Stefan-Boltzmann Law

In one sentence: A hot object gives off energy as heat and light, and doubling its temperature on the kelvin scale makes it give off 16 times as much.

$P = \varepsilon \sigma A T^4$

Read it out loud: “P equals epsilon, times sigma, times A, times T to the fourth power. P is the radiated power, the energy the surface gives off every second, measured in watts. Epsilon (EP-sih-lon) is the emissivity, a number from 0 to 1 that says how well the surface radiates. Sigma (SIG-muh) is the Stefan-Boltzmann constant, a fixed number of nature. A is the surface area in square meters. T is the absolute temperature in kelvin, multiplied by itself four times.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Powers of ten, like $10^{-8}$ and $10^{9}$: Some numbers in this formula are far too small or too large to write out in full, so they use a power of ten. The small raised number is the exponent. $10^{9}$ means 1 followed by nine zeros, which is one billion. $10^{-8}$ means 1 divided by 10 eight times, which is 0.00000001. The constant $\sigma$ is written $5.670374419 \times 10^{-8}$ , which means 5.670374419 multiplied by that tiny power of ten. This way of writing numbers is called scientific notation.
Powers of a letter, like $T^4$ , and negative powers in units: A small raised number tells you how many times to multiply a number by itself. So $T^4$ means $T \times T \times T \times T$ , and $300^4$ means $300 \times 300 \times 300 \times 300$ . A negative power means division: $\text{m}^{-2}$ means one divided by square meters, so $\text{W m}^{-2}$ means watts per square meter. The area $A$ is measured in square meters, written $\text{m}^2$ .
Greek letters $\sigma$ (sigma) and $\varepsilon$ (epsilon): Greek letters are extra names for quantities, because the Latin alphabet runs out. The letter $\sigma$ is said SIG-muh, and it names the Stefan-Boltzmann constant. The letter $\varepsilon$ is said EP-sih-lon, and it names the emissivity, a number that says how well a surface gives off radiation. The letters are only names. Each page says what its letters stand for.
Temperature in kelvin and absolute zero: Kelvin is a temperature scale that starts at absolute zero, the coldest possible temperature, where things have no heat energy left to give away. Zero kelvin is minus 273.15 degrees Celsius. Water freezes at 273.15 K and boils at 373.15 K. To convert Celsius to kelvin, add 273.15, so 27 degrees Celsius is 300.15 K. The formula needs kelvin, because only on that scale is zero a true zero.
Watts, joules and power: Energy is the ability to make something happen, and its unit is the joule, written J. Power is energy per second, and its unit is the watt, written W. One watt means one joule every second. A 10 watt LED bulb gives off 10 joules every second. The law gives power, so it tells you how many joules leave the surface each second, not the total energy.
Thermal radiation and a blackbody: Every object above absolute zero gives off radiation, which is energy carried by light waves. At ordinary temperatures most of it is infrared light, which our eyes cannot see. Radiation caused by heat is called thermal radiation. A blackbody is an ideal surface that absorbs all the light that hits it and gives off the most radiation possible at its temperature. Its emissivity is exactly 1.

What each symbol means

Symbol	Meaning
$P$	The radiated power, the total energy the surface gives off as radiation each second. It is measured in watts (W), and one watt is one joule per second. In the example below, a 1 square meter blackbody at room temperature gives off about 459 W, a little more than four 100 W light bulbs.
$\varepsilon$	The emissivity, a number from 0 to 1 that says how well a surface gives off radiation compared with a perfect blackbody. A perfect blackbody has $\varepsilon = 1$ , a dull black paint is close to 1, and a polished metal can be 0.1 or lower. Emissivity has no unit, because it is a ratio of two radiation amounts.
$\sigma$	The Stefan-Boltzmann constant, a fixed number of nature: $5.670374419 \times 10^{-8}$ W m $^{-2}$ K $^{-4}$ . Its unit reads as watts per square meter per kelvin to the fourth power. It is the same for every object, and it is built from other fixed constants of physics.
$A$	The surface area, the size of the glowing surface, in square meters (m $^2$ ). A square meter is a square one meter on each side, about the size of a small table top. Doubling the area doubles the power, because twice the surface gives off twice the energy.
$T$	The absolute temperature in kelvin (K), measured from absolute zero. Room temperature is about 300 K, and the surface of the Sun is about 5,800 K. Never put Celsius into this formula. The fourth power of a Celsius number gives a meaningless answer, because Celsius has an arbitrary zero point.

The formula piece by piece

$T$: Start with $T$ , the absolute temperature of the surface in kelvin. The bigger $T$ is, the more energy the surface gives off each second. Everything else in the formula builds on this one number, which sets how strongly the surface glows.
$T^4$: Raise $T$ to the fourth power by multiplying it by itself four times. Doubling $T$ gives $2^4 = 16$ times the result, not 2 times. This fourth power is why a small rise in temperature makes an object glow much more strongly.
$\sigma T^4$: Multiply by the constant $\sigma$ . The result is the power given off by each square meter of a perfect blackbody, in W/m $^2$ . The kelvins cancel: $\sigma$ carries K $^{-4}$ , and $T^4$ carries K $^4$ , which leaves watts per square meter.
$A \sigma T^4$: Multiply by the area $A$ , the number of square meters of glowing surface. Now the result is the power of a whole surface, in watts. Two square meters at the same temperature give twice the power of one, because each square meter gives off the same amount.
$\varepsilon \sigma A T^4$: Multiply by the emissivity $\varepsilon$ to adjust the result for a real surface. A value of 1 gives a perfect blackbody. A value of 0.5 gives half the power, at the same temperature and area.
$P = \varepsilon \sigma A T^4$: The whole formula in one sentence: the power a surface gives off as radiation equals its emissivity times the Stefan-Boltzmann constant times its area times its absolute temperature to the fourth power.

What the formula is doing, and why

A hot surface gives off energy as radiation, which is light and heat that leave in all directions. The Stefan-Boltzmann law gives the total power given off, adding up the radiation at every wavelength. The formula $P = \varepsilon \sigma A T^4$ takes three things in: the emissivity $\varepsilon$ , the area $A$ and the absolute temperature $T$ . What comes out is $P$ , the power in watts, which is the energy leaving the surface every second.

Why the fourth power? Hotter matter means its particles jiggle more violently, and they send out more light at every wavelength at once. Adding up that glow over all wavelengths gives a total that grows with $T^4$ . You do not need to do that addition to use the law. What matters is the result: a 10 percent rise in kelvin temperature raises the power by about 46 percent, because $1.1^4 \approx 1.46$ .

Why multiply by the area? Radiation leaves from the surface, so a bigger surface sends out more. Doubling the area doubles the power. The emissivity $\varepsilon$ is a ratio. A surface with $\varepsilon = 0.5$ gives off half the power of a perfect blackbody at the same temperature and area, and a polished metal can give off much less than a black surface.

Why kelvin? The fourth power only has a meaning when zero is a true zero. Kelvin starts at absolute zero, where thermal motion stops. Celsius puts zero at the freezing point of water, which is an arbitrary choice. Celsius numbers would give answers that depend on the scale you picked, so always convert to kelvin first.

When does it apply? It applies to thermal radiation from a surface at one uniform temperature, such as a hot filament, a star or a warm wall. It gives the total power over all wavelengths, not the color. It does not cover heat carried by moving air or water, which is convection, or heat passed through a solid, which is conduction. Josef Stefan found the pattern from measurements, and Ludwig Boltzmann later explained it from the laws of heat and energy.

An everyday picture

Think of a campfire, or the heat you feel from a radiator. Turning the heat up a little makes the surface give off much more energy, because the power rises with the fourth power of the temperature. The fourth power acts like a volume knob for glow: a small turn makes a big difference. Emissivity is how well the surface gives off its energy. A dull black surface radiates more than a polished metal surface at the same temperature.

Worked example: Heat given off by a 1 square meter blackbody at 300 K

Imagine a flat plate with a perfect blackbody surface, so $\varepsilon = 1$ . Its area is $A = 1.00$ m $^2$ , a square one meter on each side. Its temperature is $T = 300$ K, which is close to room temperature. We use $\sigma = 5.670374419 \times 10^{-8}$ W m $^{-2}$ K $^{-4}$ . The inputs are round numbers chosen for practice, not measurements of a real surface.

Convert the temperature first. The formula needs kelvin. We use 300 K, which is 27 degrees Celsius, and the exact value is 300.15 K. We round to 300 K so the arithmetic stays easy. The emissivity is 1 and the area is 1.00 m $^2$ , so only the temperature needs work before we calculate.
Raise the temperature to the fourth power. This means multiplying 300 by itself four times. $300 \times 300 = 9 \times 10^{4}$ , which is 90,000. Multiplying by 300 again gives $9 \times 10^{4} \times 300 = 2.7 \times 10^{7}$ . Once more gives $2.7 \times 10^{7} \times 300 = 8.1 \times 10^{9}$ K $^4$ , which is 8.1 billion. Each time we multiply the powers of ten by adding exponents.
Multiply by the constant sigma. Multiply the front numbers and the powers of ten separately. The front numbers give $5.670374419 \times 8.1 \approx 45.93$ . The powers of ten give $10^{-8} \times 10^{9} = 10^{1}$ , because the exponents add to $-8 + 9 = 1$ . Together, $45.93 \times 10^{1} = 459.3$ W/m $^2$ for each square meter.
Check the units. The constant has the unit W m $^{-2}$ K $^{-4}$ , and $T^4$ is in K $^4$ . The kelvins cancel, which leaves watts per square meter. The units tell us the answer is a power for each square meter, which is what we need before multiplying by the area.
Multiply by the emissivity and the area. The emissivity is 1 and the area is 1.00 m $^2$ , so both multiplications leave the number unchanged: $P = (1)(459.3\ \text{W/m}^2)(1.00\ \text{m}^2) = 459.3$ W. The square meters cancel, which leaves watts.
Translate the answer into everyday terms. 459 W is a little more than four 100 W light bulbs. Over one minute the surface gives off $459.3 \times 60 \approx 2.76 \times 10^{4}$ J, which is about 27,600 joules. The watts tell us how fast the energy leaves, and the joules tell us the total.
Double the temperature to see the fourth power. At 600 K, which is $2 \times 300$ , the formula gives $\sigma \times 600^4 \approx 7349$ W/m $^2$ . That is about 7,350 W/m $^2$ , which is 16 times the 459.3 W/m $^2$ at 300 K. Doubling the temperature is worth a factor of $2^4 = 16$ , not 2.
Try a less perfect surface. With emissivity 0.5 at the same 300 K and area, $P = 0.5 \times 459.3 = 229.65$ W, which is about 230 W. The surface gives off half the power of a perfect blackbody, even though its temperature and area are the same.

Answer: A perfect 1 square meter blackbody at 300 K gives off about 459 W, a little more than four 100 W light bulbs. Almost all of that is infrared light, which our eyes cannot see. The same surface at 600 K would give off about 16 times as much, and a surface with emissivity 0.5 gives off about half.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Using degrees Celsius for $T$ . Convert first: 27 degrees Celsius is 300.15 K. Putting in 27 gives a power that is far too small.
Using $T$ where the formula needs $T^4$ . Doubling $T$ multiplies the power by 16, not by 2, so small temperature changes matter a lot.
Giving the emissivity a value above 1 or below 0. Real emissivity lies between 0 and 1, and a perfect blackbody is exactly 1.
Mixing up power and energy. The law gives watts, which are joules per second. To get the energy given off over a time, multiply the power by the time in seconds.

Where you will meet it

Estimating a star’s surface temperature from its total brightness and its size, since the law links all three.
Designing spacecraft radiators, which shed heat by radiation, because space has no air to carry heat away by moving air.
Estimating heat losses from furnaces and incandescent filaments, where radiation is a large part of the heat balance.
Climate science, where Earth must give off, on average, as much radiated power as it takes in from sunlight.

Questions people ask

Why must the temperature be in kelvin?

The formula holds only when $T$ is measured from absolute zero, where thermal motion stops. Celsius and Fahrenheit set their zero points by convention, not by absolute zero, so raising their numbers to the fourth power would give answers that depend on the scale and not on the physics.

Does a hot object glow only when it is red hot?

No. Every object above absolute zero gives off thermal radiation. At room temperature it is mostly infrared, which our eyes cannot see. Heated metal first looks dull red at several hundred degrees Celsius, because by then it gives off enough visible light to see.

What is emissivity?

It is the ratio of the radiation from a surface to the radiation from a perfect blackbody at the same temperature. It runs from 0 to 1 and depends on the material and how the surface is finished.

Does this law give the color of the light?

No. It gives the total power across all wavelengths. For the peak color of an object, use Wien’s displacement law, which is a separate formula.

Planck's Law: Planck’s law says how brightly a hot object glows at each color, and the mix of colors it gives off depends on how hot the object is.
Wien's Displacement Law: Wien’s law says a hotter object glows brightest at a shorter wavelength, and its brightest wavelength equals a fixed number divided by its temperature.
Planck-Einstein Relation (E = hf): E = hf says each packet of light carries energy in proportion to its frequency: higher frequency means more energy per packet, and brightness does not change it.