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
• 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
Image from: www.crystalinks.com/andromeda.html
Andromeda: the Constellation
From: www.aer.noao.edu
Andromeda: the Constellation
From: www.astrosurf.com
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
Image from: www.astrosurf.com
•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: coolcosmos.ipac.caltech.edu
Hypothesis
Because Andromeda is the closest galaxy to
our own, we predicted that its presence
would be detectable using the SRT.
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
Data Set #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.
Data Set #2
Averages of the
raw data from
second set of
observations.
Analysis

Determined the
equation of the line
to be:
y=2.25x + 382.57

Subtracted the line
from the averaged
data.
Same data
with the slope
removed.
Emissions
from our own
galaxy
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
Graph of Data Set #3.
Averages of the
relative intensities of
all the observed
frequencies.
Analysis

Determined the
equation of the
line to be:
y=1.92x + 329.55

Subtracted the line
from the averaged
data.
Data set #3
with the slope
removed.
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.
Focal point
(0, F)
α
θ
Distance to focal point = E
Focal length
=F
(x2, y2)
Distance to vertex
=V
Vertex (0,0)
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.

y = (1/(4F))*(x-h)2+k
 y = x 2 /4.16

X
2

Y
2
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 2 2+(y 2-1.04) 2)
We let Excel do the calculations
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
2
2+y
2
2)
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?
α =cos -1 ((V 2-E 2 – F 2 )/(-2EF))
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)
Using the formula
s=2/4.16*x,
we can calculate the
slope
α
θ
Distance to focal point
=E
Focal length
=F
(x2, y2)
Distance to vertex
=V
Vertex (0,0)
Determine each angle at any given
point by finding:
the angle θ using the arctangent function
Using the formula θ = arctangent (s),
we can calculate the angle with respect
to horizon.
Focal point
(0, F)
α
θ
Focal
length
=F
Vertex
(0,0)
Distance to focal
point = E
(x2, y2)
Distance to
vertex
=V
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.
Examples of solar cooker

Parabolic
cooker
made
from
dung and mud
 The
A parabolic
Tire
Cooker
cooker
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
“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>