Black-body radiation

As the temperature decreases the peak of the black-body radiation curve moves to lower intensities andlonger wavelengths The black-body radiation graph isalso compared with the classical model of Rayleigh andJeans

Black-body radiation is the thermalelectromagnetic radiation within orsurrounding a body in thermodynamicequilibrium with its environment or emittedby a black body (an opaque and non-reflective body) It has a specific spectrumand intensity that depends only on the

The color (chromaticity) of black-body radiationdepends on the temperature of the black body thelocus of such colors shown here in CIE 1931 xy spaceis known as the Planckian locus

bodys temperature which is assumed forthe sake of calculations and theory to beuniform and constant[1][2][3][4]

The thermal radiation spontaneouslyemitted by many ordinary objects can beapproximated as black-body radiation Aperfectly insulated enclosure that is inthermal equilibrium internally containsblack-body radiation and will emit it througha hole made in its wall provided the hole issmall enough to have negligible effect uponthe equilibrium

A black-body at room temperature appearsblack as most of the energy it radiates isinfra-red and cannot be perceived by thehuman eye Because the human eye cannotperceive light waves at lower frequencies a

black body viewed in the dark at the lowestjust faintly visible temperature subjectivelyappears grey even though its objectivephysical spectrum peaks in the infraredrange[5] When it becomes a little hotter itappears dull red As its temperatureincreases further it becomes yellow whiteand ultimately blue-white

Although planets and stars are neither inthermal equilibrium with their surroundingsnor perfect black bodies black-bodyradiation is used as a first approximation forthe energy they emit[6] Black holes are near-perfect black bodies in the sense that theyabsorb all the radiation that falls on them Ithas been proposed that they emit black-body radiation (called Hawking radiation)

with a temperature that depends on themass of the black hole[7]

The term black body was introduced byGustav Kirchhoff in 1860[8] Black-bodyradiation is also called thermal radiationcavity radiation complete radiation ortemperature radiation

Black-body radiation has a characteristiccontinuous frequency spectrum thatdepends only on the bodys temperature[9]

called the Planck spectrum or Plancks lawThe spectrum is peaked at a characteristicfrequency that shifts to higher frequencieswith increasing temperature and at roomtemperature most of the emission is in the

Spectrum

infrared region of the electromagneticspectrum[10][11][12] As the temperatureincreases past about 500 degrees Celsiusblack bodies start to emit significantamounts of visible light Viewed in the darkby the human eye the first faint glowappears as a ghostly grey (the visible lightis actually red but low intensity lightactivates only the eyes grey-level sensors)With rising temperature the glow becomesvisible even when there is some backgroundsurrounding light first as a dull red thenyellow and eventually a dazzling bluish-white as the temperature rises[13][14] Whenthe body appears white it is emitting asubstantial fraction of its energy asultraviolet radiation The Sun with aneffective temperature of approximately

5800 K[15] is an approximate black bodywith an emission spectrum peaked in thecentral yellow-green part of the visiblespectrum but with significant power in theultraviolet as well

Black-body radiation provides insight intothe thermodynamic equilibrium state ofcavity radiation If each Fourier mode of theequilibrium radiation in an otherwise emptycavity with perfectly reflective walls isconsidered as a degree of freedom capableof exchanging energy then according to theequipartition theorem of classical physicsthere would be an equal amount of energyin each mode Since there are an infinitenumber of modes this implies infinite heatcapacity (infinite energy at any non-zero

temperature) as well as an unphysicalspectrum of emitted radiation that growswithout bound with increasing frequency aproblem known as the ultravioletcatastrophe Instead in quantum theory theoccupation numbers of the modes arequantized cutting off the spectrum at highfrequency in agreement with experimentalobservation and resolving the catastropheThe study of the laws of black bodies andthe failure of classical physics to describethem helped establish the foundations ofquantum mechanics

Explanation

All normal (baryonic) matter emitselectromagnetic radiation when it has atemperature above absolute zero Theradiation represents a conversion of abodys thermal energy into electromagneticenergy and is therefore called thermalradiation It is a spontaneous process ofradiative distribution of entropy

Conversely all normal matter absorbselectromagnetic radiation to some degree

Color of a black body from 800 K to 12200 K This rangeof colors approximates the range of colors of stars ofdifferent temperatures as seen or photographed in thenight sky

An object that absorbs all radiation fallingon it at all wavelengths is called a blackbody When a black body is at a uniformtemperature its emission has acharacteristic frequency distribution thatdepends on the temperature Its emission iscalled black-body radiation

The concept of the black body is anidealization as perfect black bodies do notexist in nature[16] Graphite and lamp blackwith emissivities greater than 095 howeverare good approximations to a blackmaterial Experimentally black-bodyradiation may be established best as theultimately stable steady state equilibriumradiation in a cavity in a rigid body at auniform temperature that is entirely opaque

and is only partly reflective[16] A closed boxof graphite walls at a constant temperaturewith a small hole on one side produces agood approximation to ideal black-bodyradiation emanating from the opening[17][18]

Black-body radiation has the uniqueabsolutely stable distribution of radiativeintensity that can persist in thermodynamicequilibrium in a cavity[16] In equilibrium foreach frequency the total intensity ofradiation that is emitted and reflected froma body (that is the net amount of radiationleaving its surface called the spectralradiance) is determined solely by theequilibrium temperature and does notdepend upon the shape material orstructure of the body[19] For a black body (a

perfect absorber) there is no reflectedradiation and so the spectral radiance isentirely due to emission In addition a blackbody is a diffuse emitter (its emission isindependent of direction) Consequentlyblack-body radiation may be viewed as theradiation from a black body at thermalequilibrium

Black-body radiation becomes a visibleglow of light if the temperature of the objectis high enough The Draper point is thetemperature at which all solids glow a dimred about 798 K[20] At 1000 K a smallopening in the wall of a large uniformlyheated opaque-walled cavity (let us call it anoven) viewed from outside looks red at6000 K it looks white No matter how the

perfect absorber) there is no reflectedradiation and so the spectral radiance isentirely due to emission In addition a blackbody is a diffuse emitter (its emission isindependent of direction) Consequentlyblack-body radiation may be viewed as theradiation from a black body at thermalequilibrium

Black-body radiation becomes a visibleglow of light if the temperature of the objectis high enough The Draper point is thetemperature at which all solids glow a dimred about 798 K[20] At 1000 K a smallopening in the wall of a large uniformlyheated opaque-walled cavity (let us call it anoven) viewed from outside looks red at6000 K it looks white No matter how the

oven is constructed or of what material aslong as it is built so that almost all lightentering is absorbed by its walls it willcontain a good approximation to black-bodyradiation The spectrum and therefore colorof the light that comes out will be a functionof the cavity temperature alone A graph ofthe amount of energy inside the oven perunit volume and per unit frequency intervalplotted versus frequency is called the black-body curve Different curves are obtained byvarying the temperature

The temperature of a Pāhoehoe lava flow can beestimated by observing its color The result agrees well

Two bodies that are at the sametemperature stay in mutual thermalequilibrium so a body at temperature Tsurrounded by a cloud of light attemperature T on average will emit as muchlight into the cloud as it absorbs followingPrevosts exchange principle which refersto radiative equilibrium The principle ofdetailed balance says that inthermodynamic equilibrium everyelementary process works equally in itsforward and backward sense[21][22] Prevostalso showed that the emission from a bodyis logically determined solely by its owninternal state The causal effect of

with other measurements of temperatures of lava flowsat about 1000 to 1200 degC (1830 to 2190 degF)

thermodynamic absorption onthermodynamic (spontaneous) emission isnot direct but is only indirect as it affectsthe internal state of the body This meansthat at thermodynamic equilibrium theamount of every wavelength in everydirection of thermal radiation emitted by abody at temperature T black or not is equalto the corresponding amount that the bodyabsorbs because it is surrounded by light attemperature T[23]

When the body is black the absorption isobvious the amount of light absorbed is allthe light that hits the surface For a blackbody much bigger than the wavelength thelight energy absorbed at any wavelength λper unit time is strictly proportional to the

black-body curve This means that theblack-body curve is the amount of lightenergy emitted by a black body whichjustifies the name This is the condition forthe applicability of Kirchhoffs law ofthermal radiation the black-body curve ischaracteristic of thermal light whichdepends only on the temperature of thewalls of the cavity provided that the walls ofthe cavity are completely opaque and arenot very reflective and that the cavity is inthermodynamic equilibrium[24] When theblack body is small so that its size iscomparable to the wavelength of light theabsorption is modified because a smallobject is not an efficient absorber of light oflong wavelength but the principle of strictequality of emission and absorption is

black-body curve This means that theblack-body curve is the amount of lightenergy emitted by a black body whichjustifies the name This is the condition forthe applicability of Kirchhoffs law ofthermal radiation the black-body curve ischaracteristic of thermal light whichdepends only on the temperature of thewalls of the cavity provided that the walls ofthe cavity are completely opaque and arenot very reflective and that the cavity is inthermodynamic equilibrium[24] When theblack body is small so that its size iscomparable to the wavelength of light theabsorption is modified because a smallobject is not an efficient absorber of light oflong wavelength but the principle of strictequality of emission and absorption is

always upheld in a condition ofthermodynamic equilibrium

In the laboratory black-body radiation isapproximated by the radiation from a smallhole in a large cavity a hohlraum in anentirely opaque body that is only partlyreflective that is maintained at a constanttemperature (This technique leads to thealternative term cavity radiation) Any lightentering the hole would have to reflect offthe walls of the cavity multiple times beforeit escaped in which process it is nearlycertain to be absorbed Absorption occursregardless of the wavelength of theradiation entering (as long as it is smallcompared to the hole) The hole then is aclose approximation of a theoretical black

body and if the cavity is heated thespectrum of the holes radiation (ie theamount of light emitted from the hole ateach wavelength) will be continuous andwill depend only on the temperature and thefact that the walls are opaque and at leastpartly absorptive but not on the particularmaterial of which they are built nor on thematerial in the cavity (compare withemission spectrum)

Calculating the black-body curve was amajor challenge in theoretical physicsduring the late nineteenth century Theproblem was solved in 1901 by Max Planckin the formalism now known as Plancks lawof black-body radiation[25] By makingchanges to Wiens radiation law (not to be

confused with Wiens displacement law)consistent with thermodynamics andelectromagnetism he found a mathematicalexpression fitting the experimental datasatisfactorily Planck had to assume thatthe energy of the oscillators in the cavitywas quantized ie it existed in integermultiples of some quantity Einstein built onthis idea and proposed the quantization ofelectromagnetic radiation itself in 1905 toexplain the photoelectric effect Thesetheoretical advances eventually resulted inthe superseding of classicalelectromagnetism by quantumelectrodynamics These quanta were calledphotons and the black-body cavity wasthought of as containing a gas of photonsIn addition it led to the development of

quantum probability distributions calledFermindashDirac statistics and BosendashEinsteinstatistics each applicable to a differentclass of particles fermions and bosons

The wavelength at which the radiation isstrongest is given by Wiens displacementlaw and the overall power emitted per unitarea is given by the StefanndashBoltzmann lawSo as temperature increases the glow colorchanges from red to yellow to white to blueEven as the peak wavelength moves into theultra-violet enough radiation continues tobe emitted in the blue wavelengths that thebody will continue to appear blue It willnever become invisiblemdashindeed theradiation of visible light increasesmonotonically with temperature[26] The

StefanndashBoltzmann law also says that thetotal radiant heat energy emitted from asurface is proportional to the fourth powerof its absolute temperature The law wasformulated by Josef Stefan in 1879 andlater derived by Ludwig Boltzmann Theformula E = σT4 is given where E is theradiant heat emitted from a unit of area perunit time T is the absolute temperature andσ = 5670 367 times 10minus8 Wmiddotmminus2sdotKminus4 is theStefanndashBoltzmann constant [27]

The radiance or observed intensity is not afunction of direction Therefore a blackbody is a perfect Lambertian radiator

Real objects never behave as full-ideal blackbodies and instead the emitted radiation ata given frequency is a fraction of what the

ideal emission would be The emissivity of amaterial specifies how well a real bodyradiates energy as compared with a blackbody This emissivity depends on factorssuch as temperature emission angle andwavelength However it is typical inengineering to assume that a surfacesspectral emissivity and absorptivity do notdepend on wavelength so that theemissivity is a constant This is known asthe gray body assumption

9-year WMAP image (2012) of the cosmic microwavebackground radiation across the universe[28][29]

With non-black surfaces the deviationsfrom ideal black-body behavior aredetermined by both the surface structuresuch as roughness or granularity and thechemical composition On a perwavelength basis real objects in states oflocal thermodynamic equilibrium still followKirchhoffs Law emissivity equalsabsorptivity so that an object that does notabsorb all incident light will also emit lessradiation than an ideal black body theincomplete absorption can be due to someof the incident light being transmittedthrough the body or to some of it beingreflected at the surface of the body

In astronomy objects such as stars arefrequently regarded as black bodies though

this is often a poor approximation Analmost perfect black-body spectrum isexhibited by the cosmic microwavebackground radiation Hawking radiation isthe hypothetical black-body radiationemitted by black holes at a temperaturethat depends on the mass charge and spinof the hole If this prediction is correctblack holes will very gradually shrink andevaporate over time as they lose mass bythe emission of photons and other particles

A black body radiates energy at allfrequencies but its intensity rapidly tends tozero at high frequencies (shortwavelengths) For example a black body atroom temperature (300 K) with one squaremeter of surface area will emit a photon in

the visible range (390ndash750 nm) at anaverage rate of one photon every 41seconds meaning that for most practicalpurposes such a black body does not emitin the visible range

Plancks law of black-bodyradiation

Plancks law states that[30]

where

Bν(T) is the spectral radiance (the powerper unit solid angle and per unit of area

Equations

normal to the propagation) density offrequency ν radiation per unit frequencyat thermal equilibrium at temperature Th is the Planck constantc is the speed of light in a vacuumk is the Boltzmann constantν is the frequency of the electromagneticradiationT is the absolute temperature of the body

For a black body surface the spectralradiance density (defined per unit of areanormal to the propagation) is independentof the angle of emission with respect tothe normal However this means thatfollowing Lamberts cosine law is the radiance density perunit area of emitting surface as the surfacearea involved in generating the radiance is

increased by a factor with respectto an area normal to the propagationdirection At oblique angles the solid anglespans involved do get smaller resulting inlower aggregate intensities

Wiens displacement law

Wiens displacement law shows how thespectrum of black-body radiation at anytemperature is related to the spectrum atany other temperature If we know theshape of the spectrum at one temperaturewe can calculate the shape at any othertemperature Spectral intensity can beexpressed as a function of wavelength or offrequency

A consequence of Wiens displacement lawis that the wavelength at which the intensityper unit wavelength of the radiationproduced by a black body is at a maximum is a function only of the temperature

where the constant b known as Wiensdisplacement constant is equal to2897 7729(17) times 10minus3 K m[31]

Plancks law was also stated above as afunction of frequency The intensitymaximum for this is given by

[32]

StefanndashBoltzmann law

By integrating over the frequencythe integrated radiance is

by using with

and with

being the StefanndashBoltzmann constant Theradiance is then

per unit of emitting surface

On a side note at a distance d the intensity per area of radiating surface is theuseful expression

when the receiving surface is perpendicularto the radiation

By subsequently integrating over the solidangle (where ) the StefanndashBoltzmann law is calculated stating that thepower j emitted per unit area of the surfaceof a black body is directly proportional tothe fourth power of its absolutetemperature

by using

Human-body emission

Much of a persons energy is radiated away in theform of infrared light Some materials aretransparent in the infrared but opaque to visiblelight as is the plastic bag in this infrared image(bottom) Other materials are transparent to visiblelight but opaque or reflective in the infrarednoticeable by the darkness of the mans glasses

The human body radiates energy as infraredlight The net power radiated is thedifference between the power emitted andthe power absorbed

Applying the StefanndashBoltzmann law

where A and T are the body surface areaand temperature is the emissivity and T0

is the ambient temperature

The total surface area of an adult is about 2m2 and the mid- and far-infrared emissivityof skin and most clothing is near unity as itis for most nonmetallic surfaces[33][34] Skintemperature is about 33 degC[35] but clothing

reduces the surface temperature to about28 degC when the ambient temperature is20 degC[36] Hence the net radiative heat lossis about

The total energy radiated in one day is about8 MJ or 2000 kcal (food calories) Basalmetabolic rate for a 40-year-old male isabout 35 kcal(m2middoth)[37] which is equivalentto 1700 kcal per day assuming the same 2m2 area However the mean metabolic rateof sedentary adults is about 50 to 70greater than their basal rate[38]

There are other important thermal lossmechanisms including convection andevaporation Conduction is negligible ndash the

Nusselt number is much greater than unityEvaporation by perspiration is only requiredif radiation and convection are insufficientto maintain a steady-state temperature (butevaporation from the lungs occursregardless) Free-convection rates arecomparable albeit somewhat lower thanradiative rates[39] Thus radiation accountsfor about two-thirds of thermal energy lossin cool still air Given the approximatenature of many of the assumptions this canonly be taken as a crude estimate Ambientair motion causing forced convection orevaporation reduces the relative importanceof radiation as a thermal-loss mechanism

Application of Wiens law to human-bodyemission results in a peak wavelength of

For this reason thermal imaging devices forhuman subjects are most sensitive in the7ndash14 micrometer range

The black-body law may be used toestimate the temperature of a planetorbiting the Sun

Temperature relation betweena planet and its star

Earths longwave thermal radiation intensity from

The temperature of a planet depends onseveral factors

Incident radiation from its star

Emitted radiation of the planet egEarths infrared glow

The albedo effect causing a fraction oflight to be reflected by the planet

The greenhouse effect for planets with anatmosphere

Energy generated internally by a planetitself due to radioactive decay tidalheating and adiabatic contraction due bycooling

clouds atmosphere and ground

The analysis only considers the Suns heatfor a planet in a Solar System

The StefanndashBoltzmann law gives the totalpower (energysecond) the Sun is emitting

where

is the StefanndashBoltzmann constant is the effective temperature of theSun and

The Earth only has an absorbing area equal to a twodimensional disk rather than the surface of a sphere

is the radius of the Sun

The Sun emits that power equally in alldirections Because of this the planet is hitwith only a tiny fraction of it The powerfrom the Sun that strikes the planet (at thetop of the atmosphere) is

where

is the radius of the planet and is the distance between the Sun andthe planet

Because of its high temperature the Sunemits to a large extent in the ultraviolet andvisible (UV-Vis) frequency range In this

frequency range the planet reflects afraction of this energy where is thealbedo or reflectance of the planet in theUV-Vis range In other words the planetabsorbs a fraction of the Suns lightand reflects the rest The power absorbedby the planet and its atmosphere is then

Even though the planet only absorbs as acircular area it emits equally in alldirections as a sphere If the planet were aperfect black body it would emit accordingto the StefanndashBoltzmann law

where is the temperature of the planetThis temperature calculated for the case of

the planet acting as a black body by setting is known as the effectivetemperature The actual temperature of theplanet will likely be different depending onits surface and atmospheric propertiesIgnoring the atmosphere and greenhouseeffect the planet since it is at a much lowertemperature than the Sun emits mostly inthe infrared (IR) portion of the spectrum Inthis frequency range it emits of theradiation that a black body would emitwhere is the average emissivity in the IRrange The power emitted by the planet isthen

For a body in radiative exchange equilibriumwith its surroundings the rate at which it

emits radiant energy is equal to the rate atwhich it absorbs it[40][41]

Substituting the expressions for solar andplanet power in equations 1ndash6 andsimplifying yields the estimatedtemperature of the planet ignoringgreenhouse effect TP

In other words given the assumptionsmade the temperature of a planet dependsonly on the surface temperature of the Sunthe radius of the Sun the distance between

the planet and the Sun the albedo and theIR emissivity of the planet

Notice that a gray (flat spectrum) ball where comes to the sametemperature as a black body no matter howdark or light gray

Virtual temperature of Earth

Substituting the measured values for theSun and Earth yields

[42]

[42]

[42]

[43]

With the average emissivity set to unitythe effective temperature of the Earth is

the planet and the Sun the albedo and theIR emissivity of the planet

Notice that a gray (flat spectrum) ball where comes to the sametemperature as a black body no matter howdark or light gray

Virtual temperature of Earth

Substituting the measured values for theSun and Earth yields

[42]

[42]

[42]

[43]

With the average emissivity set to unitythe effective temperature of the Earth is

or minus188 degC

This is the temperature of the Earth if itradiated as a perfect black body in theinfrared assuming an unchanging albedoand ignoring greenhouse effects (which canraise the surface temperature of a bodyabove what it would be if it were a perfectblack body in all spectrums[44]) The Earth infact radiates not quite as a perfect blackbody in the infrared which will raise theestimated temperature a few degrees abovethe effective temperature If we wish toestimate what the temperature of the Earthwould be if it had no atmosphere then wecould take the albedo and emissivity of theMoon as a good estimate The albedo and

emissivity of the Moon are about 01054[45]

and 095[46] respectively yielding anestimated temperature of about 136 degC

Estimates of the Earths average albedo varyin the range 03ndash04 resulting in differentestimated effective temperaturesEstimates are often based on the solarconstant (total insolation power density)rather than the temperature size anddistance of the Sun For example using 04for albedo and an insolation of 1400 W mminus2one obtains an effective temperature ofabout 245 K[47] Similarly using albedo 03and solar constant of 1372 W mminus2 oneobtains an effective temperature of 255K[48][49][50]

The cosmic microwave backgroundradiation observed today is the most perfectblack-body radiation ever observed innature with a temperature of about 27 K[51]

It is a snapshot of the radiation at the timeof decoupling between matter and radiationin the early universe Prior to this time mostmatter in the universe was in the form of anionized plasma in thermal though not fullthermodynamic equilibrium with radiation

According to Kondepudi and Prigogine atvery high temperatures (above 1010 K suchtemperatures existed in the very earlyuniverse) where the thermal motionseparates protons and neutrons in spite ofthe strong nuclear forces electron-positron

Cosmology

pairs appear and disappear spontaneouslyand are in thermal equilibrium withelectromagnetic radiation These particlesform a part of the black body spectrum inaddition to the electromagnetic radiation[52]

The relativistic Doppler effect causes a shiftin the frequency f of light originating from asource that is moving in relation to theobserver so that the wave is observed tohave frequency f

Doppler effect for a movingblack body

where v is the velocity of the source in theobservers rest frame θ is the anglebetween the velocity vector and theobserver-source direction measured in thereference frame of the source and c is thespeed of light[53] This can be simplified forthe special cases of objects moving directlytowards (θ = π) or away (θ = 0) from theobserver and for speeds much less than c

Through Plancks law the temperaturespectrum of a black body is proportionallyrelated to the frequency of light and onemay substitute the temperature (T) for thefrequency in this equation

For the case of a source moving directlytowards or away from the observer thisreduces to

Here v gt 0 indicates a receding source andv lt 0 indicates an approaching source

This is an important effect in astronomywhere the velocities of stars and galaxiescan reach significant fractions of c Anexample is found in the cosmic microwavebackground radiation which exhibits adipole anisotropy from the Earths motionrelative to this black-body radiation field

Balfour Stewart

In 1858 Balfour Stewart described hisexperiments on the thermal radiative

History

emissive and absorptive powers of polishedplates of various substances comparedwith the powers of lamp-black surfaces atthe same temperature[23] Stewart choselamp-black surfaces as his referencebecause of various previous experimentalfindings especially those of Pierre Prevostand of John Leslie He wrote Lamp-blackwhich absorbs all the rays that fall upon itand therefore possesses the greatestpossible absorbing power will possess alsothe greatest possible radiating power Morean experimenter than a logician Stewartfailed to point out that his statementpresupposed an abstract general principlethat there exist either ideally in theory orreally in nature bodies or surfaces thatrespectively have one and the same unique

universal greatest possible absorbingpower likewise for radiating power for everywavelength and equilibrium temperature

Stewart measured radiated power with athermo-pile and sensitive galvanometerread with a microscope He was concernedwith selective thermal radiation which heinvestigated with plates of substances thatradiated and absorbed selectively fordifferent qualities of radiation rather thanmaximally for all qualities of radiation Hediscussed the experiments in terms of rayswhich could be reflected and refracted andwhich obeyed the Stokes-Helmholtzreciprocity principle (though he did not usean eponym for it) He did not in this papermention that the qualities of the rays might

be described by their wavelengths nor didhe use spectrally resolving apparatus suchas prisms or diffraction gratings His workwas quantitative within these constraintsHe made his measurements in a roomtemperature environment and quickly so asto catch his bodies in a condition near thethermal equilibrium in which they had beenprepared by heating to equilibrium withboiling water His measurements confirmedthat substances that emit and absorbselectively respect the principle of selectiveequality of emission and absorption atthermal equilibrium

Stewart offered a theoretical proof that thisshould be the case separately for everyselected quality of thermal radiation but his

mathematics was not rigorously valid[54] Hemade no mention of thermodynamics in thispaper though he did refer to conservation ofvis viva He proposed that hismeasurements implied that radiation wasboth absorbed and emitted by particles ofmatter throughout depths of the media inwhich it propagated He applied theHelmholtz reciprocity principle to accountfor the material interface processes asdistinct from the processes in the interiormaterial He did not postulate unrealizableperfectly black surfaces He concluded thathis experiments showed that in a cavity inthermal equilibrium the heat radiated fromany part of the interior bounding surface nomatter of what material it might becomposed was the same as would have

mathematics was not rigorously valid[54] Hemade no mention of thermodynamics in thispaper though he did refer to conservation ofvis viva He proposed that hismeasurements implied that radiation wasboth absorbed and emitted by particles ofmatter throughout depths of the media inwhich it propagated He applied theHelmholtz reciprocity principle to accountfor the material interface processes asdistinct from the processes in the interiormaterial He did not postulate unrealizableperfectly black surfaces He concluded thathis experiments showed that in a cavity inthermal equilibrium the heat radiated fromany part of the interior bounding surface nomatter of what material it might becomposed was the same as would have

been emitted from a surface of the sameshape and position that would have beencomposed of lamp-black He did not stateexplicitly that the lamp-black-coated bodiesthat he used as reference must have had aunique common spectral emittancefunction that depended on temperature in aunique way

Gustav Kirchhoff

In 1859 not knowing of Stewarts workGustav Robert Kirchhoff reported thecoincidence of the wavelengths ofspectrally resolved lines of absorption andof emission of visible light Importantly forthermal physics he also observed thatbright lines or dark lines were apparent

depending on the temperature differencebetween emitter and absorber[55]

Kirchhoff then went on to consider somebodies that emit and absorb heat radiationin an opaque enclosure or cavity inequilibrium at temperature T

Here is used a notation different fromKirchhoffs Here the emitting power E(T i)

denotes a dimensioned quantity the totalradiation emitted by a body labeled by indexi at temperature T The total absorptionratio a(T i) of that body is dimensionlessthe ratio of absorbed to incident radiation inthe cavity at temperature T (In contrastwith Balfour Stewarts Kirchhoffs definitionof his absorption ratio did not refer inparticular to a lamp-black surface as the

source of the incident radiation) Thus theratio E(T i) a(T i) of emitting power toabsorption ratio is a dimensioned quantitywith the dimensions of emitting powerbecause a(T i) is dimensionless Also herethe wavelength-specific emitting power ofthe body at temperature T is denoted byE(λ T i) and the wavelength-specificabsorption ratio by a(λ T i) Again theratio E(λ T i) a(λ T i) of emitting powerto absorption ratio is a dimensionedquantity with the dimensions of emittingpower

In a second report made in 1859 Kirchhoffannounced a new general principle or lawfor which he offered a theoretical andmathematical proof though he did not offer

quantitative measurements of radiationpowers[56] His theoretical proof was andstill is considered by some writers to beinvalid[54][57] His principle however hasendured it was that for heat rays of thesame wavelength in equilibrium at a giventemperature the wavelength-specific ratioof emitting power to absorption ratio hasone and the same common value for allbodies that emit and absorb at thatwavelength In symbols the law stated thatthe wavelength-specific ratioE(λ T i) a(λ T i) has one and the samevalue for all bodies that is for all values ofindex i In this report there was no mentionof black bodies

In 1860 still not knowing of Stewartsmeasurements for selected qualities ofradiation Kirchhoff pointed out that it waslong established experimentally that fortotal heat radiation of unselected qualityemitted and absorbed by a body inequilibrium the dimensioned total radiationratio E(T i) a(T i) has one and the samevalue common to all bodies that is forevery value of the material index i[58] Againwithout measurements of radiative powersor other new experimental data Kirchhoffthen offered a fresh theoretical proof of hisnew principle of the universality of the valueof the wavelength-specific ratioE(λ T i) a(λ T i) at thermal equilibriumHis fresh theoretical proof was and still is

considered by some writers to beinvalid[54][57]

But more importantly it relied on a newtheoretical postulate of perfectly blackbodies which is the reason why one speaksof Kirchhoffs law Such black bodiesshowed complete absorption in theirinfinitely thin most superficial surface Theycorrespond to Balfour Stewarts referencebodies with internal radiation coated withlamp-black They were not the more realisticperfectly black bodies later considered byPlanck Plancks black bodies radiated andabsorbed only by the material in theirinteriors their interfaces with contiguousmedia were only mathematical surfacescapable neither of absorption nor emission

but only of reflecting and transmitting withrefraction[59]

Kirchhoffs proof considered an arbitrarynon-ideal body labeled i as well as variousperfect black bodies labeled BB It requiredthat the bodies be kept in a cavity in thermalequilibrium at temperature T His proofintended to show that the ratioE(λ T i) a(λ T i) was independent of thenature i of the non-ideal body howeverpartly transparent or partly reflective it was

His proof first argued that for wavelength λand at temperature T at thermalequilibrium all perfectly black bodies of thesame size and shape have the one and thesame common value of emissive powerE(λ T BB) with the dimensions of power

His proof noted that the dimensionlesswavelength-specific absorption ratioa(λ T BB) of a perfectly black body is bydefinition exactly 1 Then for a perfectlyblack body the wavelength-specific ratio ofemissive power to absorption ratioE(λ T BB) a(λ T BB) is again justE(λ T BB) with the dimensions of powerKirchhoff considered successively thermalequilibrium with the arbitrary non-ideal bodyand with a perfectly black body of the samesize and shape in place in his cavity inequilibrium at temperature T He arguedthat the flows of heat radiation must be thesame in each case Thus he argued that atthermal equilibrium the ratioE(λ T i) a(λ T i) was equal to E(λ T BB)which may now be denoted Bλ (λ T) a

continuous function dependent only on λ atfixed temperature T and an increasingfunction of T at fixed wavelength λ at lowtemperatures vanishing for visible but notfor longer wavelengths with positive valuesfor visible wavelengths at highertemperatures which does not depend onthe nature i of the arbitrary non-ideal body(Geometrical factors taken into detailedaccount by Kirchhoff have been ignored inthe foregoing)

Thus Kirchhoffs law of thermal radiationcan be stated For any material at allradiating and absorbing in thermodynamicequilibrium at any given temperature T forevery wavelength λ the ratio of emissivepower to absorptive ratio has one universal

source of the incident radiation) Thus theratio E(T i) a(T i) of emitting power toabsorption ratio is a dimensioned quantitywith the dimensions of emitting powerbecause a(T i) is dimensionless Also herethe wavelength-specific emitting power ofthe body at temperature T is denoted byE(λ T i) and the wavelength-specificabsorption ratio by a(λ T i) Again theratio E(λ T i) a(λ T i) of emitting powerto absorption ratio is a dimensionedquantity with the dimensions of emittingpower

In a second report made in 1859 Kirchhoffannounced a new general principle or lawfor which he offered a theoretical andmathematical proof though he did not offer

quantitative measurements of radiationpowers[56] His theoretical proof was andstill is considered by some writers to beinvalid[54][57] His principle however hasendured it was that for heat rays of thesame wavelength in equilibrium at a giventemperature the wavelength-specific ratioof emitting power to absorption ratio hasone and the same common value for allbodies that emit and absorb at thatwavelength In symbols the law stated thatthe wavelength-specific ratioE(λ T i) a(λ T i) has one and the samevalue for all bodies that is for all values ofindex i In this report there was no mentionof black bodies

In 1860 still not knowing of Stewartsmeasurements for selected qualities ofradiation Kirchhoff pointed out that it waslong established experimentally that fortotal heat radiation of unselected qualityemitted and absorbed by a body inequilibrium the dimensioned total radiationratio E(T i) a(T i) has one and the samevalue common to all bodies that is forevery value of the material index i[58] Againwithout measurements of radiative powersor other new experimental data Kirchhoffthen offered a fresh theoretical proof of hisnew principle of the universality of the valueof the wavelength-specific ratioE(λ T i) a(λ T i) at thermal equilibriumHis fresh theoretical proof was and still is

considered by some writers to beinvalid[54][57]

But more importantly it relied on a newtheoretical postulate of perfectly blackbodies which is the reason why one speaksof Kirchhoffs law Such black bodiesshowed complete absorption in theirinfinitely thin most superficial surface Theycorrespond to Balfour Stewarts referencebodies with internal radiation coated withlamp-black They were not the more realisticperfectly black bodies later considered byPlanck Plancks black bodies radiated andabsorbed only by the material in theirinteriors their interfaces with contiguousmedia were only mathematical surfacescapable neither of absorption nor emission

but only of reflecting and transmitting withrefraction[59]

Kirchhoffs proof considered an arbitrarynon-ideal body labeled i as well as variousperfect black bodies labeled BB It requiredthat the bodies be kept in a cavity in thermalequilibrium at temperature T His proofintended to show that the ratioE(λ T i) a(λ T i) was independent of thenature i of the non-ideal body howeverpartly transparent or partly reflective it was

His proof first argued that for wavelength λand at temperature T at thermalequilibrium all perfectly black bodies of thesame size and shape have the one and thesame common value of emissive powerE(λ T BB) with the dimensions of power

His proof noted that the dimensionlesswavelength-specific absorption ratioa(λ T BB) of a perfectly black body is bydefinition exactly 1 Then for a perfectlyblack body the wavelength-specific ratio ofemissive power to absorption ratioE(λ T BB) a(λ T BB) is again justE(λ T BB) with the dimensions of powerKirchhoff considered successively thermalequilibrium with the arbitrary non-ideal bodyand with a perfectly black body of the samesize and shape in place in his cavity inequilibrium at temperature T He arguedthat the flows of heat radiation must be thesame in each case Thus he argued that atthermal equilibrium the ratioE(λ T i) a(λ T i) was equal to E(λ T BB)which may now be denoted Bλ (λ T) a

continuous function dependent only on λ atfixed temperature T and an increasingfunction of T at fixed wavelength λ at lowtemperatures vanishing for visible but notfor longer wavelengths with positive valuesfor visible wavelengths at highertemperatures which does not depend onthe nature i of the arbitrary non-ideal body(Geometrical factors taken into detailedaccount by Kirchhoff have been ignored inthe foregoing)

Thus Kirchhoffs law of thermal radiationcan be stated For any material at allradiating and absorbing in thermodynamicequilibrium at any given temperature T forevery wavelength λ the ratio of emissivepower to absorptive ratio has one universal

quantitative measurements of radiationpowers[56] His theoretical proof was andstill is considered by some writers to beinvalid[54][57] His principle however hasendured it was that for heat rays of thesame wavelength in equilibrium at a giventemperature the wavelength-specific ratioof emitting power to absorption ratio hasone and the same common value for allbodies that emit and absorb at thatwavelength In symbols the law stated thatthe wavelength-specific ratioE(λ T i) a(λ T i) has one and the samevalue for all bodies that is for all values ofindex i In this report there was no mentionof black bodies

In 1860 still not knowing of Stewartsmeasurements for selected qualities ofradiation Kirchhoff pointed out that it waslong established experimentally that fortotal heat radiation of unselected qualityemitted and absorbed by a body inequilibrium the dimensioned total radiationratio E(T i) a(T i) has one and the samevalue common to all bodies that is forevery value of the material index i[58] Againwithout measurements of radiative powersor other new experimental data Kirchhoffthen offered a fresh theoretical proof of hisnew principle of the universality of the valueof the wavelength-specific ratioE(λ T i) a(λ T i) at thermal equilibriumHis fresh theoretical proof was and still is

considered by some writers to beinvalid[54][57]

But more importantly it relied on a newtheoretical postulate of perfectly blackbodies which is the reason why one speaksof Kirchhoffs law Such black bodiesshowed complete absorption in theirinfinitely thin most superficial surface Theycorrespond to Balfour Stewarts referencebodies with internal radiation coated withlamp-black They were not the more realisticperfectly black bodies later considered byPlanck Plancks black bodies radiated andabsorbed only by the material in theirinteriors their interfaces with contiguousmedia were only mathematical surfacescapable neither of absorption nor emission

but only of reflecting and transmitting withrefraction[59]

Kirchhoffs proof considered an arbitrarynon-ideal body labeled i as well as variousperfect black bodies labeled BB It requiredthat the bodies be kept in a cavity in thermalequilibrium at temperature T His proofintended to show that the ratioE(λ T i) a(λ T i) was independent of thenature i of the non-ideal body howeverpartly transparent or partly reflective it was

His proof first argued that for wavelength λand at temperature T at thermalequilibrium all perfectly black bodies of thesame size and shape have the one and thesame common value of emissive powerE(λ T BB) with the dimensions of power

His proof noted that the dimensionlesswavelength-specific absorption ratioa(λ T BB) of a perfectly black body is bydefinition exactly 1 Then for a perfectlyblack body the wavelength-specific ratio ofemissive power to absorption ratioE(λ T BB) a(λ T BB) is again justE(λ T BB) with the dimensions of powerKirchhoff considered successively thermalequilibrium with the arbitrary non-ideal bodyand with a perfectly black body of the samesize and shape in place in his cavity inequilibrium at temperature T He arguedthat the flows of heat radiation must be thesame in each case Thus he argued that atthermal equilibrium the ratioE(λ T i) a(λ T i) was equal to E(λ T BB)which may now be denoted Bλ (λ T) a

continuous function dependent only on λ atfixed temperature T and an increasingfunction of T at fixed wavelength λ at lowtemperatures vanishing for visible but notfor longer wavelengths with positive valuesfor visible wavelengths at highertemperatures which does not depend onthe nature i of the arbitrary non-ideal body(Geometrical factors taken into detailedaccount by Kirchhoff have been ignored inthe foregoing)

Thus Kirchhoffs law of thermal radiationcan be stated For any material at allradiating and absorbing in thermodynamicequilibrium at any given temperature T forevery wavelength λ the ratio of emissivepower to absorptive ratio has one universal

In 1860 still not knowing of Stewartsmeasurements for selected qualities ofradiation Kirchhoff pointed out that it waslong established experimentally that fortotal heat radiation of unselected qualityemitted and absorbed by a body inequilibrium the dimensioned total radiationratio E(T i) a(T i) has one and the samevalue common to all bodies that is forevery value of the material index i[58] Againwithout measurements of radiative powersor other new experimental data Kirchhoffthen offered a fresh theoretical proof of hisnew principle of the universality of the valueof the wavelength-specific ratioE(λ T i) a(λ T i) at thermal equilibriumHis fresh theoretical proof was and still is

considered by some writers to beinvalid[54][57]

But more importantly it relied on a newtheoretical postulate of perfectly blackbodies which is the reason why one speaksof Kirchhoffs law Such black bodiesshowed complete absorption in theirinfinitely thin most superficial surface Theycorrespond to Balfour Stewarts referencebodies with internal radiation coated withlamp-black They were not the more realisticperfectly black bodies later considered byPlanck Plancks black bodies radiated andabsorbed only by the material in theirinteriors their interfaces with contiguousmedia were only mathematical surfacescapable neither of absorption nor emission

but only of reflecting and transmitting withrefraction[59]

Kirchhoffs proof considered an arbitrarynon-ideal body labeled i as well as variousperfect black bodies labeled BB It requiredthat the bodies be kept in a cavity in thermalequilibrium at temperature T His proofintended to show that the ratioE(λ T i) a(λ T i) was independent of thenature i of the non-ideal body howeverpartly transparent or partly reflective it was

His proof first argued that for wavelength λand at temperature T at thermalequilibrium all perfectly black bodies of thesame size and shape have the one and thesame common value of emissive powerE(λ T BB) with the dimensions of power

His proof noted that the dimensionlesswavelength-specific absorption ratioa(λ T BB) of a perfectly black body is bydefinition exactly 1 Then for a perfectlyblack body the wavelength-specific ratio ofemissive power to absorption ratioE(λ T BB) a(λ T BB) is again justE(λ T BB) with the dimensions of powerKirchhoff considered successively thermalequilibrium with the arbitrary non-ideal bodyand with a perfectly black body of the samesize and shape in place in his cavity inequilibrium at temperature T He arguedthat the flows of heat radiation must be thesame in each case Thus he argued that atthermal equilibrium the ratioE(λ T i) a(λ T i) was equal to E(λ T BB)which may now be denoted Bλ (λ T) a

continuous function dependent only on λ atfixed temperature T and an increasingfunction of T at fixed wavelength λ at lowtemperatures vanishing for visible but notfor longer wavelengths with positive valuesfor visible wavelengths at highertemperatures which does not depend onthe nature i of the arbitrary non-ideal body(Geometrical factors taken into detailedaccount by Kirchhoff have been ignored inthe foregoing)

Thus Kirchhoffs law of thermal radiationcan be stated For any material at allradiating and absorbing in thermodynamicequilibrium at any given temperature T forevery wavelength λ the ratio of emissivepower to absorptive ratio has one universal

considered by some writers to beinvalid[54][57]

But more importantly it relied on a newtheoretical postulate of perfectly blackbodies which is the reason why one speaksof Kirchhoffs law Such black bodiesshowed complete absorption in theirinfinitely thin most superficial surface Theycorrespond to Balfour Stewarts referencebodies with internal radiation coated withlamp-black They were not the more realisticperfectly black bodies later considered byPlanck Plancks black bodies radiated andabsorbed only by the material in theirinteriors their interfaces with contiguousmedia were only mathematical surfacescapable neither of absorption nor emission

but only of reflecting and transmitting withrefraction[59]

Kirchhoffs proof considered an arbitrarynon-ideal body labeled i as well as variousperfect black bodies labeled BB It requiredthat the bodies be kept in a cavity in thermalequilibrium at temperature T His proofintended to show that the ratioE(λ T i) a(λ T i) was independent of thenature i of the non-ideal body howeverpartly transparent or partly reflective it was

His proof first argued that for wavelength λand at temperature T at thermalequilibrium all perfectly black bodies of thesame size and shape have the one and thesame common value of emissive powerE(λ T BB) with the dimensions of power

His proof noted that the dimensionlesswavelength-specific absorption ratioa(λ T BB) of a perfectly black body is bydefinition exactly 1 Then for a perfectlyblack body the wavelength-specific ratio ofemissive power to absorption ratioE(λ T BB) a(λ T BB) is again justE(λ T BB) with the dimensions of powerKirchhoff considered successively thermalequilibrium with the arbitrary non-ideal bodyand with a perfectly black body of the samesize and shape in place in his cavity inequilibrium at temperature T He arguedthat the flows of heat radiation must be thesame in each case Thus he argued that atthermal equilibrium the ratioE(λ T i) a(λ T i) was equal to E(λ T BB)which may now be denoted Bλ (λ T) a

continuous function dependent only on λ atfixed temperature T and an increasingfunction of T at fixed wavelength λ at lowtemperatures vanishing for visible but notfor longer wavelengths with positive valuesfor visible wavelengths at highertemperatures which does not depend onthe nature i of the arbitrary non-ideal body(Geometrical factors taken into detailedaccount by Kirchhoff have been ignored inthe foregoing)

Thus Kirchhoffs law of thermal radiationcan be stated For any material at allradiating and absorbing in thermodynamicequilibrium at any given temperature T forevery wavelength λ the ratio of emissivepower to absorptive ratio has one universal

but only of reflecting and transmitting withrefraction[59]

Kirchhoffs proof considered an arbitrarynon-ideal body labeled i as well as variousperfect black bodies labeled BB It requiredthat the bodies be kept in a cavity in thermalequilibrium at temperature T His proofintended to show that the ratioE(λ T i) a(λ T i) was independent of thenature i of the non-ideal body howeverpartly transparent or partly reflective it was

His proof first argued that for wavelength λand at temperature T at thermalequilibrium all perfectly black bodies of thesame size and shape have the one and thesame common value of emissive powerE(λ T BB) with the dimensions of power

His proof noted that the dimensionlesswavelength-specific absorption ratioa(λ T BB) of a perfectly black body is bydefinition exactly 1 Then for a perfectlyblack body the wavelength-specific ratio ofemissive power to absorption ratioE(λ T BB) a(λ T BB) is again justE(λ T BB) with the dimensions of powerKirchhoff considered successively thermalequilibrium with the arbitrary non-ideal bodyand with a perfectly black body of the samesize and shape in place in his cavity inequilibrium at temperature T He arguedthat the flows of heat radiation must be thesame in each case Thus he argued that atthermal equilibrium the ratioE(λ T i) a(λ T i) was equal to E(λ T BB)which may now be denoted Bλ (λ T) a

continuous function dependent only on λ atfixed temperature T and an increasingfunction of T at fixed wavelength λ at lowtemperatures vanishing for visible but notfor longer wavelengths with positive valuesfor visible wavelengths at highertemperatures which does not depend onthe nature i of the arbitrary non-ideal body(Geometrical factors taken into detailedaccount by Kirchhoff have been ignored inthe foregoing)

Thus Kirchhoffs law of thermal radiationcan be stated For any material at allradiating and absorbing in thermodynamicequilibrium at any given temperature T forevery wavelength λ the ratio of emissivepower to absorptive ratio has one universal

His proof noted that the dimensionlesswavelength-specific absorption ratioa(λ T BB) of a perfectly black body is bydefinition exactly 1 Then for a perfectlyblack body the wavelength-specific ratio ofemissive power to absorption ratioE(λ T BB) a(λ T BB) is again justE(λ T BB) with the dimensions of powerKirchhoff considered successively thermalequilibrium with the arbitrary non-ideal bodyand with a perfectly black body of the samesize and shape in place in his cavity inequilibrium at temperature T He arguedthat the flows of heat radiation must be thesame in each case Thus he argued that atthermal equilibrium the ratioE(λ T i) a(λ T i) was equal to E(λ T BB)which may now be denoted Bλ (λ T) a

continuous function dependent only on λ atfixed temperature T and an increasingfunction of T at fixed wavelength λ at lowtemperatures vanishing for visible but notfor longer wavelengths with positive valuesfor visible wavelengths at highertemperatures which does not depend onthe nature i of the arbitrary non-ideal body(Geometrical factors taken into detailedaccount by Kirchhoff have been ignored inthe foregoing)

Thus Kirchhoffs law of thermal radiationcan be stated For any material at allradiating and absorbing in thermodynamicequilibrium at any given temperature T forevery wavelength λ the ratio of emissivepower to absorptive ratio has one universal

continuous function dependent only on λ atfixed temperature T and an increasingfunction of T at fixed wavelength λ at lowtemperatures vanishing for visible but notfor longer wavelengths with positive valuesfor visible wavelengths at highertemperatures which does not depend onthe nature i of the arbitrary non-ideal body(Geometrical factors taken into detailedaccount by Kirchhoff have been ignored inthe foregoing)

Thus Kirchhoffs law of thermal radiationcan be stated For any material at allradiating and absorbing in thermodynamicequilibrium at any given temperature T forevery wavelength λ the ratio of emissivepower to absorptive ratio has one universal

value which is characteristic of a perfectblack body and is an emissive power whichwe here represent by Bλ (λ T) (For ournotation Bλ (λ T) Kirchhoffs originalnotation was simply e)[58][60][61][62][63][64]

Kirchhoff announced that the determinationof the function Bλ (λ T) was a problem ofthe highest importance though herecognized that there would beexperimental difficulties to be overcome Hesupposed that like other functions that donot depend on the properties of individualbodies it would be a simple functionOccasionally by historians that functionBλ (λ T) has been called Kirchhoffs(emission universal) function[65][66][67][68]

though its precise mathematical form would

not be known for another forty years till itwas discovered by Planck in 1900 Thetheoretical proof for Kirchhoffs universalityprinciple was worked on and debated byvarious physicists over the same time andlater[57] Kirchhoff stated later in 1860 thathis theoretical proof was better than BalfourStewarts and in some respects it wasso[54] Kirchhoffs 1860 paper did notmention the second law ofthermodynamics and of course did notmention the concept of entropy which hadnot at that time been established In a moreconsidered account in a book in 1862Kirchhoff mentioned the connection of hislaw with Carnots principle which is a formof the second law[69]

not be known for another forty years till itwas discovered by Planck in 1900 Thetheoretical proof for Kirchhoffs universalityprinciple was worked on and debated byvarious physicists over the same time andlater[57] Kirchhoff stated later in 1860 thathis theoretical proof was better than BalfourStewarts and in some respects it wasso[54] Kirchhoffs 1860 paper did notmention the second law ofthermodynamics and of course did notmention the concept of entropy which hadnot at that time been established In a moreconsidered account in a book in 1862Kirchhoff mentioned the connection of hislaw with Carnots principle which is a formof the second law[69]

According to Helge Kragh Quantum theoryowes its origin to the study of thermalradiation in particular to the black-bodyradiation that Robert Kirchhoff had firstdefined in 1859ndash1860[70]

Bolometer

Color temperature

Infrared thermometer

Photon polarization

Plancks law

Pyrometry

RayleighndashJeans law

Thermography

SakumandashHattori equation

See also

1 Loudon 2000 Chapter 1

2 Mandel amp Wolf 1995 Chapter 13

3 Kondepudi amp Prigogine 1998 Chapter 11

4 Landsberg 1990 Chapter 13

5 Partington JR (1949) p 466

6 Ian Morison (2008) Introduction toAstronomy and Cosmology J Wiley amp Sonsp 48 ISBN 0-470-03333-9

7 Alessandro Fabbri Joseacute Navarro-Salas(2005) Chapter 1 Introduction Modelingblack hole evaporation Imperial CollegePress ISBN 1-86094-527-9

8 From (Kirchhoff 1860) (Annalen derPhysik und Chemie) p 277 Der Beweiswelcher fuumlr die ausgesprochene Behauptung

References

hier gegeben werden soll hellip vollkommenschwarze oder kuumlrzer schwarze nennen(The proof which shall be given here for theproposition stated [above] rests on theassumption that bodies are conceivablewhich in the case of infinitely smallthicknesses completely absorb all rays thatfall on them thus [they] neither reflect nortransmit rays I will call such bodiescompletely black [bodies] or more brieflyblack [bodies]) See also (Kirchhoff 1860)(Philosophical Magazine) p 2

9 Tomokazu Kogure Kam-Ching Leung(2007) sect23 Thermodynamic equilibriumand black-body radiation The astrophysicsof emission-line stars Springer p 41ISBN 0-387-34500-0

10 Wien W (1893) Eine neue Beziehung derStrahlung schwarzer Koumlrper zum zweitenHauptsatz der WaumlrmetheorieSitzungberichte der Koumlniglich-PreuszligischenAkademie der Wissenschaften (Berlin) 18931 55ndash62

11 Lummer O Pringsheim E (1899) DieVertheilung der Energie im Spectrum desschwarzen Koumlrpers Verhandlungen derDeutschen Physikalischen Gessellschaft(Leipzig) 1899 1 23ndash41

12 Planck 1914

13 Draper JW (1847) On the production oflight by heat London Edinburgh and DublinPhilosophical Magazine and Journal ofScience series 3 30 345ndash360 [1]

14 Partington 1949 pp 466ndash467 478

15 Goody amp Yung 1989 pp 482 484

16 Planck 1914 p 42

17 Wien 1894

18 Planck 1914 p 43

19 Joseph Caniou (1999) sect422Calculation of Plancks law Passive infrareddetection theory and applications Springerp 107 ISBN 0-7923-8532-2

20 J R Mahan (2002) Radiation heattransfer a statistical approach (3rd ed)Wiley-IEEE p 58 ISBN 978-0-471-21270-6

21 de Groot SR Mazur P (1962) Non-equilibrium Thermodynamics North-HollandAmsterdam

22 Kondepudi amp Prigogine 1998 Section 94

23 Stewart 1858

24 Huang Kerson (1967) StatisticalMechanics New York John Wiley amp SonsISBN 0-471-81518-7

25 Planck Max (1901) Ueber das Gesetzder Energieverteilung im Normalspectrum[On the law of the distribution of energy inthe normal spectrum] Annalen der Physik4th series (in German) 4 (3) 553ndash563Bibcode1901AnP309553P doi101002andp19013090310

26 Landau L D E M Lifshitz (1996)Statistical Physics (3rd Edition Part 1 ed)Oxford ButterworthndashHeinemann ISBN 0-521-65314-2

27 httpswwwbritannicacomscienceStefan-Boltzmann-law

28 Gannon Megan (December 21 2012)New Baby Picture of Universe Unveiled Spacecom Retrieved December 21 2012

29 Bennett CL Larson L Weiland JLJarosk N Hinshaw N Odegard N SmithKM Hill RS Gold B Halpern MKomatsu E Nolta MR Page L SpergelDN Wollack E Dunkley J Kogut ALimon M Meyer SS Tucker GS WrightEL (December 20 2012) Nine-YearWilkinson Microwave Anisotropy Probe(WMAP) Observations Final Maps andResults 1212 5225 arXiv12125225  Bibcode2013ApJS20820B doi1010880067-0049208220

30 Rybicki amp Lightman 1979 p 22

31 httpphysicsnistgovcgi-bincuuValuebwien

32 Nave Dr Rod Wiens Displacement Lawand Other Ways to Characterize the Peak ofBlackbody Radiation HyperPhysicsProvides 5 variations of Wiens displacementlaw

33 Infrared Services Emissivity Values forCommon Materials Retrieved 2007-06-24

34 Omega Engineering Emissivity ofCommon Materials Retrieved 2007-06-24

35 Farzana Abanty (2001) Temperature ofa Healthy Human (Skin Temperature) ThePhysics Factbook Retrieved 2007-06-24

36 Lee B Theoretical Prediction andMeasurement of the Fabric Surface ApparentTemperature in a Simulated

ManFabricEnvironment System (PDF)Archived from the original (PDF) on 2006-09-02 Retrieved 2007-06-24

37 Harris J Benedict F Benedict (1918) ABiometric Study of Human BasalMetabolism Proc Natl Acad Sci USA 4(12) 370ndash3 Bibcode1918PNAS4370H doi101073pnas412370 PMC 1091498  PMID 16576330

38 Levine J (2004) Nonexercise activitythermogenesis (NEAT) environment andbiology Am J Physiol Endocrinol Metab286 (5) E675ndashE685doi101152ajpendo005622003 PMID 15102614

39 DrPhysicscom Heat Transfer and theHuman Body Retrieved 2007-06-24

40 Prevost P (1791) Meacutemoire sur lequilibredu feu Journal de Physique (Paris) vol 38pp 314-322

41 Iribarne JV Godson WL (1981)Atmospheric Thermodynamics secondedition D Reidel Publishing DordrechtISBN 90-277-1296-4 page 227

42 NASA Sun Fact Sheet

43 Cole George H A Woolfson Michael M(2002) Planetary Science The Science ofPlanets Around Stars (1st ed) Institute ofPhysics Publishing pp 36ndash37 380ndash382ISBN 0-7503-0815-X

44 Principles of Planetary Climate byRaymond T Peirrehumbert CambridgeUniversity Press (2011) p 146 FromChapter 3 which is available online here

Archived March 28 2012 at the WaybackMachine p 12 mentions that Venus black-body temperature would be 330 K in thezero albedo case but that due toatmospheric warming its actual surfacetemperature is 740 K

45 Saari J M Shorthill R W (1972) TheSunlit Lunar Surface I Albedo Studies andFull Moon The Moon 5 (1ndash2) 161ndash178Bibcode1972Moon5161S doi101007BF00562111

46 Lunar and Planetary Science XXXVII(2006) 2406

47 Michael D Papagiannis (1972) Spacephysics and space astronomy Taylor ampFrancis pp 10ndash11 ISBN 978-0-677-04000-4

48 Willem Jozef Meine Martens amp JanRotmans (1999) Climate Change anIntegrated Perspective Springer pp 52ndash55ISBN 978-0-7923-5996-8

49 F Selsis (2004) The PrebioticAtmosphere of the Earth In PascaleEhrenfreund et al Astrobiology FuturePerspectives Springer pp 279ndash280ISBN 978-1-4020-2587-7

50 Wallace JM Hobbs PV (2006)Atmospheric Science An IntroductorySurvey second edition Elsevier AmsterdamISBN 978-0-12-732951-2 exercise 46 pages119-120

51 White M (1999) Anisotropies in theCMB arXivastro-ph9903232  Bibcode1999dpfconfW

52 Kondepudi amp Prigogine 1998 pp 227ndash228 also Section 116 pages 294ndash296

53 The Doppler Effect T P Gill Logos Press1965

54 Siegel 1976

55 Kirchhoff 1860a

56 Kirchhoff 1860b

57 Schirrmacher 2001

58 Kirchhoff 1860c

59 Planck 1914 p 11

60 Chandrasekhar 1950 p 8

61 Milne 1930 p 80

62 Rybicki amp Lightman 1979 pp 16ndash17

63 Mihalas amp Weibel-Mihalas 1984 p 328

64 Goody amp Yung 1989 pp 27ndash28

Bibliography

Chandrasekhar S (1950) RadiativeTransfer Oxford University Press

Goody R M Yung Y L (1989)Atmospheric Radiation Theoretical Basis(2nd ed) Oxford University PressISBN 978-0-19-510291-8

65 Paschen F (1896) personal letter citedby Hermann 1971 p 6

66 Hermann 1971 p 7

67 Kuhn 1978 pp 8 29

68 Mehra and Rechenberg 1982 pp 26 2831 39

69 Kirchhoff amp 18621882 p 573

70 Kragh 1999 p 58

Bibliography

Chandrasekhar S (1950) RadiativeTransfer Oxford University Press

Goody R M Yung Y L (1989)Atmospheric Radiation Theoretical Basis(2nd ed) Oxford University PressISBN 978-0-19-510291-8

65 Paschen F (1896) personal letter citedby Hermann 1971 p 6

66 Hermann 1971 p 7

67 Kuhn 1978 pp 8 29

68 Mehra and Rechenberg 1982 pp 26 2831 39

69 Kirchhoff amp 18621882 p 573

70 Kragh 1999 p 58

Hermann A (1971) The Genesis ofQuantum Theory Nash CW (transl) MITPress ISBN 0-262-08047-8 a translationof Fruumlhgeschichte der Quantentheorie(1899ndash1913) Physik VerlagMosbachBaden

Kirchhoff G [27 October 1859] (1860a)Uumlber die Fraunhoferschen Linien [OnFraunhofers lines] Monatsberichte derKoumlniglich Preussischen Akademie derWissenschaften zu Berlin 662ndash665

Kirchhoff G [11 December 1859](1860b) Uumlber den Zusammenhangzwischen Emission und Absorption vonLicht und Waumlrme [On the relationbetween emission and absorption of lightand heat] Monatsberichte der Koumlniglich

Preussischen Akademie derWissenschaften zu Berlin 783ndash787

Kirchhoff G (1860c) Ueber dasVerhaumlltniss zwischen demEmissionsvermoumlgen und demAbsorptionsvermoumlgen der Koumlrper fuumlrWaumlrme and Licht [On the relationbetween bodies emission capacity andabsorption capacity for heat and light]Annalen der Physik und Chemie 109 (2)275ndash301 Bibcode1860AnP185275K doi101002andp18601850205 Translated by Guthrie F as Kirchhoff G(1860) On the relation between theradiating and absorbing powers ofdifferent bodies for light and heat Philosophical Magazine Series 4 volume20 1ndash21

Kirchhoff G (1882) [1862] Ueber dasVerhaumlltniss zwischen demEmissionsvermoumlgen und demAbsorptionsvermoumlgen der Koumlrper fuumlrWaumlrme und Licht GessamelteAbhandlungen Leipzig JohannAmbrosius Barth pp 571ndash598

Kondepudi D Prigogine I (1998)Modern Thermodynamics From HeatEngines to Dissipative Structures JohnWiley amp Sons ISBN 0-471-97393-9

Kragh H (1999) Quantum Generations aHistory of Physics in the TwentiethCentury Princeton University PressISBN 0-691-01206-7

Kuhn T S (1978) BlackndashBody Theory andthe Quantum Discontinuity Oxford

University Press ISBN 0-19-502383-8

Landsberg P T (1990) Thermodynamicsand statistical mechanics (Reprint ed)Courier Dover Publications ISBN 0-486-66493-7

Lavenda Bernard Howard (1991)Statistical Physics A ProbabilisticApproach John Wiley amp Sons pp 41ndash42ISBN 978-0-471-54607-8

Loudon R (2000) [1973] The QuantumTheory of Light (third ed) CambridgeUniversity Press ISBN 0-19-850177-3

Mandel L Wolf E (1995) OpticalCoherence and Quantum OpticsCambridge University Press ISBN 0-521-41711-2

Mehra J Rechenberg H (1982) TheHistorical Development of QuantumTheory volume 1 part 1 Springer-VerlagISBN 0-387-90642-8

Mihalas D Weibel-Mihalas B (1984)Foundations of Radiation HydrodynamicsOxford University Press ISBN 0-19-503437-6

Milne EA (1930) Thermodynamics ofthe Stars Handbuch der Astrophysik 3part 1 63ndash255

Partington JR (1949) An AdvancedTreatise on Physical Chemistry Volume 1Fundamental Principles The Properties ofGases Longmans Green and Co

Planck M (1914) [1912] The Theory ofHeat Radiation translated by Masius M

P Blakistons Sons amp Co

Rybicki G B Lightman A P (1979)Radiative Processes in Astrophysics John Wiley amp Sons ISBN 0-471-82759-2

Schirrmacher A (2001) Experimentingtheory the proofs of Kirchhoffs radiationlaw before and after Planck MuumlnchnerZentrum fuumlr Wissenschafts undTechnikgeschichte

Siegel DM (1976) Balfour Stewart andGustav Robert Kirchhoff two independentapproaches to Kirchhoffs radiationlaw Isis 67 (4) 565ndash600doi101086351669

Stewart B (1858) An account of someexperiments on radiant heat

Transactions of the Royal Society ofEdinburgh 22 1ndash20

Wien W (1894) Temperatur undEntropie der Strahlung [Temperature andentropy of radiation] Annalen der Physik288 (5) 132ndash165Bibcode1894AnP288132W doi101002andp18942880511

Kroemer Herbert Kittel Charles (1980)Thermal Physics (2nd ed) W H FreemanCompany ISBN 0-7167-1088-9

Tipler Paul Llewellyn Ralph (2002)Modern Physics (4th ed) W H FreemanISBN 0-7167-4345-0

Further reading

External links

Black-body radiation JavaScriptInteractives Black-body radiation by Fu-Kwun Hwang and Loo Kang Wee

Calculating Black-body RadiationInteractive calculator with Doppler EffectIncludes most systems of units

Color-to-Temperature demonstration atAcademoorg

Cooling Mechanisms for Human Body ndashFrom Hyperphysics

Descriptions of radiation emitted by manydifferent objects

Black-Body Emission Applet

Blackbody Spectrum by Jeff BryantWolfram Demonstrations Project 2007

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Last edited 26 days ago by an anonyhellip