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Suppose that we were to make a small hole in the wall of our enclosure,
and observe the emitted radiation. A small hole is the best approximation in
Physics to a blackbody, which is defined as an object which absorbs, and,
therefore, emits, radiation perfectly at all wavelengths.
What is the power radiated by the hole? Well, the power density inside the enclosure
can be written

(647) 
where is the mean
number of photons per unit volume whose frequencies lie
in the range to
. The radiation field inside the
enclosure is isotropic (we are assuming that the hole is sufficiently small that
it does not distort the field). It follows that the mean number of photons
per unit volume
whose frequencies lie in the specified range, and
whose directions of propagation make an angle in the range
to
with the normal to the hole, is

(648) 
where is proportional to the solid angle in the specified
range of directions,
and

(649) 
Photons travel at the velocity of light, so the power per unit area escaping from
the hole in the frequency range to
is

(650) 
where is the component of the photon velocity in the direction
of the hole.
This gives

(651) 
so

(652) 
is the power per unit area radiated by a blackbody in the frequency range
to
.
A blackbody is very much an idealization. The power
spectra of real radiating bodies
can deviate quite substantially from blackbody spectra. Nevertheless, we
can make some useful predictions using this model. The blackbody power spectrum
peaks when
. This means that the peak radiation
frequency scales linearly with the temperature of the body. In other words,
hot bodies tend to radiate at higher frequencies than cold bodies. This
result (in particular, the linear scaling) is known as Wien's displacement
law. It allows us to estimate the surface temperatures of stars from their
colours (surprisingly enough, stars are fairly good blackbodies). Table 9 shows
some stellar temperatures determined by this method (in fact,
the whole emission spectrum is fitted to a blackbody spectrum).
It can be seen that the
apparent colours (which correspond quite well to the colours of the peak radiation)
scan the whole visible spectrum, from red to blue, as the stellar surface temperatures
gradually rise.
Table 9:
Physical properties of some wellknown stars
Name 
Constellation 
Spectral Type 
Surf. Temp. ( K) 
Colour 
Antares 
Scorpio 
M 
3300 
Very Red 
Aldebaran 
Taurus 
K 
3800 
Reddish 
Sun 

G 
5770 
Yellow 
Procyon 
Canis Minor 
F 
6570 
Yellowish 
Sirius 
Canis Major 
A 
9250 
White 
Rigel 
Orion 
B 
11,200 
Bluish White 

Probably the most famous blackbody spectrum is cosmological in origin. Just after
the ``big bang'' the Universe was essentially a ``fireball,'' with the energy associated with
radiation completely dominating that associated with
matter. The early Universe was also pretty
well described by equilibrium statistical thermodynamics,
which means that the radiation had a blackbody spectrum. As the Universe expanded,
the radiation was gradually Doppler shifted to ever larger wavelengths (in other
words, the radiation did work against the expansion of the Universe, and, thereby,
lost energy), but its spectrum remained invariant. Nowadays, this primordial
radiation is detectable as a faint microwave background which pervades the
whole universe. The microwave background was discovered accidentally by Penzias
and Wilson in 1961. Until recently, it was difficult to measure the full
spectrum with any degree of
precision, because of strong microwave absorption and scattering by
the Earth's atmosphere. However, all of this changed when the COBE satellite
was launched in 1989. It took precisely nine minutes to measure the perfect
blackbody spectrum reproduced in Fig. 9.
This data can be fitted to a blackbody
curve of characteristic
temperature K. In a very real sense, this can be regarded
as the ``temperature of the Universe.''
Figure 9:
Cosmic background radiation spectrum measured by the
Far Infrared Absolute Spectrometer (FIRAS) aboard the Cosmic Background
Explorer satellite (COBE).

Next: The StefanBoltzmann law
Up: Quantum statistics
Previous: The Planck radiation law
Richard Fitzpatrick
20060202