Andromeda and the Dish

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Andromeda and the Dish

• by Angelica Vialpando
• Jennifer Miyashiro
• Zenaida Ahumada

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Overview

• Part I
• Observing Andromeda Using the SRT
• Part II
• Lesson Plan The Parabola and the Dish

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Part I

• Observing Andromeda
• Using the SRT

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Andromeda in Greek Mythology

• Her mother, Cassiopea, compared Andromedas
  beauty to that of the sea-nymphs (Nereids).
  
• This greatly angered the nymphs and the god
  Poseiden.
  
• To appease the gods, her parents tied Andromeda
  to a rock by the sea to be eaten by the sea
  monster Cetus.
  
• She was rescued by Perseus and they led a
  wonderful life.
  
• She and her family were rewarded for leading such
  a commemorative life by being placed into the
  stars by the gods.

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Andromeda the Constellation
Image from www.crystalinks.com/andromeda.html
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Andromeda the Constellation
From www.aer.noao.edu
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Andromeda the Constellation
From www.astrosurf.com
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Andromeda the Galaxy

• Closest spiral galaxy to our own Milky Way
• Observable to the naked eye (fuzzy)
• Angular size is about 2 degrees
• AKA M31
• About 2.2 million light years away
• Approximately 1.5 times the size of the Milky
  Way
  
• The most studied galaxy (other than our own)
• (Harmut and Kronberg, 2004)

Image from www.astrosurf.com
Image from coolcosmos.ipac.caltech.edu
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Hypothesis
Because Andromeda is the closest galaxy to our
own, we predicted that its presence would be
detectable using the SRT.
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Procedure

• Data collected from 830 913 a.m. on 19 July
  2004.
  
• Offset - azimuth12 degrees, elevation 3
  degrees.
  
• Central frequency 1421.85 MHz
• Number of bins 30
• Spacing 0.08 MHz

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Data Set 1
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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 917 1019
  a.m. on 19 July 2004.
  

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Data Set 2
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Averages of the raw data from second set of
observations.
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Analysis

• Determined the equation of the line to be
• y2.25x 382.57
• Subtracted the line from the averaged data.

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Same data with the slope removed.
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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.

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Analysis

• Converted the frequencies to velocities using the
  formula
  
• v -(? ?0)/ ?0 c
• Where v velocity
• ?0 1420.52 MHz
• ? observed frequencies
• c 3105 km/s

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Preliminary Conclusion

• It was unclear if the radio signals detected by
  the SRT were from M31.
  
• Peaks were not clearly defined.

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Procedural Adjustment 2

• We decided to see if a longer observation of M31
  would yield a stronger (and more noticeable)
  signal.
  
• Data collected from 1130 pm 1130 a.m.
  beginning on 19 July 2004.
  
• Offset - azimuth12 degrees, elevation 3
  degrees.
  
• Central frequency 1422.2 MHz
• Number of bins 30
• Spacing 0.08 MHz

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Graph of Data Set 3. Averages of the relative in
tensities of all the observed frequencies.
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Analysis

• Determined the equation of the line to be
• y1.92x 329.55
• Subtracted the line from the averaged data.

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Data set 3 with the slope removed.
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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.

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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

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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.

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Part II

• Lesson Plan
• The Parabola and the Dish

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The Parabola and the DishMath 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

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Resources

• NCTM Standards
• Connections and Geometry
• Student Worksheet
• The Parabola and the Dish

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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.

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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 a using the Law of Cosines
• Determine each angle at any given point by
  finding
  
• the slope with respect to the horizon
• the angle ? using the arctangent function

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Width (x value) and Height (y value) the
equation of the parabola

• The equation of a Parabola is
• y (1/(4F))(x-h)2k,
• 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?

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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.

• y (1/(4F))(x-h)2k
• ? y x 2 /4.16
• X 2
• Y 2

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Width (x value) and Height (y value) the
distances

• The distance between two points is
• Dv((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.

The simplified equation looks like
E v(x 2 2(y 2-1.04) 2) We let Excel do the ca
lculations
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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)

V v(x 2 2y 2 2)
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Width (x value) and Height (y value) the angle a
using the Law of Cosines

• Using three lengths of any triangle, we can
  determine any interior angle, by the Law of
  Cosines.
  
• a cos-1((c2-a2b2)/(-2ab))
• What will our equation for angle a be?

a cos -1 ((V 2-E 2 F 2 )/(-2EF))
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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.

X value? Y value?
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Determine each angle at any given point by
findingthe slope with respect to the horizon

• Using the formula s2/4.16x,
• we can calculate the slope

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Determine each angle at any given point by
finding the angle ? using the arctangent
function

Using the formula ? arctangent (s), we can
calculate the angle with respect to horizon.
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Example of completed worksheet

• Excel Worksheet with universal variables
• Excel Worksheet with basic setup

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Extension Given the width, find the best focal
length to roast a marshmallow

• The diameter is 1 foot wide.
• Pick a height for your dish.
• Write an equation to solve for F.
• Adjust your height until you are satisfied with
  your dimensions.
  
• Create a dish from cardboard and foil.
• Time how long it takes for your marshmallow to
  roast.
  

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Examples of solar cooker

• Parabolic cooker made from dung and mud

• The Tire Cooker

A parabolic cooker
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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.
  

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References Cited
Andromeda. Retrieved 20 July 2004. Astronomy
Education Review. l Andromeda Galaxy. Retrieved 22 July 2004.
Crystalinks Ellie Cristals Metaphysical and S
cience Website. Last Update 22 July 2004.

Frommert, Hartmut and Christine Kronberg. M 31
Spiral Galaxy M31 (NGC 224), type Sb, in Androme
da Andromeda Galaxy. Retrieved 20 July 2004.
Students for the Exploration and Development of
Space (SEDS). Last Update 18 September 2003.
M31 the Andromeda Galaxy
. Retrieved 21 July 2004. Cool
Cosmos.