Stellar radii

Physics A2 · Astronomy and cosmology. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A star gives out radiation over a range of wavelengths, with one wavelength where the intensity is greatest. Wien's displacement law says this peak wavelength is inversely proportional to the surface temperature. A hotter star has a shorter peak wavelength, so hot stars look blue and cooler stars look red.

The Stefan-Boltzmann law links the luminosity of a star to its surface area and temperature. Luminosity depends on the square of the radius and on the fourth power of the temperature, so temperature has a very strong effect.

Together they give the radius of a star: find the temperature from the peak wavelength, then use the known luminosity to find the radius.

Rules to remember

Common mistake

Students forget that temperature is raised to the fourth power, or leave the wavelength in nm. Change nm to m (× 10−9) before using λmaxT = 2.9 × 10−3 m K.

Worked example

A star has luminosity 5.0 × 1026 W and its spectrum peaks at 580 nm. Using λmaxT = 2.9 × 10−3 m K and σ = 5.67 × 10−8 W m−2 K−4, estimate its radius.

  1. T = (2.9 × 10−3) ÷ (5.8 × 10−7) = 5000 K, so T4 = 6.25 × 1014 K4.
  2. 4πσT4 = 4π × 5.67 × 10−8 × 6.25 × 1014 = 4.45 × 108 W m−2.
  3. r2 = L ÷ (4πσT4) = (5.0 × 1026) ÷ (4.45 × 108) = 1.12 × 1018 m2.
  4. r = √(1.12 × 1018) = 1.06 × 109 m.

Answer: About 1.1 × 109 m.

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