AST 101 - HW 4 - Jesse Brown - keplers-laws

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Jan 9, 2024

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Tutorial-Exercise – Kepler’s Laws In this exercise, you’ll explore Kepler’s laws of orbital motion. Remember that these exercises are not meant for you to do alone; you should work with others near you on them, and should raise your hand and ask questions as you have them. Your fourth homework assignment is included on the back of this handout. You should complete it by class time on October 12 (or 7) and put it in your TA’s mailbox. 1 Kepler’s First Law Kepler’s first law says that planets orbit the Sun in an ellipse with the Sun at one focus . An ellipse has two foci; they lie along the long axis of an ellipse, are nearer the center for ellipses that are less eccentric (stretched out), and are nearer the edge for ellipses that are more eccentric (more stretched out). Here are four orbits. 1. Two of them have two possible locations for the Sun. Draw them in. 2. One of them has only one possible location for the Sun. Draw it in. 3. One of them is not a possible shape for a planet’s orbit. Cross it out. Of the three possible orbits, which one has zero eccentricity? Which one has the highest eccentricity? The Second ellipse has the highest eccentricity The fourth Ellipse has zero eccentricity.

2 Kepler’s Second Law As we saw demonstrated a bit ago, Kepler’s second law says that an imaginary line between the Sun and a planet orbiting it sweeps out an equal amount of area in an equal amount of time. First, let’s apply this to Earth’s orbit, which is very close to a perfect circle. Pretending for now that Earth’s year is divided into twelve equal months and that its orbit is a perfect circle, here is a cartoon of Earth going around the Sun. 1. Using your pencil, shade the region that the imaginary line between Earth and the Sun sweeps out during January, and again during September. 2. Does Earth’s orbit follow Kepler’s second law? How do you know? 3. Kepler described his observations of planetary motion in terms of the “area swept out” by the line, but we can also think about the speed of Earth’s motion. Does the Earth move faster during January or during September? How do you know? Yes Earth's orbit follows Kepler's second law because the shaded areas between each month cover the same amount of area. Based on the example above the earth moves the same speed during January and September.

4. Now, let’s consider a different planet whose orbit is more eccentric. Here is a cartoon of its orbit, showing its position at the beginning of January and February, along with several other positions Shade in the area that the line connecting the Sun to the planet would sweep out during January, as you did before. 5. Then, find two other points in the orbit during which the line connecting the planet to the Sun would sweep out approximately that same area. You would like to fill in the blanks in the statement: “As it moves from point to point , the line connecting this planet to the Sun sweeps out the same area as it does during the month of January.” Remember what “area” means: this means that if you were to color both areas in with ink, you would use the same amount of ink on both. 6. Does the motion on the left through the region you’ve highlighted take more than , less than , or exactly one month? How do you know? (Look back at the text of Kepler’s second law on the front page.) 7. Does the planet cover more distance during January or during the highlighted portion you’ve shown on the left? A D If the area is actually the same, then it will take exactly one month. It covers more distance during the highlighted potion on the left.

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