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

Stefan-Boltzmann Law

• A star’s luminosity depends on its:

  • surface temperature
  • 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:

$L=4πr^{2}σT^{4}$

• 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:

  • Use Wien’s displacement law to find the surface temperature of the star
  • Use the inverse square law of flux equation to find the luminosity of the star (if given the radiant flux and stellar distance)
  • Then, use the Stefan-Boltzmann law to calculate the radius of the star

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

$λ_{max}T=2.9\times10^{-3}mK$

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

$T=\frac{2.9\times10^{-3}}{λ_{max}}=\frac{2.9\times10^{-3}}{850\times10^{-9}}=3410K$

Step 3: Write down the Stefan-Boltzmann law

$L=4πr^{2}σT^{4}$

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

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

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

$\frac{r_{B}}{r_{s}}=\frac{6.83\times10^{11}}{6.95\times10^{8}}=983$

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