Stefan–Boltzmann Law

Wien's law gives you a star's temperature. But two stars can share exactly the same surface temperature and still pump out wildly different amounts of power — because one is simply far bigger than the other. The missing piece is a law connecting temperature, size and total power output all in one equation.

Stating Stefan's law

The Stefan–Boltzmann law (often just called Stefan's law) gives the total power radiated by a black body in terms of its surface area and temperature:

where $P$ is the total power radiated (in watts), $A$ is the surface area of the radiating body (in $\text{m}^2$), $T$ is its surface temperature (in kelvin), and $\sigma$ is the Stefan–Boltzmann constant, a fixed number with the same value for every black body. Notice the very strong dependence on temperature: because power depends on $T^4$, even a modest increase in temperature produces a dramatic increase in radiated power — doubling a star's surface temperature increases its power output sixteen-fold, assuming its size stays the same.

Applying the law to a star

Stars are, to a very good approximation, spheres, so the general law above is normally applied to stars using the surface area of a sphere of radius $r$:

Substituting this into Stefan's law gives the form of the equation used throughout stellar astrophysics:

Here $P$ is usually identified with the star's luminosity $L$ from the earlier lesson on the inverse square law, since luminosity is exactly the total power radiated by the star. This version of the law — with the $4\pi r^2$ spelled out explicitly — is the one you should reach for whenever a question gives you (or asks you to find) a star's radius.

Linking luminosity, temperature and radius

Stefan's law, combined with Wien's law and the inverse square law from the previous lessons, gives astronomers a genuinely powerful toolkit. Between the three relationships, any one of a star's luminosity, surface temperature, or radius can be found if the other two are known:

• Wien's law gives temperature from the peak wavelength of the spectrum.
• The inverse square law gives luminosity from measured flux and known distance.
• Stefan's law then links luminosity, temperature and radius together, so knowing any two lets you find the third.

This three-way relationship between $L$, $T$ and $r$ is central to how stars are compared and classified. For example, two stars with the same luminosity but different surface temperatures cannot be the same size: the cooler star must have a much larger radius to make up, through sheer surface area, for radiating less power per square metre than the hotter one. This exact idea — using $L$ and $T$ together to infer relative sizes — is exactly what makes sense of the giant and dwarf regions of the Hertzsprung–Russell diagram, covered in the next lesson.