Stefan-Boltzmann Sun Power Calculator

Enter star radius and temperature to calculate emitted power using the Stefan-Boltzmann law.

Worked example — Stefan-Boltzmann Sun Power Calculator

Surface area (4πR²)	4πR²≈6.082×10¹⁸ m²
Relative Radius Squared	(1)²≈1
Emissivity	1

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Betelgeuse is cooler than the Sun and 79,000× brighter 🖖

Per square metre temperature dominates — T⁴ against R² — yet the presets here are won by radius, because real stars do not vary the two equally. Compare the tool's own two factor rows: across these four presets radius spans 3820× (0.2 to 764 R☉) while temperature spans just 3.1× (3200 to 9940 K). Betelgeuse runs cooler than the Sun, so its (T/T☉)⁴ row reads 0.135 — an outright penalty — and it still lands at 78,914 L☉ because its (R/R☉)² row reads 5.84 × 10⁵. Sirius, nearly twice as hot as the Sun, manages only 25.7. A steeper exponent loses when the other variable has four orders of magnitude to play with.

Per square metre, temperature is the stronger lever 🖖

This tool has only two inputs — how big a star is and how hot its surface is — yet they pull with very different strength. Radius enters as R², but temperature enters as T⁴, so per square metre heat is the dominant lever. Make a star twice as hot without changing its size and every square metre radiates 2⁴ = 16 times more power. That fourth power governs the flux. Whether it governs the whole star is a different question, and the answer depends on how much radius the star has to offer.

Stefan measured the Sun's heat in 1879 🖖

Josef Stefan didn't just state that flux grows as T⁴ — in 1879 he ran the law backwards. Feeding measured sunlight through σT⁴, he estimated the Sun's surface at roughly 5430 °C (about 5700 K), the first credible figure when rival guesses ranged from 1500° to millions of degrees. This calculator performs the same inversion: the flux you compute is exactly the quantity Stefan measured to take the Sun's temperature.

Problem solved in full

The Sun's total power from its size and surface temperature 5 steps
Work out the Sun's total power from nothing but its size and its surface temperature, then use that single number to predict how cold Earth ought to be. This is the tool's Sun state: R = 1 R_☉ = 6.957 × 10⁸ m, T = 5772 K, ε = 1.
1. Stefan-Boltzmann gives power per square metre, so the count of square metres has to come first. A star radiates from its photosphere, and a sphere's area needs only the radius.
  
  $A = 4 π R^{2} = 4 π (6.957 \times 1 0^{8})^{2} = 6.082 \times 1 0^{18} m^{2}$
2. Now the flux from one square metre. The fourth power is what makes this law so unforgiving: a star only 20% hotter than the Sun radiates 1.20⁴ = 2.07 times as hard from every patch of its surface.
  
  $F = ε σ T^{4} = (5.6704 \times 1 0^{- 8}) (5772)^{4} = (5.6704 \times 1 0^{- 8}) (1.110 \times 1 0^{15}) = 6.294 \times 1 0^{7} W/ m^{2}$
3. Luminosity is flux times area and nothing else, because every photon that leaves the star crosses that surface exactly once. The result is the definition of one solar luminosity, which is why the tool's comparison row reads 1.
  
  $L = A F = (6.082 \times 1 0^{18}) (6.294 \times 1 0^{7}) = 3.828 \times 1 0^{26} W$
4. Leave the star. The same power now spreads over a sphere whose radius is Earth's distance, so only the area in the denominator changes. Satellites above the atmosphere measure 1361 W/m², and that agreement is the check on all three steps above.
  
  $S = \frac{L}{4 π d ^{2}} = \frac{3.828 \times 1 0 ^{26}}{4 π ( 1.496 \times 1 0 ^{11} ) ^{2}} = 1361 W/ m^{2}$
5. Earth intercepts sunlight across a disc of area πR_⊕² but radiates from its entire sphere, 4πR_⊕². The planet's radius cancels, the factor of 4 does not, and clouds and ice reflect roughly 30% straight back out before any of it is absorbed.
  
  $π R_{\oplus}^{2} (1 - a) S = 4 π R_{\oplus}^{2} σ T_{eq}^{4} ⟹ T_{eq} = 4 \frac{( 1 - 0.30 ) ( 1361 )}{4 σ} = 254.6 K$
Answer

The tool prints 6.082 × 10¹⁸ m², 6.294 × 10⁷ W/m² and 3.828 × 10²⁶ W. Everything after that is the same law used twice more, and it ends at 254.6 K. Earth's actual mean surface temperature is 288 K, so the prediction is wrong by 33 K, and it is wrong in a specific direction: too cold, always. That gap is the greenhouse effect, and it is the one quantity in the chain that σT⁴ cannot supply, because the law describes what a surface radiates and says nothing about what the air above it does with the radiation on the way out. A calculation this crude landing within 12% of a planet is the reason it is the first thing anyone computes about an exoplanet.

$T_{eq} = 4 \frac{( 1 - a ) S}{4 σ} = 254.6 K, T_{\oplus} = 288 K, Δ T = 33 K$

References (3)

Betelgeuse’s radius and temperature, the preset behind the first block: M. Joyce et al., "Standing on the Shoulders of Giants: New Mass and Distance Estimates for Betelgeuse through Combined Evolutionary, Asteroseismic, and Hydrodynamic Simulations with MESA." The Astrophysical Journal 902(1), 63, 2020.
The solar radius, luminosity and temperature the tool normalises to: A. Prša et al., "Nominal Values for Selected Solar and Planetary Quantities: IAU 2015 Resolution B3." The Astronomical Journal 152(2), 41, 2016.
The value of σ: E. Tiesinga, P. J. Mohr, D. B. Newell and B. N. Taylor, "CODATA recommended values of the fundamental physical constants: 2018." Reviews of Modern Physics 93(2), 025010, 2021.

By the 42Math team · Updated 2026-08-17

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Sun - R = 1 R☉ and T = 5772 K, so both factor rows read exactly 1 and the luminosity is 1 L☉. Everything else here is measured against it.
Sirius A - At 9940 K it is nearly twice the Sun's temperature, worth a factor of 8.80 on its own. With 1.711 R☉ squared on top, it reaches 25.7 L☉.
Betelgeuse - 764 R☉ at 3500 K: the temperature row is a penalty of 0.135 and the radius row is 5.84 × 10⁵, so it still lands at 78,914 L☉.
Red Dwarf - 0.2 R☉ at 3200 K, so both rows sit below 1 and multiply down to 0.004 L☉. The only preset where neither lever helps.