The Stefan–Boltzmann law describes the power radiated from a
black body
A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The name "black body" is given because it absorbs all colors of light. A black body ...
in terms of its
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
. Specifically, the Stefan–Boltzmann law states that the total
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
radiated per unit
surface area of a
black body
A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The name "black body" is given because it absorbs all colors of light. A black body ...
across
all wavelengths per unit
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
(also known as the black-body ''
radiant emittance
In radiometry, radiant exitance or radiant emittance is the radiant flux emitted by a surface per unit area, whereas spectral exitance or spectral emittance is the radiant exitance of a surface per unit frequency or wavelength, depending on whether ...
'') is directly
proportional to the fourth power of the black body's
thermodynamic temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
''T'':
:
The
constant of proportionality
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...
''σ'', called the
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 inc ...
, is derived from other known
physical constants.
Since 2019, the value of the constant is
:
where ''k'' is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
, ''h'' is
Planck's constant, and ''c'' is
the speed of light in a vacuum. The
radiance
In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Radiance is used to characterize diffuse emission and reflection of electromagnetic radiati ...
from a specified angle of view (watts per square metre per
steradian
The steradian (symbol: sr) or square radian is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radian ...
) 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 emissivity of the surface of a material is its effectiveness in emitting energy as thermal radiation. Thermal radiation is electromagnetic radiation that most commonly includes both visible radiation (light) and infrared radiation, which is n ...
,
:
:
The radiant emittance
has
dimensions
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
of
energy flux Energy flux is the rate of transfer of energy through a surface. The quantity is defined in two different ways, depending on the context:
# Total rate of energy transfer (not per unit area); SI units: W = J⋅s−1.
# Specific rate of energy transfe ...
(energy per unit time per unit area), and the
SI units
The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
of measure are
joule
The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force applied ...
s per second per square metre, or equivalently,
watt
The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James Wa ...
s per square metre. The SI unit for absolute temperature ''T'' is the
kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...
. ''
'' is the
emissivity
The emissivity of the surface of a material is its effectiveness in emitting energy as thermal radiation. Thermal radiation is electromagnetic radiation that most commonly includes both visible radiation (light) and infrared radiation, which is n ...
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
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may a ...
radiated from an object, multiply by its surface area,
:
:
Wavelength- and subwavelength-scale particles,
[
] metamaterial
A metamaterial (from the Greek word μετά ''meta'', meaning "beyond" or "after", and the Latin word ''materia'', meaning "matter" or "material") is any material engineered to have a property that is not found in naturally occurring materials. ...
s, and other nanostructures are not subject to ray-optical limits and may be designed to exceed the Stefan–Boltzmann law.
History
In 1864,
John Tyndall presented measurements of the infrared emission by a platinum filament and the corresponding color of the filament.
The proportionality to the fourth power of the absolute temperature was deduced by
Josef Stefan
Josef Stefan ( sl, Jožef Štefan; 24 March 1835 – 7 January 1893) was an ethnic Carinthian Slovene physicist, mathematician, and poet of the Austrian Empire.
Life and work
Stefan was born in an outskirt village of St. Peter (Slovene: ; to ...
(1835–1893) in 1877 on the basis of Tyndall's experimental measurements, in the article ''Über die Beziehung zwischen der Wärmestrahlung und der Temperatur'' (''On the relationship between thermal radiation and temperature'') in the ''Bulletins from the sessions'' of the Vienna Academy of Sciences.
A derivation of the law from theoretical considerations was presented by
Ludwig Boltzmann
Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodyn ...
(1844–1906) in 1884, drawing upon the work of
Adolfo Bartoli
Adolfo Bartoli (19 March 1851 – 18 July 1896) was an Italian physicist, who is best known for introducing the concept of radiation pressure from thermodynamical considerations.
Born in Florence, Bartoli studied physics and mathematics at the Un ...
.
Bartoli in 1876 had derived the existence of
radiation pressure
Radiation pressure is the mechanical pressure exerted upon any surface due to the exchange of momentum between the object and the electromagnetic field. This includes the momentum of light or electromagnetic radiation of any wavelength that is a ...
from the principles of
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
. Following Bartoli, Boltzmann considered an ideal
heat engine
In thermodynamics and engineering, a heat engine is a system that converts heat 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 ...
using electromagnetic radiation instead of an ideal gas as working matter.
The law was almost immediately experimentally verified.
Heinrich Weber in 1888 pointed out deviations at higher temperatures, but perfect accuracy within measurement uncertainties was confirmed up to temperatures of 1535 K by 1897.
The law, including the theoretical prediction of the
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 inc ...
as a function of the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
, the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
and
Planck's constant, is a
direct consequence of
Planck's law as formulated in 1900.
As of the
2019 redefinition of SI base units, which fixes the values of the Boltzmann constant ''k'', the Planck constant ''h'', and the speed of light ''c'', the Stefan–Boltzmann constant is exactly
:
:σ = .
Examples
Temperature of the Sun
With his law, Stefan also determined the temperature of the
Sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
's surface. He inferred from the data of
Jacques-Louis Soret
Jacques-Louis Soret (30 June 1827 – 13 May 1890) was a Swiss chemist and spectroscopist. He studied both spectroscopy and electrolysis. He held the chairs of chemistry (1873-1887) and medical physics (1887-1890) at the University of Genev ...
(1827–1890) that the energy flux density from the Sun is 29 times greater than the energy flux density of a certain warmed metal
lamella
Lamella (plural lamellae) means a small plate or flake in Latin, and in English may refer to:
Biology
* Lamella (mycology), a papery rib beneath a mushroom cap
* Lamella (botany)
* Lamella (surface anatomy), a plate-like structure in an animal
...
(a thin plate). A round lamella was placed at such a distance from the measuring device that it would be seen at the same
angular diameter
The angular diameter, angular size, apparent diameter, or apparent size is an angular distance describing how large a sphere or circle appears from a given point of view. In the vision sciences, it is called the visual angle, and in optics, it is ...
as the Sun. Soret estimated the temperature of the lamella to be approximately 1900
°C
The degree Celsius is the unit of temperature on the Celsius scale (originally known as the centigrade scale outside Sweden), one of two temperature scales used in the International System of Units (SI), the other being the Kelvin scale. The ...
to 2000 °C. Stefan surmised that ⅓ of the energy flux from the Sun is absorbed by the
Earth's atmosphere
The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing for ...
, so he took for the correct Sun's energy flux a value 3/2 times greater than Soret's value, namely 29 × 3/2 = 43.5.
Precise measurements of atmospheric
absorption were not made until 1888 and 1904. The temperature Stefan obtained was a median value of previous ones, 1950 °C and the absolute thermodynamic one 2200 K. As 2.57
4 = 43.5, it follows from the law that the temperature of the Sun is 2.57 times greater than the temperature of the lamella, so Stefan got a value of 5430 °C or 5700 K. This was the first sensible value for the temperature of the Sun. Before this, values ranging from as low as 1800 °C to as high as 13,000,000 °C were claimed. The lower value of 1800 °C was determined by
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 Sciences (elected 1837).
Biography
He studied sciences at the École n ...
(1790–1868) in 1838 using the
Dulong–Petit law
The Dulong–Petit law, a thermodynamic law proposed by French physicists Pierre Louis Dulong and Alexis Thérèse Petit, states that the classical expression for the molar specific heat capacity of certain chemical elements is constant for tempe ...
. Pouillet also took just half the value of the Sun's correct energy flux.
Temperature of stars
The temperature of
star
A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
s other than the Sun can be approximated using a similar means by treating the emitted energy as a
black body
A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The name "black body" is given because it absorbs all colors of light. A black body ...
radiation.
So:
:
where ''L'' is the
luminosity
Luminosity is an absolute measure of radiated electromagnetic power (light), the radiant power emitted by a light-emitting object over time. In astronomy, luminosity is the total amount of electromagnetic energy emitted per unit of time by a st ...
, ''σ'' is the
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 inc ...
, ''R'' is the stellar radius and ''T'' is the
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 surface temperature ...
. This formula can then be rearranged to calculate the temperature:
: