Homework due Oct. 4 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 &times; 10-11 m3/kg2. Given that:
𝐺𝑀
𝑔 = 2𝐸
𝑟
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
F
 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, m2r, where 
The angular velocity can be related to the period, T, (time necessary to complete one revolution)
by
T = 2/
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