The rules: Collaborative work is encouraged. This homework can be done in consultation with your fellow classmates, the AI, or the professor. However, everyone must submit their own solutions to get credit, and all help should be acknowledged (a single sentence mentioning the others in your study group is sufficient). Show your work.
1. Just the facts [5 pts]
(a) [1 pts]
True or False, The sun never shines on the "far side" of the moon.
(b) [1 pts]
True or False, Earth's atmosphere is composed mostly of nitrogen.
(c) [1 pts]
True or False, Geologists obtain most of their information about Earth's
mantle by drilling deep into our planet's interior.
(d) [2 pts]
Observations of S- and P- waves have confirmed that the Earth's core is ________
and the outer core is _________.
2. Earth, Wind, and Fire [29 pts]
(a) [4 pts] On average, the temperature beneath the Earth's crust increases at a rate of 20o C per kilometer. At what depth would water boil? (Assume the surface temperature is 20o C and ignore the effect of the pressure of overlying rock on the boiling point of water.)
(b) [5 pts] Find the radius of the Earth's core relative to the total radius if the core density is 10000 kg/m3, the mantle density is 4500 kg/m3, and the average density is 5500 kg/m3.
(c) [5 pts] Estimate the age of a rock sample in which 25% of the 238U nuclei have decayed into 206Pb.
(d) [5 pts] A rock sample has a 40Ar-to-40K ratio of 18, a 87Sr-to-87Rb ratio of 0.082, and a 206Pb- to-238U ratio of 1.3. What is the age of the rock?
(e) [5 pts] In the Earth's exosphere (700 km above the Earth's surface), the temperature can reach 2000 K. Estimate the lifetime of water vapor there by comparing its mean velocity with the Earth's escape velocity.
(f) [5 pts] The summit of Mauna Kea, Hawaii (4200 m above sea level) is considered one of the world's premier observing sites. It is especially good as an infrared site because of the low water vapor and carbon dioxide content in the atmosphere above this altitude. (Water and carbon dioxide have many infrared wavelength absorption bands.) However, working at these altitudes can be hard on astronomers who are not acclimated to the lower abundance of molecular oxygen. To understand both of these effects, calculate the scale height for H2O, CO2, and O2 in the Earth's atmosphere.
3. A Little TLC (Temperature-Luminosity Calculations) [16 pts]
(a) [12 pts] (i) The mean albedo of Mercury is 0.06. Calculate and compare Mercury's surface temperature at local noon at perihelion and at aphelion. (ii) Calculate the peak wavelength at which the surface emits at noon for both perihelion and aphelion. (iii) At local midnight, Mercury's surface temperature is 125 K. Calculate the peak wavelength at which the surface emits at midnight.
(b) [4 pts] What is the luminosity of a star which radiates like a black body at a temperature of 7200 K and has a radius of 9.05 x 105 km?