Stefan-Boltzmann law
/ SHTEF-an BOLTS-man /
Stand near a campfire and step back: even a small increase in the fire's temperature makes it feel dramatically hotter on your face. The Stefan-Boltzmann law captures this, and tells you exactly how much: it says how much total power a warm surface radiates, summed over all wavelengths, depending on its temperature. The striking part is how steeply that power climbs with temperature.
The law states that the power radiated per square meter of surface is proportional to the fourth power of the temperature. The fourth power is the key: double the temperature and the surface radiates not twice but sixteen times as much energy (two to the fourth). Triple it and the output jumps eighty-one-fold. To get a whole object's total power (its luminosity), multiply this per-area output by the object's surface area, so luminosity depends on both temperature (to the fourth power) and size (the surface area). A small, very hot object can out-shine a large, cool one, and vice versa.
This law is the bridge from a star's surface to its total power output. Combined with a temperature from Wien's law and a measured size, it gives a star's luminosity — its intrinsic brightness — which is what we ultimately want to know about any star. It explains why a tiny, blazing-hot white dwarf can be faint (little surface area) while a cool but enormous red supergiant can be one of the brightest stars in the sky (vast surface area). It also governs the energy balance of planets and the cooling of stellar remnants.
Two stars have the same size, but one is twice as hot. The hotter one is not twice as luminous but sixteen times as luminous, because power scales with temperature to the fourth power. This is why even a modest temperature difference translates into a huge difference in brightness.
Radiated power climbs with the fourth power of temperature — a tiny temperature rise means a huge power rise.
The law gives power per unit area; total luminosity also depends on surface area, so size matters as much as temperature. The fourth-power rule applies to an ideal blackbody — real surfaces emit slightly less, captured by an 'emissivity' factor.