What this calculator does
Luminosity is the total power a star radiates in every direction, and it depends on just two things: how big the star is, and how hot its surface is. The Stefan-Boltzmann law says every unit of a hot surface radiates power proportional to the fourth power of its temperature, so a star's total output is that rate multiplied by its entire surface area, 4πR².
Because both variables matter, a star does not need to be especially hot to be extremely luminous, and vice versa. A red giant can outshine a much hotter but far smaller star simply because its enormous surface area more than makes up for its cooler temperature, which is why luminosity and temperature are plotted against each other on the Hertzsprung-Russell diagram rather than temperature being used alone.
The formula
The formula is L = 4πR²σT⁴, where R is the star's radius, σ is the Stefan-Boltzmann constant, a fixed physical value of 5.670374×10⁻⁸ W/(m²·K⁴), and T is the surface temperature in kelvin. Radius is entered in multiples of the Sun's radius for convenience, then converted to metres before the area is calculated. The result is also expressed as a multiple of the Sun's own luminosity, since that is a far more intuitive scale than watts for comparing stars.
| Term | Meaning |
|---|---|
| L | Luminosity: total radiated power, in watts. |
| R | Stellar radius, entered here as a multiple of the Sun's radius (695,700 km). |
| σ | The Stefan-Boltzmann constant, 5.670374×10⁻⁸ W/(m²·K⁴), a fixed physical constant. |
| T | Surface temperature in kelvin. |
The inputs explained
| Field | What to enter |
|---|---|
| Star radius (in solar radii) | The star's radius as a multiple of the Sun's radius. Enter 1 for a Sun-sized star, 0.1 for a small red dwarf, or 100 for a large giant. |
| Surface temperature (K) | Surface temperature in kelvin. The Sun's surface is about 5,778 K; cooler red stars sit closer to 3,000 to 4,000 K, and hot blue stars can exceed 20,000 K. |
When to use it
Estimating how a star compares to the Sun
Entering a star's known radius and temperature and reading off the result in solar luminosities gives an immediate sense of scale, since a value of 100 means the star radiates 100 times the Sun's power.
Seeing why temperature dominates the formula
Because luminosity depends on temperature to the fourth power but radius only to the second, a modest increase in surface temperature changes total output far more than the same proportional increase in size.
Working through classroom Stefan-Boltzmann problems
Astrophysics courses commonly set problems giving a star's radius and temperature and asking for luminosity in solar units, which is exactly what this calculator returns directly.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How luminosity changes with surface temperature, at 1 solar radius
A Sun-sized star across a range of surface temperatures.
| Surface temperature | In solar luminosities |
|---|---|
| 3,000 K | 0.073 L☉ |
| 4,500 K | 0.369 L☉ |
| 5,778 K | 1.004 L☉ |
| 6,500 K | 1.608 L☉ |
| 8,000 K | 3.690 L☉ |
| 10,000 K | 9.009 L☉ |
How luminosity changes with radius, at the Sun's surface temperature
A star at the Sun's own temperature, across a range of radii.
| Radius (solar radii) | In solar luminosities |
|---|---|
| 0.5 R☉ | 0.251 L☉ |
| 1 R☉ | 1.004 L☉ |
| 2 R☉ | 4.017 L☉ |
| 5 R☉ | 25.104 L☉ |
| 10 R☉ | 100.416 L☉ |
| 20 R☉ | 401.665 L☉ |
Questions
Why is radius entered in solar radii rather than kilometres?
Stellar radii are almost always quoted relative to the Sun in astronomy, since the numbers involved in kilometres or metres are unwieldy. The calculator converts to metres internally before applying the formula.
Why does temperature affect luminosity so much more than radius?
The Stefan-Boltzmann law raises temperature to the fourth power but radius only to the second, through the surface area term. A star twice as hot radiates about sixteen times the power at the same size, whereas a star twice the radius radiates about four times the power at the same temperature.
What is the Sun's actual luminosity?
About 3.828×10²⁶ watts, the figure this calculator uses as the reference point for expressing other stars' output as a multiple of the Sun's.
Does this formula work for any star?
It works for any object radiating as a near-ideal blackbody, which is a good approximation for the vast majority of stars. It does not account for extended, non-spherical structures such as an accretion disc, or for a star with very unusual surface emissivity.
For the physics of blackbody emission at a single wavelength rather than total output, see the blackbody radiation calculator.