Luminosity Calculator — Stellar Power via Stefan-Boltzmann Law
Enter a star's radius (in solar radii) and effective surface temperature (in Kelvin) to calculate its total luminosity using the Stefan-Boltzmann law. Results are given in watts and in solar luminosities (L☉).
R☉
K
Total power radiated relative to the Sun (1 L☉ = 3.828 × 10²⁶ W)
- 1
(R / R☉)² — radius factor
1 ² = 1 - 2
(T / T☉)⁴ — temperature factor
(5,778 ÷ 5,778)⁴ = 1 - 3
L / L☉ = (R/R☉)² × (T/T☉)⁴
1 × 1 = 1.0042Equivalent to the full Stefan-Boltzmann formula since both sides scale identically with R and T.
How does this calculator work?
Luminosity L = 4πR²σT⁴, where σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴. Enter radius in solar radii and temperature in Kelvin. The Sun (R = R☉, T = 5 778 K) gives L = 3.828 × 10²⁶ W = 1 L☉. Temperature has four times the exponent of radius, so hotter stars are far more luminous than larger ones.
Formula
How this is calculated
A star radiates energy from its surface like a blackbody. The Stefan-Boltzmann law relates the total power radiated per unit area to the fourth power of the absolute temperature: P/A = σT⁴. Integrating over the full spherical surface area 4πR² gives the total luminosity L = 4πR²σT⁴. Because T appears to the fourth power, temperature has a much stronger effect than radius: doubling T multiplies luminosity by 16, while doubling R only multiplies it by 4.
The result is expressed in watts and also in solar luminosities (L☉ = 3.828 × 10²⁶ W) for easy comparison. The bolometric magnitude is computed from the standard relation M_bol = 4.74 − 2.5 log₁₀(L/L☉), where 4.74 is the IAU 2015 nominal solar bolometric absolute magnitude. This is the bolometric (total-power) magnitude, not the visual magnitude.
This model assumes the star is a perfect blackbody with a single uniform surface temperature. Real stars are not perfectly uniform (sunspots, limb darkening, rotation, magnetic activity), so this is an approximation. It also does not account for interstellar absorption, which reduces the apparent brightness for distant stars.
Frequently asked questions
Stellar luminosity is the total power radiated by a star across all wavelengths of the electromagnetic spectrum, measured in watts. The Sun's luminosity (L☉) is 3.828 × 10²⁶ W — roughly 4 × 10²⁶ joules of energy emitted every second.
In L = 4πR²σT⁴, R appears squared while T appears to the fourth power. Doubling the radius (keeping T fixed) increases L by a factor of 4. Doubling the temperature (keeping R fixed) increases L by 2⁴ = 16. Even modest temperature differences between stars produce enormous luminosity differences.
For main-sequence stars it is quite accurate (within a few percent) when the effective temperature and radius are measured reliably. For giant and supergiant stars the assumptions break down somewhat because their extended, low-density atmospheres deviate more from an ideal blackbody. The biggest uncertainty in practice is usually the measurement of R and T themselves.
Also known as
TG we-Calculate Editorial Team. (2026). Luminosity Calculator — Stellar Power via Stefan-Boltzmann Law [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/luminosity-calculator
TG we-Calculate Editorial Team. "Luminosity Calculator — Stellar Power via Stefan-Boltzmann Law." TG we-Calculate. 2026. https://we-calculate.com/calculator/luminosity-calculator.
TG we-Calculate Editorial Team, "Luminosity Calculator — Stellar Power via Stefan-Boltzmann Law," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/luminosity-calculator
@misc{wecalculate_luminosity_calculator, title = {Luminosity Calculator — Stellar Power via Stefan-Boltzmann Law}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/luminosity-calculator}}, year = {2026}, note = {TG we-Calculate} }
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