Stellar Luminosity Calculator (Stefan–Boltzmann)
Compute a star’s total power output from its radius and surface temperature using the Stefan–Boltzmann law.
How it works
L = 4π·R²·σ·T⁴ · σ = 5.670×10⁻⁸ W·m⁻²·K⁻⁴
The Stefan–Boltzmann law says a blackbody radiates power proportional to its surface area and the fourth power of its temperature. A star treated as a sphere therefore has luminosity L = 4πR²σT⁴. Because temperature is raised to the fourth power, a modest rise in surface temperature causes a huge jump in output — a star twice as hot as the Sun (at equal size) shines 16 times as brightly.
Worked example
For the Sun, R = 1 R☉ = 6.957×10⁸ m and T = 5,772 K: L = 4π × (6.957×10⁸)² × 5.670×10⁻⁸ × 5,772⁴ ≈ 3.83×10²⁶ W — precisely one solar luminosity, as it should be.
Sources: Stefan–Boltzmann law — Wikipedia
Frequently asked questions
Why is temperature raised to the fourth power?
It comes from integrating blackbody emission over all wavelengths (the Stefan–Boltzmann law). The strong T⁴ dependence means small temperature differences produce large luminosity differences between stars.
What is a solar radius?
One solar radius (R☉) is the Sun’s radius, about 6.957×10⁸ m or 696,000 km. Entering the radius in these units lets you compare stars to the Sun directly.
Does this assume the star is a perfect blackbody?
Yes — it uses the effective temperature, the temperature of a blackbody that would emit the same total power. Real stars are close enough that this gives excellent luminosity estimates.
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This calculator is for educational purposes. Double-check important results before acting on them.