Andromeda and the Dish by Angelica Vialpando



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Andromeda and the Dish


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 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

  • Signals not strong enough



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

  • Calculating Angles with 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:

  • the equation of the parabola

  • 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

  • The equation of a Parabola is

  • 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

  • Excel Worksheet – with basic setup



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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