Stefan-Boltzmann Law & Stellar Radii | CIE A Level Physics Revision Notes 2025

Stefan-Boltzmann Law

A star’s luminosity depends on two factors:
- Its surface temperature
- Its radius
The relationship between these is known as 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

It is equal to:

$L = 4 π r^{2} σ T^{4}$

Where:
- L = luminosity of the star (W)
- r = radius of the star (m)
- σ = the Stefan-Boltzmann constant
- T = surface temperature of the star (K)

Estimating the Radius of Stars

The radius of a star can be estimated by combining Wien’s displacement law and the Stefan–Boltzmann law
The procedure for this is as follows:
- Using Wien’s displacement law to find the surface temperature of the star
- Using the inverse square law of flux equation to find the luminosity of the star (if given the radiant flux and stellar distance)
- Then, using the Stefan-Boltzmann law, the stellar radius can be obtained

Summary of Stellar Equations (1), downloadable AS & A Level Physics revision notes

Worked example

Betelgeuse is our nearest red giant star.

It has a luminosity of 4.49 × 10³¹ W and emits radiation with a peak wavelength of 850 nm.

Calculate the ratio of the radius of Betelgeuse r_B to the radius of the Sun r_s.

Radius of the sun r_s = 6.95 × 10⁸ m.

Answer:

Step 1: Write down Wien’s displacement law

$λ_{m a x} T = 2.9 \times 10^{- 3} m K$

Step 2: Rearrange Wien’s displacement law to find the surface temperature of Betelgeuse

$T = \frac{2.9 \times 10^{- 3}}{λ_{m a x}} = \frac{2.9 \times 10^{- 3}}{850 \times 10^{- 9}} = 3410 K$

Step 3: Write down the Stefan-Boltzmann law

$L = 4 π r^{2} σ T^{4}$

Step 4: Rearrange for r and calculate the stellar radius of Betelgeuse

$r_{B} = \sqrt{\frac{L}{4 π σ T^{4}}} = \sqrt{\frac{4.49 \times 10^{31}}{4 π \times (5.67 \times 10^{- 8}) \times {(3410)}^{4}}} = 6.83 \times 10^{11} m$

Step 5: Calculate the ratio $\frac{r_{B}}{r_{s}}$

$\frac{r_{B}}{r_{s}} = \frac{6.83 \times 10^{11}}{6.95 \times 10^{8}} = 983$

Therefore, the radius of Betelgeuse is about 1000 times larger than the Sun’s radius

Stefan-Boltzmann Law & Stellar Radii (CIE A Level Physics)

Revision Note