Determination of Stellar Radii (DP IB Physics: HL): Revision Note

Betelguese 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: 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