Look closely at the night sky and the stars are not all white: Betelgeuse is orange-red, Rigel blue-white, the Sun yellow-white. Those colors are thermometers. A hot, dense object (a star, a stove coil, you) gives off a smooth rainbow of light called thermal or blackbody radiation, spread across the electromagnetic spectrum, and its shape depends only on temperature.
Why it works
In a hot object, atoms and electrons jostle with energies set by the temperature, and the light they give off carries those energies. Hotter means more energetic jostling, so more energetic photons, which have shorter wavelengths. The curve of brightness against wavelength has one peak, and the peak moves to shorter wavelengths in exact proportion as the object heats up. This is Wien's law: $$\lambda_{\text{peak}} = \frac{2.90 \times 10^{6}\ \text{nm·K}}{T}.$$ The temperature must be in kelvins, counted from absolute zero (see temperature scales).
Key idea
Wien's law: $\lambda_{\text{peak}} = \dfrac{2.90 \times 10^{6}\ \text{nm·K}}{T}$. Hotter objects peak at shorter wavelengths, so blue-white stars are the hottest and red stars the coolest.
Worked examples
Worked example: The Sun
The Sun's surface is about $5800$ K. Where does its spectrum peak?
- $\lambda_{\text{peak}} = \frac{2.90 \times 10^{6}}{5800} = 500$ nm.
- That is blue-green, in the middle of the visible range. The Sun gives off plenty of every visible color, so the mix looks white from space.
Answer: $500$ nm.
Worked example: A red star
Betelgeuse's spectrum peaks near $806$ nm. How hot is its surface?
- $T = \frac{2.90 \times 10^{6}}{806} = 3600$ K.
- $806$ nm is just past red, in the infrared: Betelgeuse gives off more red than blue light, so it looks orange-red.
Answer: About $3600$ K.
Worked example: You glow too
Your skin is about $37^\circ$C. Where does your thermal glow peak?
- Convert: $T = 37 + 273 = 310$ K.
- $\lambda_{\text{peak}} = \frac{2.90 \times 10^{6}}{310} = 9.4 \times 10^{3}$ nm, about $9.4\ \mu$m: infrared, which is what thermal cameras see.
Answer: About $9.4\ \mu$m, in the infrared.
Common mistake
Everyday language calls red "hot" and blue "cold", but for glowing objects it is the other way around. A stove coil glows dull red at about $1000$ K; a welding arc glows blue-white at several thousand kelvins. Check with Wien's law: a shorter peak wavelength (blue) needs a larger $T$.