The Stefan–Boltzmann law, also known as Stefan's law, describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time (also known as the black-body radiant exitance or emissive power), , is directly proportional to the fourth power of the black body's thermodynamic temperature T:

The constant of proportionality σ, called the Stefan–Boltzmann constant or Stefan's constant, derives from other known constants of nature. The value of the constant is
where k is the Boltzmann constant, h is Planck's constant, and c is the speed of light in a vacuum. Thus at 100 K the energy flux is 5.67 W/m2, at 1000 K 56,700 W/m2, etc. The radiance (watts per square metre per steradian) is given by
A body that does not

absorb all incident radiation (sometimes known as a grey body) emits less total energy than a black body and is characterized by an emissivity, :
The irradiance has dimensions of energy flux (energy per time per area), and the SI units of measure are joules per second per square metre, or equivalently, watts per square metre. The SI unit for absolute temperature T is the kelvin. is the emissivity of the grey body; if it is a perfect blackbody, . In the still more general (and realistic) case, the emissivity depends on the wavelength, .
To find the total power radiated from an object, multiply by its surface area, :
Wavelength- and subwavelength-scale particles, metamaterials, and other nanostructures are not subject to ray-optical limits and may be designed to exceed the Stefan–Boltzmann law.

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  • 1. [Stefan–Boltzmann constant] The Stefan–Boltzmann constant (also Stefan's constant), a physical constant denoted by the Greek letter σ(sigma), is the constant of proportionality in the Stefan–Boltzmann law: "the total intensity (physics) radiated over all wavelengths increases as the temperature increases", of a black body which is proportional to the fourth power of the thermodynamic temperature. The theory of
  • 2. [Wien's displacement law] Wien's displacement law states that the black body radiation curve for different temperatures peaks at a wavelength inversely proportional to the temperature. The shift of that peak is a direct consequence of the Planck radiation law which describes the spectral brightness of black body radiation as a function of wavelength at any given temperature. However
  • 3. [Planck's law] Planck's law describes the electromagnetic radiation emitted by a black body in thermal equilibrium at a definite temperature. The law is named after Max Planck, who originally proposed it in 1900. It is a pioneering result of modern physics and quantum theory.
  • 4. [Emissivity] The emissivity of the surface of a material is its effectiveness in emitting energy as thermal radiation. Thermal radiation is light, but for objects near room temperature this light is infrared and isn't visible to human eyes. The thermal radiation from very hot objects (see photograph) is easily visible to the eye. Quantitatively, emissivity is
  • 5. [Black body] A black body (also blackbody) is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. A white body is one with a "rough surface [that] reflects all incident rays completely and uniformly in all directions."
  • 6. [Joseph Stefan] Joseph Stefan (Slovene: Jožef Štefan) (24 March 1835 – 7 January 1893) was an ethnic Carinthian Slovene physicist, mathematician, and poet of the Austrian Empire.
  • 7. [Dulong–Petit law] The Dulong–Petit law, a thermodynamic rule proposed in 1819 by French physicists Pierre Louis Dulong and Alexis Thérèse Petit, states the classical expression for the molar specific heat capacity of certain chemical elements. Experimentally the two scientists had found that the heat capacity per weight (the mass-specific heat capacity) for a number of elements was
  • 8. [Black-body radiation] Black-body radiation is the type of electromagnetic radiation within or surrounding a body in thermodynamic equilibrium with its environment, or emitted by a black body (an opaque and non-reflective body) held at constant, uniform temperature. The radiation has a specific spectrum and intensity that depends only on the temperature of the body.
  • 9. [Radiance] In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a surface, per unit solid angle per unit projected area, and spectral radiance is the radiance of a surface per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. These are directional
  • 10. [Thermodynamic temperature] Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics.
    Thermodynamic temperature is defined by the third law of thermodynamics in which the theoretically lowest temperature is the null or zero point. At this point, absolute zero, the particle constituents of matter have minimal motion and can become no
  • 11. [Ludwig Boltzmann] Ludwig Eduard Boltzmann (February 20, 1844 – September 5, 1906) was an Austrian physicist and philosopher whose greatest achievement was in the development of statistical mechanics, which explains and predicts how the properties of atoms (such as mass, charge, and structure) determine the physical properties of matter (such as viscosity, thermal conductivity, and diffusion).
  • 12. [Claude Pouillet] Claude Servais Mathias Pouillet (16 February 1790 – 14 June 1868) was a French physicist and a professor of physics at the Sorbonne and member of the French Academy of Science. Succeeding Dulong, he became the fourth holder of the chair of physics at Polytechnique for a brief period of time (1830-1831) but chose to resign, citing health reasons. He was succeeded by César Despretz in 1831 and Gabriel Lamé in 1832.
  • 13. [Boltzmann constant] The Boltzmann constant (kB or k), named after Ludwig Boltzmann, is a physical constant relating energy at the individual particle level with temperature. It is the gas constant R divided by the Avogadro constant NA:
  • 14. [Hawking radiation] Hawking radiation is black body radiation that is predicted to be released by black holes, due to quantum effects near the event horizon. It is named after the physicist Stephen Hawking, who provided a theoretical argument for its existence in 1974, and sometimes also after Jacob Bekenstein, who predicted that black holes should have a finite, non-zero temperature and entropy.
  • 15. [Proportionality (mathematics)] In mathematics, two variables are proportional if a change in one is always accompanied by a change in the other, and if the changes are always related by use of a constant multiplier. The constant is called the coefficient of proportionality or proportionality constant.
  • 16. [Planck constant] The Planck constant (denoted h, also called Planck's constant) is a physical constant that is the quantum of action in quantum mechanics. Published in 1900, it originally described the proportionality constant between the energy, E, of a charged atomic oscillator in the wall of a black body, and the frequency, ν, of its associated electromagnetic
  • 17. [Polylogarithm] In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Lis(z) of order s and argument z. Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or rational functions. In quantum statistics, the polylogarithm function appears as the closed
  • 18. [Black hole thermodynamics] In physics, black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. As the study of the statistical mechanics of black body radiation led to the advent of the theory of quantum mechanics, the effort to understand the statistical mechanics of black holes has had a deep impact upon the understanding of quantum gravity, leading to the formulation of the holographic principle.
  • 19. [John Tyndall] John Tyndall FRS (2 August 1820 – 4 December 1893) was a prominent 19th century physicist. His initial scientific fame arose in the 1850s from his study of diamagnetism. Later he made discoveries in the realms of infrared radiation and the physical properties of air. Tyndall also published more than a dozen science books which
  • 20. [Heat engine] In thermodynamics, a heat engine is a system that converts heat or thermal energy to mechanical energy, which can then be used to do mechanical work. It does this by bringing a working substance from a higher state temperature to a lower state temperature. A heat "source" generates thermal energy that brings the working substance
  • 21. [Charles Soret] Charles Soret (born 23 September 1854, Geneva, Switzerland; died 4 April 1904) was a Swiss physicist and chemist. He is known for his work on thermodiffusion (the so-called Soret effect).
  • 22. [Absorption (electromagnetic radiation)] In physics, absorption of electromagnetic radiation is the way in which the energy of a photon is taken up by matter, typically the electrons of an atom. Thus, the electromagnetic energy is transformed into internal energy of the absorber, for example thermal energy. The reduction in intensity of a light wave propagating through a medium
  • 23. [Riemann zeta function] The Riemann zeta function or Euler–Riemann zeta function, ζ(s), is a function of a complex variable s that analytically continues the sum of the infinite series
  • 24. [Kelvin] The kelvin is a unit of measure for temperature based upon an absolute scale. It is one of the seven base units in the International System of Units (SI) and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at
  • 25. [Luminosity] In astronomy, luminosity is the total amount of energy emitted by a star, galaxy, or other astronomical object per unit time. It is related to the brightness, which is the luminosity of an object in a given spectral region.
  • 26. [Effective temperature] The effective temperature of a body such as a star or planet is the temperature of a black body that would emit the same total amount of electromagnetic radiation. Effective temperature is often used as an estimate of a body's temperature when the body's emissivity curve (as a function of wavelength) is not known.
  • 27. [Albedo] Albedo (/ælˈbd/), or reflection coefficient, derived from Latin albedo "whiteness" (or reflected sunlight) in turn from albus "white", is the diffuse reflectivity or reflecting power of a surface.
  • 28. [Speed of light] The speed of light in vacuum, commonly denoted c, is a universal physical constant important in many areas of physics. Its value is exactly 7008299792458000000♠299792458 metres per second (≈7008300000000000000♠3.00×108 m/s), as the length of the metre is defined from this constant and the international standard for time. According to special relativity, c is the maximum
  • 29. [Solar radius] Solar radius is a unit of distance used to express the size of stars in astronomy equal to the current radius of the Sun:
    The solar radius is approximately 695,500 kilometres (432,450 miles), which is about 10 times the average radius of Jupiter, 110 times the radius of the Earth, and 1/215th of an astronomical unit,
  • 30. [Dimensionless physical constant] In physics, a dimensionless physical constant, sometimes called fundamental physical constant, is a physical constant that is dimensionless – having no units attached, having a numerical value that is the same under all possible systems of units. A common example is the fine-structure constant α, with approximate value 6997729735257000000♠7.29735257×10−3.
  • 31. [Methods of contour integration] In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.
    One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods.
  • 32. [Joule] The joule (/ˈl/), symbol J, is a derived unit of energy in the International System of Units. It is equal to the energy transferred (or work done) to an object when a force of one newton acts on that object in the direction of its motion through a distance of one metre (1 newton metre
  • 33. [Spherical coordinate system] In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.
  • 34. [International System of Units] The International System of Units (French: Système International d'Unités, SI) is the modern form of the metric system, and is the most widely used system of measurement. It comprises a coherent system of units of measurement built on seven base units. It defines twenty-two named units, and includes many more unnamed coherent derived units. The
  • 35. [Black hole] A black hole is a geometrically defined region of spacetime exhibiting such strong gravitational effects that nothing—including particles and electromagnetic radiation such as light—can escape from inside it. The theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole. The boundary of the region from which no
  • 36. [Thermodynamics] Thermodynamics is a branch of physics concerned with heat and temperature and their relation to energy and work. It defines macroscopic variables, such as internal energy, entropy, and pressure, that partly describe a body of matter or radiation. It states that the behavior of those variables is subject to general constraints, that are common to
  • 37. [Metamaterial] Metamaterials are materials engineered to have properties that have not yet been found in nature. They are made from assemblies of multiple elements fashioned from conventional materials such as metals or plastics. The materials are usually arranged in repeating patterns, often at microscopic or smaller scales that are smaller than the wavelengths of the phenomena
  • 38. [Water vapor] Water vapor, or water vapour or aqueous vapor, is the gaseous phase of water. It is one state of water within the hydrosphere. Water vapor can be produced from the evaporation or boiling of liquid water or from the sublimation of ice. Unlike other forms of water, water vapor is invisible. Under typical atmospheric conditions,
  • 39. [Celsius] Celsius, historically known as centigrade, is a scale and unit of measurement for temperature. As an SI derived unit, it is used by most countries in the world. It is named after the Swedish astronomer Anders Celsius (1701–1744), who developed a similar temperature scale. The degree Celsius (°C) can refer to a specific temperature on
  • 40. [Light] Light is electromagnetic radiation within a certain portion of the electromagnetic spectrum. The word usually refers to visible light, which is visible to the human eye and is responsible for the sense of sight. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), or 6993400000000000000♠400×10−9 m to 6993700000000000000♠700×10−9 m,
  • 41. [Energy] In physics, energy is a property of objects which can be transferred to other objects or converted into different forms, but cannot be created or destroyed. The "ability of a system to perform work" is a common description, but it is difficult to give one single comprehensive definition of energy because of its many forms.
  • 42. [Methane] Methane (/ˈmɛθn/ or /ˈmθn/) is a chemical compound with the chemical formula CH4 (one atom of carbon and four atoms of hydrogen). It is the simplest alkane and the main component of natural gas. The relative abundance of methane on Earth makes it an attractive fuel, though capturing and storing it poses challenges due to
  • 43. [Time] Time is a measure in which events can be ordered from the past through the present into the future, and also the measure of durations of events and the intervals between them. Time is often referred to as the fourth dimension, along with the three spatial dimensions.
  • 44. [Sun] The Sun (in Greek: Helios, in Latin: Sol) is the star at the center of the Solar System and is by far the most important source of energy for life on Earth. It is a nearly perfect spherical ball of hot plasma, with internal convective motion that generates a magnetic field via a dynamo process.
  • 45. [Taylor series] In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
  • 46. [Atmosphere of Earth] The atmosphere of Earth is the layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention (greenhouse effect), and reducing temperature extremes between day and night (the diurnal temperature variation).
  • 47. [Greenhouse gas] A greenhouse gas (sometimes abbreviated GHG) is a gas in an atmosphere that absorbs and emits radiation within the thermal infrared range. This process is the fundamental cause of the greenhouse effect. The primary greenhouse gases in Earth's atmosphere are water vapor, carbon dioxide, methane, nitrous oxide, and ozone. Without greenhouse gases, the average temperature
  • 48. [Differentiable function] In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. As a result, the graph of a differentiable function must have a (non-vertical) tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or cusps.
  • 49. [Convex hull] In mathematics, the convex hull or convex envelope of a set X of points in the Euclidean plane or Euclidean space is the smallest convex set that contains X. For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around X.
  • 50. [Watt] The watt (symbol: W) is a derived unit of power in the International System of Units (SI), named after the Scottish engineer James Watt (1736–1819). The unit is defined as joule per second and can be used to express the rate of energy conversion or transfer with respect to time. It has dimensions of L2MT−3.
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