Serway & Jewett — Physics and Measurement: Problem 19

Your roommate is playing a video game from the latest Star Wars movie while you are studying physics. Distracted by the noise, you go to see what is on the screen. The game involves trying to fly a spacecraft through a crowded field of asteroids in the asteroid belt around the Sun. You say to him, “Do you know that the game you are playing is very unrealistic? The asteroid belt is not that crowded and you don’t have to maneuver through it like that!” Distracted by your statement, he accidentally allows his spacecraft to strike an asteroid, just missing the high score. He turns to you in disgust and says, “Yeah, prove it.” You say, “Okay, I’ve learned recently that the highest concentration of asteroids is in a doughnut-shaped region between the Kirkwood gaps at radii of 2.06 AU and 3.27 AU from the Sun. There are an estimated $10^{9}$ asteroids of radius 100 m or larger, like those in your video game, in this region . . .” Finish your argument with a calculation to show that the number of asteroids in the space near a spacecraft is tiny. (An astronomical unit—AU—is the mean distance of the Earth from the Sun: $1 \text{ AU} = 1.496 \times 10^{11} \text{ m}$.)