Planetary Temperature Worksheet Template
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4351 PS5
1
Earth 351 - Forming a Habitable Planet
Name _______________________________
Problem Set #5 - Planetary Temperature
Due: Mon., May 2, 2016
1.
What planetary minimum and maximum surface temperatures to be in the habitable zone?
2. One way to compare planets is via their blackbody or effective temperatures. To calculate this,
we assume that the solar radiation per unit area at a distance D from a star with luminosity L is
L/4πD
2
. Thus a planet of radius r absorbs solar power at a rate proportional to the ratio of its
cross sectional area πr
2
times a term (1-A) where A is the albedo, or fraction of solar radiation
reflected by the atmosphere. Thus the power absorbed is
Pabs = πr
2
(1-A) L/4πD
2
If the planet radiates this power as an ideal black body, then
σT
Prad= 4πr
2
4
bb
where σ is the Stefan-Boltzman constant (5.7 x 10
-8
J s
-1
m
-2
K
-4
) and T
is the blackbody
bb
temperature in degrees Kelvin.
a. Assuming a planet radiates all the solar power it absorbs, derive an expression for its blackbody
temperature. How does this depend on the planet’s radius?
b. Compute blackbody temperatures for the four terrestrial planets, assuming they have no
atmosphere (A=0),and L=3.9x10
26
W. The planets’ distances from the sun are given in the table
in Astronomical Units (1 AU= 1.5 x 10
8
km). Do everything in SI units and convert the final
result to degrees C. This is easiest with a spreadsheet or program.
c. The table also gives actual blackbody temperatures (i.e. including the albedo) and surface
temperatures.
D (AU)
black body T deg C
surface T dec C
mercury
0.39
169
167
venus
0.72
-42
464
earth
1
-19
15
mars
1.52
-63
-65
On one plot, plot these two temperatures and that from part 2b. for each planet as a function of
their distance from the sun. The plot should have three temperatures for each planet.
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