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Andromeda and the Dish. 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.

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andromeda and the dish
Andromeda and the Dish

by Angelica Vialpando

Jennifer Miyashiro

Zenaida Ahumada

overview
Overview

Part I:

“Observing Andromeda Using the SRT”

Part II:

Lesson Plan “The Parabola and the Dish”

part i

Part I

“Observing Andromeda

Using the SRT”

andromeda in greek mythology
Andromeda in Greek Mythology
  • Her mother, Cassiopea, compared Andromeda’s 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.
andromeda the constellation
Andromeda: the Constellation

Image from: www.crystalinks.com/andromeda.html

andromeda the constellation1
Andromeda: the Constellation

From: www.aer.noao.edu

andromeda the constellation2
Andromeda: the Constellation

From: www.astrosurf.com

andromeda the galaxy
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

hypothesis
Hypothesis

Because Andromeda is the closest galaxy to our own, we predicted that its presence would be detectable using the SRT.

procedure
Procedure
  • Data collected from 8:30 – 9:13 a.m. on 19 July 2004.
  • Offset - azimuth:12 degrees, elevation: 3 degrees.
  • Central frequency: 1421.85 MHz
  • Number of bins: 30
  • Spacing: 0.08 MHz
procedural adjustment 1
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
Analysis
  • Determined the equation of the line to be:

y=2.25x + 382.57

  • Subtracted the line from the averaged data.
slide18

Emissions from our own galaxy

Same data with the slope removed.

analysis1
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.
analysis2
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
Preliminary Conclusion
  • It was unclear if the radio signals detected by the SRT were from M31.
    • Peaks were not clearly defined.
procedural adjustment 2
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
slide24

Graph of Data Set #3.

Averages of the relative intensities of all the observed frequencies.

analysis3
Analysis
  • Determined the equation of the line to be:

y=1.92x + 329.55

  • Subtracted the line from the averaged data.
analysis4
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
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
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

Part II

Lesson Plan

“The Parabola and the Dish”

the parabola and the dish math modeling on excel
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
resources
Resources
  • NCTM Standards

Connections and Geometry

  • Student Worksheet

The Parabola and the Dish

the problem

Focal point

(0, F)

Distance to focal point = E

α

Focal length

= F

(x2, y2)

θ

Distance to vertex

= V

Vertex (0,0)

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
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 αusing 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
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.

y = (1/(4F))*(x-h)2+k

 y = x 2 /4.16

X 2

Y 2

Width (x value) and Height (y value): x and y values
width x value and height y value the distances
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.

The simplified equation looks like

E= √(x 22+(y 2-1.04) 2)

We let Excel do the calculations

width x value and height y value the distances1
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 = √(x 22+y 22)

width x value and height y value the angle using the law of cosines
Width (x value) and Height (y value): the angle αusing 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 αbe?

α=cos -1 ((V 2-E 2 – F 2 )/(-2EF))

width x value and height y value the angle using the law of cosines1
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?

determine each angle at any given point by finding the slope with respect to the horizon

Focal point

(0, F)

Distance to focal point = E

α

Focal length

= F

(x2, y2)

θ

Distance to vertex

= V

Vertex (0,0)

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 using the arctangent function

Focal point

(0, F)

Distance to focal point = E

α

Focal length

= F

(x2, y2)

θ

Distance to vertex

= V

Vertex (0,0)

Determine each angle at any given point by finding:the angle θusing the arctangent function

Using the formula θ = arctangent (s), we can calculate the angle with respect to horizon.

example of completed worksheet
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
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.

Extension:Given the width, find the best focal length to roast a marshmallow
examples of solar cooker
Examples of solar cooker
  • Parabolic cooker made from dung and mud
  • The Tire Cooker

A parabolic cooker

acknowledgements
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
References Cited
  • “Andromeda.” Retrieved: 20 July 2004. Astronomy Education
    • Review. <http://aer.noao.edu/about.html>
  • “Andromeda Galaxy.” Retrieved: 22 July 2004. Crystalinks:
    • Ellie Cristal’s Metaphysical and Science Website. Last Update: 22 July 2004. <www.crystalinks.com/andromeda.html>
  • Frommert, Hartmut and Christine Kronberg. “M 31 Spiral
    • Galaxy M31 (NGC 224), type Sb, in Andromeda Andromeda Galaxy.” Retrieved: 20 July 2004. Students for the Exploration and Development of Space (SEDS). Last Update: 18 September 2003. <http://www.seds.org>
  • “M31 the Andromeda Galaxy.” Retrieved 21 July 2004. Cool
    • Cosmos. <http://coolcosmos.ipac.caltech.edu>