Black-Body Radiation

Alright. You want me to take this dry, academic text and… inject it with my particular brand of life. To make it… interesting. Fine. Don't expect sunshine and rainbows. This is about the cold, hard facts, delivered with a precision that might sting.

Consider this less of a rewrite and more of an… illumination. Because light, as you know, is often annoying.

Thermal Electromagnetic Radiation

As the temperature of a black body decreases, the emitted thermal radiation diminishes in intensity, and its peak wavelength shifts towards longer wavelengths. For comparison, the classical Rayleigh–Jeans law and its associated ultraviolet catastrophe are shown.

Black-body radiation is the thermal electromagnetic radiation that exists within, or surrounds, a body that is in a state of thermodynamic equilibrium with its environment. It is emitted by an idealized black body – a theoretical object that is perfectly opaque and reflects nothing. This radiation possesses a specific continuous spectrum, the shape of which is dictated solely by the body's temperature.[1] [2] [3] [4]

Imagine a perfectly insulated enclosure, maintained at a uniform temperature. Within this space, the black-body radiation is in perfect equilibrium. If a small aperture is created in the wall of this enclosure – small enough not to disturb the equilibrium – the radiation that escapes through it will closely approximate ideal black-body radiation. Many real-world objects, while not perfect, emit thermal radiation that can be reasonably approximated by this model.

It's worth noting that even celestial bodies like planets and stars, including our own Earth and the Sun, are not perfect black bodies, nor are they in strict thermal equilibrium with their surroundings. Nevertheless, black-body radiation serves as a remarkably effective first approximation for understanding the energy they emit.[5]

The term "black body" itself was first coined by Gustav Kirchhoff in 1860.[6] This phenomenon is also known by other, perhaps more descriptive, names: thermal radiation, cavity radiation, complete radiation, or temperature radiation. Each hints at its origin and nature.

Theory

Spectrum

Blacksmiths, with their practiced eyes, can gauge the temperature of their workpieces simply by observing the color of the glow. It’s a practical, visceral understanding of this very phenomenon.

The color chart of a blacksmith’s glow, while useful, stops at the melting point of steel. Beyond that, things get… more intense.

Black-body radiation emits a spectrum of electromagnetic frequencies that is distinctively continuous. Its precise shape is a direct consequence of the body's temperature,[8] a relationship codified as Planck's law. As the temperature rises, the peak of this spectrum shifts towards higher frequencies. At room temperature, the bulk of this emission lies in the infrared portion of the electromagnetic spectrum, invisible to our eyes but certainly felt as heat.[9] [10] [11]

However, as temperatures climb past approximately 500 degrees Celsius, black bodies begin to emit light that we can see. In darkness, the initial faint glow appears as a "ghostly" grey. This is misleading; the light is actually red, but at such low intensity that it only triggers the eye's achromatic (grey-level) sensors. As the temperature escalates, the glow becomes perceptible even in ambient light, progressing from a dull red to yellow, and finally to a "dazzling bluish-white" at extreme temperatures.[12] [13] When an object appears white, it's radiating a significant portion of its energy in the ultraviolet range. Our Sun, with an effective temperature hovering around 5800 K,[14] is a decent approximation of a black body. Its emission spectrum peaks in the yellow-green region of the visible spectrum, but it also pumps out considerable energy in the ultraviolet.

The study of black-body radiation offers profound insights into the nature of thermodynamic equilibrium, particularly concerning the radiation confined within a cavity.

Black Body

A black body is, in essence, a perfect absorber. All electromagnetic radiation that strikes it, regardless of wavelength or angle, is absorbed. This is the idealized definition, of course. In reality, perfect black bodies are theoretical constructs. Materials like graphite and lamp black, with emissivities exceeding 0.95, come remarkably close.

The most practical and stable realization of black-body radiation is found within a cavity in a solid object, maintained at a uniform temperature. This cavity must be entirely opaque and only minimally reflective. A simple yet effective example is a closed box with walls made of graphite, kept at a constant temperature, and featuring a small hole. The radiation escaping from this opening closely mimics ideal black-body radiation.[15] [16] [17]

Black-body radiation represents the most stable state of radiative intensity achievable in a cavity in thermodynamic equilibrium.[15] Crucially, at thermal equilibrium, the intensity of radiation emitted and reflected by any body, for any given frequency, is solely a function of its temperature. It is independent of the body's shape, material composition, or internal structure.[18] For a true black body, which absorbs all incident radiation, there is no reflection. Therefore, its spectral radiance is purely a measure of its emission. Furthermore, a black body is a diffuse emitter, meaning its radiation intensity is uniform in all directions.

When a black body reaches a sufficiently high temperature, its emission becomes visible light.[19] The Draper point, approximately 798 K, is the temperature at which all solids begin to emit a dim red glow. At 1000 K, a tiny hole in a uniformly heated, opaque cavity appears red when viewed from the outside. At 6000 K, it would appear white. The spectrum, and thus the color, of the light emitted from such a cavity is determined only by its temperature. A graph plotting spectral radiation intensity against frequency (or wavelength) is known as the blackbody curve. Each curve represents a different temperature.

The color of Pāhoehoe lava flows, for instance, can be used to estimate their temperature, and these estimates align well with other measurements for flows between 1,000 and 1,200 °C (1,830 to 2,190 °F).

When a body is perfectly black, its absorption is complete – it absorbs all light that falls upon it. For a black body significantly larger than the wavelength of light, the energy absorbed per unit time at any given wavelength is directly proportional to the blackbody curve. This is why the blackbody curve accurately represents the energy emitted by a black body. This principle is central to Kirchhoff's law of thermal radiation, stating that the emitted radiation is characteristic of thermal light, dependent only on the temperature of the cavity walls, provided they are opaque, not overly reflective, and the cavity is in thermodynamic equilibrium.[21] When the black body is small, comparable in size to the wavelength of light, its absorption characteristics change. It becomes less efficient at absorbing long wavelengths. However, the fundamental principle of emission and absorption being equal in thermodynamic equilibrium always holds.

In laboratory settings, black-body radiation is approximated using a small hole in a larger cavity – a hohlraum – within an opaque, partially reflective body maintained at a constant temperature. This is why it's sometimes called cavity radiation. Any light entering the hole undergoes multiple reflections off the cavity walls, making absorption highly probable. This absorption occurs irrespective of the wavelength of the incoming radiation, as long as it's small compared to the hole's size. Thus, the hole acts as a near-perfect black body, and its emitted radiation spectrum depends solely on the cavity's temperature and its opaque, absorptive walls, not on the specific materials used.

The observed intensity, or radiance, from such a source is uniform in all directions. This means a black body is a perfect Lambertian radiator.

Real objects deviate from this ideal. The radiation they emit at any given frequency is only a fraction of what a perfect black body would emit. The emissivity of a material quantifies this. It's the ratio of the radiation emitted by a real object to that emitted by a perfect black body at the same temperature. This emissivity can vary with temperature, emission angle, and wavelength. In engineering, it's common to simplify this by assuming spectral emissivity and absorptivity are constant, leading to the "gray body" assumption.

The image shows a nine-year WMAP view (from 2012) of the cosmic microwave background radiation permeating the universe.[22] [23]

With surfaces that aren't perfectly black, deviations from ideal black-body behavior are influenced by surface structure (like roughness or granularity) and chemical composition. However, objects in a state of local thermodynamic equilibrium still adhere to Kirchhoff's Law: emissivity equals absorptivity. This means an object that doesn't absorb all incident light will also emit less radiation than an ideal black body. This incomplete absorption can occur because some light is transmitted through the object or reflected from its surface.

In astronomy, stars are often treated as black bodies, though this is frequently a rough approximation. The cosmic microwave background radiation exhibits a nearly perfect black-body spectrum. Hawking radiation is the hypothetical black-body radiation theorized to be emitted by black holes, with a temperature dependent on the hole's mass, charge, and spin. If this theory holds true, black holes would gradually shrink and evaporate over vast timescales by emitting photons and other particles.

A black body radiates energy across all frequencies, but its intensity drops sharply at high frequencies (short wavelengths). For instance, a black body at room temperature (300 K) with a surface area of one square meter would emit a photon in the visible range (390–750 nm) only once every 41 seconds, on average. For practical purposes, this means such an object emits virtually no visible light.[24]

The study of black-body radiation and the inability of classical physics to accurately describe it were crucial in laying the groundwork for quantum mechanics.

Additional explanations

Classical physics, when applied to radiation in an empty cavity with perfectly reflective walls, encountered a significant problem. If each Fourier mode of the equilibrium radiation is treated as a degree of freedom that can exchange energy, the equipartition theorem dictates that each mode should possess an equal share of energy. Since there are infinitely many modes, this implies an infinite heat capacity and a non-physical spectrum of emitted radiation that increases without bound as frequency rises, predicting infinite emission power. This anomaly is known as the ultraviolet catastrophe. Furthermore, the classical model failed to explain the experimentally observed peak in the emission spectra (see also Wien's law).

The solution arrived with the quantum treatment. In this approach, the energy of the modes is quantized. This means energy can only exist in discrete, integer multiples of a fundamental unit. This quantization naturally attenuates the spectrum at high frequencies, aligning with experimental observations and resolving the catastrophe. Modes with energy less than the fundamental thermal energy of the substance itself were not considered, and due to quantization, modes requiring infinitesimally small amounts of energy were excluded.

Consequently, for shorter wavelengths, fewer modes are permitted (requiring more energy than $h\nu$ to be occupied). This explains why the emitted energy decreases at wavelengths shorter than the observed peak of emission, a finding that supports the experimental data.

It's important to recognize the interplay of two factors shaping the blackbody curve. Firstly, shorter wavelengths are associated with a greater number of modes. This contributes to the rise in spectral radiance as one moves from longer wavelengths towards the peak. Secondly, however, at shorter wavelengths, more energy is required to reach the threshold for occupying each mode. The greater the energy needed to excite a mode, the lower the probability of that mode being occupied. As wavelength decreases, this probability plummets, leading to fewer occupied modes and, subsequently, a decrease in spectral radiance at very short wavelengths (to the left of the peak). The combination of these two effects produces the characteristic shape of the curve.[25]

Deriving the blackbody curve was a significant challenge in theoretical physics in the late 19th century. The breakthrough came in 1901 with Max Planck and his formulation, now known as Planck's law of blackbody radiation.[26] By modifying Wien's radiation law (distinct from Wien's displacement law) to be consistent with the principles of thermodynamics and electromagnetism, he developed a mathematical expression that accurately fit the experimental data. Planck's crucial assumption was that the energy of the oscillators within the cavity was quantized, existing only in integer multiples of a basic unit. Albert Einstein, building on this concept, proposed the quantization of electromagnetic radiation itself in 1905 to explain the photoelectric effect. These foundational advances ultimately led to the replacement of classical electromagnetism with quantum electrodynamics. The quanta of light were named photons, and the blackbody cavity was conceptually viewed as containing a gas of photons. This work also contributed to the development of quantum probability distributions, specifically Fermi–Dirac statistics and Bose–Einstein statistics, each applicable to different classes of particles: fermions and bosons, respectively.

The wavelength at which the radiation intensity is at its maximum is described by Wien's displacement law. The total power radiated per unit area is given by the Stefan–Boltzmann law. Consequently, as temperature increases, the glow color shifts from red to yellow, then white, and finally blue. As the peak wavelength moves into the ultraviolet and beyond, a portion of the spectrum remains in the visible range, and its intensity increases, leading to a blue appearance. The radiation never truly disappears; in fact, the intensity of visible light increases monotonically with temperature.[27]

The Stefan–Boltzmann law states that the total radiant energy emitted by a black body surface per unit time is directly proportional to the fourth power of its absolute temperature. This law was formulated by Josef Stefan in 1879 and later derived by Ludwig Boltzmann. The equation is given as $E = \sigma T^4$ , where $E$ represents the radiant energy emitted per unit area per unit time (power per unit area), $T$ is the absolute temperature, and $\sigma$ is the Stefan–Boltzmann constant, approximately $5.670367 \times 10^{-8} \, \text{W} \cdot \text{m}^{-2} \cdot \text{K}^{-4}$ .

Equations

Planck's law of blackbody radiation

• Main article: Planck's law

Planck's law states that:

$B_{\nu }(T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}},$

where:

• $B_{\nu }(T)$ is the spectral radiance at thermal equilibrium at temperature $T$ , measured in units of power per (unit solid angle * unit area normal to propagation * unit frequency). • $h$ is the Planck constant. • $c$ is the speed of light in a vacuum. • $k$ is the Boltzmann constant. • $\nu$ is the frequency of the electromagnetic radiation. • $T$ is the absolute temperature of the body.

For the surface of a black body, the spectral radiance density (defined per unit area normal to the direction of propagation) is independent of the emission angle $\theta$ relative to the normal. However, this implies that, according to Lambert's cosine law, $B_{\nu }(T)\cos \theta$ represents the radiance density per unit area of the emitting surface. As the emitting surface area increases by a factor of $1/\cos \theta$ for oblique angles relative to an area normal to the propagation direction, and the solid angle subtended decreases, the aggregate intensity is lower.

The emitted energy flux density, or irradiance, $B_{\nu }(T,E)$ , is related to the photon flux density, $b_{\nu }(T,E)$ , by:

$B_{\nu }(T,E)=Eb_{\nu }(T,E)$

The equivalent equation in terms of $\omega$ is:

$B_{\omega }(T)={\frac {\hbar \omega ^{3}}{4\pi ^{3}c^{2}}}{\frac {1}{e^{\frac {\hbar \omega }{kT}}-1}}.$

Wien's displacement law

• Main article: Wien's displacement law

Wien's displacement law describes the relationship between the spectrum of black-body radiation at any given temperature and the spectrum at any other temperature. Knowing the spectral shape at one temperature allows for the calculation of its shape at any other temperature. Spectral intensity can be expressed as a function of either wavelength or frequency.

A consequence of Wien's displacement law is that the wavelength at which the intensity per unit wavelength of the radiation produced by a black body reaches a local maximum, $\lambda_{\text{peak}}$ , is solely dependent on the temperature:

$\lambda_{\text{peak}}={\frac {b}{T}},$

where the constant $b$ , known as Wien's displacement constant, is approximately $2.897771955 \times 10^{-3} \, \text{m} \cdot \text{K}$ .[32] At a typical room temperature of 293 K (20 °C), the maximum intensity occurs at a wavelength of 9.9 μm.

Planck's law was also presented earlier as a function of frequency. The frequency at which the intensity reaches its maximum is given by:

$\nu_{\text{peak}}=T \times 5.879... \times 10^{10} \, \text{Hz} / \mathrm{K}.$

In dimensionless form, the maximum occurs when $e^x(1-x/3)=1$ , where $x = h\nu / kT$ . The approximate numerical solution for $x$ is $2.82$ . At a typical room temperature of 293 K (20 °C), this corresponds to a frequency of 17 THz.

Stefan–Boltzmann law

• Main article: Stefan–Boltzmann law

By integrating $B_{\nu }(T)\cos(\theta)$ over all frequencies, the radiance $L$ (units: power / [area × solid angle]) is obtained:

$L=\int _{0}^{\infty }B_{\nu }(T)\cos(\theta )d\nu ={\frac {2\pi ^{5}}{15}}{\frac {k^{4}T^{4}}{c^{2}h^{3}}}{\frac {\cos(\theta )}{\pi }}=\sigma T^{4}{\frac {\cos(\theta )}{\pi }}}$

This integration utilizes the result $\int _{0}^{\infty}dx\,{\frac {x^{3}}{e^{x}-1}}={\frac {\pi ^{4}}{15}}$ , where $x\equiv {\frac {h\nu }{kT}}$ . The constant $\sigma \equiv {\frac {2\pi ^{5}}{15}}{\frac {k^{4}}{c^{2}h^{3}}} \approx 5.670373 \times 10^{-8} \, \text{W} \cdot \text{m}^{-2} \cdot \text{K}^{-4}$ is the Stefan–Boltzmann constant.

As a side note, at a distance $d$ , the intensity $dI$ per area $dA$ of a radiating surface is given by the useful expression:

$dI=\sigma T^{4}{\frac {\cos \theta }{\pi d^{2}}}dA$

when the receiving surface is perpendicular to the direction of radiation.

By subsequently integrating $L$ over the solid angle $\Omega$ for all azimuthal angles (0 to $2\pi$ ) and polar angles $\theta$ from 0 to $\pi/2$ , we arrive at the Stefan–Boltzmann law: the power emitted per unit area of a black body surface is directly proportional to the fourth power of its absolute temperature:

$j^{\star }=\sigma T^{4},$

The integral used is $\int \cos \theta \,d\Omega =\int _{0}^{2\pi }\int _{0}^{\pi /2}\cos \theta \sin \theta \,d\theta \,d\phi =\pi$ .

Applications

Human-body emission

The human body radiates energy primarily as infrared light. The net power radiated is the difference between the power emitted and the power absorbed:

$P_{\text{net}}=P_{\text{emit}}-P_{\text{absorb}}.$

Applying the Stefan–Boltzmann law, this becomes:

$P_{\text{net}}=A\sigma \varepsilon \left(T^{4}-T_{0}^{4}\right),$

where $A$ is the surface area of the body, $T$ is its temperature, $\varepsilon$ is the emissivity, and $T_0$ is the ambient temperature.

An average adult has a surface area of about 2 m². The emissivity of skin and most clothing in the mid- and far-infrared ranges is close to unity, a common characteristic of non-metallic surfaces.[34] [35] While skin temperature is around 33 °C,[36] clothing can reduce the effective surface temperature to about 28 °C when the ambient temperature is 20 °C.[37] Consequently, the net radiative heat loss is approximately 100 W.

The total energy radiated in a day amounts to about 8 MJ, or 2000 kcal (food calories). The basal metabolic rate for a 40-year-old male is roughly 35 kcal/(m²·h), which translates to about 1700 kcal per day for a 2 m² area. However, the average metabolic rate for sedentary adults is typically 50% to 70% higher than their basal rate.[38] [39]

Other significant mechanisms of thermal loss include convection and evaporation. Conduction is generally negligible, as the Nusselt number is significantly greater than unity. Evaporation via perspiration becomes necessary only when radiation and convection are insufficient to maintain a steady state temperature (though evaporation from the lungs occurs continuously). Free convection rates are comparable to, though slightly lower than, radiative rates.[40] Therefore, in cool, still air, radiation accounts for about two-thirds of thermal energy loss. Given the approximations involved, this is a rough estimate. Factors like air movement (forced convection) or evaporation can reduce the relative importance of radiation.

Applying Wien's law to human-body emission yields a peak wavelength of:

$\lambda_{\text{peak}}=\mathrm {\frac {2.898\times 10^{-3}~K\cdot m}{305~K}} =\mathrm {9.50~\mu m} .$

This is why thermal imaging devices designed for human subjects are most sensitive in the 7–14 micrometer range.

Temperature relation between a planet and its star

• Main article: Planetary equilibrium temperature

The blackbody law can be employed to estimate the temperature of a planet orbiting a star.

The image depicts Earth's longwave thermal radiation intensity, originating from clouds, the atmosphere, and the ground.

A planet's temperature is influenced by several factors:

The incident radiation from its star.
The planet's own emitted radiation (e.g., Earth's infrared glow).
The albedo effect, causing a fraction of incoming light to be reflected.
The greenhouse effect in planets possessing an atmosphere.
Internal energy sources within the planet, such as radioactive decay, tidal heating, and adiabatic contraction due to cooling.

For simplicity, the following analysis focuses solely on the heat received from the Sun by a planet within a Solar System.

The Stefan–Boltzmann law describes the total power (energy per second) emitted by the Sun:

$P_{\rm {S\ emt}}=4\pi R_{\rm {S}}^{2}\sigma T_{\rm {S}}^{4}}$ where:

$\sigma$ is the Stefan–Boltzmann constant.
$T_{\rm {S}}$ is the effective temperature of the Sun.
$R_{\rm {S}}$ is the radius of the Sun.

The Sun radiates this power uniformly in all directions. Consequently, the planet intercepts only a small fraction of this energy. The power from the Sun that reaches the planet (at the top of its atmosphere) is:

$P_{\rm {SE}}=P_{\rm {S\ emt}}\left({\frac {\pi R_{\rm {E}}^{2}}{4\pi D^{2}}}\right)$ where:

$R_{\rm {E}}$ is the radius of the planet.
$D$ is the distance between the Sun and the planet.

Due to its high temperature, the Sun emits predominantly in the ultraviolet and visible (UV-Vis) frequency range. The planet reflects a fraction $\alpha$ of this energy, where $\alpha$ is its albedo or UV-Vis reflectivity. In essence, the planet absorbs a fraction $1-\alpha$ of the Sun's light and reflects the rest. The power absorbed by the planet and its atmosphere is:

$P_{\rm {abs}}=(1-\alpha )\,P_{\rm {SE}}}$

Although the planet absorbs energy over a cross-sectional area of $\pi R^2$ , it emits radiation in all directions. The total surface area is $4\pi R^2$ . If the planet acted as a perfect black body, its emission would follow the Stefan–Boltzmann law:

$P_{\rm {emt\,bb}}=4\pi R_{\rm {E}}^{2}\sigma T_{\rm {E}}^{4}}$ where $T_{\rm {E}}$ is the planet's temperature. This temperature, calculated by setting $P_{\rm {abs}}=P_{\rm {emt\,bb}}$ , is known as the effective temperature. The actual surface temperature of a planet will likely differ, depending on its surface and atmospheric characteristics.

Ignoring the atmosphere and the greenhouse effect, and assuming the planet emits primarily in the infrared (IR) spectrum (due to its much lower temperature compared to the Sun), it emits $\overline{\varepsilon}$ of the radiation that a black body would. Here, $\overline{\varepsilon}$ represents the average emissivity in the IR range. The power emitted by the planet is then:

$P_{\rm {emt}}={\overline {\epsilon }}\,P_{\rm {emt\,bb}}}$

For an object in radiative exchange equilibrium with its surroundings, the rate at which it emits radiant energy equals the rate at which it absorbs it:

$P_{\rm {abs}}=P_{\rm {emt}}}$

Substituting the expressions for solar and planetary power and simplifying yields the estimated temperature of the planet, $T_P$ , ignoring the greenhouse effect:

$T_{P}=T_{S}{\sqrt {\frac {R_{S}{\sqrt {\frac {1-\alpha }{\overline {\varepsilon }}}}}{2D}}}}$

This equation reveals that, under these assumptions, a planet's temperature depends only on the Sun's surface temperature, the Sun's radius, the Sun-planet distance, the planet's albedo, and its IR emissivity.

Notably, a gray ball (with a flat spectral response) where $(1-\alpha) = \overline{\varepsilon}$ will reach the same temperature as a black body, regardless of its shade of gray.

Effective temperature of Earth

Using the measured values for the Sun and Earth:

$T_{\rm {S}} = 5772 \, \text{K}$ ,[43]
$R_{\rm {S}} = 6.957 \times 10^8 \, \text{m}$ ,[43]
$D = 1.496 \times 10^{11} \, \text{m}$ ,[43]
$\alpha = 0.309 \,$ [44]

Assuming an average emissivity $\overline{\varepsilon}$ of unity, the effective temperature of the Earth is calculated as:

$T_{\rm {E}}=254.356 \, \text{K}$ or -18.8 °C.

This is the temperature the Earth would have if it radiated as a perfect black body in the infrared, assuming a constant albedo and disregarding greenhouse effects (which can elevate a body's surface temperature above its black-body equivalent across all spectrums[45]). In reality, Earth's infrared radiation deviates slightly from that of a perfect black body, which would raise the estimated temperature by a few degrees. If we were to estimate Earth's temperature without an atmosphere, the albedo and emissivity of the Moon could serve as a reasonable approximation. The Moon's albedo and emissivity are approximately 0.1054[46] and 0.95[47] respectively, yielding an estimated temperature of about 1.36 °C.

Estimates for Earth's average albedo range from 0.3 to 0.4, leading to variations in estimated effective temperatures. Often, calculations are based on the solar constant (total solar irradiance) rather than the Sun's temperature, size, and distance. For instance, using an albedo of 0.4 and an insolation of 1400 W m⁻², the effective temperature is approximately 245 K.[48] Similarly, with an albedo of 0.3 and a solar constant of 1372 W m⁻², the effective temperature is calculated to be around 255 K.[49] [50] [51]

Cosmology

The cosmic microwave background radiation observed today is the most precise example of black-body radiation found in nature, with a temperature of approximately 2.7 K.[52] It represents a "snapshot" of the radiation at the time of decoupling between matter and radiation in the early universe. Prior to this epoch, most matter in the universe existed as an ionized plasma in thermal, though not complete thermodynamic, equilibrium with radiation.

At extremely high temperatures (above $10^{10}$ K), such as those present in the very early universe where thermal motion overcomes the strong nuclear forces binding protons and neutrons, electron-positron pairs spontaneously form and annihilate, existing in thermal equilibrium with electromagnetic radiation. These particles, in addition to the electromagnetic radiation, contribute to the black body spectrum.[53]

A black body at room temperature (23 °C (296 K; 73 °F)) emits radiation primarily in the infrared spectrum, which is invisible to the human eye.[54] However, certain reptiles can perceive it. As the temperature rises to about 500 °C (773 K; 932 °F), the emission spectrum intensifies and extends into the visible range, causing the object to glow a dull red. With further increases in temperature, it emits progressively more orange, yellow, green, and eventually blue light (and beyond violet, into the ultraviolet).

Light bulb

Tungsten filament light bulbs produce a continuous black body spectrum with a color temperature around 2,700 K (2,430 °C; 4,400 °F), and also radiate a significant amount of energy in the infrared. More efficient modern fluorescent and LED lights do not exhibit a continuous black body spectrum. Instead, they emit light directly or utilize phosphors that emit specific, narrow spectral lines.

The color (specifically, the chromaticity) of blackbody radiation shifts inversely with the temperature of the black body. The path traced by these colors, often visualized in CIE 1931 x,y space, is known as the Planckian locus.

History

The notion of a black body was first introduced by Isaac Newton in query 6 of his work Opticks, where he pondered, "Do not black Bodies conceive heat more easily from Light than those of other Colours do, by reason that the Light falling on them is not reflected outwards, but enters into the Bodies, and is often reflected and refracted within them, until it be stifled and lost?"[55] [56] [57] In his initial memoir, Augustin-Jean Fresnel (1788–1827), responding to an interpretation he derived from a French translation of Newton's Opticks, suggested that Newton envisioned light particles traversing space unimpeded by the pervasive caloric medium. Fresnel refuted this (a view Newton never held) by arguing that a black body illuminated would indefinitely increase in heat.[58]

Balfour Stewart

In 1858, Balfour Stewart detailed experiments comparing the thermal radiative emissive and absorptive powers of polished plates of various substances with those of lamp-black surfaces at the same temperature.[59] Stewart selected lamp-black surfaces as his reference, drawing upon prior experimental observations, notably those of Pierre Prevost and John Leslie. He posited, "Lamp-black, which absorbs all the rays that fall upon it, and therefore possesses the greatest possible absorbing power, will possess also the greatest possible radiating power." Stewart's assertion was based on a general principle: the existence of a body or surface with the maximum possible absorbing and radiating power across all wavelengths and at any equilibrium temperature.

Stewart's research focused on selective thermal radiation, investigating plates that selectively radiated and absorbed different wavelengths. He analyzed these phenomena in terms of rays that could be reflected and refracted, adhering to the Stokes-Helmholtz reciprocity principle. His work did not account for wavelength-dependent ray properties and did not employ tools like prisms or diffraction gratings. Within these constraints, his measurements were quantitative. He conducted his experiments in a room-temperature environment and worked swiftly to capture his bodies in a state near their prepared thermal equilibrium.

Gustav Kirchhoff

In 1859, Gustav Robert Kirchhoff reported the remarkable coincidence of wavelengths between spectrally resolved lines of absorption and emission in visible light. Crucially for thermal physics, he also observed that bright lines or dark lines appeared depending on the temperature difference between the emitter and the absorber.[60]

Kirchhoff then extended his investigation to bodies emitting and absorbing thermal radiation within an opaque enclosure or cavity, maintained in equilibrium at a temperature $T$ .

Here, a notation different from Kirchhoff's is employed. The emitting power $E(T, i)$ denotes a dimensional quantity representing the total radiation emitted by a body indexed by $i$ at temperature $T$ . The total absorption ratio $a(T, i)$ of that body is dimensionless, representing the ratio of absorbed to incident radiation within the cavity at temperature $T$ . (Unlike Balfour Stewart's definition, Kirchhoff's absorption ratio did not specifically refer to a lamp-black surface as the source of incident radiation.) Consequently, the ratio $E(T, i) / a(T, i)$ of emitting power to absorptivity is a dimensional quantity with the dimensions of emitting power, as $a(T, i)$ is dimensionless. Furthermore, the wavelength-specific emitting power of the body at temperature $T$ is denoted by $E(\lambda, T, i)$ , and the wavelength-specific absorption ratio by $a(\lambda, T, i)$ . Again, the ratio $E(\lambda, T, i) / a(\lambda, T, i)$ of emitting power to absorptivity is a dimensional quantity with the dimensions of emitting power.

In a subsequent report in 1859, Kirchhoff announced a new general principle or law, for which he provided a theoretical and mathematical proof, though he did not present quantitative measurements of radiation powers.[61] His theoretical proof has been considered invalid by some scholars then and now.[62] [63] However, his principle has endured: for heat rays of the same wavelength, in equilibrium at a given temperature, the wavelength-specific ratio of emitting power to absorptivity is constant for all bodies that emit and absorb at that wavelength. In symbolic terms, the law stated that the ratio $E(\lambda, T, i) / a(\lambda, T, i)$ is the same for all bodies. This report did not mention black bodies.

In 1860, still unaware of Stewart's measurements concerning selective radiation qualities, Kirchhoff pointed out that it had been experimentally established that for total heat radiation emitted and absorbed by a body in equilibrium, the dimensional total radiation ratio $E(T, i) / a(T, i)$ possesses a single, common value for all bodies.[64] Again, without presenting measurements of radiative powers or other new experimental data, Kirchhoff offered a fresh theoretical proof of his principle of the universality of the value of the wavelength-specific ratio $E(\lambda, T, i) / a(\lambda, T, i)$ at thermal equilibrium. This new theoretical proof has also been considered invalid by some scholars.[62] [63]

More significantly, it relied on a new theoretical postulate of "perfectly black bodies," which is the origin of the term Kirchhoff's law. These hypothetical black bodies exhibited complete absorption at their infinitely thin, superficial surface. They mirrored Balfour Stewart's reference bodies, which possessed internal radiation and were coated with lamp-black. They were not the more realistic perfectly black bodies later conceptualized by Planck. Planck's black bodies radiated and absorbed energy solely through the material within their interiors; their interfaces with contiguous media were merely mathematical surfaces, incapable of absorption or emission, only reflection and transmission with refraction.[65]

Kirchhoff's proof involved an arbitrary non-ideal body labeled $i$ and various perfect black bodies labeled BB, all placed within a cavity in thermal equilibrium at temperature $T$ . His proof aimed to demonstrate that the ratio $E(\lambda, T, i) / a(\lambda, T, i)$ is independent of the nature $i$ of the non-ideal body, regardless of whether it is partially transparent or partially reflective.

His proof first asserted that for a given wavelength $\lambda$ and temperature $T$ , at thermal equilibrium, all perfectly black bodies of the same size and shape share a single, common value for their emissive power $E(\lambda, T, \text{BB})$ . This value has the dimensions of power. His proof noted that the dimensionless wavelength-specific absorptivity $a(\lambda, T, \text{BB})$ of a perfectly black body is, by definition, exactly 1. Therefore, for a perfectly black body, the wavelength-specific ratio of emissive power to absorptivity, $E(\lambda, T, \text{BB}) / a(\lambda, T, \text{BB})$ , is simply $E(\lambda, T, \text{BB})$ , which has the dimensions of power. Kirchhoff then considered thermal equilibrium with an arbitrary non-ideal body and a perfectly black body of identical size and shape, both placed within the equilibrium cavity at temperature $T$ . He argued that the heat radiation flows must be identical in both cases. Consequently, he concluded that at thermal equilibrium, the ratio $E(\lambda, T, i) / a(\lambda, T, i)$ equals $E(\lambda, T, \text{BB})$ , which can now be denoted as $B_\lambda(\lambda, T)$ . $B_\lambda(\lambda, T)$ is a continuous function, dependent only on $\lambda$ at a fixed temperature $T$ , and increases with $T$ at a fixed wavelength $\lambda$ . It diminishes to zero at low temperatures for visible wavelengths. This finding is independent of the specific nature $i$ of the arbitrary non-ideal body (geometrical factors, thoroughly analyzed by Kirchhoff, have been omitted here).

Thus, Kirchhoff's law of thermal radiation can be stated as follows: For any material, radiating and absorbing in thermodynamic equilibrium at any given temperature $T$ , and for every wavelength $\lambda$ , the ratio of emissive power to absorptivity has a single, universal value, characteristic of a perfect black body. This value represents the emissive power, which we denote here as $B_\lambda(\lambda, T)$ . (Kirchhoff's original notation for what we represent as $B_\lambda(\lambda, T)$ was simply $e$ .)[64] [66] [67] [68] [69] [70]

Kirchhoff announced that determining the function $B_\lambda(\lambda, T)$ was a problem of utmost importance, acknowledging the experimental challenges involved. He surmised that, like other functions independent of individual body properties, it would likely be a simple function. This function, $B_\lambda(\lambda, T)$ , has occasionally been referred to by historians as "Kirchhoff's (emission, universal) function,"[71] [72] [73] [74] although its precise mathematical form was not discovered until Planck's work in 1900, forty years later. The theoretical underpinnings of Kirchhoff's universality principle were explored and debated by various physicists during this period and beyond.[63] Kirchhoff later stated in 1860 that his theoretical proof was superior to Balfour Stewart's, and in some aspects, it was.[62] Kirchhoff's 1860 paper made no mention of the second law of thermodynamics, nor did it include the concept of entropy, which had not yet been established. In a more detailed account in a book published in 1862, Kirchhoff did note the connection between his law and Carnot's principle, a form of the second law.[75]

According to Helge Kragh, "Quantum theory owes its origin to the study of thermal radiation, in particular to the 'blackbody' radiation that Robert Kirchhoff had first defined in 1859–1860."[76]

Doppler effect

The relativistic Doppler effect causes a shift in the frequency $f$ of light originating from a source moving relative to an observer. The observed frequency $f'$ is given by:

$f'=f{\frac {1-{\frac {v}{c}}\cos \theta }{\sqrt {1-v^{2}/c^{2}}}},$

where $v$ is the velocity of the source in the observer's rest frame, $\theta$ is the angle between the velocity vector and the observer-source direction (measured in the source's frame), and $c$ is the speed of light.[77] This simplifies for sources moving directly towards ( $\theta = \pi$ ) or away from ( $\theta = 0$ ) the observer, and for speeds much less than $c$ .

Given that Planck's law relates the temperature spectrum of a black body proportionally to the frequency of light, temperature ( $T$ ) can be substituted for frequency in this equation.

For a source moving directly towards or away from the observer, the equation becomes:

$T'=T{\sqrt {\frac {c-v}{c+v}}}.$

Here, $v > 0$ indicates a receding source, and $v < 0$ indicates an approaching source.

This effect is significant in astronomy, where the velocities of stars and galaxies can approach a considerable fraction of $c$ . A prime example is the cosmic microwave background radiation, which exhibits a dipole anisotropy resulting from Earth's motion relative to this blackbody radiation field.