Stefan-Boltzmann Luminosity Calculator Formula

Understand the math behind the stefan-boltzmann luminosity calculator. Each variable explained with a worked example.

Use the Stefan-Boltzmann Luminosity Calculator

Formulas Used

Luminosity

luminosity = 4 * pi * pow(radius, 2) * 5.670e-8 * pow(temperature, 4)

In Solar Luminosities

luminosity_solar = 4 * pi * pow(radius, 2) * 5.670e-8 * pow(temperature, 4) / 3.828e26

Variables

VariableDescriptionDefault
radiusStellar Radius(m)695700000
temperatureSurface Temperature(K)5778

How It Works

Stefan-Boltzmann Law

The total power radiated by a spherical blackbody:

L = 4π R² σ T⁴

  • σ = 5.670 × 10⁻⁸ W m⁻² K⁻⁴ (Stefan-Boltzmann constant)
  • R = stellar radius
  • T = surface (effective) temperature

    • Luminosity depends on radius squared and temperature to the fourth power.

    Worked Example

    The Sun: R = 6.957e8 m, T = 5778 K.

    radius = 695700000temperature = 5778
    1. 01L = 4π R² σ T⁴
    2. 02R² = (6.957e8)² = 4.840e17
    3. 034π R² = 6.079e18 m²
    4. 04T⁴ = 5778⁴ = 1.115e15
    5. 05σ T⁴ = 5.670e-8 × 1.115e15 = 6.322e7
    6. 06L = 6.079e18 × 6.322e7 ≈ 3.843e26 W ≈ 1.00 L_sun

    Ready to run the numbers?

    Open Stefan-Boltzmann Luminosity Calculator