Stellar Luminosity & Radius Calculator

The Stefan-Boltzmann law states that a blackbody radiates power proportional to the fourth power of its temperature. For a spherical star of radius R and surface temperature T, the total luminosity is L = 4 pi R squared times sigma times T to the fourth, where sigma = 5.67 x 10^-8 W per square metre per K^4. Dividing by the solar expression gives L/L_sun = (R/R_sun)^2 times (T/T_sun)^4.

The IAU 2015 nominal solar luminosity is L☉ = 3.828 x 10^26 watts. This is the total electromagnetic power radiated by the Sun across all wavelengths. It is used as the standard unit for stellar luminosity because it keeps the numbers compact. Sirius A, for example, is about 25 L☉ rather than 9.57 x 10^27 W.

Use the formula L/L☉ = (R/R☉)^2 x (T/T☉)^4 where T☉ = 5778 K and L☉ = 3.828 x 10^26 W. Enter the radius in solar radii and the temperature in Kelvin in Find Luminosity mode. For Sirius A with R = 1.711 R☉ and T = 9940 K, the luminosity works out to about 25.6 L☉, consistent with the measured value.

Because the Stefan-Boltzmann law gives radiated power per unit area as sigma times T^4. When you integrate over the entire spherical surface of a star (area = 4 pi R^2), both the area factor (R^2) and the flux factor (T^4) multiply together. Temperature therefore has a far greater influence on luminosity than radius: doubling T increases luminosity 16-fold at constant radius, while doubling R increases it only 4-fold.

The IAU 2015 nominal solar radius is R☉ = 695,700 km, approximately 109 times the radius of Earth. This value is used as the reference unit in stellar astronomy. Red supergiants like Betelgeuse can reach 700 R☉ (about 485 million km), while white dwarfs are typically 0.01 R☉ (roughly 7000 km, similar to Earth).

Because luminosity scales as T^4, even a moderate temperature increase produces a large luminosity boost. An A-type star at 8500 K is about (8500/5778)^4 = 4.68 times more luminous per unit area than the Sun. A star twice the solar radius at 8500 K would therefore be 4 x 4.68 = 18.7 times more luminous than the Sun in total.

O-type stars are the hottest, with surface temperatures above 30,000 K. They appear blue-white and are extremely luminous, often 100,000 times more luminous than the Sun. The Morgan-Keenan sequence from hot to cool is O, B, A, F, G, K, M. The Sun is a G-type star at 5778 K. M-type red dwarfs are the coolest main-sequence stars, typically below 3700 K.

The formula applies to any object that radiates approximately as a blackbody, including white dwarfs. A typical white dwarf at T = 25,000 K and R = 0.01 R☉ has L = (0.01)^2 x (25000/5778)^4 = 0.0001 x 70.1 = 0.007 L☉. Neutron stars are trickier because their radiation is often non-thermal and the radius definition is complicated by general relativistic effects.

Luminosity is the total power output of a star, independent of distance. Apparent brightness (flux) is how bright the star appears from Earth and depends on both luminosity and distance via the inverse-square law: F = L / (4 pi d^2). Two stars with equal luminosity at different distances will have very different apparent brightnesses. Astronomers measure apparent magnitude and use the distance modulus to convert to absolute magnitude, which is proportional to luminosity.

Direct methods include interferometry (for nearby giants like Betelgeuse) and transit timing (when a planet or companion eclipses the star). Indirect methods use the Stefan-Boltzmann law in reverse: measure the luminosity from the distance and apparent magnitude, measure the temperature from spectral fitting, then solve for R = R☉ times sqrt(L/L☉) times (T☉/T)^2. This calculator uses that same relationship in Find Radius mode.

The formula assumes the star radiates as a perfect blackbody with a uniform effective temperature over its photosphere. Real stars have limb darkening, star spots, and atmospheric absorption, so the effective temperature T_eff is a luminosity-weighted average. For most main-sequence stars the blackbody approximation is accurate to within a few percent. It becomes less accurate for very extended or irregular objects like luminous blue variables.

The solar effective temperature is defined by rearranging the Stefan-Boltzmann law: T_eff = (L☉ / (4 pi R☉^2 sigma))^0.25. Inserting the measured values L☉ = 3.828 x 10^26 W and R☉ = 695,700 km gives T_eff = 5778 K. This is not the temperature at the Sun's core (about 15 million K) or corona (over 1 million K) but the temperature of the visible photosphere that determines the total radiated flux.