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:

Andromeda: the Constellation From:

Andromeda: the Constellation From:

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: 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 Offset - azimuth:12 degrees, elevation: 3 degrees. Central frequency: 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  Subtracted the line from the averaged data.

Same data with the slope removed. Emissions from our own galaxy

Analysis  Narrowed the range of frequencies ( to ).  Determined the slope of this range of frequencies: y = 0.019x –  Removed slope of this range of frequencies from the data.

Analysis Converted the frequencies to velocities using the formula: v = -(υ – υ 0 )/ υ 0 *c Where v = velocity υ 0 = MHz υ = observed frequencies c = 3*10 5 km/s

Preliminary Conclusion  It was unclear if the radio signals detected by the SRT were from M31. Peaks were not clearly defined.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  Offset - azimuth:12 degrees, elevation: 3 degrees.  Central frequency: 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  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= x 2 – x –  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 Connections and Geometry  Student Worksheet The Parabola and the Dish 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. Distance to vertex = V Distance to focal point = E (x 2, y 2 )Focal length = F θ α Focal point (0, F) 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= 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 x 2 and y 2.  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=√((x 2 -x 1 ) 2 + (y 2 -y 1 ) 2 ) (x 2,y 2 ) is any point on the parabola and (x 1,y 1 ) 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 like E= √(x 2 2 +(y ) 2 ) We let Excel do the calculations

Width (x value) and Height (y value): the distances What would the equation be for V? Hint: (x 2,y 2 ) is any point on the parabola and (x 1,y 1 ) 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 ((c 2 -a 2 –b 2 )/(-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 Using the formula s=2/4.16*x, we can calculate the slope Distance to vertex = V Distance to focal point = E (x 2, y 2 )Focal length = F θ α Focal point (0, F) 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. Distance to vertex = V Distance to focal point = E (x 2, y 2 )Focal length = F θ α Focal point (0, F) Vertex (0,0)

Example of completed worksheet  Excel Worksheet – with universal variables Excel Worksheet – with universal variables Excel Worksheet – with universal variables  Excel Worksheet – with basic setup Excel Worksheet – with basic setup 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 Tire Cooker A parabolic 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 Astronomy Education Review. “Andromeda Galaxy.” Retrieved: 22 July Crystalinks: Ellie Cristal’s Metaphysical and Science Website. Last Update: 22 July Frommert, Hartmut and Christine Kronberg. “M 31 Spiral Galaxy M31 (NGC 224), type Sb, in Andromeda Andromeda Galaxy.” Retrieved: 20 July Students for the Exploration and Development of Space (SEDS). Last Update: 18 September “M31 the Andromeda Galaxy.” Retrieved 21 July Cool Cosmos.