The luminosity L of a star is the total radiant power it emits, in watts. The Stefan-Boltzmann law (Stefan's law) gives the luminosity of a black body:L = 4πr2σT4 where r is the radius of the star, T is its surface temperature in kelvin, and σ is the Stefan constant, 5.67 × 10−8 W m−2 K−4. The term 4πr2 is the star's surface area, so σT4 is the power emitted per square metre of surface.Luminosity therefore depends on two things: size (L ∝ r2) and temperature (L ∝ T4). The fourth power means temperature has a very strong effect: doubling T at constant radius increases L by 24 = 16 times.This explains the HR diagram. Red giants are cool but very luminous, so they must be very large. White dwarfs are hot but dim, so they must be very small.Worked example: the Sun has a radius of 7.0 × 108 m and a surface temperature of 5800 K. Calculate its luminosity.
L = 4π × (7.0 × 108)2 × 5.67 × 10−8 × 58004
L = 6.16 × 1018 × 6.42 × 107 = 4.0 × 1026 WRatio questions are often quicker. For two stars:L1/L2 = (r1/r2)2 × (T1/T2)4
L = 4π × (7.0 × 108)2 × 5.67 × 10−8 × 58004
L = 6.16 × 1018 × 6.42 × 107 = 4.0 × 1026 WRatio questions are often quicker. For two stars: