Stefan–Boltzmann Law
Luminosity of a star from radius and temperature.
Formula
Variables
Example
The Sun’s huge luminosity comes from its surface radiating at ~5800 K over a vast area.
How it works
A star’s luminosity is the total power it emits as thermal radiation from its surface. The Stefan–Boltzmann law comes from treating the star as an approximate blackbody, so each square metre of its surface radiates an amount proportional to T^4; multiplying by the surface area 4πR^2 gives the total output. The strong fourth-power dependence means temperature matters far more than size: a modest increase in surface temperature can greatly increase luminosity.
When to use it
Use this relation when you know, or can estimate, a star’s radius and effective surface temperature and want its total radiated power. It is also useful for comparing stars, estimating stellar sizes from luminosity and temperature, or checking whether an object behaves approximately like a blackbody.
Worked example, step by step
A star has radius R = 2.0 × 10^9 m and surface temperature T = 7000 K. Step 1: Write the formula: L = 4πR^2σT^4 Step 2: Substitute values with σ = 5.67 × 10^-8 W m^-2 K^-4 L = 4π(2.0 × 10^9)^2(5.67 × 10^-8)(7000)^4 Step 3: Compute the area term (2.0 × 10^9)^2 = 4.0 × 10^18 4πR^2 = 4π × 4.0 × 10^18 ≈ 5.03 × 10^19 m^2 Step 4: Compute the temperature term 7000^4 = 2.401 × 10^15 K^4 Step 5: Multiply everything L ≈ (5.03 × 10^19)(5.67 × 10^-8)(2.401 × 10^15) L ≈ 6.84 × 10^27 W Final answer: The star’s luminosity is about 6.8 × 10^27 watts.
Common mistakes to avoid
- Using the star’s diameter instead of its radius in 4πR^2
- Forgetting that temperature must be in kelvin, not Celsius
- Mixing up luminosity (total power in watts) with intensity or surface flux
Did You Know?
This law lets astronomers gauge a star's size from its temperature and total brightness.
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