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The total power radiated per unit area by a blackbody at all
frequencies is given by

(653) 
or

(654) 
where
. The above integral can easily be looked up in
standard mathematical tables. In fact,

(655) 
Thus, the total power radiated per unit area by a blackbody is

(656) 
This dependence of the radiated power is called the StefanBoltzmann law,
after Josef Stefan, who first obtained it
experimentally, and Ludwig Boltzmann, who first derived it theoretically.
The parameter

(657) 
is called the StefanBoltzmann constant.
We can use the StefanBoltzmann law to estimate the temperature of the Earth
from first principles. The Sun is a ball of glowing gas of radius
km and surface temperature
K. Its
luminosity is

(658) 
according to the StefanBoltzmann law. The Earth is a globe of radius
km located an average distance
km
from the Sun. The Earth intercepts an amount of energy

(659) 
per second from the Sun's radiative output: i.e., the power output of the Sun
reduced by the ratio of the solid angle subtended by the Earth at the Sun to
the total solid angle . The Earth absorbs this energy, and then
reradiates it at longer wavelengths. The luminosity of the Earth is

(660) 
according to the StefanBoltzmann law,
where is the average temperature of the Earth's surface.
Here, we are ignoring
any surface
temperature variations between polar and equatorial regions, or between day
and night. In steadystate, the luminosity of the Earth must balance the radiative
power input from the Sun,
so equating and we arrive at

(661) 
Remarkably, the ratio of the Earth's surface temperature to that of the Sun depends
only on the EarthSun distance and the solar radius. The above expression
yields
K or C (or F). This is slightly on the cold side, by a few
degrees, because of the
greenhouse action of the Earth's atmosphere,
which was neglected in our
calculation. Nevertheless, it is quite encouraging that such a crude calculation
comes so close to the correct answer.
Next: Conduction electrons in a
Up: Quantum statistics
Previous: Blackbody radiation
Richard Fitzpatrick
20060202