Exploring Orbits. Introduction


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1 Exploring Orbits Introduction In 1609, the German mathematician and astronomer Johannes Kepler deciphered a major puzzle of the solar system. The strange backandforth motions of the planets in the sky were nearly impossible to predict until Kepler figured out the true shapes of the planetary orbits around the sun. Orbits were always believed to be circular, but Kepler used mathematics to discover they were actually elliptical. An ellipse is an oval that is characterized by two quantities. The first quantity is the width of the ellipse, which is called the major axis. The second quantity is called the eccentricity, which is a measure of how stretched out the ellipse is. Eccentricity is defined by the distance between two mathematically determined points within the ellipse called the foci. For planets, one focus of their orbital ellipse is the sun. The other is an empty point in space. The orbit of each planet is an ellipse, but each planet s elliptical orbit has different major axes and eccentricities. Once Kepler understood the proper nature of orbits, the movements of the planets in the sky could be predicted with precision. In the investigation, you will draw ellipses, calculate their eccentricities, observe an interesting property of ellipses, and compare the ellipses you draw with the orbital eccentricities of Earth and other planets in the solar system. So, in this investigation we will find out what the elliptical orbits of the planets look like. 1
2 Figure 1 Focal Length Sun Major Axis PreLab Discussion Read the entire investigation before completing the prelab questions 1. In your opinion, are the shapes of the planet s orbits more circular or more elongated? 2. What is the independent variable for the ellipses you will draw? 3. What is the dependent variable for the ellipses you will draw? 4. What purpose do the two pushpins serve in this investigation? 2
3 Materials 3 sheets of paper Heavy corrugated cardboard 2 pushpins Ruler String (30 cm long) 5 colored pencils Tape Calculator Procedure 1. Fold a sheet of paper in half lengthwise. Flatten it out again. 2. Place the paper on the cardboard and measure 5cm from the center of the page. Place one pushpin at the 5cm mark. Measure 5 cm to the other side of the center of the paper and place a pushpin there. The push pins should be 10 cm apart. CAUTION: Be careful when handling the pins; they can puncture skin. 3. Label one of the pushpins as the sun. 4. Tie the string in a loop and place it around the pins. Using one of the colored pencils, gently pull the string tight. Keep the string tight without pulling the pins out of the cardboard. Carefully drag the pencil around the pins to draw an ellipse, as shown on figure Using the same colored pencil, draw a circle around the pin that is not labeled as the sun. Remove the pin. 6. Use the metric ruler to measure the length of the major axis and focal length. Record these values in Data Table 1. 3
4 7. Reposition the second pin so that it is now 8.0 cm from the other pin. Repeat steps 4 through 6, using a different colored pencil. 8. Repeat step 7, using distances of 6.0 cm, 4.0 cm, and 2.0 cm between the pins. As the focal length for each ellipse becomes smaller, you may need to tape additional sheets of paper above and below the original sheet of paper to draw the entire ellipse. 9. The eccentricity for each ellipse is calculated by dividing the focal length by the length of the major axis Eccentricity = focal length Major axis Calculate the eccentricity for each ellipse. Record the values in Data Table Label each ellipse on your diagram with its matching eccentricity. Observations Data Table 1 Ellipse (Color) Major Axis (cm) Focal Length (cm) Eccentricity Part B: Observing Properties of Ellipses 11. Choose one of the ellipses you made on your diagram and label the foci A and B respectively. 12. Choose a point anywhere on the ellipse and label it C. 4
5 13. Measure the length of the lines AC and BC in centimeters. Record your measurements in Data Table Repeat steps 12 and 13, placing point C at three different points on the ellipse. Carefully measure lines AC and BC and record your measurements in Data Table 2. Data Table 2 Position of C Length of AC Length of BC Length of AC + BC Analysis and Conclusions 1. Compare the following values for planetary eccentricities to those you calculated for your ellipses. What can you state about the orbits of the various planets? Planet Eccentricity Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto
6 2. What shape would you make if both pushpins were placed at a single central point? What would be the focal length and eccentricity of the shape? 3. What did you discover in Part B about the sums of lines AC and BC for your ellipse? Generalize your findings as a basic law of ellipses. 4. What body in the solar system do you think is one focus of the moon s orbit? 6
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