Homework 10 - Rings

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Apr 3, 2024

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ASTR 1010 - Fall 2022 Written Homework 10 20 pts Total Due Friday, December 2 at 11:59pm on Canvas How quickly do particles in Saturn’s rings orbit? A good question was raised at Fiske this week: “For the particles of ice, rock, and dust that make up Saturn’s rings, how quickly do they orbit the planet?” We discussed that the orbits of these particles follow Kepler’s Laws and made a guess that they would orbit on the order of hours, but let’s check this estimate. The image below shows Saturn’s rings. Note that the rings that you are used to seeing in telescope images extend from the C-ring to the A-ring. (The inner D-ring and outer F-G-E- rings are faint and diffuse.) Measurements indicate the inner edge of the C-ring is 74,500 km (or 7.45 x 10 7 m) from Saturn’s center and the outer edge of the A-ring is 137,000 km (or 1.37 x 10 8 m).
1. If Saturn was the Same Mass as the Sun (12 pts total). First, let’s (incorrectly) assume that Saturn had the same mass as the Sun. In this case, the ring particles would follow Kepler’s 3 rd Law as we discussed earlier for planets orbiting the Sun; that is, the square of the orbital period (in years) would equal the cube of the distance to the center of Saturn (in AU). Below is the same statement as a formula: [࠵? (࠵?࠵? ࠵?࠵?࠵?࠵?࠵?)] ! = 1.0 ࠵?࠵?࠵?࠵? ! ࠵?࠵? " 4 ∗ [࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? (࠵?࠵? ࠵?࠵?)] " a. (4 pts) First, convert the inner C-ring distance and the outer A-ring distance into AU. Show your work below. b. (4 pts) Using the inner C-ring distance in AU, how quickly would the inner particles orbit in years? How fast is this in seconds? Show your work. c. (4 pts) Using the outer A-ring distance in AU, how quickly would the outer particles orbit in years? How fast is this in seconds? Show your work.
2. Using Saturn’s Actual Mass (8 pts total). That’s pretty fast!!! In reality, Saturn is less massive than the Sun so we need to use a slightly modified formula for Kepler’s Law which accounts for this lower mass: [࠵? (࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?)] ! = 1.04 ࠵? 10 #$% ࠵? ! ࠵? " 4 ∗ [࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? (࠵?࠵? ࠵?)] " a. (4 pts) Using this equation (which properly accounts for Saturn’s gravity), how quickly do the inner particles orbit in seconds? How fast is this in hours? Show your work. b. (4 pts) How quickly do the outer particles orbit in seconds? How fast is this in hours? Show your work. So the guess that was provided in class was reasonably close, but now we know. Note that dating back to the time of Newton, planetary scientists have been able to measure the masses of planets with moons by measuring the moon’s distance to the planet and the moon’s orbital period. Knowing the ratio between these two values lets us determine the mass of the planet!!
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