Star Luminosity Calculator

Stellar luminosity from the Stefan–Boltzmann law

A star behaves roughly like a blackbody, which is why its total power output (luminosity) obeys the Stefan–Boltzmann law: L = 4π·R²·σ·T⁴. Here R is the radius, T the effective surface temperature, and σ ≈ 5.67·10⁻⁸ W·m⁻²·K⁻⁴. The handy unit to work in is the Sun's luminosity, L☉ = 3.828·10²⁶ W. Example: feed in the solar values R = 1 R☉ and T = 5778 K and you get L = 1 L☉ back. For comparison, Sirius A sits near 25 L☉ and Vega near 40 L☉. Betelgeuse reaches about 100,000 L☉, a red supergiant where the sheer size of the radius wins out over its cooler temperature.

Applications

Luminosity sits at the heart of astrophysics and of the Hertzsprung–Russell diagram, which plots it against temperature to sort stars into evolutionary stages such as the main sequence, giants and white dwarfs. It matters in exoplanet searches too: the host star's luminosity marks out the habitable zone and you need it to make sense of transit-method signals from missions like Kepler and TESS. Pair it with an apparent magnitude and the distance-modulus equation turns it into a distance estimate.

FAQ

Why temperature to the fourth power? Integrate the blackbody emissive power over every wavelength and it comes out proportional to T⁴, which falls straight out of Planck's law.

Is the star really a blackbody? Close enough. A real spectrum carries absorption lines from the photosphere, yet the integrated luminosity still tracks Stefan–Boltzmann well.

Why is Betelgeuse so bright despite being cool (~3500 K)? Its radius runs to roughly 1000 R☉, and that R² factor more than makes up for the shortfall in T⁴.

How do I measure R and T? For R you turn to interferometry or limb-darkening models; for T you read it off spectral lines and photometric colors.