Stefan-Boltzmann Luminosity Calculator Formula
Understand the math behind the stefan-boltzmann luminosity calculator. Each variable explained with a worked example.
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) / 3.828e26Variables
| Variable | Description | Default |
|---|---|---|
radius | Stellar Radius(m) | 695700000 |
temperature | Surface Temperature(K) | 5778 |
How It Works
Stefan-Boltzmann Law
The total power radiated by a spherical blackbody:
L = 4π R² σ T⁴
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
- 01L = 4π R² σ T⁴
- 02R² = (6.957e8)² = 4.840e17
- 034π R² = 6.079e18 m²
- 04T⁴ = 5778⁴ = 1.115e15
- 05σ T⁴ = 5.670e-8 × 1.115e15 = 6.322e7
- 06L = 6.079e18 × 6.322e7 ≈ 3.843e26 W ≈ 1.00 L_sun
Frequently Asked Questions
Why is the T^4 dependence so powerful?
A star twice as hot is 16 times as luminous (for the same size). Temperature dominates brightness for stars of similar radius.
Can this determine a star's radius?
Yes. If you know L and T, solve for R = √(L / (4πσT⁴)). This is a standard method for finding stellar radii.
Is this exact for real stars?
Stars are not perfect blackbodies, but the formula gives an excellent approximation using the effective temperature.
Ready to run the numbers?
Open Stefan-Boltzmann Luminosity Calculator