ICW 05 - Planetary Orbits

Worksheet — Planetary Orbits Page 1 of 7 Astronomy BPHYS 101 Worksheet — Planetary Orbits Identify Yourself Background Material Question 1. Draw a line connecting each law on the left with a description of it on the right. Kepler’s 1 st Law Planets with larger orbits take more time to complete an orbit. Kepler’s 2 nd Law Only a force acting on an object can change its motion. Kepler’s 3 rd Law Objects interact each other with forces of equal magnitude. Newton’s 1 st Law Planets move faster when close to the Sun. Newton’s 2 nd Law The acceleration of an object is proportional to the force on it. Newton’s 3 rd Law Planets orbit the Sun in elliptical paths. Question 2. When Kepler’s 3 rd Law is written as 𝑝 2 = ? 3 ( 𝑝 measured in years, ? measured in AU), what is it applicable to? A Any object orbiting our Sun. B Any object orbiting any star. C Any object orbiting any other object. Two of an ellipse’s properties define its eccentricity. The eccentricity ( 𝑒 ) is the ratio of the distance of a focal point from the center ( ? ) to the length of the semi-major axis ( ? ). 𝑒 = ? ? Because the eccentricity is the ratio of two distances, it is dimensionless (it has no units). Name 1: Name 3: Name 2: Name 4:

Worksheet — Planetary Orbits Page 2 of 7 Question 3. What is the approximate eccentricity of this ellipse? A 0.25 B 0.50 C 0.75 D 0.90 Question 4. For a planet in an elliptical orbit to “sweep out equal areas in equal amounts of time” , what must be true? A It moves slowest when near the Sun. B It moves fastest when near the Sun. C It moves at the same speed at all times. D It has an orbital eccentricity of zero (circular orbit). Question 5. If a planet is twice as far from the sun at aphelion than at perihelion, how does the strength of the gravitational force on the planet when it is at aphelion compare to when the planet is at perihelion? A The gravitational force is four times the strength at aphelion than it is at perihelion. B The gravitational force is two times the strength at aphelion than it is at perihelion. C The gravitational force is the same at aphelion as it is at perihelion. D The gravitational force is half the strength at aphelion than it is at perihelion. E The gravitational force is one quarter the strength at aphelion than it is at perihelion. Kepler’s 1st Law Launch the NAAP Planetary Orbit Simulator . Tip: You can change the value of a slider by clicking on the slider bar or by entering a number in the value box. • Open the [K epler’s 1 st Law] tab if it is not already (it’s open by default). • Enable all five check boxes. • The white dot is the “simulated planet”. You can click on it and drag it around. ? ? ?

Worksheet — Planetary Orbits Page 4 of 7 Question 9. For the ellipse with ? = 20.0 AU and 𝑒 = 0.500 , can you find a point in the orbit where 𝑟 1 and 𝑟 2 are equal? Sketch the ellipse, the location of this point, and 𝑟 1 and 𝑟 2 in the space below. Question 10. What is the value of the sum of 𝑟 1 and 𝑟 2 and how does it relate to the ellipse properties? (Properties include eccentricity, semi-major axis length, and semi-minor axis length). Is this relationship true for all ellipses? Kepler’s 2nd Law • Use the [clear optional features] button to remove the 1st Law features. • Open the [Kepler's 2nd Law] tab. • Press the [start sweeping] button. Adjust the semi-major 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.

Worksheet — Planetary Orbits Page 5 of 7 Question 11. Erase all sweeps and create an ellipse with ? = 1.00 AU and 𝑒 = 0.000 . Set the fractional sweep size to one-twelfth of the period. Drag the sweep segment around. Does its size or shape change? Question 12. Leave the semi-major axis at ? = 1.00 AU and change the eccentricity to 𝑒 = 0.500 . 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?) Question 13. What eccentricity in the simulator gives the greatest variation of sweep segment shape? Question 14. Halley’s comet has a semi -major axis of about 18.5 AU, a period of 76 years, and an eccentricity of about 0.97 (so Halley’s orbit cannot be shown in this simulator.) The orbit of Halley’s Comet, the Earth’s Orbit, and the Sun are shown in th e diagram below (not exactly to scale). Based upon what you know about Kepler’s 2 nd Law, explain why we can only see the comet for about 6 months every orbit (76 years)?

Worksheet — Planetary Orbits Page 7 of 7 • Click the boxes to show both the velocity and the acceleration vectors of the planet. Observe the direction and length of the arrows. The length is proportional to the magnitude of the vector. Question 18. The acceleration vector always pointing toward what object? Question 19. Create an ellipse with ? = 6.00 AU and 𝑒 = 0.500 . For each marked location in the diagram below, indicate if the speed is increasing or decreasing (with an arrow). Assume the orbital motion is counterclockwise ⟲ . A B C D E F G H speed { (↑) increasing (↓) decreasing ↑ Question 20. Where do the maximum and minimum values of velocity occur in the orbit? Question 21. Can you describe a general rule, which identifies where in the orbit velocity is increasing and where it is decreasing? A C E F G H D B