Estimating Stellar Radius and Temperature

We cannot measure the size of a distant star directly, but combining the laws from the previous lessons lets us calculate it.Step 1, temperature: find the peak wavelength of the star's spectrum and use Wien's law, T = 2.898 × 10⁻³ ÷ λ_max.Step 2, luminosity: measure the intensity I received at Earth and find the distance d (for example by parallax). Then L = 4πd²I.Step 3, radius: rearrange the Stefan-Boltzmann law, L = 4πr²σT⁴:r = √(L ÷ 4πσT⁴)Worked example: a star has λ_max = 830 nm and L = 4.5 × 10³¹ W. Find T and r.T = 2.898 × 10⁻³ ÷ 8.30 × 10⁻⁷ = 3.5 × 10³ Kr = √(4.5 × 10³¹ ÷ (4π × 5.67 × 10⁻⁸ × 3500⁴)) = 6.5 × 10¹¹ mThat is nearly 1000 times the Sun's radius: a cool, very luminous star must be a red supergiant.Comparing stars: since r ∝ √L / T², two stars of equal luminosity where one is twice as hot have radii in the ratio 1 : 4, the hotter star being smaller.