The Stefan-Boltzmann law and how stars shine
What the L = 4 pi r squared sigma T to the fourth formula means, why radius and temperature drive luminosity, and how to read solar units.
Where the formula comes from
The Stefan-Boltzmann law says that a blackbody radiates power per unit area equal to sigma times its temperature raised to the fourth power. A star is close enough to a blackbody that we can treat its whole surface this way. Multiplying that surface flux by the total surface area of a sphere, 4 pi r squared, gives the star's luminosity: L equals 4 pi r squared sigma T to the fourth. The constant sigma, about 5.67 times ten to the minus eight, ties the two together in SI units.
Radius and temperature pull in different ways
The equation shows two levers. Luminosity grows with the square of the radius, so a star twice as wide is four times as luminous at the same temperature. It grows with the fourth power of temperature, a far steeper relationship, so temperature is usually the deciding factor. That is why a small, blistering white dwarf and a huge, cool red giant can end up with wildly different luminosities despite one being much larger than the other.
Reading the answer in watts and solar units
The raw result is a power in watts, and for stars that number is astronomically large, often written in scientific notation. Dividing by the Sun's luminosity of 3.828 times ten to the twenty-six watts rescales it into solar luminosities, a far friendlier unit. A value of 25 solar luminosities means the star pours out twenty-five times the Sun's power. This calculator shows both forms so you can quote whichever suits your audience.
What the model leaves out
Treating a star as a uniform blackbody is an idealisation. Real photospheres darken toward the edge, a phenomenon called limb darkening, and their spectra carry absorption lines that a perfect blackbody would not. Stellar winds, rotation, and magnetic activity nudge the true output as well. For most comparisons the blackbody estimate is within a few percent, but it should be read as a clean approximation rather than an exact measurement.