5.11.7 Stefan's Law

Stefan's Law

An objects luminosity depends on two factors:
- Its surface temperature
- Its surface area
The relationship between these is known as Stefan's Law or the Stefan-Boltzmann Law, which states:

The total energy emitted by a black body per unit area per second is proportional to the fourth power of the absolute temperature of the body

So Stefan's Law shows that the luminosity of a star is directly proportional:
- To its radius L ∝ r²
- To its surface area L ∝ 4πr²
- To its surface absolute temperature L ∝ T⁴

L = 4πr²σT⁴

Where:
- L = luminosity of the star (W)
- r = radius of the star
- σ = the Stefan-Boltzmann constant
- T = surface temperature of the star (K)
The surface area of a star (or other spherical object) can be calculated using: A = 4πr²
- Where r = radius of the star

The surface temperature of Proxima Centuri, the nearest star to Earth, is 3000 K and its luminosity is 6.506 × 10²³ W.

Calculate the radius of Proxima Centuri in kilometres and show your working clearly.

Step 1: List the known quantities:

- Surface temperature, T = 3000 K
- Luminosity, L = 6.506 × 10²³ W
- Stefan's constant, σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴

Step 2: Write down Stefan's Law

L = 4πr²σT⁴

Step 3: Rearrange the equation for r

$r = \sqrt{\frac{L}{4 {πσT}^{4}}}$

Step 4: Substitute into the equation

$r = \sqrt{\frac{6.506 \times 10^{23}}{4 π \times (5.67 \times 10^{- 8}) \times 3000^{4}}}$ = 106 173 971 m

Step 5: Write the final answer to the correct amount of significant figures

Remember to convert temperatures into Kelvin.

Check the values you obtain in your calculations. Do they make sense? Do they fit in with the magnitudes of other stars/objects that you know about?