**Radiance: Integrating the Planck Equation**

Above we considered three
different spectral units: frequency, ν,
(Hz), wavelength, λ, (μm)
and wavenumber, σ, (cm^{-1}). We derived expressions for the spectral
radiances *L*_{ν}* _{
}*,

We will perform the integration
of *L*_{ν} over all frequencies, ν* *:

. (19)

(Here we used the result Σ *n*^{-4} =
ζ (4) = π*
*^{4}/90, where ζ is
the Reimann zeta function) The
total radiated power per unit area, called the *radiant emittance* or *radiant exitance*, *M*,
can be found by further integrating with respect to solid angle over the
hemisphere into which the surface radiates. A source whose radiance is independent of angle is called *Lambertian* (from Lambert’s cosine law of reflection). This is implicitly assumed for an ideal
blackbody, and is a good approximation for many real sources. For a Lambertian source, *M* is related to *L* by

so

. (20)

This is the Stefan-Boltzmann law, and the quantity in
parentheses is the Stefan-Boltzmann constant. A common mistake in deriving this result is to assume the
factor is 2π
rather than π,
because there are 2π* *steradians in the hemisphere, but this neglects the
cosθ
reduction from Lambert’s cosine law.

Similarly, the total photon radiance is found by integrating *
L _{σ}^{P}*
over all frequencies:

(21)

Recall ζ is the Riemann zeta function. ζ(3) ≈ 1.202056903159594
is also known as Apéry’s constant.
Integrating *L _{σ}^{P}*
over the hemisphere (again assuming a Lambertian source) gives

(22)

This is the Stefan Boltzmann law for photon radiant
emittance. Notice that the total
photon flux is proportional to *T ^{ }*

** **