Wien's displacement law
- Wien’s displacement law relates the observed wavelength of light from a star to its surface temperature, it states:
The black body radiation curve for different temperatures peaks at a wavelength which is inversely proportional to the temperature
- This relation can be written as:
- Where
- = the maximum wavelength emitted by the star at the peak intensity (m)
- = thermodynamic temperature at the surface of the star (K)
- A black body is an object which:
- absorbs all the radiation that falls on it, and is also a good emitter
- does not reflect or transmit any radiation
- A black-body is a theoretical object, however, stars are the best approximation there is
- The radiation emitted from a black-body has a characteristic spectrum that is determined by the temperature alone
The intensity-wavelength graph shows how thermodynamic temperature links to the peak wavelength for four different stars
- The constant of proportionality in Wien's law is given by
- This equation tells us the higher the temperature of a body:
- the shorter the wavelength at the peak intensity
- the greater the intensity of the radiation at each wavelength
- This means that hotter stars tend to be white or blue and cooler stars tend to be red or yellow
Table to compare surface temperature and star colour
Colour of star | Temperature / K |
blue | > 33 000 |
blue-white | 10 000 – 30 000 |
white | 7500 – 10 000 |
yellow-white | 6000 – 7500 |
yellow | 5000 – 6000 |
orange | 3500 – 5000 |
red | < 3500 |
Worked example
The spectrum of the star Rigel in the constellation of Orion peaks at a wavelength of 263 nm, while the spectrum of the star Betelgeuse peaks at a wavelength of 828 nm.
Which of these two stars is cooler, Betelgeuse or Rigel?
Answer:
Step 1: Write down Wien’s displacement law
Step 2: Rearrange for temperature T
Step 3: Calculate the surface temperature of each star
Rigel:
Betelgeuse:
Step 4: Write a concluding sentence
- Betelgeuse has a surface temperature of 3500 K, therefore, it is much cooler than Rigel
The Orion Constellation; cooler stars, such as Betelgeuse, appear red or yellow, while hotter stars, such as Rigel, appear white or blue
Examiner Tip
Note that the temperature used in Wien’s Law is in Kelvin (K). Remember to convert from oC if the temperature is given in degrees in the question before using the Wien’s Law equation.