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Stefon-Boltzmann Law

Origin

The law was empirically discovered by Josef Stefan (1879) and theoretically derived by Ludwig Boltzmann (1884) using classical thermodynamics.

Statement

The total energy radiated per unit surface area of a Black Body per unit time is proportional to the fourth power of its absolute temperature

$ j = \sigma T^4 $

$j$ : Radiated power per unit area (in W/m², also called radiant exitance)

$\sigma$ : Stefan-Boltzmann constant = $5.670 \times 10^{-8}$ W·m⁻²·K⁻⁴

$T$ : Absolute temperature in Kelvin (K).

Juxtaposition

There is no material (yet) that is a Perfect Black Body (perfectly emit/absorb radiated power), thus the introduction of Emissivity into the equation.

$\epsilon $
New Formula (imposing the Real World)

$ j = \epsilon \sigma T^4 $

Emissivity

$ \epsilon = 1 $
- A perfect black body (or absorber/emitter)
- Is able to equally emit and absorb the maximum possible thermal radiation at a given temperature
$ \epsilon = 0 $
- A perfect white body (or reflector)
- Is able to reflect all incoming radiation perfectly
$ 0 \leq \epsilon \leq 1 $
- Thus, emissivity of genuine, real world material should fall into this range.

Example Application

In SpaceX's Starship, the emissivity of the black tiles (that are a close perfect black body ~0.9 $\epsilon$) prevents heating the steel interior (shiny white body) as it radiates nearly 500 kilowatts of energy per square meter out and away into space
- essentially it is really close to shedding as much heat as it takes.