Supplement 2.3: The Stefan-Boltzmann Law (1/2)

Derivation from Planck's radiation law

The energy density U

The spectral distribution of the energy density of the radiation field for black bodies as a function of frequency is:

u_{f} = \frac{8 π f^{2}}{c^{3}} \frac{h f}{exp {h f / k T} - 1}

It is integrated over all frequencies to obtain the spectrally integrated energy density $U$ :

U = \int_{f = 0}^{\infty} u_{f} d f

what means:

U = \frac{8 π h}{c^{3}} \int_{f = 0}^{\infty} \frac{f^{3}}{exp {h f / k T} - 1} d f

The integration becomes more clear by introducing a new variable:

h f / k T = x

With the replacements

f = \frac{k T}{h} x

and

d f = \frac{k T}{h} d x

one has:

U = \frac{8 π k^{4}}{c^{3} h^{3}} T^{4} \int_{x = 0}^{\infty} \frac{x^{3}}{e^{x} - 1} d x

The integral cannot be solved elementarily. It becomes:

\int_{x = 0}^{\infty} \frac{x^{3}}{e^{x} - 1} d x = \frac{π^{4}}{15}

...for mathematicians ↓

In function theory it is shown, that this results from the Riemann zeta function $ς (ν)$ and the gamma function $Γ (ν)$ with the argument $ν = 4$ . See for example I.S. Gradshteyn & I.M. Ryzhik, Table of Integrals, Series and Products (Academic Press), in section 3.4.1.1, equation 1:

\int_{x = 0}^{\infty} \frac{x^{ν - 1}}{e^{x} - 1} d x = Γ (ν) ς (ν) with Re ν > 1

with the Riemann zeta function

ς (ν) = \sum_{n = 1}^{\infty} n^{- ν} with Re ν > 1

and the gamma function

Γ (ν) = (ν - 1)! with ν \in ℕ

In particular one has:

ς (4) = 1 + \frac{1}{2^{4}} + \frac{1}{3^{4}} + ... = \frac{π^{4}}{90}

and

Γ (4) = 3 \cdot 2 \cdot 1 = 6

Hereby it follows:

U = \frac{8 π^{5} k^{4}}{15 c^{3} h^{3}} T^{4}

The specific emission M

The relation of energy and specific emission of an isotropic radiation field (what means, no direction of propagation is preferred) reads:

U = \frac{4}{c} M

It follows for the specific emission of a black body:

M = \frac{2 π^{5} k^{4}}{15 c^{2} h^{3}} T^{4} = σ T^{4}

With $h = 6.6 \cdot 10^{- 34} Js$ , $k = 1.38 \cdot 10^{- 23} J/K$ and $c = 3 \cdot 10^{10} m/s$ one obtaines the Stefan-Boltzmann constant:

σ = 5.670 \cdot 10^{- 8} \frac{W}{m^{2} K^{4}}

The radiance L

The relation of energy density and radiance of an isotropic radiation field reads:

U = \frac{4 π}{c} L

Hereby becomes:

L = \frac{σ}{π} T^{4}

Understanding Spectra from the Earth

Supplement 2.3: The Stefan-Boltzmann Law (1/2)

Derivation from Planck's radiation law

The energy density U

The specific emission M

The radiance L