Wien's Displacement Law

In Lesson 2 we met the Stefan–Boltzmann law, which tells us how much power a star radiates per unit area. That law ties together luminosity, radius and temperature. But it does not, on its own, give us a way to measure the temperature of a star directly. We still need some independent observation to fix T.

That independent measurement comes from the shape of a star's spectrum — specifically, from the wavelength at which its emission is brightest. The law that connects this peak wavelength to temperature is Wien's displacement law, and it is the second of the two great empirical laws of stellar radiation that you meet in OCR Module 5.5.

This lesson states the law, shows how to apply it, and explains how it works alongside Stefan–Boltzmann (and alongside spectroscopy in general) to turn a star's spectrum into a set of physical properties.

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Why Stars Have Different Colours

If you look at a clear night sky with a small telescope or even with the naked eye, you will notice that bright stars are not all the same colour. Sirius is brilliant white-blue. Rigel in Orion is a hotter blue. Betelgeuse is visibly orange-red. Antares is deep red. Capella is yellow. These colours are real, not optical artefacts, and they tell us directly about the stars' surface temperatures.

The reason stars differ in colour is that a hot black body's emission peaks at a different wavelength depending on its temperature. A cool star emits most strongly in the red or infrared and appears red-orange; a hot star emits most strongly in the blue or ultraviolet and appears blue-white. Wien's displacement law quantifies this shift.

graph LR
    S1[Cool star<br/>T ≈ 3000 K<br/>red] -. peak ~966 nm .-> I[Infrared]
    S2[Sun<br/>T ≈ 5800 K<br/>yellow-white] -. peak ~500 nm .-> V[Visible green]
    S3[Hot star<br/>T ≈ 15000 K<br/>blue-white] -. peak ~193 nm .-> U[Ultraviolet]

The Sun's spectrum actually peaks in the green, near 500 nm — which is where the human eye is most sensitive, unsurprisingly, since our eyes evolved under sunlight. A hotter star like Sirius (approx 9900 K) peaks in the near ultraviolet. A cooler star like Betelgeuse peaks in the near infrared and emits comparatively little visible light of any colour other than red-orange.

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Statement Of Wien's Displacement Law

Wien's displacement law states that the wavelength λ_max at which a black body's emission is brightest is inversely proportional to its absolute temperature:

λ_max × T = constant

The constant, known as Wien's displacement constant, is

λ_max × T = 2.90 × 10⁻³ m K

This value appears on the OCR data sheet. Note the units carefully: metre-kelvin. Not millimetres, not nanometres — when you substitute into the formula, temperature is in kelvin and wavelength is in metres. Converting to and from nanometres is one of the most common sources of error on exam questions.

Rearranging for temperature:

T = (2.90 × 10⁻³) / λ_max

Or for peak wavelength:

λ_max = (2.90 × 10⁻³) / T

Exam Tip: Always convert λ_max to metres before dividing. If you are given λ_max = 500 nm, you must write it as 500 × 10⁻⁹ m or 5.00 × 10⁻⁷ m in the calculation.

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Worked Example 1: The Sun's Surface Temperature

The Sun's emission peaks at approximately λ_max = 500 nm (visible green). Use Wien's law to calculate the Sun's surface temperature.

Step 1. Convert to metres:

λ_max = 500 × 10⁻⁹ m = 5.00 × 10⁻⁷ m

Step 2. Apply Wien's law:

T = (2.90 × 10⁻³) / (5.00 × 10⁻⁷)
  = 5800 K

This matches the accepted value of the Sun's effective surface temperature, T_☉ = 5800 K. The fact that this simple law gives the right answer for the Sun is one of the reasons astrophysicists trust it for other stars.

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Worked Example 2: Temperature Of A Red Star

Betelgeuse's spectrum peaks at about λ_max = 828 nm (near infrared). Calculate its surface temperature.

λ_max = 8.28 × 10⁻⁷ m
T = (2.90 × 10⁻³) / (8.28 × 10⁻⁷)
  = 3500 K

This matches the spectroscopically determined temperature of Betelgeuse to the precision of the data. Betelgeuse looks red because its peak emission is in the near infrared, well to the red side of the visible peak of a Sun-like star.

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Worked Example 3: Peak Wavelength Of A Hot Star

A B-type main-sequence star has a surface temperature of T = 15 000 K. Where does its emission peak?

λ_max = (2.90 × 10⁻³) / 15 000
      = 1.93 × 10⁻⁷ m
      = 193 nm

This is in the ultraviolet, well outside the visible range. The star still emits visible light, of course — the Planck curve is broad — but its peak output is in the UV. This is why the hottest stars appear blue-white rather than deep blue: we only see the blue tail of a spectrum whose maximum is in the UV, beyond the reach of our eyes.

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The "Displacement" In "Displacement Law"

The name "Wien's displacement law" refers to the fact that, as temperature rises, the peak of the black body spectrum shifts (is displaced) to shorter wavelengths. It does not mean the peak moves around on an individual star — a star at fixed temperature has a fixed λ_max. But if you plotted the spectra of many stars at different temperatures, the peak of the curve would slide to the left (shorter λ) as T increases.

This inverse relationship is the mathematical content of the law:

λ_max ∝ 1/T

The product λ_max × T is a constant for all black bodies.

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Stellar Colour Chart

The following table gives typical values for main-sequence stars of different spectral types. You will not be asked to memorise these in an exam, but they are useful for intuition and for pattern-matching on unseen questions.