Physics · Astrophysics · A2 Level

Wien's Displacement Law

A star's surface temperature is not something you can walk up to and measure with a thermometer — it has to be read off its light. Wien's displacement law is the tool that does exactly this: it turns the single number "where does the spectrum peak?" directly into a temperature in kelvin.

7 min read

Stating the law

Recall from the previous lesson that a black body's emission curve — intensity plotted against wavelength — has a single peak, and that this peak shifts to shorter wavelengths as temperature increases. Wien's displacement law makes this relationship exact:

$$\lambda_{max} T = \text{constant} = 2.9\times10^{-3}\ \text{m K}$$

Here $\lambda_{max}$ is the wavelength (in metres) at which the black body's emission curve reaches its peak intensity, and $T$ is the black body's surface temperature in kelvin. The product of these two quantities is always the same fixed number, roughly $2.9\times10^{-3}\ \text{m K}$, for any black body radiator, whatever its size or composition.

A graph of intensity against wavelength showing three black-body emission curves for different temperatures. The hottest curve has the tallest peak, shifted furthest to the left (shorter wavelength); the coolest curve has the lowest, broadest peak, shifted furthest to the right (longer wavelength). A dashed diagonal line traces Wien's law, connecting the peak of each curve to show that the peak wavelength decreases as temperature increases. — Black-body curves for three different temperatures. As temperature rises, the peak of the curve grows taller and shifts to shorter wavelength — the relationship described exactly by Wien's displacement law.

What "displacement" means here

The law is called a displacement law because it describes how the peak of the curve is displaced (shifted) along the wavelength axis as temperature changes. Since the product $\lambda_{max}T$ is constant, $\lambda_{max}$ and $T$ must be inversely proportional to one another — as one goes up, the other must come down to keep their product fixed.

A very hot star, with a surface temperature of tens of thousands of kelvin, has its emission peak shifted to short wavelengths — often into the blue or ultraviolet part of the spectrum, giving the star a blue-white appearance.
A cooler star, with a surface temperature of only a few thousand kelvin, has its emission peak at much longer wavelengths — typically in the red or near-infrared, giving the star a noticeably red or orange appearance.

This is why star colour is such a direct, visual clue to surface temperature: what the eye perceives as "redder" or "bluer" starlight is really a rough read-out of where that star's black-body curve happens to peak.

Using the law to find a temperature

In practice, Wien's law is used the other way round from how it's often first introduced: astronomers record a star's spectrum, identify the wavelength at which its intensity is greatest, and substitute that peak wavelength into the law to calculate the star's surface temperature — all without ever needing to visit the star or measure its temperature directly.

Worked exampleSurface temperature from a peak wavelength

A star's spectrum peaks at $\lambda_{max} = 4.83\times10^{-7}\ \text{m}$ (483 nm, in the blue-green). Find its surface temperature.

Rearrange Wien's law to make $T$ the subject.

$T = \dfrac{2.9\times10^{-3}}{\lambda_{max}}$

Divide the Wien constant by the peak wavelength.

$T = \dfrac{2.9\times10^{-3}}{4.83\times10^{-7}} \approx 6.0\times10^{3}\ \text{K}$

Evaluate — a surface temperature close to that of the Sun.

T ≈ 6000 K

Worked examplePeak wavelength from a known temperature

A hot young star has a surface temperature of $T = 2.9\times10^{4}\ \text{K}$. Find $\lambda_{max}$ and identify which part of the spectrum it lies in.

Rearrange Wien's law to make $\lambda_{max}$ the subject.

$\lambda_{max} = \dfrac{2.9\times10^{-3}}{T} = \dfrac{2.9\times10^{-3}}{2.9\times10^{4}} = 1.0\times10^{-7}\ \text{m}$

Divide the Wien constant by the temperature.

$1.0\times10^{-7}\ \text{m} = 100\ \text{nm}$, which lies in the ultraviolet.

Compare to the visible range (roughly 400–700 nm) — this peak lies well below it.

λ_max ≈ 100 nm, UV

The key idea

Wien's law lets you determine a star's surface temperature from a single feature of its spectrum — the peak wavelength — without needing to know anything else about the star, such as its distance, size or luminosity. That independence from distance is what makes it such a powerful and widely used tool in stellar classification.

Key takeaways

Wien's displacement law states $\lambda_{max}T = \text{constant} = 2.9\times10^{-3}\ \text{m K}$, linking a black body's peak emission wavelength to its temperature.
$\lambda_{max}$ and $T$ are inversely proportional: hotter black bodies peak at shorter wavelengths, cooler ones at longer wavelengths.
A hot, blue-white star has its emission peak in the blue or ultraviolet; a cool, red star has its peak in the red or infrared.
In practice, the law is used to find a star's surface temperature from the measured peak wavelength of its spectrum.
The relationship can be rearranged either way: to find $T$ from a known $\lambda_{max}$, or to predict $\lambda_{max}$ (and hence the star's apparent colour) from a known $T$.
Determining temperature this way needs no knowledge of the star's distance, size, or total luminosity — only the shape of its spectrum.

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Black Body Radiation

Stefan–Boltzmann Law