# 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 radiated over all wavelengths increases as the temperature increases", of a black body which is proportional to the fourth power of the thermodynamic temperature.[1] The theory of thermal radiation lays down the theory of quantum mechanics, by using physics to relate to molecular, atomic and sub-atomic levels. Slovenian physicist Josef Stefan formulated the constant in 1879, and it was later derived in 1884 by Austrian physicist Ludwig Boltzmann.[2] The equation can also be derived from Planck's law, by integrating over all wavelengths at a given temperature, which will represent a small flat black body box.[3] "The amount of thermal radiation emitted increases rapidly and the principal frequency of the radiation becomes higher with increasing temperatures".[4] The Stefan–Boltzmann constant can be used to measure the amount of heat that is emitted by a blackbody, which absorbs all of the radiant energy that hits it, and will emit all the radiant energy. Furthermore, the Stefan–Boltzmann constant allows for temperature (K) to be converted to units for intensity (W m−2), which is power per unit area.

The value of the Stefan–Boltzmann constant is given in SI units by

σ = 5.670367(13)×10−8 W⋅m−2⋅K−4.[5]

In cgs units the Stefan–Boltzmann constant is:

σ5.6704×10−5 erg cm−2 s−1 K−4.

In thermochemistry the Stefan–Boltzmann constant is often expressed in cal cm−2 day−1 K−4:

σ11.7×10−8 cal cm−2 day−1 K−4.

In US customary units the Stefan–Boltzmann constant is:[6]

σ1.714×10−9 BTU hr−1 ft−2 °R−4.

The value of the Stefan–Boltzmann constant is derivable as well as experimentally determinable; see Stefan–Boltzmann law for details. It can be defined in terms of the Boltzmann constant as:

${\displaystyle \sigma ={\frac {2\pi ^{5}k_{\rm {B}}^{4}}{15h^{3}c^{2}}}={\frac {\pi ^{2}k_{\rm {B}}^{4}}{60\hbar ^{3}c^{2}}}=5.670367(13)\,\times 10^{-8}\ {\textrm {J}}\,{\textrm {m}}^{-2}\,{\textrm {s}}^{-1}\,{\textrm {K}}^{-4}}$

where:

The CODATA recommended value is calculated from the measured value of the gas constant:

${\displaystyle \sigma ={\frac {2\pi ^{5}R^{4}}{15h^{3}c^{2}N_{\rm {A}}^{4}}}={\frac {32\pi ^{5}hR^{4}R_{\infty }^{4}}{15A_{\rm {r}}({\rm {e}})^{4}M_{\rm {u}}^{4}c^{6}\alpha ^{8}}}}$

where:

Dimensional formula: [ M1 T−3 Θ−4]

A related constant is the radiation constant (or radiation density constant) a which is given by:[7]

${\displaystyle a={\frac {4\sigma }{c}}=7.5657\times 10^{-15}{\textrm {erg}}\,{\textrm {cm}}^{-3}\,{\textrm {K}}^{-4}=7.5657\times 10^{-16}{\textrm {J}}\,{\textrm {m}}^{-3}\,{\textrm {K}}^{-4}.}$
Log-log graphs of peak emission wavelength and radiant exitance vs black-body temperature – red arrows show that 5780 K black bodies have 501 nm peak wavelength and 63.3 MW/m² radiant exitance

## References

1. ^ Krane, Kenneth (2012). Modern Physics. John Wiley & Sons. p. 81.
2. ^ "Stefan-Boltzmann Law". Encyclopædia Britannica.
3. ^ Halliday & Resnick (2014). Fundamentals of Physics (10th Ed). John Wiley and Sons. p. 1166.
4. ^ Eisberg, Resnick, Robert, Robert (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd Ed) (PDF). John Wiley & Sons. Archived from the original (PDF) on 2014-02-26.
5. ^ "CODATA Value: Stefan-Boltzmann constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
6. ^ Heat and Mass Transfer: a Practical Approach, 3rd Ed. Yunus A. Çengel, McGraw Hill, 2007
7. ^ Radiation constant from ScienceWorld
Boltzmann (disambiguation)

Ludwig Boltzmann was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics.

Boltzmann may also refer to:

24712 Boltzmann, a main-belt asteroid

Boltzmann constant

Boltzmann (crater), an old lunar crater

Boltzmann distribution

Boltzmann equation

Boltzmann's entropy formula

Boltzmann relation

Stefan–Boltzmann law

Stefan–Boltzmann constant

Cloud forcing

Cloud forcing (sometimes described as cloud radiative forcing or cloud radiative effect) is, in meteorology, the difference between the radiation budget components for average cloud conditions and cloud-free conditions. Much of the interest in cloud forcing relates to its role as a feedback process in the present period of global warming.

Effect of radiation on perceived temperature

The “radiation effect” results from radiation heat exchange between human bodies and surrounding surfaces, such as walls and ceilings. It may lead to phenomena such as houses feeling cooler in the winter and warmer in the summer at the same temperature. For example, in a room in which air temperature is maintained at 22° Celsius at all times, but in which the inner surfaces of the house is estimated to be an average temperature of 10° Celsius in the winter or 25° Celsius in the summer, heat transfer from the surfaces to the individual will occur, resulting in a difference in the perceived temperature.

We can observe and compare the rate of radiation heat transfer between a person and the surrounding surfaces if we first make a few simplifying assumptions:

The heat exchange in the environment is in a “steady state”, meaning that there is a constant flow of heat either into or out of the house.

The person is completely surrounded by the interior surfaces of the room.

Heat transfer by convection is not considered.

The walls, ceiling, and floor are all at the same temperature.For an average person, the outer surface area is 1.4 m², the surface temperature is 30° Celsius, and the emissivity (ε) is 0.95. Emissivity is the ability of a surface to emit radiant energy compared to that of a black body at the same temperature.

We will be using the following equation to find out how much heat is lost by a person standing in the same room in summertime as compared to the winter, at exactly the same thermostat reading temperature:

Q ̇=εσA_s (T_s^4-T_surr^4)

Where Q ̇ is the rate of heat loss (W), ε is the emissivity (or the ability of an objects surface to emit energy by radiation) of a person, σ is the Stefan-Boltzmann constant (5.670x〖10〗^(-8 )W/m2∙K4), As is the surface area of a person, Ts is the surface temperature of a person (K), and Tsurr is the surface temperature of the walls, ceiling, and floor (K). Please note that this equation is only valid for an object standing in a completely enclosed room, box, etc.In the winter, the amount of heat loss from a person, when the inner surfaces of the room were 10 degrees Celsius, was found to be 152 Watts.

(Q ̇=(0.95)(5.67x〖10〗^(-8) )(1.4)[(30+273)^4-(10+273)^4 ]=152) ̇

In the summer, the amount of heat loss from a person, when the inner surfaces of the room were 25 degrees Celsius, was found to be 40.9 Watts.

(Q ̇=(0.95)(5.67x〖10〗^(-8) )(1.4)[(30+273)^4-(25+273)^4 ]=40.9) ̇

Thermal radiation is the form of radiation emitted by bodies because of their temperature.

It differs from other forms of electromagnetic radiation such as x-rays, gamma rays, microwaves, radio waves, and television rays that are not related to temperature. Scientists have found that all bodies at a temperature above absolute zero emit thermal radiation. People are constantly radiating their body heat, but at different rates. From these values, the rate of heat loss from a person is almost four times as large in the winter than in the summer, which explains the “chill” we feel in the winter even if the thermostat setting is kept the same.

HD 85512

HD 85512 is a solitary K-type main-sequence star located approximately 36 light-years away in the constellation Vela. It is approximately one billion years older than the Sun. It is extremely chromospherically inactive, only slightly more active than Tau Ceti. The star is known to host one low-mass planet.

Heat current

A heat current is a kinetic exchange rate between molecules, relative to the material in which the kinesis occurs. It is defined as ${\displaystyle {\frac {dQ}{dt}}}$, where ${\displaystyle Q}$ is heat and ${\displaystyle t}$ is time.

For conduction, heat current is defined by Fourier's law as

${\displaystyle {\frac {\partial Q}{\partial t}}=-k\oint _{S}{{\overrightarrow {\nabla }}T\cdot \,{\overrightarrow {dS}}}}$

where

${\displaystyle {\big .}{\frac {\partial Q}{\partial t}}{\big .}}$ is the amount of heat transferred per unit time [W] and
${\displaystyle {\overrightarrow {dS}}}$ is an oriented surface area element [m2]

The above differential equation, when integrated for a homogeneous material of 1-D geometry between two endpoints at constant temperature, gives the heat flow rate as:

${\displaystyle {\big .}{\frac {\Delta Q}{\Delta t}}=-kA{\frac {\Delta T}{\Delta x}}}$

where

A is the cross-sectional surface area,
${\displaystyle \Delta T}$ is the temperature difference between the ends,
${\displaystyle \Delta x}$ is the distance between the ends.

For thermal radiation, heat current is defined as

${\displaystyle W=\sigma \cdot A\cdot T^{4}}$

where the constant of proportionality ${\displaystyle \sigma }$ is the Stefan–Boltzmann constant, ${\displaystyle A}$ is the radiating surface area, and ${\displaystyle T}$ is temperature.

Heat current can also be thought of as the total phonon distribution multiplied by the energy of one phonon, times the group velocity of the phonons. The phonon distribution of a particular phonon mode is given by the Bose-Einstein factor, which is dependent on temperature and phonon energy.

Josef Stefan

Josef Stefan (Slovene: Jožef Štefan; 24 March 1835 – 7 January 1893) was an ethnic Carinthian Slovene physicist, mathematician, and poet of the Austrian Empire.

Lacaille 9352

Lacaille 9352 (Lac 9352) is a star in the southern constellation of Piscis Austrinus. With an apparent visual magnitude of 7.34, this star is too faint to be viewed with the naked eye even under excellent seeing conditions. Parallax measurements place it at a distance of about 10.74 light-years (3.29 parsecs) from Earth. It is the eleventh closest star system to the Solar System and is the closest star in the constellation Piscis Austrinus. The ChView simulation shows that its closest neighbour is the EZ Aquarii

triple star system at about 4.1 ly from Lacaille 9352.

List of letters used in mathematics and science

the symbol of time is seconds and minutes.Latin and Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities.

Some common conventions:Intensive quantities in physics are usually denoted with minusculeswhile extensive are denoted with capital letters.

Most symbols are written in italics.

Vectors can be denoted in boldface.

Sets of numbers are typically bold or blackboard bold.

List of scientific constants named after people

This is a list of physical and mathematical constants named after people.Eponymous constants and their influence on scientific citations have been discussed in the literature.

• Reduced Planck constant or Dirac constant (${\displaystyle h}$-bar, ħ) – Max Planck, Paul Dirac
List of scientists whose names are used in physical constants

Some of the constants used in science are named after great scientists. By this convention, their names are immortalised. Below is the list of the scientists whose names are used in physical constants.

List of things named after Ludwig Boltzmann

This refers to a list of things named after physicist Ludwig Eduard Boltzmann (February 20, 1844 – September 5, 1906)

Luminosity

In astronomy, luminosity is the total amount of energy emitted per unit of time by a star, galaxy, or other astronomical object. As a term for energy emitted per unit time, luminosity is synonymous with power.In SI units luminosity is measured in joules per second or watts. Values for luminosity are often given in the terms of the luminosity of the Sun, L⊙. Luminosity can also be given in terms of the astronomical magnitude system: the absolute bolometric magnitude (Mbol) of an object is a logarithmic measure of its total energy emission rate, while absolute magnitude is a logarithmic measure of the luminosity within some specific wavelength range or filter band.

In contrast, the term brightness in astronomy is generally used to refer to an object's apparent brightness: that is, how bright an object appears to an observer. Apparent brightness depends on both the luminosity of the object and the distance between the object and observer, and also on any absorption of light along the path from object to observer. Apparent magnitude is a logarithmic measure of apparent brightness.

Opacity (optics)

Opacity is the measure of impenetrability to electromagnetic or other kinds of radiation, especially visible light. In radiative transfer, it describes the absorption and scattering of radiation in a medium, such as a plasma, dielectric, shielding material, glass, etc. An opaque object is neither transparent (allowing all light to pass through) nor translucent (allowing some light to pass through). When light strikes an interface between two substances, in general some may be reflected, some absorbed, some scattered, and the rest transmitted (also see refraction). Reflection can be diffuse, for example light reflecting off a white wall, or specular, for example light reflecting off a mirror. An opaque substance transmits no light, and therefore reflects, scatters, or absorbs all of it. Both mirrors and carbon black are opaque. Opacity depends on the frequency of the light being considered. For instance, some kinds of glass, while transparent in the visual range, are largely opaque to ultraviolet light. More extreme frequency-dependence is visible in the absorption lines of cold gases. Opacity can be quantified in many ways; for example, see the article mathematical descriptions of opacity.

Different processes can lead to opacity including absorption, reflection, and scattering.

Planetary equilibrium temperature

The planetary equilibrium temperature is a theoretical temperature that a planet would be at when considered simply as if it were a black body being heated only by its parent star. In this model, the presence or absence of an atmosphere (and therefore any greenhouse effect) is not considered, and one treats the theoretical black body temperature as if it came from an idealized surface of the planet.

Other authors use different names for this concept, such as equivalent blackbody temperature of a planet, or the effective radiation emission temperature of the planet. Similar concepts include the global mean temperature, global radiative equilibrium, and global-mean surface air temperature, which includes the effects of global warming.

The first and second radiation constants c1 and c2 – see Planck's Law

The radiation density constant a – see Stefan–Boltzmann constant

Standard asteroid physical characteristics

For the majority of numbered asteroids, almost nothing is known apart from a few physical parameters and orbital elements and some physical characteristics are often only estimated. The physical data is determined by making certain standard assumptions.

Stefan–Boltzmann law

The Stefan–Boltzmann 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 ${\displaystyle j^{\star }}$ (also known as the black-body radiant emittance) is directly proportional to the fourth power of the black body's thermodynamic temperature T:

${\displaystyle j^{\star }=\sigma T^{4}.}$

The constant of proportionality σ, called the Stefan–Boltzmann constant, is derived from other known physical constants. The value of the constant is

${\displaystyle \sigma ={\frac {2\pi ^{5}k^{4}}{15c^{2}h^{3}}}=5.670373\times 10^{-8}\,\mathrm {W\,m^{-2}K^{-4}} ,}$

where k is the Boltzmann constant, h is Planck's constant, and c is the speed of light in a vacuum. The radiance (watts per square metre per steradian) is given by

${\displaystyle L={\frac {j^{\star }}{\pi }}={\frac {\sigma }{\pi }}T^{4}.}$

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, ${\displaystyle \varepsilon <1}$:

${\displaystyle j^{\star }=\varepsilon \sigma T^{4}.}$

The radiant emittance ${\displaystyle j^{\star }}$ 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. ${\displaystyle \varepsilon }$ is the emissivity of the grey body; if it is a perfect blackbody, ${\displaystyle \varepsilon =1}$. In the still more general (and realistic) case, the emissivity depends on the wavelength, ${\displaystyle \varepsilon =\varepsilon (\lambda )}$.

To find the total power radiated from an object, multiply by its surface area, ${\displaystyle A}$:

${\displaystyle P=Aj^{\star }=A\varepsilon \sigma T^{4}.}$

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.