The radius of a distant star cannot be measured directly, but it can be estimated by combining Wien's displacement law and Stefan's law.Step 1: measure the peak wavelength λmax of the star's spectrum and use Wien's law to find its surface temperature:T = 2.9 × 10−3 / λmax Step 2: find the star's luminosity L (for example, from its measured intensity at the Earth and its distance).Step 3: rearrange Stefan's law to find the radius:r = √(L / 4πσT4) Worked example: a star has a luminosity of 4.0 × 1027 W and its spectrum peaks at 290 nm. Estimate its radius.
T = 2.9 × 10−3 / 2.9 × 10−7 = 1.0 × 104 K
4πσT4 = 4π × 5.67 × 10−8 × (1.0 × 104)4 = 7.13 × 109 W m−2
r2 = 4.0 × 1027 / 7.13 × 109 = 5.61 × 1017 m2
r = 7.5 × 108 m (about the same as the Sun's radius)Ratio method: comparing a star with the Sun, r/r☉ = √(L/L☉) × (T☉/T)2. This avoids needing σ.The result is only an estimate, because real stars are not perfect black bodies and the peak wavelength is hard to measure precisely.
T = 2.9 × 10−3 / 2.9 × 10−7 = 1.0 × 104 K
4πσT4 = 4π × 5.67 × 10−8 × (1.0 × 104)4 = 7.13 × 109 W m−2
r2 = 4.0 × 1027 / 7.13 × 109 = 5.61 × 1017 m2
r = 7.5 × 108 m (about the same as the Sun's radius)Ratio method: comparing a star with the Sun, r/r☉ = √(L/L☉) × (T☉/T)2. This avoids needing σ.The result is only an estimate, because real stars are not perfect black bodies and the peak wavelength is hard to measure precisely.