Q0: What would be Earth’s equilibrium temperature1 without an
atmosphere?
Thermal equilibrium means that: Thermal energy in = Thermal energy out
Definition: Power = rate of change of energy = energy/time, so
Thermal equilibrium also means that: Power in = Power out
Power absorbed by Earth = Power emitted by Earth
(a) Power emitted by Earth = intensity of thermal radiation * total surface area
Intensity of thermal radiation for blackbody radiator = T4 = power/area, where the
Boltzmann constant = 5.67 x 10-8 W/m2K4 and T = effective radiating temperature (in
Kelvin).
Earth’s surface area = ______ = 4 R2Earth , where REarth = radius of the Earth
So the Power emitted by Earth = ______ = 4 R2Earth  T4.
[0.1]
(b) Power absorbed by Earth = Power received from Sun
= Intensity of solar radiation * area of Earth radiated
Area of Earth irradiated by the Sun = half the Earth at a time, more intense near the
equator and less intense near the poles = disk area = ______ = R2Earth
Intensity of solar radiation2 at Earth’s location = Solar “constant” S = 1370 W/m2
So the Power received from Sun = _______ =  R2Earth S.
[0.2]
Now put it all together:
1
Modified from Kump, Kasten, and Crane, The Earth System, 2d Ed., Pearson Prentice Hall 2004
This can be measured by satellites outside Earth’s atmosphere, or it can be calculated from first principles,
treating the Sun as a blackbody radiator and knowing the distance between the Sun and the Earth
2
Thermal equilibrium: Power emitted by Earth = Power absorbed by Earth
equation [1] = equation [2]
4 R2Earth  T4 =  R2Earth S
Simplify by eliminating common terms:
4  T4 = S
We can also look at this as a balance of radiation intensities:

 T4 = S/4
[0.3]
Intensity of radiation emitted by Earth = Intensity of solar radiation received
We can illustrate this energy balance schematically:
Finally, we solve for the effective radiating temperature T of the Earth, in this very
simple model:
1370 mW2
S
T 4
4
 279 K ~  6C
4
4  5.67 108 mW2 K 4
Reality check: This is very cold! And we have neglected a big effect – Earth’s
atmosphere reflects a significant amount of the Sun’s incoming radiation. What do you
predict we will find for the Earth’s equilibrium temperature if we take cloud reflection
into account? A warmer or colder planet?