Star Luminosity Calculator
Calculate a star's total luminosity from its radius and surface temperature using the Stefan–Boltzmann law, in watts and multiples of the Sun.
كيفية استخدام هذه الحاسبة
- 1Enter the star's radius, in solar radii, kilometres, or metres.
- 2Enter its surface temperature in kelvin.
- 3Read its luminosity relative to the Sun and in watts, plus its likely colour.
طريقة الحساب
The Stefan–Boltzmann law
L = 4π R² σ T⁴ L = luminosity (watts), R = radius (m), T = surface temperature (K) σ = 5.670 × 10⁻⁸ W·m⁻²·K⁻⁴ (Stefan–Boltzmann constant) the Sun: L = 3.828 × 10²⁶ W, T = 5,772 K, R = 6.96 × 10⁸ m
A star radiates like a hot ball, and the Stefan–Boltzmann law says each square metre of its surface emits power proportional to the fourth power of its temperature. Multiply that by the star's total surface area, 4πR², and you get its luminosity — the total energy it pours out every second. The fourth-power temperature term is what makes hot stars so extraordinarily bright: it dominates the radius, so a small hot star can outshine a large cool one.
مثال محلول
For the Sun — radius 6.96 × 10⁸ m, temperature 5,772 K: L = 4π R² × σ × T⁴ ≈ 3.83 × 10²⁶ watts, which by definition is 1 solar luminosity. A star the same size at twice the temperature would radiate 2⁴ = 16 times as much power.
Star Luminosity Calculator: الدليل الكامل
Luminosity is intrinsic; brightness is not
Luminosity is the total power a star actually emits — a fixed property of the star itself, like the wattage of a bulb. Apparent brightness is how bright it looks from Earth, which also depends on distance: a dim nearby star can outshine a brilliant distant one. Confusing the two is one of the oldest mistakes in astronomy, and untangling them is what lets us map the real properties of stars.
The Stefan–Boltzmann law gives the intrinsic luminosity from just two numbers — how big a star is and how hot its surface is. Combined with a measured apparent brightness, this then reveals the distance, because brightness falls off with the square of distance. This chain is how astronomers gauge the scale of the galaxy, using stars of known luminosity as 'standard candles'.
Why the fourth power of temperature matters so much
Luminosity depends on the square of the radius but the fourth power of the temperature, and that difference in exponents has dramatic consequences. Doubling a star's size makes it four times as luminous; doubling its temperature makes it sixteen times as luminous. Temperature, not size, is usually the bigger lever on how much light a star produces.
This is why the hottest stars are so staggeringly bright. A blue supergiant at 30,000 K blazes with the output of hundreds of thousands of Suns, while a cool red dwarf at 3,000 K emits a thousandth of the Sun's power despite not being a thousand times smaller. The fourth-power law is the reason the two extremes of the stellar zoo differ in luminosity by a factor of billions.
Reading a star's colour and place in its life
Temperature also sets a star's colour, through Wien's law: hotter surfaces peak at shorter, bluer wavelengths, cooler ones at longer, redder wavelengths. This is why blue stars are hot and red stars are cool — the reverse of the everyday association of red with heat. A star's colour is therefore a direct readout of its surface temperature, which is why astronomers classify stars by colour at all.
Plotting luminosity against temperature for many stars produces the Hertzsprung–Russell diagram, one of the most important tools in astronomy. Most stars fall along a diagonal 'main sequence' where they spend their stable lives fusing hydrogen; giants and dwarfs sit off it. The Stefan–Boltzmann calculation above places a star on that diagram, revealing not just how bright it is but roughly what kind of star, and what stage of life, it represents.
الأسئلة الشائعة
What is the Stefan–Boltzmann law?
It states that the power radiated per unit area of a hot surface is proportional to the fourth power of its absolute temperature. For a star, multiplying by the total surface area 4πR² gives its luminosity: L = 4π R² σ T⁴.
What is the difference between luminosity and brightness?
Luminosity is the total power a star emits — an intrinsic property. Apparent brightness is how bright it appears from Earth, which also depends on distance. A luminous star far away can look dimmer than a faint one nearby. This calculator gives intrinsic luminosity.
Why are blue stars hotter than red stars?
Because of Wien's law: hotter surfaces radiate most strongly at shorter (bluer) wavelengths, cooler ones at longer (redder) wavelengths. So a blue star is hot (perhaps 20,000–30,000 K) and a red star is cool (around 3,000 K) — opposite to the everyday link between red and heat.
How much brighter is a hotter star of the same size?
Luminosity scales with temperature to the fourth power, so a star twice as hot as another of the same radius is 2⁴ = 16 times as luminous. This is why the hottest stars vastly outshine cooler ones even at modest sizes.