## 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
## Andromeda: the Constellation
## Andromeda: the Constellation
## Andromeda: the Galaxy
## Hypothesis
## Procedure
## 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 ## Determined the equation of the line to be: ## y=2.25x + 382.57 ## Subtracted the line from the averaged data.
## 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
## Analysis ## Determined the equation of the line to be: ## y=1.92x + 329.55 ## Subtracted the line from the averaged data.
## 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
## 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
## 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.
## Fix it Up ## Our job is to repair the ## Width (x value) and Height (y value) by finding: ## 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 ## 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.
## 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.
## 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) -
## 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?
## 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.
## 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 θ using the arctangent function
## Example of completed worksheet ## Excel Worksheet – with universal variables
## Extension: Given the width, find the best focal length to roast a marshmallow
## Examples of solar cooker -
**Parabolic cooker made from dung and mud **
## 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
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