Chapter 12, Problem 49EAP

Be sure to show all calculations clearly and state your final answers in complete sentences.

1. Adding Up Asteroids. It’s estimated that there are a million asteroids 1 kilometer across or larger. If a million asteroids 1 kilometer across were all combined into one object, how big would it be? How many 1-kilometer asteroids would it take to make an object as large as Earth? (Hint: You can assume they’re spherical. The expression for the volume of a sphere is jirr³, where r is the radius.)
2. Impact Energies. A relatively small impact crater 20 kilometers in diameter could be made by a comet 2 kilometers in diameter traveling at 30 kilometers per second (30,000 m/s).
3. Assume that the comet has a total mass of 4.2 x 10'? kilograms. What is its total kinetic energy? (Hint: The kinetic energy is equal to Jmiz², where m is ihe comet’s mass and v is its speed. If you use mass in kilograms and velocity in m/s, the answer for kinetic energy will have units of joules.)
4. Convert your answer from part a to an equivalent in megatons of TNT, the unit used for nuclear bombs. Comment on the degree of devastation the impact of such a comet could cause if it struck a populated region on Earth. (Hint: One megaton of TNT releases 4.2 X 10^ls joules of energy.)
5. The "Near Miss" of Toaiatis. The 5-kilometcr asteroid Toutatis paBsed a mere 1.5 million kilometers from Earth in 2004. Suppose Toutatis were destined to pass somewhere within 1.5 million kilometers of Earth. Calculate the probability that this "somewhere" would have meant that it slammed into Earth. Based on your result, do you think it is fair to call the 2004 passage a “near miss"? Explain. (Hint: You can calculate the probability by considering an imaginary dartboard of radius 1.5 million kilometers in which the bull's-eye has Earth’s radius, 6378 kilometers.)
6. Room to Roam. It's estimated that there are a trillion comets in the Oort cloud, which extends out to about 50,000 AU. What is the total volume of the Oort cloud, in cubic AU? How much space does each comet have in cubic AU, on average? Take the cube root of the average volume per comet to find the comets’ typical spacing in AU. (Hints: For the purpose of this calculation, you can assume the Oort cloud fills the whole sphere out to 50,000 AU. The volume of a sphere is given by ywr³, where ris the radius.)
7. Comet Temperatures. Find the “no greenhouse" temperatures for a comet at distances from the Sun of 50,000 AU (in the Oort cloud), 3 AU, and 1 AU (see Mathematical Insight 10.1). Assume that the comet reflects 3% of the incoming sunlight. At which location will the temperature be high enough for water ice to vaporize (about 150 K)? How do your results explain comet anatomy? Explain.
8. Comet Dust Accumulation. A few hundred tons of comet dust arc added to Earth daily from the millions of meteors that enter our atmosphere. Estimate the time it would take for Earth to get 0.1 % heavier at this rate. Is this mass accumulation significant for Earth as a planet? Explain.