25.1.4 Stefan-Boltzmann Law & Stellar Radii

• 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 = 4πr²σT⁴

• 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)

• 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

Step 1: Write down Wien’s displacement law

λ_maxT = 2.9 × 10^-3 m K

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

Step 3: Write down the Stefan-Boltzmann law

L = 4πr²σT⁴

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

Step 5: Calculate the ratio r_B / r_s

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