Audited —·Last updated 27 Jul 2026·6 citations·Tier 1·0 uses

Stefan-Boltzmann Law Calculator

Calculate black-body radiation power with the Stefan-Boltzmann law. Enter temperature, area, and emissivity for step-by-step results using CODATA constants.

Stefan-Boltzmann Law Calculator

Total radiated power
63,200,699.7348
Radiated power per unit area
63,200,699.7348

Background.

The Stefan-Boltzmann law calculator computes the total electromagnetic power radiated by a black body or gray body from its surface temperature, area, and emissivity. The law is one of the pillars of thermal radiation physics and underpins fields from astrophysics to climate science to engineering thermodynamics. Every star, every incandescent filament, every radiant heater, and every planetary energy balance calculation relies on the fourth-power relationship between temperature and radiated power that this tool implements.

The law was established experimentally by Josef Stefan in 1879, who showed that the total radiation from a hot body varied as the fourth power of its absolute temperature. Stefan's student Ludwig Boltzmann derived the same result theoretically in 1884 by combining thermodynamics with Maxwell's electromagnetic theory, applying Carnot-cycle reasoning to a radiation-filled cavity. The resulting constant sigma, now called the Stefan-Boltzmann constant, is not a measured empirical constant in modern physics; it is computed exactly from other fundamental constants via sigma = 2 pi^5 k^4 / (15 h^3 c^2), where k is the Boltzmann constant, h is the Planck constant, and c is the speed of light. The CODATA 2018 recommended value is 5.670374419 x 10^-8 W per square meter per kelvin to the fourth.

The calculator serves a broad user base. Astrophysicists use it to estimate stellar luminosities from observed temperatures and radii. Entering the Sun's effective temperature of 5,778 K and its radius of 6.96 x 10^8 m yields a luminosity of approximately 3.9 x 10^26 W, which matches the accepted solar constant integrated over a sphere at Earth's orbital distance. Engineers designing radiative cooling systems for spacecraft use the law to size radiator panels that must dissipate waste heat in the vacuum of space, where convection is absent. Climate scientists apply it to calculate the effective radiating temperature of planets, including Earth, by equating absorbed solar irradiance to emitted thermal radiation.

The fourth-power dependence makes the law highly nonlinear and therefore consequential. Doubling the absolute temperature of a black body increases its radiated power by a factor of sixteen. This sensitivity explains why a star twice as hot as the Sun is not merely brighter but dramatically more luminous. It also explains why small changes in Earth's average surface temperature produce significant changes in outgoing longwave radiation, a feedback mechanism central to climate models. Conversely, cooling a body by a few percent yields a large reduction in radiated power, which is why infrared cameras can detect temperature variations of fractions of a degree.

The emissivity parameter epsilon allows the calculator to handle real materials, which are not perfect black bodies. A polished metal surface may have an emissivity of 0.05, meaning it radiates only 5% as much power as a black body at the same temperature. Human skin has an emissivity of about 0.98, making it an excellent approximation to a black body in the infrared. By adjusting epsilon, users can model everything from reflective thermal shields to dark absorptive coatings, making the tool applicable across materials science and engineering design. The Stefan-Boltzmann law thus bridges idealized theory and practical engineering, offering quantitative predictions for thermal management in electronics, architecture, and spacecraft.

What is stefan-boltzmann law calculator?

The Stefan-Boltzmann law states that the total power radiated per unit area of a black body in thermal equilibrium is proportional to the fourth power of its absolute temperature. A black body is an idealized object that absorbs all incident electromagnetic radiation and emits radiation with a characteristic spectrum determined solely by its temperature. The proportionality constant is the Stefan-Boltzmann constant sigma.

For real surfaces, which reflect or transmit some incident radiation, the law is modified by the emissivity epsilon, a dimensionless factor between 0 and 1 that measures how closely the surface approximates a black body. The total radiated power is then P = epsilon sigma A T^4. Emissivity depends on temperature, wavelength, and surface condition, but for many engineering calculations a single broadband value is sufficient.

The law applies to any object in thermal equilibrium that emits thermal radiation. It is valid across the entire electromagnetic spectrum, integrating over all wavelengths. For temperatures below about 10^5 K, the emitted spectrum peaks in the infrared or visible range, and relativistic corrections to the law are negligible. At higher temperatures, quantum electrodynamic and pair-production effects become relevant, but such regimes are encountered only in astrophysical contexts such as neutron star surfaces and supernova cores.

How to use this calculator.

  1. Enter the surface temperature of the body in kelvin.
  2. Enter the total surface area in square meters.
  3. Enter the emissivity of the surface, or leave it at 1.0 for an ideal black body.
  4. The calculator computes the radiant exitance j = epsilon sigma T^4 in W/m^2.
  5. The calculator multiplies by area to obtain total radiated power P in watts.
  6. Review both outputs to assess heat dissipation or energy balance.
  7. For spherical bodies, area can be computed as 4 pi r^2 and entered separately.

The formula.

P = ε σ A T⁴
P · radiated powerε · emissivityσ · Stefan–Boltzmann constantA · surface areaT · absolute temperature

The Stefan-Boltzmann law can be derived by integrating Planck's law of black-body radiation over all wavelengths and solid angles. Planck's law gives the spectral radiance B_lambda(T) as a function of wavelength lambda and temperature T. Integrating over all wavelengths from zero to infinity and over a hemisphere yields the total radiant exitance j = sigma T^4. The definite integral of x^3/(e^x - 1) dx from 0 to infinity equals pi^4/15, producing the factor of pi^4/15 that enters the theoretical expression for sigma.

The exact modern value of the Stefan-Boltzmann constant is not measured directly but is computed from the defining relation sigma = 2 pi^5 k^4 / (15 h^3 c^2). Here k is the Boltzmann constant, h is the Planck constant, and c is the speed of light. Because the kelvin was redefined in 2019 in terms of the Boltzmann constant, and because h and c are defined exactly, the value of sigma is known exactly in SI units to the precision with which k is known. The CODATA 2018 recommended value carries a relative uncertainty of 3.7 x 10^-7, dominated by the uncertainty in the Boltzmann constant at that time. The 2019 SI redefinition fixed the numerical values of h, c, and k exactly, making sigma an exact computable constant in the modern SI.

The fourth-power temperature dependence has profound physical consequences. In astrophysics, it determines the mass-luminosity relation for main-sequence stars and sets the timescale for stellar cooling. In engineering, it governs the design of high-temperature furnaces, where radiative heat transfer dominates over conduction and convection above roughly 800 K. It also explains why the night sky is dark: the cosmic microwave background, at 2.725 K, radiates only about 3 x 10^-6 W/m^2, far below the threshold of human vision. If the universe were hotter, Olbers' paradox would be far more severe.

The emissivity correction epsilon is essential for practical applications because no real material is a perfect black body. Kirchhoff's law of thermal radiation states that, at thermal equilibrium, the emissivity of a surface equals its absorptivity at the same wavelength. A surface that is a good absorber is therefore a good emitter, and a perfect reflector (absorptivity zero) emits no thermal radiation. This reciprocity is why emergency blankets use reflective coatings to minimize radiative heat loss and why black-anodized heat sinks maximize it.

A worked example.

Example

To estimate the Sun's luminosity using the Stefan-Boltzmann law, treat the photosphere as a spherical black body with radius 6.96 x 10^8 meters and effective temperature 5,778 kelvin. First compute the surface area: 4 pi times the square of the radius equals 4 x 3.14159 x 4.844 x 10^17, which is 6.087 x 10^18 square meters. Next, raise the temperature to the fourth power: 5,778 squared is 33,785,284, and squaring again gives 1.14145 x 10^15 K^4. Multiply by the Stefan-Boltzmann constant 5.670374419 x 10^-8 to obtain a radiant exitance of 6.4724 x 10^7 watts per square meter. Finally, multiply by the surface area: 6.4724 x 10^7 times 6.087 x 10^18 equals 3.939 x 10^26 watts. This result is within a few percent of the accepted solar luminosity of 3.828 x 10^26 W; the small discrepancy reflects the Sun's non-black-body spectrum and the fact that 5,778 K is a wavelength-averaged effective temperature rather than a true surface temperature.

area6,087,000,000,000,000,000
emissivity1
temperature5,778

Frequently asked questions.

What is a black body and why is it important to this law?
A black body is an idealized object that absorbs all incident electromagnetic radiation regardless of wavelength or angle, and emits thermal radiation with a spectrum determined solely by its temperature. No real object is a perfect black body, but the concept is fundamental because it sets an upper bound on the radiative efficiency of any surface. The Stefan-Boltzmann law applies exactly to black bodies; for real materials, the emissivity factor epsilon corrects the result downward. Cavities with small apertures approximate black bodies closely because radiation entering the aperture undergoes multiple reflections and is almost completely absorbed.
Why does radiated power depend on the fourth power of temperature?
The T^4 dependence arises from the integration of Planck's spectral distribution over all wavelengths. Planck's law contains a term proportional to nu^3/(e^(h nu / kT) - 1) for frequency nu. Changing variables to x = h nu / kT shows that the total integrated power scales as T^4 times a dimensionless integral that evaluates to pi^4/15. Physically, two factors of T come from the increase in photon energy per mode (h nu proportional to T), and two more come from the increase in the number of modes accessible at higher temperature (the phase-space volume expands as T^3, and one factor is absorbed into the Stefan-Boltzmann constant). This steep dependence makes temperature the dominant control knob for thermal radiation.
Can this calculator be used for non-black bodies?
Yes. By entering an emissivity value less than 1.0, the calculator adjusts the black-body result to approximate the behavior of real materials. Most metals have emissivities between 0.02 and 0.3, depending on surface polish and oxide layer. Human skin, water, and most organic materials have emissivities above 0.9. For precise calculations, emissivity should be measured at the relevant temperature and wavelength, because epsilon varies across the spectrum. The calculator uses a single broadband value, which is adequate for many engineering estimates but may require refinement for high-precision thermal analysis.
What is the difference between the Stefan-Boltzmann law and Wien's displacement law?
The Stefan-Boltzmann law gives the total power radiated across all wavelengths, integrating the entire spectrum into a single number. Wien's displacement law gives the wavelength at which the spectral radiance peaks: lambda_max = b/T, where b is Wien's displacement constant (2.897771955 x 10^-3 m K). A star at 5,778 K peaks in the visible yellow-green, while a human body at 310 K peaks in the infrared at 9.4 micrometers. The two laws are complementary: Stefan-Boltzmann tells you how much is radiated; Wien tells you where in the spectrum the radiation is concentrated.
How is the Stefan-Boltzmann constant related to other fundamental constants?
The constant is defined exactly by sigma = 2 pi^5 k^4 / (15 h^3 c^2), where k is the Boltzmann constant, h is the Planck constant, and c is the speed of light. In the 2019 revision of the SI, h, c, and k were assigned fixed numerical values, which means sigma is now an exact derived constant in the modern SI framework. Prior to 2019, the value was determined experimentally with a relative uncertainty near 3.7 x 10^-7. The exact theoretical relationship reflects the deep connection between statistical mechanics, quantum theory, and electromagnetism that underlies thermal radiation.
Why do stars with higher surface temperatures have much greater luminosities?
Luminosity depends on both surface area and temperature via L = 4 pi R^2 sigma T^4. Main-sequence stars follow a mass-luminosity relation because more massive stars have larger radii and higher core temperatures. The T^4 term means that a modest increase in surface temperature produces a dramatic increase in luminosity. A star twice as hot as the Sun with the same radius would be sixteen times more luminous. A blue supergiant at 30,000 K radiates about 270 times more power per unit area than the Sun. Combined with radii hundreds of times larger, this explains why such stars can be millions of times more luminous than the Sun despite containing only tens of solar masses.
Can the law be applied to cooler objects like planets?
Yes, provided the object is in thermal equilibrium and emits thermal radiation. Earth radiates approximately 240 W/m^2 of outgoing longwave radiation, corresponding to an effective emitting temperature of about 255 K when treated as a black body. The calculator confirms this: sigma times (255)^4 = 5.670374419 x 10^-8 x 4.228 x 10^9 = 239.7 W/m^2. This is the baseline energy balance calculation used in climate science, though the actual system is complicated by greenhouse gases, clouds, and atmospheric convection. The Stefan-Boltzmann law remains the starting point for all such analyses.
What happens if the temperature is entered in Celsius instead of kelvin?
The Stefan-Boltzmann law requires absolute temperature in kelvin. Using Celsius values would produce dramatically incorrect results, especially for temperatures near or below the freezing point of water. For example, a surface at 27 C (300 K) radiates sigma times (300)^4 = 459 W/m^2, but entering 27 directly would yield sigma times (27)^4 = 0.003 W/m^2, an error of five orders of magnitude. The calculator assumes kelvin input; users must convert from Celsius by adding 273.15. For high-temperature astrophysical objects, the difference is negligible, but for terrestrial and biological applications the offset is essential.
Does the law apply to objects that are not in thermal equilibrium?
The Stefan-Boltzmann law in its standard form applies to objects in local thermodynamic equilibrium, where a single temperature characterizes the radiation field and the emitting material. Non-equilibrium situations such as lasers, fluorescent lamps, or the cosmic microwave background during recombination require more sophisticated treatments. However, many practical systems are close enough to equilibrium that the law provides excellent approximations. A tungsten filament in an incandescent bulb, for instance, is not in perfect equilibrium with its surroundings, but its spectral output is still well approximated by a black body at the filament temperature, corrected by the wavelength-dependent emissivity of tungsten.
How does emissivity vary with temperature and wavelength?
Emissivity is generally a function of both temperature and wavelength, written epsilon(lambda, T). Metals tend to have low emissivity in the infrared but higher emissivity at shorter wavelengths; this is why polished steel glows red when hot even though it reflects most room-temperature infrared radiation. Non-metals and rough surfaces tend to have high, relatively constant emissivity across the thermal infrared. For the broadband calculations performed by this calculator, an average emissivity over the relevant wavelength range is used. When precise radiative heat transfer calculations are required, engineers perform wavelength-by-wavelength integrations using spectral emissivity data from materials databases such as those maintained by NIST.

References& sources.

  1. [1]Stefan, J. (1879). Über die Beziehung zwischen der Wärmestrahlung und der Temperatur. Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften 79:391-428.
  2. [2]Boltzmann, L. (1884). Ableitung des Stefan'schen Gesetzes, betreffend die Abhängigkeit der Wärmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie. Annalen der Physik 258(6):291-294.
  3. [3]NIST (2024). CODATA Recommended Values of the Fundamental Physical Constants: 2018.
  4. [4]BIPM (2019). The International System of Units (SI Brochure), 9th ed.
  5. [5]Carroll, B.W. and Ostlie, D.A. (2017). An Introduction to Modern Astrophysics, 2nd ed. Cambridge University Press. Chapter 9: The Interiors of Stars.
  6. [6]Planck, M. (1901). Über das Gesetz der Energieverteilung im Normalspectrum. Annalen der Physik 309(3):553-563.
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