B.1.14—Black-body radiation

Black-body emission

A black body is an ideal surface that emits electromagnetic radiation according to its absolute temperature. Its total emitted power, or luminosity, is modelled by

$$L=\sigma AT^4$$

Read the variables

$$A$$ is the emitting surface area, $$T$$ is absolute temperature in kelvin and $$\sigma$$ is the Stefan–Boltzmann constant. The equation gives total power emitted, not the brightness received by a particular observer.

Use proportional reasoning

At fixed area, doubling T multiplies L by $$2^4=16$$. At fixed temperature, doubling the emitting area doubles L. The fourth-power dependence makes temperature especially important.

Worked comparison from local Question Bank row 29005

Treat Mars at $200\,\mathrm{K}$ and Earth at $300\,\mathrm{K}$ as black bodies. For equal emitting area,

$$\frac{L_{Mars}}{L_{Earth}}=\left(\frac{200}{300}\right)^4=0.198\approx0.20$$

Mars emits about one fifth as much power per unit area in this ideal model.

Common trap

Do not use Celsius in the fourth-power term, and do not confuse luminosity with apparent brightness, which also depends on distance.