Andromeda and the DishAndromeda and the Dish

by by Angelica VialpandoAngelica VialpandoJennifer MiyashiroJennifer MiyashiroZenaida AhumadaZenaida Ahumada

OverviewOverview

Part I:Part I:

““Observing Andromeda Using the SRT”Observing Andromeda Using the SRT”

Part II:Part II:

Lesson Plan “The Parabola and the Lesson Plan “The Parabola and the Dish”Dish”

Part IPart I

““Observing Andromeda Observing Andromeda

Using the SRT”Using the SRT”

Andromeda in Greek MythologyAndromeda 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 ConstellationAndromeda: the Constellation

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

Andromeda: the ConstellationAndromeda: the Constellation

From: www.aer.noao.edu

Andromeda: the ConstellationAndromeda: the Constellation

From: www.astrosurf.com

Andromeda: the GalaxyAndromeda: 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

HypothesisHypothesis

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

ProcedureProcedure

•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 #1Procedural Adjustment #1 Because the intensities Because the intensities

appeared higher at the appeared higher at the edges of our observed edges of our observed frequencies, we frequencies, we increased the frequency increased the frequency range by increasing the range by increasing the number of bins to 50.number of bins to 50.

Second set of data Second set of data collected from 9:17 – collected from 9:17 – 10:19 a.m. on 19 July 10:19 a.m. on 19 July 2004.2004.

Data Set #2

Averages of the raw data from second set of observations.

AnalysisAnalysis

Determined the Determined the equation of the line to equation of the line to be:be:

y=2.25x + 382.57y=2.25x + 382.57

Subtracted the line from Subtracted the line from the averaged data.the averaged data.

Same data with the slope removed.

Emissions from our own galaxy

AnalysisAnalysis

Narrowed the range of Narrowed the range of frequencies (1421.0 to frequencies (1421.0 to 1423.4).1423.4).

Determined the slope of Determined the slope of this range of this range of frequencies:frequencies:

y = 0.019x – 2.766y = 0.019x – 2.766

Removed slope of this Removed slope of this range of frequencies range of frequencies from the data.from the data.

AnalysisAnalysis

Converted the frequencies to Converted the frequencies to velocities using the formula:velocities using the formula:

v = -(v = -(υ – υυ – υ00)/ υ)/ υ00 *c *c

Where v = velocityWhere v = velocity

υυ00 = 1420.52 MHz= 1420.52 MHz

υ = observed frequenciesυ = observed frequenciesc = 3*10c = 3*1055 km/s km/s

Preliminary ConclusionPreliminary Conclusion

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

defined.defined.

Procedural Adjustment #2Procedural Adjustment #2 We decided to see if a longer We decided to see if a longer

observation of M31 would yield a observation of M31 would yield a stronger (and more noticeable) signal.stronger (and more noticeable) signal.

Data collected from 11:30 pm ~ 11:30 Data collected from 11:30 pm ~ 11:30 a.m. beginning on 19 July 2004.a.m. beginning on 19 July 2004.

Offset - azimuth:12 degrees, elevation: Offset - azimuth:12 degrees, elevation: 3 degrees.3 degrees.

Central frequency: 1422.2 MHzCentral frequency: 1422.2 MHz

Number of bins: 30Number of bins: 30

Spacing: 0.08 MHzSpacing: 0.08 MHz

Graph of Data Set #3.

Averages of the relative intensities of all the observed frequencies.

AnalysisAnalysis

Determined the Determined the equation of the line equation of the line to be:to be:

y=1.92x + 329.55y=1.92x + 329.55

Subtracted the line Subtracted the line from the averaged from the averaged data.data.

Data set #3 with the slope removed.

AnalysisAnalysis

Determined the Determined the equation of a “good fit” equation of a “good fit” parabola to be:parabola to be:

y= 0.0029xy= 0.0029x22 – 0.0867x – 0.067 – 0.0867x – 0.067

Subtracted the parabola Subtracted the parabola from the averaged data.from the averaged data.

ConclusionConclusion

The data did not The data did not definitively show a pattern definitively show a pattern that would indicate M31.that would indicate M31.

Possible reasons a signal Possible reasons a signal was not observed:was not observed:

SRT not properly aimed at SRT not properly aimed at M31M31

Signals not strong enoughSignals not strong enough

Looking AheadLooking AheadFuture investigations could explore:Future investigations could explore:

Calibration of SRT to insure correct Calibration of SRT to insure correct

offsetsoffsets

Frequency patterns observed when the Frequency patterns observed when the

SRT is aimed at “nothing” as compared SRT is aimed at “nothing” as compared

to when aimed at the M31to when aimed at the M31

The lower-limits of signal intensity The lower-limits of signal intensity

detectable by the SRT.detectable by the SRT.

Part IIPart II

Lesson Plan Lesson Plan

““The Parabola and the Dish”The Parabola and the Dish”

The Parabola and the DishThe Parabola and the DishMath Modeling on ExcelMath Modeling on Excel

In this problem we will review and apply:In this problem we will review and apply: Properties of ParabolaProperties of Parabola Calculating Distance on Coordinate PlaneCalculating Distance on Coordinate Plane Calculating an Angle of a TriangleCalculating an Angle of a Triangle

(with 3 lengths)(with 3 lengths) Properties of SlopeProperties of Slope Calculating Angles with SlopeCalculating Angles with Slope

ResourcesResources NCTM StandardsNCTM Standards

Connections and GeometryConnections and Geometry

Student WorksheetStudent Worksheet

The Parabola and the DishThe Parabola and the Dish

The ProblemThe Problem You have found a You have found a

mangled Small mangled Small Radio Telescope. Radio Telescope. All that could be All that could be determined is that determined is that the the focal length is focal length is 1.04 m1.04 m and the and the angle from the angle from the focal point to the focal point to the edge of the dish is edge of the dish is 66 degrees.66 degrees.

Distance to vertex= V

Distance to focal point = E

(x2, y2)Focal length= Fθ

α

Focal point(0, F)

Vertex (0,0)

Fix it UpFix it Up

Our job is to repair the Our job is to repair the Width (x value) and Height (y value) by finding:Width (x value) and Height (y value) by finding: the equation of the parabolathe equation of the parabola x and y valuesx and y values the distances E and Vthe distances E and V the angle the angle αα using the Law of Cosinesusing the Law of CosinesDetermine each angle at any given point by Determine each angle at any given point by

finding:finding: the slope with respect to the horizonthe slope with respect to the horizon the angle the angle θθ using the arctangent function using the arctangent function

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

The equation of a Parabola is The equation of a Parabola is y = (1/(4F))*(x-h)y = (1/(4F))*(x-h)22+k,+k,where F is the focal length andwhere F is the focal length and(h,k) is the vertex.(h,k) is the vertex.

To simplify this equation let the vertex be (0,0) To simplify this equation let the vertex be (0,0) and substitute the focal length, and substitute the focal length, F= 1.04.F= 1.04.

What is your new equation?What is your new equation?

Width (x value) and Height (y value): Width (x value) and Height (y value): x and y valuesx and y values

The new equation The new equation of the dish isof the dish is

You can use Excel You can use Excel to calculate the x to calculate the x and y valuesand y values

which will be which will be

referred to as xreferred to as x2 2

and yand y22..

y = (1/(4F))*(x-h)y = (1/(4F))*(x-h)22+k+k

y = x y = x 2 2 /4.16/4.16

X X 22

Y Y 22

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

The distance between two points isThe distance between two points is D=√((xD=√((x22-x-x11))2 2 + (y+ (y22-y-y11))22))

(x(x22,y,y2 2 ) is any point on the parabola and ) is any point on the parabola and

(x(x11,y,y1 1 ) is the focal point (0,F) =(0,1.04)) is the focal point (0,F) =(0,1.04)

Lets write the equation to find the length of for E.Lets write the equation to find the length of for E.

The simplified equation looks likeThe simplified equation looks like

E= E= √(x √(x 22 22+(y +(y 22-1.04) -1.04) 22))

We let Excel do the calculationsWe let Excel do the calculations

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

What would the What would the equation be for V? equation be for V?

Hint: Hint: (x(x22,y,y2 2 ) is any point on ) is any point on the parabola and the parabola and

(x(x11,y,y1 1 ) is the vertex (0, ) is the vertex (0, 0)0)

V = V = √(x √(x 22 22+y +y 22 22))

Width (x value) and Height (y value): Width (x value) and Height (y value): the angle the angle αα using the Law of Cosinesusing the Law of Cosines

Using three lengths of any Using three lengths of any triangle, we can triangle, we can determine any interior determine any interior angle, by the Law of angle, by the Law of Cosines.Cosines.

αα =cos=cos-1-1((c((c22-a-a22–b–b22)/(-2ab)))/(-2ab))

What will our equation What will our equation for angle for angle αα be? be?

αα =cos =cos -1-1 ((V ((V 22-E -E 2 2 – F – F 22 )/(-2EF)) )/(-2EF))

Width (x value) and Height (y value): Width (x value) and Height (y value): the angle using the Law of Cosinesthe angle using the Law of Cosines

At 66 degrees find the corresponding x At 66 degrees find the corresponding x and y values. These are the optimal and y values. These are the optimal dimensions for this telescope.dimensions for this telescope.

X value?

Y value?

Determine each angle at any given point Determine each angle at any given point by finding:by finding:

the slope with respect to the horizonthe slope with respect to the horizon

Using the formula Using the formula s=2/4.16*x, s=2/4.16*x, we can calculate the we can calculate the slopeslope

Distance to vertex= V

Distance to focal point = E

(x2, y2)Focal length= Fθ

α

Focal point(0, F)

Vertex (0,0)

Determine each angle at any given Determine each angle at any given point by finding:point by finding:

the angle the angle θθ using the arctangent function using the arctangent function

Using the formula Using the formula θθ = arctangent (s), = arctangent (s), we can calculate the angle with respect we can calculate the angle with respect

to horizon.to horizon.

Distance to vertex= V

Distance to focal point = E

(x2, y2)Focal length= F

θ

α

Focal point(0, F)

Vertex (0,0)

Example of completed worksheetExample of completed worksheet

Excel Worksheet – with universal variExcel Worksheet – with universal variablesables

Excel Worksheet – with basic setupExcel Worksheet – with basic setup

Extension:Extension: Given the width, find the best focal length to Given the width, find the best focal length to

roast a marshmallow roast a marshmallow The diameter is 1 foot wide.The diameter is 1 foot wide. Pick a height for your dish.Pick a height for your dish. Write an equation to solve for F.Write an equation to solve for F. Adjust your height until you are Adjust your height until you are

satisfied with your dimensions.satisfied with your dimensions. Create a dish from cardboard and Create a dish from cardboard and

foil.foil. Time how long it takes for your Time how long it takes for your

marshmallow to roast. marshmallow to roast.

Examples of solar cookerExamples of solar cooker

Parabolic cooker made from dung and mudParabolic cooker made from dung and mud The Tire CookerThe Tire Cooker A parabolic cookerA parabolic cooker

AcknowledgementsAcknowledgements

We would like to thank:We would like to thank:

Mark Claussen for spending time with us to Mark Claussen for spending time with us to crunch all the numbers.crunch all the numbers.

Robyn Harrison for cheerfully meeting with us Robyn Harrison for cheerfully meeting with us at totally unreasonable times to point the at totally unreasonable times to point the telescope.telescope.

Lisa Young for encouraging us to explore Lisa Young for encouraging us to explore things and patiently explaining what we things and patiently explaining what we were looking at.were looking at.

References CitedReferences 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>