Stefan's law gives the total power emitted (the luminosity) by a black body:P = σAT4 where P is power in W, A is the surface area in m2, T is the surface temperature in K and σ is the Stefan constant, 5.67 × 10-8 W m-2 K-4. For a spherical star of radius r, A = 4πr2.Because of the T4, power depends very strongly on temperature: doubling T increases P by 24 = 16. At the same temperature, P ∝ r2, so a star 3 times the radius emits 9 times the power. This is how we know giant stars are huge: they are cool but very luminous.The power spreads out over a sphere, so the intensity received at distance d follows the inverse square law:I = P / 4πd2 Worked example: the Sun has r = 6.96 × 108 m and T = 5800 K.
A = 4π(6.96 × 108)2 = 6.09 × 1018 m2
P = 5.67 × 10-8 × 6.09 × 1018 × 58004 = 3.9 × 1026 W
At Earth, d = 1.5 × 1011 m:
I = 3.9 × 1026 / (4π × (1.5 × 1011)2) = 1.4 × 103 W m-2Combining Wien's law (for T) and Stefan's law (for P) lets astronomers work out a star's radius.
A = 4π(6.96 × 108)2 = 6.09 × 1018 m2
P = 5.67 × 10-8 × 6.09 × 1018 × 58004 = 3.9 × 1026 W
At Earth, d = 1.5 × 1011 m:
I = 3.9 × 1026 / (4π × (1.5 × 1011)2) = 1.4 × 103 W m-2Combining Wien's law (for T) and Stefan's law (for P) lets astronomers work out a star's radius.