Stellar Luminosity Calculator

A star radiates energy as a blackbody, so its luminosity depends on both its size and temperature. The larger and hotter a star, the more energy it emits.

Stefan-Boltzmann Law:

L = 4πR²σT⁴

Where:

• L = luminosity (watts)
• R = radius (meters)
• σ = Stefan-Boltzmann constant = 5.67 × 10⁻⁸ W/m²/K⁴
• T = surface temperature (Kelvin)

In solar units (much easier to use):

L/L☉ = (R/R☉)² × (T/T☉)⁴

Where L☉ = 3.828 × 10²⁶ W, R☉ = 695,700 km, T☉ = 5,772 K.

Why 5,772 and not the 5,778 you usually see. Those three solar constants have to be consistent with each other, because the first one is what the other two produce when you put them through the Stefan-Boltzmann law. The International Astronomical Union fixed L☉ and R☉ at the values above in 2015, and the effective temperature that reproduces them is 5,772 K. The older, more widely quoted 5,778 K predates that and is 0.4% too high for this pair, which is enough to make a calculator hand back 1.0042 L☉ for a star that is definitionally exactly one.

Why temperature matters so much: Because of the T⁴ term, temperature has an enormous effect. A star twice as hot as the Sun radiates 2⁴ = 16 times more power per unit area. A star 10 times hotter radiates 10,000 times more per unit area.

Example stars:

• Sun: R = 1 R☉, T = 5,772 K, L = 1 L☉
• Sirius A: R = 1.71 R☉, T = 9,940 K, L ≈ 25.7 L☉
• Rigel: R ≈ 78 R☉, T ≈ 12,100 K, L ≈ 117,500 L☉
• Betelgeuse: R ≈ 1,000 R☉, T ≈ 3,500 K, L ≈ 135,000 L☉. Note that Betelgeuse’s radius is genuinely uncertain; modern estimates cluster nearer 764 R☉, which would put it closer to 79,000 L☉ (large but cool)
• Proxima Centauri: R ≈ 0.15 R☉, T ≈ 3,042 K, L ≈ 0.0017 L☉

This formula shows why giant cool stars can still be very luminous. Their enormous surface area more than compensates for the lower temperature.

Turning the answer into an absolute magnitude. Luminosity and absolute magnitude are the same statement on two different scales, so the calculator gives both:

M_bol = 4.74 - 2.5 × log₁₀(L / L☉)

Note the 4.74. The absolute magnitude calculator uses 4.83 for the Sun, and both figures are right: 4.83 is the Sun’s magnitude in the visual band, while 4.74 is its bolometric magnitude, covering every wavelength. The Stefan-Boltzmann law counts all the energy a star radiates, ultraviolet and infrared included, so a bolometric zero point is the one that belongs here. For a G-type star the two are close. For Betelgeuse, which pours most of its output into the infrared, the gap runs to nearly two magnitudes.