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Name: ________________________ AST 223 Planetary Science Homework #3: Gravity, Density, and Terrestrial Planets 1. (4 points) Assume that the Earth is a perfect sphere with a radius, r, of 6,300 km and that acceleration due to gravity, g, on the surface of the earth is 9.81 m/s2. The gravitational constant, G, is 6.67 × 10-11 m3/kg2. Given that: πΊπ π = 2πΈ π Calculate ME , which is the mass of the Earth. 2. (2 points) Calculate the mean density of the Earth based on your answer to question 1. 3. (2 points) In class, we discussed the gravitational force of the Earth acting on a satellite is: GmM Fο½ ο½ ma , r2 where m is the mass of the satellite and r is the distance between the centers of mass of the two bodies. For a satellite in a circular orbit around the Earth this force is balanced by the outward centrifugal force acting on the satellite, mο·2r, where ο·ο The angular velocity can be related to the period, T, (time necessary to complete one revolution) by T = 2ο°/ο· From this information derive an equation that relates the period, T, to orbital radius, r, and the mass of the Earth, ME. 1 4. (4 points) Given that the moon has a period of 27.3 days, and the mass of the Earth ME you calculated in #1, calculate the orbital radius of the moon using the equation derived in problem 3. 5. A newly-discovered planet has the same mean density as the Earth, but has a radius 2 times as large as the Earth. a. (2 points) If a woman has a mass of 50 kg on the Earth, what is her mass on this planet? b. (2 points) What is the value of g on the surface of this planet? c. (2 points) How much would the woman weigh on this planet? d. (2 points) What velocity would be required for a satellite to orbit this planet just above the its surface? (On Earth, the satellite would have to have a velocity of about 8000 m/s in order to orbit just above the surface.) 2 3 6. Interiors. Suppose we discover two new planets -- Nacnud and Retep. They are both believed to have a core of rock surrounded by a mantle of ice. They are the same size but Retep has a mean density of 2.8 g/cm3 and Nacnud has a density of 1.4 g/cm3. a. (3 points) Which has the thinner mantle of ice? Explain why. b. (3 points) Which is expected to have greater radioactivity heating the interior? Explain why. c. Ydnil has the same mean density (2.8 g/cm3) as Retep but Ydnil has a radius half that of Retep. Which planet will have cooled faster? Explain. d. What would be the evidence that you would look for on the surfaces of Retep and Ydnil to confirm your hypothesis of which planet had cooled first? 4