stefan boltzmann constant — from formula anatomyThe Stefan–Boltzmann constant (σ) is the proportionality factor in the Stefan–Boltzmann law, which states that the total power radiated per unit area from a blackbody is proportional to the fourth power of its absolute temperature: P/A = σT⁴. Its value is approximately 5.670374419 × 10⁻⁸ W·m⁻²·K⁻⁴. This constant bridges macroscopic thermal radiation with fundamental physics, as it can be expressed in terms of the Boltzmann constant, Planck's constant, and the speed of light.
The law was empirically formulated by Josef Stefan in 1879 and theoretically derived by Ludwig Boltzmann in 1884 using thermodynamics. The precise value of the constant was later refined through quantum theory by Max Planck in 1900.
The constant emerges from integrating Planck's law of blackbody radiation over all frequencies and solid angles. The derivation involves quantum statistics: photons are bosons, and their energy distribution follows the Bose–Einstein distribution. The integral yields a universal number that depends only on fundamental constants, making σ a fixed property of nature rather than a material-dependent parameter.
It is used in astrophysics to estimate stellar luminosities and temperatures, in engineering for heat transfer calculations, and in climate science to model Earth's energy balance.
The Stefan–Boltzmann constant is exactly related to the radiation constant 'a' by a = 4σ/c, where c is the speed of light, and its value is known to better than one part per million.
| L | variable | Luminosity (total power radiated) — luminosity watt |
| R | variable | Radius of the radiating body — radius metre |
| T | variable | Effective temperature of the surface — temperature kelvin |
| \sigma | constant | Proportionality constant relating blackbody radiated power to temperature — stefan boltzmann constant = 5.670374419e-8 watt per metre squared per kelvin to the fourth |
| T_eff | variable | Effective temperature — temperature kelvin |
| F_lambda | variable | Spectral flux density per unit wavelength — spectral flux density watt per metre cubed |
| lambda | variable | Wavelength — wavelength metre |
| sigma | constant | Relates total blackbody flux to temperature — stefan boltzmann constant = 5.670374419e-8 watt per metre squared per kelvin to the fourth |
| ∫ | operator | Integration over all wavelengths — integral |
| T(r) | variable | disk temperature at radius r — temperature kelvin |
| r | variable | radial distance from the central object — radius metre |
| M | variable | mass of the central object — mass kilogram |
| \dot{M} | variable | mass accretion rate — mass flow rate kilogram per second |
| G | constant | Constant of universal gravitation. — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| \sigma | constant | Constant relating blackbody radiated power to temperature. — stefan boltzmann constant = 5.670374419e-8 watt per square metre per kelvin to the fourth |
| F_{\text{total}} | variable | total radiative flux emitted per unit area — radiative flux watt per square metre |
| T | variable | absolute temperature of the radiating body — temperature kelvin |
| \sigma | constant | Stefan-Boltzmann constant relating temperature to radiated power — stefan boltzmann constant = 5.670374419e-8 watt per square metre per kelvin to the fourth |
