Astronomy of Stars Galaxies and the Cosmos

PHYS-1050 Kepler, Newton, and Planetary Orbits Solutions Spring 2013 - Press the “start sweeping” button. Adjust the semimajor axis and animation rate so that the planet moves at a reasonable speed. - Adjust the size of the sweep using the “adjust size” slider. - Click and drag the sweep segment around. Note how the shape of the sweep segment changes, but the area does not. - Add more sweeps. Erase all sweeps with the “erase sweeps” button. - The “sweep continuously” check box will cause sweeps to be created continuously when sweeping. Test this option. 1. Erase all sweeps and create an ellipse with a = 1 AU and e = 0. Set the fractional sweep size to one-twelfth of the period. Drag the sweep segment around. Does its size or shape change? Solution: It’s a circle, so neither the shape nor the size change as the sweep segment is dragged around the orbit. 2. Leave the semi-major axis at a = 1 AU and change the eccentricity to e =0 . 5. Drag the sweep segment around and note that its size and shape change. Where is the sweep segment the “skinniest?” Where is it the “fattest?” Where is the planet when it is sweeping out each of these segments? (What names do astronomers use for these positions?) Solution: “Skinniest” is when the planet is farthest from the star: aphelion. “Fattest” is when the planet is closest to the star: perihelion. 3. Halley’s comet has a semimajor axis of about 18.5 AU, a period of 76 years, and an eccentricity of about 0.97 (so Halleys orbit cannot be shown in this simulator). The orbit of Halleys Comet, the Earths Orbit, and the Sun are shown in Figure 3 (not exactly to scale). Based upon what you know about Kepler’s Second Law, explain why we can only see the comet for about 6 months every orbit (76 years)? Solution: For the six months when it’s swinging through perihelion, it’s moving at its fastest in the inner portion of the solar system. We can see it just ﬁne. But when it moves out of the inner solar system, it gets harder to see, but it also slows down. So it spends the largest amount of its orbital period far away from us, moving very slowly, where we can’t see it too well. Figure 3: The orbit of Halley’s Comet, the Earth’s orbit, and the sun. 4.3 Kepler’s Third Law - Use the “clear optional features” button to remove the Second Law features. - Open the Kepler’s Third Law tab. 1. Use the simulator to complete Table 1. 2. As the size of a planet’s orbit increases, what happens to its period? Solution: Its period increases: a = 3 √ P 2 . 3. Start with the Earth’s orbit and change the eccentricity to 0.6. Does changing the eccentricity change the period of the planet? 5/7

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