Radiation

Every body at temperature $T > 0$ K emits electromagnetic radiation: this is thermal emission, and unlike conduction and convection it requires no material medium — it works even in a vacuum (this is how the Sun’s heat reaches us). For a perfect black body, the power emitted per unit area follows the Stefan-Boltzmann law:

Principle — Stefan-Boltzmann law

$\ev{\frac{P}{A} = \sigma\,T^4}$ where $\sigma = 5.67\cdot 10^{-8}\;\text{W}/(\text{m}^2\text{K}^4)$. For a real body an emissivity $\varepsilon \le 1$ is introduced, and the emitted power becomes $P = \varepsilon\,\sigma\,A\,T^4$.

The dependence on the fourth power of the absolute temperature is the dominant feature: here $T$ must be in kelvin. A body not only emits but also absorbs radiation from its surroundings; what matters is therefore the net balance.

Radiative balance

A body at temperature $T$ immersed in a surrounding at $T_\text{amb}$ exchanges a net power $P_\text{net} = \varepsilon\,\sigma\,A\,(T^4 - T_\text{amb}^4)$ For $T \approx T_\text{amb}$ the formula linearises to $P_\text{net} \approx 4\,\varepsilon\,\sigma\,A\,T^3\,(T - T_\text{amb})$.

Why hot stars are so luminous

The $T^4$ factor explains why hot stars are billions of times more luminous than cold ones: doubling the temperature multiplies the power radiated per unit surface by $2^4 = 16$. Small temperature differences produce enormous differences in luminosity.

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Topics: Termologia Concepts: Trasmissione del calore

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