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Background: Copernicus and his model of the heliocentric solar system were vital in reshaping people’s perceptions of the universe. However, Copernicus’ model was not any better at predicting the position of the planets than Ptolemy’s geocentric model. Kepler solved this problem by proposing that each planet has an elliptical orbit rather than a circular orbit.
ellipse: an oval circular shape determined by the position of two points, each called a focus (foci is the plural form.) If you take any point on the ellipse, the sum of the distance to the focus points is constant, with the size of the ellipse determined by the sum of these two distances. The sum of these distances is equal to the length of the major axis (the longest diameter of the ellipse.) A circle is a special case of an ellipse.
major/minor axis: these are the longest and shortest diameters of an ellipse.
foci (focus points): these are the two points that define an ellipse.
eccentricity: a measure of how much a shape deviates from a circle.
1) on a sheet of graph paper, draw six sets of axes. Each axis should resemble an enlarged plus sign and be labeled from -5 to +5 on both the “x” and “y” axes. Label the sets of axes “Ellipse 1” through “Ellipse 6.”
2) look at the Ellipse Data Table. The top row gives the coordinates (x,y) of the two foci of each ellipse. For example: for ellipse 1, note that the x-coordinates are ±4.5. Therefore, the first x-coordinate is +4.5 and the second is -4.5. The y-coordinate for both points is 0. Use “+” signs to plot the foci for Ellipse 1 on the appropriate set of axes.
3) use the data table to plot the points for ellipse 1. After plotting the points, complete the ellipse by connecting the points with a smooth curve.
4) repeat the above steps for each of the other ellipses.
5) the major axis is the line that passes through the two foci and connects the two farthest ends of the ellipse. For each ellipse, find and record the length (L) of the major axis. Find the distance (D) between the foci. Then use the formula in the data table to calculate the eccentricity.
Analysis and Conclusions:
1) Study ellipses 1-3. Which ellipse looks most like a circle? Which ellipse looks least like a circle? Describe the relationship between how circular an ellipse appears and its eccentricity.
2) Which ellipse has the same eccentricity as ellipse 1? How do the shapes of these two ellipses compare? Which ellipse has the same eccentricity as ellipse 3? How do these two ellipses compare?
3) If two ellipses have the same shape, which of the following must be equal: distance between foci, length of the major axis, and/or eccentricity? Give evidence for your answer.
4) What geometric shape would result if both foci were located at point (0,0) of the graph? What would be the eccentricity of such an ellipse?
5) The “Eccentricities of the Planets” table shows the orbital eccentricity of the planets. Compare the eccentricities of your ellipses with the eccentricities of the planets. Which of your ellipses are the best models for the ellipses of the planets orbits?
6) Which planet has the most circular orbit? Which planet has the least circular orbit? Explain your answer.
7) Many comets have eccentricities of close to 1. Describe the shape of such an orbit. Which of your ellipses is most similar to a comet’s orbit?
8) The orbit of Mars has an eccentricity of 0.093. The distance between the two foci is 0.283 AU. The closest Mars gets to the sun during its orbit 1.83 AU. What is the farthest Mars gets from the Sun? Hint: remember that the sun is located at one of the foci.
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