Homework # 5, FRS 120
Due in Monday April 9
100 points total
1. Blowing away an atmosphere (40 points)
During the period of heavy bombardment in the Earth’s early history, it is postulated that the impact of a
large enough planetesimal could have stripped the planet of its early atmosphere. In this problem, you will
calculate how large a planetesimal is required.
First we will calculate the mass of the planet’s atmosphere. The densest part of the Earth’s atmosphere is
10 km thick; let us approximate it as uniform density over that thickness, and negligible at higher altitudes.
Atmospheric pressure at sea level is about one kilogram per square centimeter; this is just the weight due to
Earth’s gravity of the air in a tall skinny rectangle over each square centimeter.
a. (10 points)
atmosphere.
Use the information given to calculate the density (grams per cubic centimeter) of the
b. (5 points) Calculate the total mass of the atmosphere. Express your answer in both grams and tons
(1 ton = 106 grams).
c. (10 points) To blow this atmosphere away, you need to heat the molecules up enough to escape
the Earth’s gravitational field. Estimate the temperature to which the atmosphere needs to be heated (here,
assume that the atmosphere is principally O2 and N2 ). Explain your reasoning in full.
d. (10 points) Consider an incoming planetesimal of mass M . Argue coherently that the velocity of
the planetesimal relative to the Earth is roughly the same as the orbital speed of the Earth around the Sun
(30 km/s). Assuming that all the kinetic energy of the planetesimal goes into heating the atmosphere (which
is probably not a correct approximation in detail), what is the minimum mass M to completely remove the
atmosphere from the planet? Express your answer in both grams and tons. Big hint: it may be easier to do
this problem without using the results of part (c).
e. (5 points) Approximating the planetesimal as a sphere, calculate its radius (in kilometers), assuming
that it is mostly made of water ice (density 1 gram per cubic centimeter). Now repeat the calculation assuming
the planetesimal to be made of iron (density 7 grams per cubic centimeter).
2. Global warming (30 points)
Without greenhouse warming, the equilibrium temperature of the Earth would be
s
TEarth = T (1 − A)1/4
R
,
2d
as we discussed in class. The albedo of the Earth is A ≈ 0.3. In the presence of a greenhouse gas (say, CO2 ),
this equation is modified to:
s
R
TEarth = T (1 − A)1/4
(1 + x)1/4 ,
2d
where x is the “optical depth” of CO2 ; all you need to know is that x is proportional to the density of CO2
in the Earth’s atmosphere (to keep things simple, we are assuming here that CO2 is the only greenhouse gas
in the atmosphere).
a. (10 points)
Calculate the value of x for today’s Earth, where the mean temperature is TEarth ≈ 300K.
b. (10 points) There is roughly 7 times as much carbon dioxide locked up in (as yet unburned) fossil
fuels as there is in the Earth’s atmosphere. If all those fossil fuels were burned, releasing all that CO2 , what
would the new equilibrium temperature of the Earth be?
c. (10 points) How much carbon dioxide needs to be added to the Martian atmosphere to raise its mean
temperature to above freezing? Assume that Mars has the same albedo as Earth. Express your answer as a
ratio of the density of Martian carbon dioxide to that on Earth.
3. Snow Ball Earth (30 points)
The albedo of Antarctic snow is 0.8. For this problem lets approximate the Earth albedo as a ”step
function”:
A(T ) = 0.8
T < 273K
A(T ) = 0.3
T ≥ 273K
a. (10 points) Plot two curves on a graph of temperature (T ) versus flux, F :
2 (1 − A(T ))/2d2
• (1) the energy flux hitting the Earth, T4 R
• (2) the energy radiated by the Earth: T 4 /(1 + x)
For these plots, set x = 0.6
b (10 points) The curves should intersect at three points. Describe why two of them are stable and one is
unstable.
c (10 points) When the Earth is in the snowball phase, volcanos continue to add CO2 to the atmosphere.
However, it is not removed by weathering. Find the value of x, such that the greenhouse effect ends the
snowball phase (Hint: set T = 273K and find the value of x such that the the energy flux from the Sun
absorbed by a snow covered Earth equals the energy radiated by the Earth). Compute the equiibrium
temperature after the snow melts with this value of x.