Black-Body Radiation and Planck's Hypothesis
Heat a cavity, pierce it with a small hole, and the light that emerges has a spectrum fixed by the temperature alone — not by the material of the walls, not by their colour, not by the shape of the cavity. Kirchhoff proved this in 1860 [Kirchhoff:1860a] and thereby handed nineteenth-century physics a single universal function of frequency and temperature to compute. Thermodynamics and Maxwell's electrodynamics — the two most secure theories then available (Classical Thermodynamics and The Maxwell Equations) — between them fixed the integral of that function [Stefan:1879] [Boltzmann:1884], its behaviour under a change of temperature [Wien:1893], and, empirically, its short-wavelength tail [Wien:1896]. What they could not do was fix the function itself. Applied without reservation they returned a spectral density growing without bound at high frequency, and the divergence was no technical subtlety: it is the equipartition theorem of Kinetic Theory of Gases applied to a system with infinitely many modes [Rayleigh:1900] [Jeans:1905].
This chapter opens Part VIII — The Transition to Quantum Physics because the resolution — Planck's assumption of December 1900 that the energy exchanged between the cavity walls and a mode of frequency \(\nu\) comes in elements \(h\nu\) [Planck:1900b] [Planck:1901] — is the first quantitative evidence for quantization found anywhere in nature, and because it was forced by measurement rather than by theory: the Reichsanstalt spectra of Lummer and Pringsheim [Lummer:1899] and the far-infrared points of Rubens and Kurlbaum [Rubens:1900] falsified Wien's distribution before any successor to it existed. The chapter then follows the new constant \(h\) out of the cavity: into the specific heats of solids [Einstein:1907a] [Debye:1912a], into the statistics of indistinguishable quanta [Bose:1924], and into the present-day SI, where its numerical value is fixed by definition [BIPM:2019]. The photon itself is not here. Planck quantized the exchange and not the field, and said so; the light quantum belongs to The Photon: Photoelectric and Compton Effects.
Black-Body Radiation and Planck's Hypothesis: all derivations of this chapter are pending.
Cavity radiation and Kirchhoff's theorem
Emissive and absorptive power
[Reserved: Kirchhoff's law of thermal radiation [Kirchhoff:1860a] — at thermal equilibrium the ratio of the emissive power to the absorptive power of any body is a universal function \(K(\nu,T)\) of frequency and temperature. The proof is a thermodynamic one: two bodies exchanging radiation inside a common enclosure must not be able to drive a heat flow between equal temperatures, on pain of violating the second law of Classical Thermodynamics. Definition of a black body as the limiting absorber, and Kirchhoff's own realization of it as a small aperture in a large isothermal cavity. Consequence: the spectrum of the aperture is a property of temperature alone, so a laboratory can measure a function of nature rather than of a material.]
The radiation emerging from a small aperture in an isothermal enclosure has a spectral distribution fixed by the temperature \(T\) and the frequency \(\nu\) alone. It does not depend on the material of the walls, on their surface finish or colour, or on the size and shape of the cavity. Equivalently, for every body in equilibrium with that radiation the ratio of the spectral emissive power \(e_{\nu}\) to the spectral absorptivity \(a_{\nu}\) is one and the same universal function of frequency and temperature,
numerically equal to the emissive power of a perfect absorber [Kirchhoff:1860a].
Derivation. Place two bodies inside an evacuated enclosure whose walls are held at a common temperature \(T\), and let them exchange energy only by radiation. Interpose a filter transmitting only a narrow band \(\dd\nu\) about \(\nu\); a filter is a passive optical element and can do no work, so equilibrium must hold in that band separately. Let \(I_{\nu}\) be the spectral flux each body receives from the enclosure. Body \(j\) emits \(e_{\nu}^{(j)}\dd\nu\) and absorbs \(a_{\nu}^{(j)}I_{\nu}\dd\nu\); if the two were unequal the body would heat or cool while its surroundings stayed at \(T\), so that heat would pass spontaneously between objects at the same temperature, which the second law of Classical Thermodynamics forbids. Hence \(e_{\nu}^{(j)}=a_{\nu}^{(j)}I_{\nu}\) for each body, and
The two bodies were arbitrary, so the ratio is a function of \(\nu\) and \(T\) only, which is Equation (68.1). Repeating the argument with two cavities at the same temperature, joined through the same narrow-band filter, shows that \(I_{\nu}\) itself cannot depend on the walls: were the two fluxes unequal at any frequency, energy would flow in that band between enclosures at equal temperature. Finally a body with \(a_{\nu}=1\) at every frequency — a black body — emits exactly \(K(\nu,T)\), so the universal function is realized by the aperture of any isothermal cavity, whatever it is made of.
∎The cavity as a thermodynamic system
[Reserved: the electromagnetic field in a cavity treated as a thermodynamic body, with energy density \(u(T)\), spectral density \(u_{\nu}(\nu,T)\) and the relation \(I=cu/4\) between energy density and the flux through the aperture. Radiation pressure \(p=u/3\), obtained from the Maxwell stress tensor of The Maxwell Equations and used thermodynamically by Boltzmann [Boltzmann:1884] following Bartoli's cycle; the adiabatic invariance of \(\nu/T\) under slow expansion of a perfectly reflecting cavity. The Maxwell relations that carry the argument are those of [Callen:1985], and the same system reappears as the photon gas of Quantum Statistics.]
What classical thermodynamics could settle
The Stefan–Boltzmann law
[Reserved: Stefan's empirical finding that the total radiated flux scales as the fourth power of the absolute temperature [Stefan:1879], extracted from Tyndall's platinum-wire data; Boltzmann's derivation of the same law from thermodynamics plus the radiation pressure \(p=u/3\) [Boltzmann:1884], the first result in physics obtained by putting Maxwell's field through a Carnot cycle. The law \(I=\sigma T^{4}\) for a black body, and \(I=\varepsilon\sigma T^{4}\) for a grey one; the Stefan–Boltzmann constant, which in the present SI is not measured but computed, \(\sigma=2\pi^{5}k_{\text{B}} ^{4}/15h^{3}c^{2}=5.670374419\times 10^{-8}\) in SI units [BIPM:2019] [Mohr:2025]. Worked instances: effective temperatures of stars (Stellar Structure and Nucleosynthesis) and the planetary equilibrium temperature.]
The total power radiated per unit area by a black surface, integrated over all frequencies, is proportional to the fourth power of its absolute temperature,
Stefan extracted the fourth power from Tyndall's measurements on heated platinum wire [Stefan:1879]; a body of emissivity \(\varepsilon<1\) radiates \(\varepsilon\sigma T^{4}\). In the present SI the constant \(\sigma\) is no longer measured but computed from the defining constants: it is \(5.670374419\times 10^{-8}\) in SI units of watt per square metre per kelvin to the fourth power [BIPM:2019] [Mohr:2025].
Derivation. Treat the radiation filling an isothermal cavity as a thermodynamic fluid. By Phenomenon 68.1 its energy density \(u\) is a function of temperature alone, so \(U=u(T)V\); take from Section 68.1.2 the isotropic radiation pressure \(p=u/3\). For any simple system the energy equation of state reads
a Maxwell relation of Classical Thermodynamics. Its left side is \(u\) and its right side is \(Tu'(T)/3-u/3\), so
whose solution is \(u=aT^{4}\) with \(a\) a constant of integration. The flux escaping through a small aperture is \(M=cu/4\), the factor \(\tfrac{1}{4}\) coming from averaging \(c\cos\vartheta\) over the outward hemisphere, so \(M=\left(ac/4\right)T^{4}\): this is Equation (68.2) with \(\sigma=ac/4\). Thermodynamics fixes the exponent and leaves \(a\) undetermined. Only Phenomenon 68.5 computes it, and that it then agrees with the measured \(\sigma\) is the first quantitative success of the quantum hypothesis.
∎Wien's displacement law
[Reserved: Wien's thermodynamic argument from the adiabatic compression of a cavity [Wien:1893], which forces the universal function into the scaling form \(u_{\nu}=\nu^{3}f(\nu/T)\) without determining \(f\). Corollaries: the wavelength of the maximum satisfies \(\lambda_{\max}T=b\) with \(b=2.897771955\times 10^{-3}\,\mathrm{m}\,\mathrm{K}\), and the frequency of the maximum a different constant — the two maxima are not at the same place, because \(u_{\nu}\dd\nu\) and \(u_{\lambda} \dd\lambda\) are different densities, a point worth a remark since it is a standing source of error. Consistency of the scaling form with Section 68.2.1 on integration.]
The spectrum of a black body keeps its shape as the temperature is changed and merely slides along the wavelength axis: the wavelength at which the spectral radiance is greatest is inversely proportional to the absolute temperature,
where the constant \(b\) is \(2.897771955\times 10^{-3}\,\mathrm{m}\,\mathrm{K}\) in the present SI [Wien:1893] [Mohr:2025]. This is why a body glows first dull red and then white as it is heated, and why the temperature of a furnace or of a star can be read from the colour of its continuum alone. The maximum of the same spectrum plotted against frequency lies at a different place and obeys a different constant, because \(u_{\nu}\dd\nu\) and \(u_{\lambda}\dd\lambda\) are densities with respect to different variables; the two maxima do not correspond.
Wien's adiabatic argument: a slow compression of a perfectly reflecting cavity leaves the entropy unchanged while shifting every mode frequency, which forces the universal function of Kirchhoff's theorem into the scaling form given by \(\nu\) cubed times a function of \(\nu/T\) alone; and the maximisation of the corresponding wavelength density, which turns that scaling form into a fixed product of peak wavelength and temperature.
Wien's distribution law
[Reserved: Wien's 1896 distribution \(u_{\nu}\propto\nu^{3} \ee^{-a\nu/T}\) [Wien:1896], obtained from a Maxwell–Boltzmann analogy between emitting molecules and radiation rather than from thermodynamics, and therefore a guess of a different order from Section 68.2.2. Its excellent agreement with the visible and near-infrared measurements available before 1899, and its status until then as the accepted law. That it is the high-frequency limit of Planck's law is shown in Section 68.5.4, which is why its early success was not evidence of its truth.]
The classical field theory and its failure
Counting the modes of a cavity
[Reserved: the electromagnetic normal modes of a rectangular cavity with conducting walls, from the boundary-value problem of Electromagnetic Waves and Optics; the density of modes \(8\pi\nu^{2}/c^{3}\) per unit volume per unit frequency, with the factor of two for the two transverse polarizations counted in Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism. The result is independent of the shape of the cavity, by Weyl's asymptotic law for the eigenvalue counting function; the mode count is the same object that will reappear for phonons in Phonons and Lattice Dynamics and for the free-particle density of states in Statistical Mechanics.]
The Rayleigh–Jeans law
[Reserved: equipartition (Kinetic Theory of Gases) assigns \(k_{\text{B}}T\) to each mode, giving \(u_{\nu}=8\pi\nu^{2}k_{\text{B}} T/c^{3}\). Rayleigh's short note of June 1900 states the \(\nu^{2}T\) form and, doubting it at high frequency, inserts an exponential cut-off by hand [Rayleigh:1900]; Jeans supplies the correct numerical factor and argues the classical case in full [Jeans:1905]. Honest history: Rayleigh's factor was eight times too large and Jeans held for some years that the cavity was simply not in equilibrium, so “Rayleigh–Jeans law” names a result neither author asserted in the form now taught. The law agrees with the far infrared measurements of Section 68.4.3 to within their uncertainty.]
The ultraviolet catastrophe
[Reserved: the integral of the Rayleigh–Jeans density over frequency diverges, so a cavity at any temperature would hold infinite energy and no thermal equilibrium between matter and the field would exist. The name is Ehrenfest's [Ehrenfest:1911a], coined a decade after the fact, and the section should say so: the divergence played no part in Planck's reasoning and is a retrospective diagnosis. What is genuinely diagnosed is that the failure is not a failure of electrodynamics or of statistical mechanics separately, but of equipartition applied to a system with unboundedly many degrees of freedom — exactly the feature that returns as the ultraviolet divergences of Quantum Electrodynamics and Renormalization.]
The measurements that decided the matter
Realizing a black body
[Reserved: the electrically heated cavity radiator built at the Physikalisch-Technische Reichsanstalt by Lummer and Kurlbaum [Lummer:1898], the first laboratory object whose emission could be claimed to be Kirchhoff's universal function; its use as the primary radiometric standard and the ancestry of present-day radiation thermometry (Measurement, SI Units, and the Theory of Errors). Apparatus detail belongs here: aperture-to-cavity ratio, isothermality of the walls, bolometric and thermopile detection, and the calibration chain.]
Lummer–Pringsheim and the failure of Wien's law
[Reserved: the spectra measured by Lummer and Pringsheim between \(300\,\mathrm{K}\) and \(1650\,\mathrm{K}\) [Lummer:1899] out to about \(8\,\mu\mathrm{m}\); systematic deviations from Wien's law [Wien:1896] appearing at long wavelength and high temperature and growing beyond any plausible experimental error. This is the moment the accepted law died, and it died to data, not to an argument. The observations to be tabulated when written: wavelength, temperature, measured spectral radiance and its uncertainty, with the residual against both Wien and Planck.]
Rubens–Kurlbaum in the far infrared
[Reserved: the residual-ray (Reststrahlen) technique that isolated wavelengths of \(24\,\mu\mathrm{m}\) and \(51.2\,\mu\mathrm{m}\) by repeated reflection from fluorite and rock salt, and the finding of Rubens and Kurlbaum that at those wavelengths the radiance is proportional to \(T\) rather than to Wien's exponential [Rubens:1900]. Rubens communicated the result to Planck privately on 7 October 1900; the interpolation of Section 68.5.1 was written the same evening and read to the Physikalische Gesellschaft on 19 October. The chapter should be explicit that this is the causal link, since it is the clearest case in physics of a measurement dictating a formula.]
Far beyond the maximum of the spectrum the radiance of a black body is proportional to the absolute temperature itself, not to Wien's exponential in \(1/T\). Rubens and Kurlbaum isolated \(24\,\mu\mathrm{m}\) and \(51.2\,\mu\mathrm{m}\) by repeated reflection from fluorite and rock salt and found the radiance at those wavelengths to follow \(T\) across the whole accessible range of furnace temperatures — a departure from the accepted distribution [Wien:1896] far larger than any plausible experimental error [Rubens:1900]. Deviations of the same sign, smaller because the wavelengths were shorter, had already been reported between \(300\,\mathrm{K}\) and \(1650\,\mathrm{K}\) out to about \(8\,\mu\mathrm{m}\) [Lummer:1899].
The low-frequency limit of the cavity spectral density: that each mode carries the equipartition energy given by Boltzmann's constant times the temperature, so that the spectral density becomes proportional to the square of the frequency times the temperature, and the radiance at fixed long wavelength is therefore linear in \(T\).
Planck's law
The interpolation of October 1900
[Reserved: Planck's route through the entropy of an oscillator rather than through its energy; the second derivative \(\pp^{2}S/\pp U^{2}\), which is proportional to \(1/U^{2}\) in Wien's limit and to \(1/U\) in the Rayleigh–Jeans limit, and the simplest expression reducing to both, \(\pp^{2}S/\pp U^{2}=-\alpha/(U(\beta+U))\) [Planck:1900a]. Integrating twice returns the law that bears his name. This step is interpolation and nothing more — Planck called it a lucky guess — and the chapter should present it as such, because the physics enters only at the next step.]
The combinatorial derivation of December 1900
[Reserved: Planck's derivation of 14 December 1900 [Planck:1900b]: distribute \(P\) indivisible energy elements \(\varepsilon\) over \(N\) resonators, count the distributions, apply \(S=k\log W\) — an equation Planck wrote here for the first time, naming \(k\) after Boltzmann — and demand consistency with Wien's displacement law Section 68.2.2, which forces \(\varepsilon=h\nu\). The resulting spectral density \(u_{\nu}\propto\nu^{3}/(\ee^{h\nu/k_{\text{B}}T}-1)\); fits to the Reichsanstalt data yielding the first values of \(h\) and \(k_{\text{B}}\) and, through \(k_{\text{B}}\), the first good value of Avogadro's number and of the elementary charge.]
The spectral energy density of the radiation filling an isothermal cavity is
a single expression that reproduces the measured spectrum at every frequency and every temperature at which it has been examined, from the far infrared of Phenomenon 68.4 to the visible [Lummer:1899] [Rubens:1900] [Planck:1901]. It contains one constant of nature not present in classical physics. Planck read the two constants \(h\) and \(k_{\text{B}}\) off the Reichsanstalt data of 1900, obtaining \(6.55\times 10^{-34}\,\mathrm{J}\,\mathrm{s}\) and \(1.346\times 10^{-23}\,\mathrm{J}/\mathrm{K}\) respectively [Planck:1901]; in the present SI both are fixed by definition, at \(6.62607015\times 10^{-34}\,\mathrm{J}\,\mathrm{s}\) and \(1.380649\times 10^{-23}\,\mathrm{J}/\mathrm{K}\) [BIPM:2019], so his 1901 values were already right to about one percent.
Derivation. Take the electromagnetic mode density of Section 68.3.1, \(8\pi\nu^{2}/c^{3}\) modes per unit volume per unit frequency, and Planck's hypothesis [Planck:1900b] that a resonator of frequency \(\nu\) exchanges energy with the field only in whole elements \(\varepsilon\), so that its accessible energies are \(E_{n}=n\varepsilon\) with \(n=0,1,2,\dots\). In equilibrium at temperature \(T\) the level \(n\) carries statistical weight \(\ee^{-n\varepsilon/k_{\text{B}}T}\), and writing \(x:=\ee^{-\varepsilon/k_{\text{B}}T}<1\) the mean energy of one resonator is
both geometric series converging because \(x<1\). Multiplying by the number of modes gives the spectral density
Nothing so far fixes \(\varepsilon\). That it must be proportional to the frequency is forced by Phenomenon 68.3: only \(\varepsilon=h\nu\), with \(h\) a universal constant, makes the last expression equal to \(\nu^{3}\) times a function of \(\nu/T\) alone, which is the scaling Wien's thermodynamic argument establishes. Substituting \(\varepsilon=h\nu\) yields Equation (68.5).
The two limits are immediate. For \(h\nu\gg k_{\text{B}}T\) the exponential dominates and Equation (68.5) becomes \(\left(8\pi h\nu^{3}/c^{3}\right)\ee^{-h\nu/k_{\text{B}}T}\), which is Wien's distribution [Wien:1896] — so its early success in the visible was no evidence of its truth. For \(h\nu\ll k_{\text{B}}T\), expanding \(\ee^{h\nu/k_{\text{B}}T}-1\simeq h\nu/k_{\text{B}}T\) gives \(8\pi\nu^{2}k_{\text{B}}T/c^{3}\), the equipartition result [Rayleigh:1900] [Jeans:1905] and the observed proportionality to \(T\) of Phenomenon 68.4. Both classical results are limits of one formula, and neither is valid outside its own limit.
∎The 1901 Annalen paper
[Reserved: the considered exposition [Planck:1901], with the two universal constants \(h=6.55\times 10^{-34}\,\mathrm{J}\,\mathrm{s}\) and \(k=1.346\times 10^{-23}\,\mathrm{J}/\mathrm{K}\) as Planck determined them, to be compared in a table with the present exact SI values \(h=6.62607015\times 10^{-34}\,\mathrm{J}\,\mathrm{s}\) and \(k_{\text{B}}= 1.380649\times 10^{-23}\,\mathrm{J}/\mathrm{K}\) [BIPM:2019]; the agreement to a percent from 1901 data is itself an observation worth making. Also Planck's system of natural units, proposed in the same period from \(h\), \(c\) and \(G\) alone.]
Limits, moments and consistency
[Reserved: the two limits of Planck's law — \(h\nu\gg k_{\text{B}}T\) recovering Wien [Wien:1896], \(h\nu\ll k_{\text{B}} T\) recovering Rayleigh–Jeans [Rayleigh:1900] [Jeans:1905] — and its integral, which reproduces the Stefan–Boltzmann law with \(\sigma\) now computed rather than measured, a genuine prediction and the first quantitative success of the hypothesis. The maximum reproduces Wien's displacement constant. Every classical result of Section 68.2 is thereby subsumed, and the constant \(b\) and the constant \(\sigma\) cease to be independent empirical numbers.]
What exactly was quantized
Energy elements
[Reserved: the statement \(\varepsilon=h\nu\) as a restriction on the exchange of energy between the field and a material resonator, which is what Planck asserted [Planck:1900b], against the stronger reading that the field itself is granular, which he resisted for a decade and which is Einstein's (The Photon: Photoelectric and Compton Effects). The distinction is not scholastic: it is the difference between a property of matter and a property of light, and only the second is established by the experiments of the next chapter. Modern restatement: the mean energy of a quantum harmonic oscillator, \(h\nu/(\ee^{h\nu/k_{\text{B}}T}-1)\), derived properly in Elementary Quantum Systems and Statistical Mechanics.]
The reception, 1900–1911
[Reserved: honest history of a result nobody believed. Planck's law was accepted as a formula and rejected as a hypothesis; the turning point was the first Solvay Council of 1911 and, before it, the independent evidence from specific heats [Einstein:1907a]. Lorentz and Jeans argued for years that the cavity was simply not in equilibrium [Jeans:1905]; Ehrenfest's analysis [Ehrenfest:1911a] isolated what in the hypothesis is indispensable. The section states plainly which claims were settled by evidence and which by exhaustion.]
Out of the cavity: specific heats
The Einstein solid
[Reserved: Einstein's transfer of the quantization hypothesis from cavity walls to the vibrations of a crystal [Einstein:1907a]; \(3N\) independent oscillators of a single frequency \(\nu_{\text{E}}\), molar heat capacity falling below the Dulong–Petit value \(3R\) below \(T\approx h\nu_{\text{E}}/k_{\text{B}}\) and vanishing as \(T\to0\). This was the first prediction of the quantum hypothesis outside radiation, and it was confirmed by Nernst's low-temperature calorimetry — the evidence that converted the physics community. Its known defect: the exponential fall is too fast against the measured one.]
The molar heat capacity of a crystalline solid is close to the Dulong–Petit value \(3R\) at ordinary temperatures, falls steadily as the solid is cooled, and tends to zero as \(T\to0\). The fall sets in around a temperature characteristic of the material, so that a hard, light crystal such as diamond is already far below \(3R\) at room temperature while lead is not. Nernst's calorimetry established the behaviour across many substances down to liquid-hydrogen temperatures [Nernst:1911]; in insulating crystals the low-temperature approach to zero follows \(C\propto T^{3}\) [Debye:1912a]. Classical equipartition (Kinetic Theory of Gases) gives \(3R\) at every temperature and cannot produce any of this.
The quantum theory of lattice heat capacities: the mean energy of a quantised oscillator applied to the vibrational degrees of freedom of a crystal, which gives the exponential freeze-out of a single Einstein frequency; and the replacement of that single frequency by the elastic-continuum mode spectrum with a cut-off preserving the total mode count, which produces the observed cubic low-temperature law.
Debye's improvement and the phonon
[Reserved: Debye's replacement of the single frequency by the elastic-continuum mode spectrum, with the cut-off that keeps the mode count at \(3N\) [Debye:1912a]; the resulting \(T^{3}\) law at low temperature, which fits the data, and the Debye temperature as the single material parameter. The mode counting is that of Section 68.3.1 with sound in place of light; the full lattice-dynamical treatment is Phonons and Lattice Dynamics, and the electronic contribution that Debye's theory omits is Electrons in Solids: Band Theory.]
The statistics behind the law
Bose's derivation
[Reserved: Bose's 1924 derivation of Planck's law from the counting of indistinguishable quanta in phase-space cells, with no appeal to classical electrodynamics at any point [Bose:1924] — the first derivation in which the law follows from the statistics of the field rather than from the mechanics of the walls. Einstein translated the paper, had it published, and extended the counting to material particles, from which follow Bose–Einstein statistics and condensation (Quantum Statistics and Experiment: Bose–Einstein Condensation). Forward pointer: the indistinguishability at issue is the symmetrization postulate of Identical Particles.]
Detailed balance and the Einstein coefficients
[Reserved: Einstein's 1917 rederivation [Einstein:1917b], in which the equilibrium of a two-level atom with the radiation field requires three processes — absorption, spontaneous emission and stimulated emission — and the ratios of their rates are fixed by demanding Planck's law at equilibrium. The argument also yields the momentum \(h\nu/c\) transferred in each elementary process, which is the photon momentum of The Photon: Photoelectric and Compton Effects. Stimulated emission is the principle of the laser (Quantum Optics and the Photon), and the coefficients are computed from first principles in Approximation Methods.]
Black bodies in nature
The cosmic microwave background
[Reserved: the most exact black-body spectrum known, measured by the FIRAS instrument on COBE with residuals below one part in \(10^{4}\) of the peak and a temperature \(T=2.7255\,\mathrm{K}\) [Fixsen:2009]; the absence of any spectral distortion as a constraint on energy release in the early universe, and the present-day cosmological parameters that rest on it [Aghanim:2020]. The measurement itself is Experiment: The Cosmic Microwave Background and the cosmology Evidence-Based Cosmology; what belongs here is only that Kirchhoff's universal function is realized in nature to a precision no laboratory cavity approaches.]
Observed from above the atmosphere, the whole sky carries a nearly isotropic radiation field whose spectrum is that of a black body whose temperature \(T\) is \(2.72548(57)\,\mathrm{K}\), with no departure from Equation (68.5) detected anywhere across the peak [Fixsen:2009]. It is the most accurately black-body spectrum known, and no laboratory cavity approaches it: the universe as a whole realizes Kirchhoff's universal function better than any apparatus built to realize it.
Why the expanding universe leaves a Planck spectrum Planckian: thermal equilibrium of matter and radiation before recombination fixes the form, and the redshift of every mode by a common factor rescales the spectral density into another Planck spectrum with the temperature inversely proportional to the scale factor, so that no spectral distortion is generated by the expansion itself. The limits on distortions then become limits on energy release after equilibrium was lost.
Stellar spectra and radiation thermometry
[Reserved: stellar continua as approximate Planck spectra, the effective temperature as the parameter of the fit, and the departures — absorption lines, limb darkening, non-grey opacity — that carry the astrophysics of Stellar Structure and Nucleosynthesis; the Sun at about \(5772\,\mathrm{K}\). On the metrological side, radiation thermometry as a primary method of realizing the kelvin, and the role of \(h\) as a defining constant of the present SI [BIPM:2019] [Mohr:2025], so that Planck's law now enters the definition of the units in which it is verified (Measurement, SI Units, and the Theory of Errors).]