5.11.8 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

Step 1: List the known quantities

• • Luminosity of Betelgeuse, L = 4.49 × 10³¹ W
  • Peak wavelength of Betelgeuse, λ_max = 850 nm = 850 × 10⁻⁹ m
  • Radius of the Sun, r_s = 6.95 × 10⁸ m

Step 2: Write down Wien’s displacement law

λ_maxT = 2.9 × 10⁻³ m K

Step 3: Rearrange Wien’s displacement law to find the surface temperature of Betelguese

Step 4: Write down the Stefan-Boltzmann law

L = 4πr²σT⁴

Step 5: Rearrange for r and calculate the stellar radius of Betelguese

Step 6: Calculate the ratio r_B / r_s

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