Exploring Orbits

Name ______Class ______Date ______

Exploring Orbits Introduction

In 1609, the German mathematician and astronomer Johannes Kepler deciphered a major puzzle of the solar system. The strange back-and forth 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 this 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.

Problem What do the elliptical orbits of the planets look like?

Pre-Lab Discussion Read the entire investigation. Then work with a partner to answer the following questions.

1. Predicting: Each planet’s orbit is shaped like an ellipse. Predict whether the shapes of the planet’s orbits will be more circular or more elongated.

2. Applying Concepts: What is the one thing that the elliptical orbits of all planets, asteroids, and most comets have in common?

3. Controlling Variables What is the independent variable for the ellipses you will draw?

4. Controlling Variables What are the dependent variables for the ellipses you will draw?

5. Designing Experiments What purpose do the two pushpins serve in this investigation?

Materials (per pair of students) 3 sheets of paper heavy corrugated cardboard (~50 cm X 60 cm) 2 pushpins metric ruler string, 30 cm long 5 colored pencils cellophane tape calculator

Procedure

Part A: Drawing Ellipses and Calculating Eccentricity 1. Fold a sheet of paper in half lengthwise. Flatten it out again. 2. Place the paper on the cardboard and measure 5 cm from the center of the page. Place one pushpin at the 5 cm mark. Measure 5 cm to the other side of the center of the paper and place a pushpin there. The pushpins 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 in Figure 2. 5. 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. 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:

Calculate the eccentricity for each ellipse. Record the values in Data Table 1. A perfect circle has an eccentricity of zero, while more and more elongated ellipses have higher eccentricities ≤1.

10. Label each ellipse on your diagram with its matching eccentricity. Observations 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.” 13. Measure the length of the lines AC and BC in centimeters. Record your measurements in Data Table 2. 14. 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.

Analysis and Conclusions

1. Compare and Contrast: 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?

2. Inferring: What shape would you make if both pushpins were placed at a single central point? What would be the focal length and eccentricity of this shape?

3. How does the position of the focus points affect the ellipse?

4. Observing: 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.”

5. The semi-major axis of the Earth's elliptical orbit around the sun is 150 million kilometers. What is the average distance between the Earth and the sun?

6. What is the average distance between the focus and any point along the elliptical path of the ellipses you drew?

7. True or False? The distance from the Earth to the sun is the same no matter where the Earth is on its orbital path. Explain why or why not:

8. Inferring: What body in the solar system do you think is one focus of the moon’s orbit?

9. Drawing Conclusions Did your results from this activity confirm your original prediction? Explain why or why not.

10. The average distance between the Earth and the sun is 1 astronomical unit. How many meters is 1 astronomical unit?

Extension

Kepler published his first two laws in 1609. The first law states: “The orbit of every planet is an ellipse with the Sun at one of the foci.” The second explains why planets move at different speeds at different points in their orbit: “A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.” This is easiest to see in a diagram: Kepler continued his work on planetary motions, and in 1619 published “Harmony of the Spheres” in which he showed that there is a relationship between a planet’s distance from the Sun and the time it takes that planet to go around the Sun (its orbital period). Kepler’s third law states “The squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axis of the orbits.”

(Where P is orbital period and a is the semi-major axis) This law works very well when we use years as the unit for P and astronomical units (AU, the Earth-Sun semi-major axis) as the unit for a.

11. The table below shows several objects in the solar system with their eccentricity and semi- major axis listed. Use the semi-major axis values to calculate the orbital period for each object (in years). Record your answers in the table provided.