B.1 · IB Physics
Stefan–Boltzmann law and star sizes
Choose a star's surface temperature and radius to see its luminosity, its black-body spectrum and peak wavelength, its colour, and how it compares with the Sun.
P = σAT⁴ with A = 4πR²
σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴
Wien: λmax T = 2.90 × 10⁻³ m K
R☉ = 6.96 × 10⁸ m, T☉ = 5778 K
σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴
Wien: λmax T = 2.90 × 10⁻³ m K
R☉ = 6.96 × 10⁸ m, T☉ = 5778 K
Radius R
Surface area A
Luminosity P
P compared with Sun
(R/R☉)² × (T/T☉)⁴
Peak wavelength λmax
Colour
Try this
- Keep R = 1 R☉ and raise T from about 5800 K to 11600 K. Check that P rises by a factor of 16 and λmax halves.
- Select Betelgeuse: it is cooler than the Sun yet far more luminous. Use the ratio row to see which factor wins.
- Set T = 3000 K and find the radius that gives the same luminosity as the Sun (P compared with Sun ≈ 1).
Exam tip: when comparing two stars, write P₁/P₂ = (R₁/R₂)²(T₁/T₂)⁴ so that σ and 4π cancel, and remember that the radius is squared but the temperature is raised to the fourth power.