can use Excel to calculate the x and y values which will be referred to as x2 and y2.
Width (x value) and Height (y value): the distances The distance between two points is D=√((x2-x1)2 + (y2-y1)2) (x2,y2 ) is any point on the parabola and (x1,y1 ) is the focal point (0,F) =(0,1.04) Lets write the equation to find the length of for E.
Width (x value) and Height (y value): the distances What would the equation be for V? Hint: (x2,y2 ) is any point on the parabola and (x1,y1 ) is the vertex (0, 0)
Width (x value) and Height (y value): the angle α using the Law of Cosines Using three lengths of any triangle, we can determine any interior angle, by the Law of Cosines. α =cos-1((c2-a2–b2)/(-2ab)) What will our equation for angle α be?
Width (x value) and Height (y value): the angle using the Law of Cosines At 66 degrees find the corresponding x and y values. These are the optimal dimensions for this telescope.
Determine each angle at any given point by finding: the slope with respect to the horizon Using the formula s=2/4.16*x, we can calculate the slope
Determine each angle at any given point by finding: the angle θ using the arctangent function
Example of completed worksheet Excel Worksheet – with universal variables
Extension: Given the width, find the best focal length to roast a marshmallow
Examples of solar cooker Parabolic cooker made from dung and mud
Acknowledgements We would like to thank: Mark Claussen for spending time with us to crunch all the numbers. Robyn Harrison for cheerfully meeting with us at totally unreasonable times to point the telescope. Lisa Young for encouraging us to explore things and patiently explaining what we were looking at.
References Cited
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