Stefan-Boltzmann Law

The luminosity L of a star is the total power it emits as electromagnetic radiation, in watts. For a black body it is given by the Stefan-Boltzmann law:L = σAT⁴where σ is the Stefan-Boltzmann constant, 5.67 × 10⁻⁸ W m⁻² K⁻⁴, A is the surface area in m² and T is the surface temperature in kelvin.A star is a sphere, so A = 4πr² andL = 4πr²σT⁴Because L depends on T⁴, temperature has a huge effect: doubling the temperature multiplies the luminosity by 2⁴ = 16. Luminosity is also proportional to r², so a star three times the radius at the same temperature is 9 times as luminous. This is why cool red giants can still be very luminous: they are enormous.Worked example: the Sun has radius 6.96 × 10⁸ m and surface temperature 5800 K. Find its luminosity.A = 4π(6.96 × 10⁸)² = 6.09 × 10¹⁸ m²L = 5.67 × 10⁻⁸ × 6.09 × 10¹⁸ × 5800⁴ = 3.9 × 10²⁶ WAlways convert temperatures to kelvin before using the law, and take care to raise T (not σAT) to the fourth power.