Star colour — the Stefan-Boltzmann and Wien laws

What just happened?

A star’s colour reveals its temperature. Red stars are cool (~3000 K), yellow-white ones like the Sun are ~5800 K, and the hottest shine blue (20000 K and above). It is not a matter of “paint colour” — it is the physics of thermal radiation.

Wien’s law:

— the radiation peak falls at a wavelength inversely proportional to temperature. The hotter the star, the shorter the $\lambda_{max}$ and the bluer the colour. The Wien constant $b = 2{,}9\cdot10^{-3}$ m·K.

Invariant — constant product:

For every star, the product of the peak wavelength and the temperature gives the same constant. The panel shows that as $T$ rises, $\lambda_{max}$ falls exactly so that their product does not change.

Stefan-Boltzmann law:

Luminosity (radiated power) grows with the square of the radius and with the fourth power of the temperature. Doubling $R$ gives a 4× larger $L$, but doubling $T$ — a full 16×. That is why hot, large stars are so blindingly bright. $\sigma = 5{,}67\cdot10^{-8}$ W/(m²·K⁴).

Why you need this for the exam:

Colour → temperature (Wien), temperature and size → luminosity (Stefan-Boltzmann). The trick: from the colour read $T$, substitute into $\lambda_{max}=b/T$ or $L=4\pi R^2\sigma T^4$. Remember the fourth power on $T$ — that is the most common trap.

Remember: a hotter star = shorter $\lambda_{max}$ = bluer (λ_max·T = b), and luminosity jumps with T⁴ and R².