ASTR330 Homework #1

Homework hint: All problems in the text have answers at the back of the book! Some problems require just a written response, while others ask you to calculate something. Please write up all answers clearly, completely, and as succinctly as possible. You can work with others, but your final answers must be written up on your own.

Chapters 1 and 2

Question 1. In Figure 1.52 on page 347, we learn that the densest of all of the planets and moons in the Solar System is ... Earth! Earth is so dense because of its large iron core.
a) Why then is Mercury less dense, when its iron core is larger in proportion to its size than Earth's (see Fig. 2.17)?
b) Jupiter's density implies that it does not have a large iron core. Why doesn't it?

Question 2. a) Calculate the average bulk density of the planets Venus, Mars, and the Moon in the same way that it is done for Earth in Question 2.1 on page 32. The data that you need and the correct answers can be found in Tables A1 and A2 in the back of the book.
b) Assuming that the three objects are made from uncompressed iron, with a density of 7900kg/m3, and rock (assume 2500-3000kg/m3), would you expect iron cores larger or smaller (relative to the object's radius) than Earth's in each case? Compare with Fig. 2.17 and discuss.
c) Estimate the mass of the Earth's crust, its mantle, and its core. Use reasonable numbers given in Fig. 2.4 and state your assumptions!

Question 3. On page 66, the author states the radioactive elements U and Th are concentrated in the Earth's crust. What about Potassium (K), the most important heat-generating element? Note that unlike U and Th, where all isotopes are radioactive, K has both stable and radioactive isotopes. Drawing information from the chapter, determine if K is concentrated, and if so, where. Consider the Earth's core, mantle, and crust. Back up your argument quantitatively.

Question 4. Check out the spidergram plot in Fig. 2.28 on page 350.
a) Please explain why the elements Rb, K, Na, and Zn are depleted in the mantle.
b) Some of these elements are not depleted in the crust - from your answer to question 2c, estimate what the comparable curve for (crust + mantle) would look like on Fig. 2.28. Are these elements depleted in the outer half of the Earth?

Question 5. This problem will explore why there is no natural radioactive Aluminum 26 on Earth (or anywhere else in the Solar System! - see box 2.7 on pg. 64).
a) First, from Table 2.2, we estimate that the Earth is about 1% Aluminum by weight. Given that the weight of an Aluminum 27 atom is 4.5*10-26kg, roughly how many Aluminum 27 atoms does Earth have?
b) Now assume that the Earth originally had an equal number of Aluminum 26, which is radioactive with a 0.73 million year halflife. About how many halflives would it take for only 1 part in 1000 of the 26Al to remain? How much time is this?
c) About how much time did it take for all 26Al to decay away to nearly the last atom? Can you say how long it took the last atom to decay? Compare these numbers to the age of the Earth and discuss.

Question 6. How much heat was released in massive collisions with early Earth? Go to the Solar System Collisons program and simulate some major collisions!
a) Start with a 1 cm rocky particle moving at 20 km/s. Start making a table recording i) what happens, ii) how often it happens, and iii) energy released if appropriate. Now increase the size to 10 cm and run again. Continue to increase the impactor size by factors of 10 until you reach 9999 km.
b) Work out an estimate for the amount of energy released by radioactivity in the Earth over its entire history. Use today's present heat production rates given on page 64, and multiply your final answer by 3 to correct for the fact that there was more radioactive heating of Earth in the past. What size impactor releases a similar amount of energy? To convert energy between Megatons of TNT and the SI unit Joules, use 1 Megaton = 4.2*1015 Joules.

Question 7. Go to the Scientific Notation program. Run it until you are sure that you can do this type of problem in your sleep (or on an exam!). What happens when you get a problem right?
b) Do the same with the Working with Equations program. Check off the equations for v, g, E, and H which we may meet in this class; E gives the amount of energy released in the impacts of Question 6. What is in the help file? If you have trouble doing either of these problems, be sure to talk to me or the grader.

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