The Stefan Boltzmann Law
The Stefan – Boltzmann Law links the total power radiated per unit area of a black body to the absolute temperature of the body:
whereis the Stefan – Boltzmann constant.
Sincewhereis the luminosity of the radiating body andis it's area, we have
For a star, which we can often approximate as a sphere of radiuswe can write
Example: The Sun has a radius ofand a surface temperature of around 5800K.
Often we can determine the luminosity of a star by other means. For example, the luminosity of a cepheid variable, a star with variable brightness, is directly related to the period of the variation, so by measuring the period we can determine the luminosity. We can measure the temperature using Wien's law for example, then use the Stefan – Boltzmann law to find the radius of the star.
The fourth power makes high temperature stars much more luminous than stars of the same size but lower temperatures. A star with of a certain temperature has 2^4 =16 times the luminosity of a similarly sized star with half the temperature.