Why does the power of radiation emitted by a black body increase to the fourth power of temperature?

Why specifically the fourth power? What is the physics involved? Why if you double the temperature does that thermal radiation increase 16 fold? Does anyone know? References would be great too. Thanks!Why does the power of radiation emitted by a black body increase to the fourth power of temperature?
A not very physical explanation would be that this is just the result of integrating Planck's law over all frequencies.

Another similarly unphysical explanation is by dimensional analysis:

The power P of radiation emitted by a black body can only depend on the Boltzmann constant k, Planck's constant h, the speed of light, the temperature T and the surface area A (if there is a universal law that doesn't depend on the properties of certain particles etc.). As P also has to be linear in A, the only combination of these quantities with the correct dimension contains T to the fourth power.



The most physical explanation i know is to use the relation between radiation pressure P and the energy density of the electromagnetic field u

P = u/3

together with some thermodynamic Maxwell relations (see eg. http://en.wikipedia.org/wiki/Stefan%E2%8鈥?/a> ).