Wien’s Law and Stefan’s Law

Stars radiate approximately as black bodies — perfect emitters whose spectrum depends only on temperature. Two laws describe that spectrum, and together they let a star's temperature and size be found from light alone.

Wien's displacement law — the wavelength at which emission peaks is inversely proportional to the absolute temperature:

λ_max T = 2.90 × 10⁻³ m K

What it means — hotter objects peak at shorter wavelengths. This is why a heated metal glows dull red, then orange, then white as its temperature rises: the peak shifts through the visible spectrum.

Example: the Sun's spectrum peaks at about 500 nm, so T = (2.90 × 10⁻³) ÷ (500 × 10⁻⁹) = 5800 K. Notice what this achieves — a surface temperature measured from 150 million kilometres away, using nothing but the colour of the light.

A useful consequence — a blue star is hotter than a red one. Colour is a direct read-out of surface temperature, which is why astronomers describe stars by colour at all.

Stefan's law — the total power radiated by a black body depends on its surface area and the fourth power of its temperature:

L = 4π r² σ T⁴

where σ is the Stefan constant, 5.67 × 10⁻⁸ W m⁻² K⁻⁴, and L is the luminosity, the total power output in watts.

The fourth power is dramatic — doubling a star's temperature multiplies its power output by sixteen, for the same size. A small increase in temperature produces a very large increase in luminosity, which is why the main sequence spans such an enormous range of brightness for a modest range of temperature.

Using the two together — Wien's law gives T from the peak wavelength, and Stefan's law then gives the radius r from the measured luminosity. This is how the sizes of stars are determined: not by imaging them, since almost all are far too distant to resolve, but by inference from their spectra.

Luminosity is not brightness — luminosity is the power a star actually emits; apparent brightness is what reaches us, reduced by the inverse square law over the distance. Two stars of identical luminosity look very different if one is ten times further away.