Alexander Fufaev
My name is Alexander FufaeV and here I write about:

Wien's Displacement Law: Temperature of a Black Body

Wien's displacement law describes the relationship between the temperature of a black body (an idealized object that absorbs and emits all rays that hit it) and the wavelength of the maximum of its spectral radiation. The law was formulated by Wilhelm Wien in 1893 and can be expressed as a formula as follows:

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Here, \( \class{blue}{\lambda_{\mathrm{max}}} \) stands for the wavelength of the maximum of the spectral intensity distribution, \( T \) for the absolute temperature of the black body in Kelvin, and \( 2897.8 \,\cdot\, 10^{-6} \, \mathrm{m} \mathrm{K} \) is the Wien constant. The Wien constant comes from the derivation using Planck's radiation law.

The interpretation of this law states that at higher temperatures the black body emits radiation with shorter wavelengths, while at lower temperatures the emitted radiation has wavelengths in the longer range of the spectrum. This means that the color of the emitted radiation changes with increasing temperature.

Note that Wien's displacement law is only an approximation formula for short wavelengths!

How to Determine the Temperature of the Sun without Touching it?

To calculate the temperature \( T \) of the Sun from Earth, one must first find out what light the Sun emits. The Sun emits a polychromatic light, that is, infrared light, visible light, and even X-rays and gamma rays. You need to figure out the wavelengths \( \lambda \) of each of these radiations and with what intensity ("how bright"?) they are emitted. If you plot all wavelengths with their intensities in a diagram, you get a spectral intensity distribution of the sun. From this distribution, you have to read off the wavelength \( \class{blue}{\lambda_{\text{max}}} \) which has the highest intensity (maximum of the function).

Radiation maximum of the sun occurs at the blue light.

Then you can calculate the temperature of the sun with the Wien's displacement law. In the case of the sun, the maximum is within the blue light spectrum at the wavelength \( \lambda_{\text{max}} = 490 \, \mathrm{nm} = 490 \cdot 10^{-9}\, \mathrm{m} \). This results in a temperature of \( T = 5900 \, \mathrm{K}\) or \( 5630\, ^{\circ}\mathrm{C} \).