About the Stellar Luminosity Calculator
This stellar luminosity calculator applies the Stefan–Boltzmann law to find the total power a star radiates from its radius and surface (effective) temperature. Enter the radius in solar radii and the temperature in kelvin, and it returns the luminosity in solar luminosities (L☉) and watts, the absolute bolometric magnitude, the wavelength where the star’s light peaks and its approximate spectral class.
Students use it for stellar astrophysics problems and Hertzsprung–Russell diagram work; amateur astronomers and writers use it to understand why a cool red supergiant can outshine a hot white star thousands of times over. Add a distance in parsecs to see the apparent bolometric magnitude, and the result also gives the Earth-equivalent orbital distance where a planet would receive the same energy as Earth.
Stars are treated as spherical black bodies, which is the definition of effective temperature, so the luminosity is exact for the inputs given. Bolometric magnitudes include all wavelengths and differ from visual (V-band) magnitudes, especially for very hot or very cool stars.
With the default inputs, the luminosity is 1 L☉. Change any value above to recalculate instantly.
How to use the stellar luminosity calculator
- 1Enter the star’s radius in solar radii.
- 2Enter its effective temperature in kelvin.
- 3Optionally enter its distance in parsecs.
- 4Read the luminosity in solar units and watts.
- 5Check the bolometric magnitude, peak wavelength and spectral class.
Formula and method
The Stefan–Boltzmann law says each square metre of a black body radiates σT⁴ watts, so a sphere of radius R emits L = 4πR²σT⁴. Luminosity rises with the square of radius but the fourth power of temperature: doubling temperature multiplies output by 16. The calculator uses σ = 5.670374419 × 10⁻⁸ W·m⁻²·K⁻⁴ and the IAU 2015 nominal solar values R☉ = 695,700 km, L☉ = 3.828 × 10²⁶ W and T☉ = 5,772 K.
Absolute bolometric magnitude uses the IAU zero point M_bol☉ = 4.74; apparent magnitude adds the distance modulus 5·log₁₀(d/10 pc). Peak wavelength comes from Wien’s law λ = b/T, and the spectral class from standard Harvard temperature ranges (approximate).
- L
- Luminosity (total radiated power)
- R
- Stellar radius
- T
- Effective surface temperature (K)
- σ
- Stefan–Boltzmann constant, 5.670374419 × 10⁻⁸ W·m⁻²·K⁻⁴
- M_bol
- Absolute bolometric magnitude
Worked examples
The Sun
4π × (6.957 × 10⁸ m)² × 5.6704 × 10⁻⁸ × 5,772⁴ ≈ 3.828 × 10²⁶ W = 1 L☉, by construction of the IAU nominal values. Its light peaks near 502 nm, in the blue-green.
Sirius A (1.711 R☉, 9,940 K, 2.637 pc)
1.711² = 2.93 and (9,940/5,772)⁴ = 8.80, so L ≈ 25.7 L☉. Its bolometric apparent magnitude is −1.68, brighter than its visual −1.46 because much of its light is ultraviolet (peak 292 nm).
A red supergiant (800 R☉, 3,600 K) at 168 pc
Despite a cool 3,600 K surface, the enormous 800 R☉ radius gives 640,000 × 0.151 ≈ 96,800 L☉ — similar to Betelgeuse. Its output peaks in the infrared at about 805 nm.
A small red dwarf (0.154 R☉, 3,042 K)
0.1542² × (3,042/5,772)⁴ ≈ 0.0018 L☉ — over 500 times dimmer than the Sun, so an Earth-like planet would have to orbit at only 0.043 AU.
Frequently asked questions
How do you calculate the luminosity of a star?+
Use the Stefan–Boltzmann law L = 4πR²σT⁴. In solar units this simplifies to L/L☉ = (R/R☉)² × (T/5,772 K)⁴, so a star twice the Sun’s radius at the same temperature is 4 times as luminous.
What is the luminosity of the Sun?+
The IAU nominal solar luminosity is 3.828 × 10²⁶ watts, defined in 2015 Resolution B3. It corresponds to an absolute bolometric magnitude of 4.74 and an effective temperature of 5,772 K for a radius of 695,700 km.
Why can a cool star be more luminous than a hot one?+
Luminosity depends on surface area as well as temperature. Red supergiants are hundreds of times wider than the Sun, so even with surfaces near 3,500 K their enormous area makes them tens of thousands of times more luminous.
What is the difference between luminosity and apparent brightness?+
Luminosity is the total power a star emits, independent of distance. Apparent brightness is what reaches us, which falls with the square of distance — so a nearby dim star can look brighter than a distant luminous one.
What is bolometric magnitude?+
It is a magnitude that counts radiation at all wavelengths, not just visible light. For very hot or very cool stars a large share of the energy is ultraviolet or infrared, so their bolometric and visual magnitudes can differ by several magnitudes.