Black body radiation and stellar luminosity

​​In a nutshell

A black body is an idealised (hypothetical) object which absorbs all electromagnetic radiation it is exposed to and radiates (at thermal equilibrium) a distinct wavelength distribution. By using Wien's and Stefan's law, a star's peak wavelength, luminosity, radius and temperature can be calculated. 

Equations

Constants

Variable definitions 

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Black body radiation

A black body is an idealised (hypothetical) object that absorbs all electromagnetic radiation that it is exposed to. When a black body is at a constant temperature (thermal equilibrium), it will emit black body radiation.  

Specific temperatures give a characteristic wavelength distribution. This means that there is a different peak wavelength, where intensity of electromagnetic radiation is at its greatest, for different temperatures.

Stars can be modelled as approximate black bodies. Therefore, the wavelength distribution can give vital insights to stars, such as their temperature. Below is a graph to show wavelength distributions of black bodies at different absolute temperatures. 

1. Intensity 2. Wavelength ($$nm$$) 3. Black body at $$6000 \ K$$​​ 4. Black body at $$5000 \ K$$​​ 5. Black body at $$4000 \ K$$​​ 6. Black body at $$3000 \ K$$​​

Wien's displacement law 

Wien's displacement law states that a black body's peak wavelength is inversely proportional to its absolute temperature.

​$$\lambda_{max} \propto \frac{1}{T}$$​​

This means that: 

​$$\lambda _ {max} T = constant$$​​

This constant is Wien's constant and has a value of $$2.90 \times 10^{-3} \, m \, K$$. 

Example

The Sun has a temperature of $$5800 \, K$$​. Calculate the peak wavelength of the Sun.

Firstly, write down the known values:

$$T = 5800 \, K \newline Wien \space constant = 2.90 \times 10^{-3} \, m \, K$$ ​

Next, write down the equation needed and rearrange for $$\lambda _ {max}$$:

$$\lambda _ {max} T = constant$$

$$\lambda_{max} = \frac{constant}{T}$$​​

Substitute values into rearranged equation:

$$\lambda_{max} = \frac{2.90 \times 10^{-3} }{5800}$$​​

Calculate final answer and include units:

$$\lambda _ {max} = 5.0 \times 10 ^{-7}\,m \newline\lambda _ {max} =500 \, nm$$​​

The Sun has a peak wavelength of $$\underline{500 \, nm}$$.

Wien's law can also be used to calculate the ratio of two stars peak wavelengths and absolute temperatures.

Example

Betelgeuse is a red supergiant with a peak wavelength of $$850 \, nm$$ and has an absolute temperature of $$3400 \, K$$. Rigel, another star, has an absolute temperature of $$12 \, 500 \, K$$. Calculate the peak wavelength for Rigel.

Firstly, write down known values and convert units where needed:

$$\lambda_{max \, betelgeuse} = 850 \, nm = 8.5 \times 10^{-7} \, m \newline T_{betelgeuse} = 3500 \, K \newline T_{rigel} = 12 \, 500 \, K$$

Equate Betelgeuse and Rigel using Wien's law (as Wien's constant is the same for both equations):

​$$\lambda_{max \, betelgeuse} \, T_{betelgeuse} = \lambda_{max \, rigel} \, T_{rigel}$$​

Rearrange for $$\lambda_{max \, rigel}$$:

​$$\lambda_{max \, rigel} = \frac{\lambda_{max \, betelgeuse} \, T_{betelgeuse}}{T_{rigel}}$$​

​$$\lambda_{max \, rigel} =\frac{8.5 \times 10 ^{-7} \times 3400 }{12 \, 500}$$

Substitute values into rearranged equation:

$$\lambda_{max \, rigel} =\frac{8.5 \times 10 ^{-7} \times 3400 }{12 \, 500}$$

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Calculate final answer and include units:

$$\lambda_{max \, rigel} = 2.3 \times 10^{-7}\, m \newline\lambda_{max \, rigel} = 230 \, nm$$​​

The peak wavelength for the star Rigel is $$\underline{230 \, nm}$$.

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Stefan's law

Stefan's law states that the luminosity of a star is directly proportional to:

• ​$$(radius \space of \space a \space star)^2 = r^2$$
• ​$$surface \space area \space of \space a \space star = 4\pi r^2$$
  
• $$(a \space stars \space absolute \space surface \space temperature)^4 = T^4$$

Therefore, the equation for Stefan's law is:

Where the constant is Stephan's constant $$\sigma = 5.67 × 10^{–8} \, W\,m^{–2}\,K^{–4}$$.

Example

​​Sirius B has an absolute temperature of $$25 \, 000 \, K$$ and a radius of $$5 \, 900 \, km$$. Calculate the luminosity of Sirius B.

​Firstly, write down known values and convert units where needed:

​$$T = 25 \, 000 \, K \newline r = 5900 \, km = 5.9 \times 10^6 \, m \newline\sigma = 5.67 × 10^{–8} \, W\,m^{–2}\,K^{–4}$$​​

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Next, write down the equation needed:

$$L = 4 \pi r^2\sigma T^4$$

Substitute values into equation:

$$L = 4 \pi \times (5.9 \times 10^6)^2 \times 5.67 \times 10 ^{-8} \times (25 \, 000)^4$$

Calculate final answer and include units:

$$L = 9.8 \times 10^{24} \, W$$

Sirius B has a luminosity of $$\underline{9.8 \times 10^{24} \, W}$$.