Black-body radiation

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As the temperature decreases, the peak of the black-body radiation curve moves to lower intensities and longer wavelengths. The black-body radiation graph is also compared with the classical model of Rayleigh and Jeans.

The color (chromaticity) of black-body radiation depends on the temperature of the black body; the locus of such colors, shown here in CIE 1931 x,y space, is known as the Planckian locus.

Black-body radiation is the thermal electromagnetic radiation within or surrounding a body in thermodynamic equilibrium with its environment, or emitted by a black body (an opaque and non-reflective body). It has a specific spectrum and intensity that depends only on the body's temperature, which is assumed for the sake of calculations and theory to be uniform and constant.^[1][2][3][4]

The thermal radiation spontaneously emitted by many ordinary objects can be approximated as black-body radiation. A perfectly insulated enclosure that is in thermal equilibrium internally contains black-body radiation and will emit it through a hole made in its wall, provided the hole is small enough to have negligible effect upon the equilibrium.

A black-body at room temperature appears black, as most of the energy it radiates is infra-red and cannot be perceived by the human eye. Because the human eye cannot perceive color at very low light intensities, a black body, viewed in the dark at the lowest just faintly visible temperature, subjectively appears grey (but only because the human eye is sensitive only to black and white at very low intensities - in reality, the frequency of the light in the visible range would still be red, although the intensity would be too low to discern as red), even though its objective physical spectrum peaks in the infrared range.^[5] When it becomes a little hotter, it appears dull red. As its temperature increases further it eventually becomes blue-white.

Although planets and stars are neither in thermal equilibrium with their surroundings nor perfect black bodies, black-body radiation is used as a first approximation for the energy they emit.^[6] Black holes are near-perfect black bodies, in the sense that they absorb all the radiation that falls on them. It has been proposed that they emit black-body radiation (called Hawking radiation), with a temperature that depends on the mass of the black hole.^[7]

The term black body was introduced by Gustav Kirchhoff in 1860. Black-body radiation is also called thermal radiation, cavity radiation, complete radiation or temperature radiation.

Spectrum

Black-body radiation has a characteristic, continuous frequency spectrum that depends only on the body's temperature,^[8] called the Planck spectrum or Planck's law. The spectrum is peaked at a characteristic frequency that shifts to higher frequencies with increasing temperature, and at room temperature most of the emission is in the infrared region of the electromagnetic spectrum.^[9][10][11]

As the temperature increases past about 500 degrees Celsius, black bodies start to emit significant amounts of visible light. Viewed in the dark by the human eye, the first faint glow appears as a "ghostly" grey (the visible light is actually red, but low intensity light activates only the eye's grey-level sensors). With rising temperature, the glow becomes visible even when there is some background surrounding light: first as a dull red, then yellow, and eventually a "dazzling bluish-white" as the temperature rises.^[12][13] When the body appears white, it is emitting a substantial fraction of its energy as ultraviolet radiation. The Sun, with an effective temperature of approximately 5800 K,^[14] is an approximate black body with an emission spectrum peaked in the central, yellow-green part of the visible spectrum, but with significant power in the ultraviolet as well.

Black-body radiation provides insight into the thermodynamic equilibrium state of cavity radiation. If each Fourier mode of the equilibrium radiation in an otherwise empty cavity with perfectly reflective walls is considered as a degree of freedom capable of exchanging energy, then, according to the equipartition theorem of classical physics, there would be an equal amount of energy in each mode. Since there are an infinite number of modes this implies infinite heat capacity (infinite energy at any non-zero temperature), as well as an unphysical spectrum of emitted radiation that grows without bound with increasing frequency, a problem known as the ultraviolet catastrophe. Instead, in quantum theory the occupation numbers of the modes are quantized, cutting off the spectrum at high frequency in agreement with experimental observation and resolving the catastrophe. The study of the laws of black bodies and the failure of classical physics to describe them helped establish the foundations of quantum mechanics.

Explanation

Color of a black body from 800 K to 12200 K. This range of colors approximates the range of colors of stars of different temperatures, as seen or photographed in the night sky.

All normal (baryonic) matter emits electromagnetic radiation when it has a temperature above absolute zero. The radiation represents a conversion of a body's thermal energy into electromagnetic energy, and is therefore called thermal radiation. It is a spontaneous process of radiative distribution of entropy.

Conversely all normal matter absorbs electromagnetic radiation to some degree. An object that absorbs all radiation falling on it, at all wavelengths, is called a black body. When a black body is at a uniform temperature, its emission has a characteristic frequency distribution that depends on the temperature. Its emission is called black-body radiation.

The concept of the black body is an idealization, as perfect black bodies do not exist in nature.^[15] Graphite and lamp black, with emissivities greater than 0.95, however, are good approximations to a black material. Experimentally, black-body radiation may be established best as the ultimately stable steady state equilibrium radiation in a cavity in a rigid body, at a uniform temperature, that is entirely opaque and is only partly reflective.^[15] A closed box of graphite walls at a constant temperature with a small hole on one side produces a good approximation to ideal black-body radiation emanating from the opening.^[16][17]

Black-body radiation has the unique absolutely stable distribution of radiative intensity that can persist in thermodynamic equilibrium in a cavity.^[15] In equilibrium, for each frequency the total intensity of radiation that is emitted and reflected from a body (that is, the net amount of radiation leaving its surface, called the spectral radiance) is determined solely by the equilibrium temperature, and does not depend upon the shape, material or structure of the body.^[18] For a black body (a perfect absorber) there is no reflected radiation, and so the spectral radiance is due entirely to emission. In addition, a black body is a diffuse emitter (its emission is independent of direction). Consequently, black-body radiation may be viewed as the radiation from a black body at thermal equilibrium.

Black-body radiation becomes a visible glow of light if the temperature of the object is high enough. The Draper point is the temperature at which all solids glow a dim red, about 798 K.^[19][20] At 1000 K, a small opening in the wall of a large uniformly heated opaque-walled cavity (let us call it an oven), viewed from outside, looks red; at 6000 K, it looks white. No matter how the oven is constructed, or of what material, as long as it is built so that almost all light entering is absorbed by its walls, it will contain a good approximation to black-body radiation. The spectrum, and therefore color, of the light that comes out will be a function of the cavity temperature alone. A graph of the amount of energy inside the oven per unit volume and per unit frequency interval plotted versus frequency, is called the black-body curve. Different curves are obtained by varying the temperature.

The temperature of a Pāhoehoe lava flow can be estimated by observing its color. The result agrees well with other measurements of temperatures of lava flows at about 1,000 to 1,200 °C (1,830 to 2,190 °F).

Two bodies that are at the same temperature stay in mutual thermal equilibrium, so a body at temperature T surrounded by a cloud of light at temperature T on average will emit as much light into the cloud as it absorbs, following Prevost's exchange principle, which refers to radiative equilibrium. The principle of detailed balance says that in thermodynamic equilibrium every elementary process works equally in its forward and backward sense.^[21][22] Prevost also showed that the emission from a body is logically determined solely by its own internal state. The causal effect of thermodynamic absorption on thermodynamic (spontaneous) emission is not direct, but is only indirect as it affects the internal state of the body. This means that at thermodynamic equilibrium the amount of every wavelength in every direction of thermal radiation emitted by a body at temperature T, black or not, is equal to the corresponding amount that the body absorbs because it is surrounded by light at temperature T.^[23]

When the body is black, the absorption is obvious: the amount of light absorbed is all the light that hits the surface. For a black body much bigger than the wavelength, the light energy absorbed at any wavelength λ per unit time is strictly proportional to the black-body curve. This means that the black-body curve is the amount of light energy emitted by a black body, which justifies the name. This is the condition for the applicability of Kirchhoff's law of thermal radiation: the black-body curve is characteristic of thermal light, which depends only on the temperature of the walls of the cavity, provided that the walls of the cavity are completely opaque and are not very reflective, and that the cavity is in thermodynamic equilibrium.^[24] When the black body is small, so that its size is comparable to the wavelength of light, the absorption is modified, because a small object is not an efficient absorber of light of long wavelength, but the principle of strict equality of emission and absorption is always upheld in a condition of thermodynamic equilibrium.

In the laboratory, black-body radiation is approximated by the radiation from a small hole in a large cavity, a hohlraum, in an entirely opaque body that is only partly reflective, that is maintained at a constant temperature. (This technique leads to the alternative term cavity radiation.) Any light entering the hole would have to reflect off the walls of the cavity multiple times before it escaped, in which process it is nearly certain to be absorbed. Absorption occurs regardless of the wavelength of the radiation entering (as long as it is small compared to the hole). The hole, then, is a close approximation of a theoretical black body and, if the cavity is heated, the spectrum of the hole's radiation (i.e., the amount of light emitted from the hole at each wavelength) will be continuous, and will depend only on the temperature and the fact that the walls are opaque and at least partly absorptive, but not on the particular material of which they are built nor on the material in the cavity (compare with emission spectrum).

Calculating the black-body curve was a major challenge in theoretical physics during the late nineteenth century. The problem was solved in 1901 by Max Planck in the formalism now known as Planck's law of black-body radiation.^[25] By making changes to Wien's radiation law (not to be confused with Wien's displacement law) consistent with thermodynamics and electromagnetism, he found a mathematical expression fitting the experimental data satisfactorily. Planck had to assume that the energy of the oscillators in the cavity was quantized, i.e., it existed in integer multiples of some quantity. Einstein built on this idea and proposed the quantization of electromagnetic radiation itself in 1905 to explain the photoelectric effect. These theoretical advances eventually resulted in the superseding of classical electromagnetism by quantum electrodynamics. These quanta were called photons and the black-body cavity was thought of as containing a gas of photons. In addition, it led to the development of quantum probability distributions, called Fermi–Dirac statistics and Bose–Einstein statistics, each applicable to a different class of particles, fermions and bosons.

The wavelength at which the radiation is strongest is given by Wien's displacement law, and the overall power emitted per unit area is given by the Stefan–Boltzmann law. So, as temperature increases, the glow color changes from red to yellow to white to blue. Even as the peak wavelength moves into the ultra-violet, enough radiation continues to be emitted in the blue wavelengths that the body will continue to appear blue. It will never become invisible—indeed, the radiation of visible light increases monotonically with temperature.^[26] The Stefan-Boltzmann law also says that the total radiant heat energy emitted from a surface is proportional to the fourth power of its absolute temperature. The law was formulated by Josef Stefan in 1879 and later derived by Ludwig Boltzmann. The formula E = σT⁴ is given, where E is the radiant heat emitted from a unit of area in one second, T is the temperature in Kelvin, and sigma (σ) is the Stefan-Boltzmann constant, which is equal to 5.670367 x 10 ^-8 watts per meter² per K⁴.^[27]

The radiance or observed intensity is not a function of direction. Therefore, a black body is a perfect Lambertian radiator.

Real objects never behave as full-ideal black bodies, and instead the emitted radiation at a given frequency is a fraction of what the ideal emission would be. The emissivity of a material specifies how well a real body radiates energy as compared with a black body. This emissivity depends on factors such as temperature, emission angle, and wavelength. However, it is typical in engineering to assume that a surface's spectral emissivity and absorptivity do not depend on wavelength, so that the emissivity is a constant. This is known as the gray body assumption.

9-year WMAP image (2012) of the cosmic microwave background radiation across the universe.^[28][29]

With non-black surfaces, the deviations from ideal black-body behavior are determined by both the surface structure, such as roughness or granularity, and the chemical composition. On a "per wavelength" basis, real objects in states of local thermodynamic equilibrium still follow Kirchhoff's Law: emissivity equals absorptivity, so that an object that does not absorb all incident light will also emit less radiation than an ideal black body; the incomplete absorption can be due to some of the incident light being transmitted through the body or to some of it being reflected at the surface of the body.

In astronomy, objects such as stars are frequently regarded as black bodies, though this is often a poor approximation. An almost perfect black-body spectrum is exhibited by the cosmic microwave background radiation. Hawking radiation is the hypothetical black-body radiation emitted by black holes, at a temperature that depends on the mass, charge, and spin of the hole. If this prediction is correct, black holes will very gradually shrink and evaporate over time as they lose mass by the emission of photons and other particles.

A black body radiates energy at all frequencies, but its intensity rapidly tends to zero at high frequencies (short wavelengths). For example, a black body at room temperature (300 K) with one square meter of surface area will emit a photon in the visible range (390–750 nm) at an average rate of one photon every 41 seconds, meaning that for most practical purposes, such a black body does not emit in the visible range.^[30]

Equations

Planck's law of black-body radiation

Planck's law states that^[31]

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

where

B_ν(T) is the spectral radiance (the power per unit solid angle and per unit of area normal to the propagation) density of frequency ν radiation per unit frequency at thermal equilibrium at temperature T.

h is the Planck constant;

c is the speed of light in a vacuum;

k is the Boltzmann constant;

ν is the frequency of the electromagnetic radiation;

T is the absolute temperature of the body.

For a black body surface the spectral radiance density (defined per unit of area normal to the propagation) is independent of the angle ${\displaystyle \theta }$ of emission with respect to the normal. However, this means that, following Lambert's cosine law, ${\displaystyle B_{\nu }(T)\cos \theta }$ is the radiance density per unit area of emitting surface as the surface area involved in generating the radiance is increased by a factor ${\displaystyle 1/\cos \theta }$ with respect to an area normal to the propagation direction. At oblique angles, the solid angle spans involved do get smaller, resulting in lower aggregate intensities.

Wien's displacement law

Wien's displacement law shows how the spectrum of black-body radiation at any temperature is related to the spectrum at any other temperature. If we know the shape of the spectrum at one temperature, we can calculate the shape at any other temperature. Spectral intensity can be expressed as a function of wavelength or of 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 is at a maximum, ${\displaystyle \lambda _{\max }}$, is a function only of the temperature:

${\displaystyle \lambda _{\max }={\frac {b}{T}},}$

where the constant b, known as Wien's displacement constant, is equal to 2.8977729(17)×10⁻³ K m.^[32]

Planck's law was also stated above as a function of frequency. The intensity maximum for this is given by

${\displaystyle \nu _{\max }=T\times 58.8\ \mathrm {GHz} /\mathrm {K} }$.^[33]

Stefan–Boltzmann law

By integrating ${\displaystyle B_{\nu }(T)}$ over the frequency the integrated radiance ${\displaystyle L}$ is

${\displaystyle L={\frac {2\pi ^{5}}{15}}{\frac {k^{4}T^{4}}{c^{2}h^{3}}}{\frac {1}{\pi }}=:\sigma T^{4}{\frac {1}{\pi }}}$

by using ${\displaystyle \int _{0}^{\infty }dx\,{\frac {x^{3}}{e^{x}-1}}={\frac {\pi ^{4}}{15}}}$ with ${\displaystyle x\equiv {\frac {h\nu }{kT}}}$ and with ${\displaystyle \sigma \equiv {\frac {2\pi ^{5}}{15}}{\frac {k^{4}}{c^{2}h^{3}}}=5.670373\times 10^{-8}{\frac {W}{m^{2}K^{4}}}}$ being the Stefan–Boltzmann constant. The radiance ${\displaystyle L}$ is then

${\displaystyle \sigma T^{4}{\frac {\cos \theta }{\pi }}}$

per unit of emitting surface.

On a side note, at a distance d, the intensity ${\displaystyle dI}$ per area ${\displaystyle dA}$ of radiating surface is the useful expression

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

when the receiving surface is perpendicular to the radiation.

By subsequently integrating over the solid angle ${\displaystyle \Omega }$ (where ${\displaystyle \theta <\pi /2}$) the Stefan–Boltzmann law is calculated, stating that the power j* emitted per unit area of the surface of a black body is directly proportional to the fourth power of its absolute temperature:

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

by using

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

Human-body emission

Much of a person's energy is radiated away in the form of infrared light. Some materials are transparent in the infrared, but opaque to visible light, as is the plastic bag in this infrared image (bottom). Other materials are transparent to visible light, but opaque or reflective in the infrared, noticeable by the darkness of the man's glasses.

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

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

Applying the Stefan–Boltzmann law,

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

where A and T are the body surface area and temperature, ${\displaystyle \varepsilon }$ is the emissivity, and T₀ is the ambient temperature.

The total surface area of an adult is about 2 m², and the mid- and far-infrared emissivity of skin and most clothing is near unity, as it is for most nonmetallic surfaces.^[34][35] Skin temperature is about 33 °C,^[36] but clothing reduces the surface temperature to about 28 °C when the ambient temperature is 20 °C.^[37] Hence, the net radiative heat loss is about

${\displaystyle P_{\text{net}}=100~{\text{W}}.}$

The total energy radiated in one day is about 8 MJ, or 2000 kcal (food calories). Basal metabolic rate for a 40-year-old male is about 35 kcal/(m²·h),^[38] which is equivalent to 1700 kcal per day, assuming the same 2 m² area. However, the mean metabolic rate of sedentary adults is about 50% to 70% greater than their basal rate.^[39]

There are other important thermal-loss mechanisms, including convection and evaporation. Conduction is negligible – the Nusselt number is much greater than unity. Evaporation by perspiration is only required if radiation and convection are insufficient to maintain a steady-state temperature (but evaporation from the lungs occurs regardless). Free-convection rates are comparable, albeit somewhat lower, than radiative rates.^[40] Thus, radiation accounts for about two-thirds of thermal energy loss in cool, still air. Given the approximate nature of many of the assumptions, this can only be taken as a crude estimate. Ambient air motion, causing forced convection, or evaporation reduces the relative importance of radiation as a thermal-loss mechanism.

Application of Wien's law to human-body emission results in a peak wavelength of

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

For this reason, thermal imaging devices for human subjects are most sensitive in the 7–14 micrometer range.

Temperature relation between a planet and its star

The black-body law may be used to estimate the temperature of a planet orbiting the Sun.

Earth's longwave thermal radiation intensity, from clouds, atmosphere and ground

The temperature of a planet depends on several factors:

• Incident radiation from its star
• Emitted radiation of the planet, e.g., Earth's infrared glow
• The albedo effect causing a fraction of light to be reflected by the planet
• The greenhouse effect for planets with an atmosphere
• Energy generated internally by a planet itself due to radioactive decay, tidal heating, and adiabatic contraction due by cooling.

The analysis only considers the Sun's heat for a planet in a Solar System.

The Stefan–Boltzmann law gives the total power (energy/second) the Sun is emitting:

The Earth only has an absorbing area equal to a two dimensional disk, rather than the surface of a sphere.

${\displaystyle P_{\rm {S\ emt}}=4\pi R_{\rm {S}}^{2}\sigma T_{\rm {S}}^{4}\qquad \qquad (1)}$

where

${\displaystyle \sigma \,}$ is the Stefan–Boltzmann constant,

${\displaystyle T_{\rm {S}}\,}$ is the effective temperature of the Sun, and

${\displaystyle R_{\rm {S}}\,}$ is the radius of the Sun.

The Sun emits that power equally in all directions. Because of this, the planet is hit with only a tiny fraction of it. The power from the Sun that strikes the planet (at the top of the atmosphere) is:

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

where

${\displaystyle R_{\rm {E}}\,}$ is the radius of the planet and

${\displaystyle D\,}$ is the distance between the Sun and the planet.

Because of its high temperature, the Sun emits to a large extent in the ultraviolet and visible (UV-Vis) frequency range. In this frequency range, the planet reflects a fraction ${\displaystyle \alpha }$ of this energy where ${\displaystyle \alpha }$ is the albedo or reflectance of the planet in the UV-Vis range. In other words, the planet absorbs a fraction ${\displaystyle 1-\alpha }$ of the Sun's light, and reflects the rest. The power absorbed by the planet and its atmosphere is then:

${\displaystyle P_{\rm {abs}}=(1-\alpha )\,P_{\rm {SE}}\qquad \qquad (3)}$

Even though the planet only absorbs as a circular area ${\displaystyle \pi R^{2}}$, it emits equally in all directions as a sphere. If the planet were a perfect black body, it would emit according to the Stefan–Boltzmann law

${\displaystyle P_{\rm {emt\,bb}}=4\pi R_{\rm {E}}^{2}\sigma T_{\rm {E}}^{4}\qquad \qquad (4)}$

where ${\displaystyle T_{\rm {E}}}$ is the temperature of the planet. This temperature, calculated for the case of the planet acting as a black body by setting ${\displaystyle P_{\rm {abs}}=P_{\rm {emt\,bb}}}$, is known as the effective temperature. The actual temperature of the planet will likely be different, depending on its surface and atmospheric properties. Ignoring the atmosphere and greenhouse effect, the planet, since it is at a much lower temperature than the Sun, emits mostly in the infrared (IR) portion of the spectrum. In this frequency range, it emits ${\displaystyle {\overline {\epsilon }}}$ of the radiation that a black body would emit where ${\displaystyle {\overline {\epsilon }}}$ is the average emissivity in the IR range. The power emitted by the planet is then:

${\displaystyle P_{\rm {emt}}={\overline {\epsilon }}\,P_{\rm {emt\,bb}}\qquad \qquad (5)}$

For a body in radiative exchange equilibrium with its surroundings, the rate at which it emits radiant energy is equal to the rate at which it absorbs it:^[41][42]

${\displaystyle P_{\rm {abs}}=P_{\rm {emt}}\qquad \qquad (6)}$

Substituting the expressions for solar and planet power in equations 1–6 and simplifying yields the estimated temperature of the planet, ignoring greenhouse effect, T_P:

${\displaystyle T_{P}=T_{S}{\sqrt {\frac {R_{S}{\sqrt {\frac {1-\alpha }{\overline {\varepsilon }}}}}{2D}}}\qquad \qquad (7)}$

In other words, given the assumptions made, the temperature of a planet depends only on the surface temperature of the Sun, the radius of the Sun, the distance between the planet and the Sun, the albedo and the IR emissivity of the planet.

Notice that a gray ( flat spectrum ) ball where ${\displaystyle ({1-\alpha })={\overline {\varepsilon }}}$ comes to the same temperature as a black body no matter how dark or light gray .

Virtual temperature of Earth

Substituting the measured values for the Sun and Earth yields:

${\displaystyle T_{\rm {S}}=5778\ \mathrm {K} ,}$^[43]

${\displaystyle R_{\rm {S}}=6.96\times 10^{8}\ \mathrm {m} ,}$^[43]

${\displaystyle D=1.496\times 10^{11}\ \mathrm {m} ,}$^[43]

${\displaystyle \alpha =0.306\ }$^[44]

With the average emissivity ${\displaystyle {\overline {\varepsilon }}}$ set to unity, the effective temperature of the Earth is:

${\displaystyle T_{\rm {E}}=254.356\ \mathrm {K} }$

or −18.8 °C.

This is the temperature of the Earth if it radiated as a perfect black body in the infrared, assuming an unchanging albedo and ignoring greenhouse effects (which can raise the surface temperature of a body above what it would be if it were a perfect black body in all spectrums^[45]). The Earth in fact radiates not quite as a perfect black body in the infrared which will raise the estimated temperature a few degrees above the effective temperature. If we wish to estimate what the temperature of the Earth would be if it had no atmosphere, then we could take the albedo and emissivity of the Moon as a good estimate. The albedo and emissivity of the Moon are about 0.1054^[46] and 0.95^[47] respectively, yielding an estimated temperature of about 1.36 °C.

Estimates of the Earth's average albedo vary in the range 0.3–0.4, resulting in different estimated effective temperatures. Estimates are often based on the solar constant (total insolation power density) rather than the temperature, size, and distance of the Sun. For example, using 0.4 for albedo, and an insolation of 1400 W m⁻², one obtains an effective temperature of about 245 K.^[48] Similarly using albedo 0.3 and solar constant of 1372 W m⁻², one obtains an effective temperature of 255 K.^[49][50][51]

Cosmology

The cosmic microwave background radiation observed today is the most perfect black-body radiation ever observed in nature, with a temperature of about 2.7 K.^[52] It is a "snapshot" of the radiation at the time of decoupling between matter and radiation in the early universe. Prior to this time, most matter in the universe was in the form of an ionized plasma in thermal, though not full thermodynamic, equilibrium with radiation.

According to Kondepudi and Prigogine, at very high temperatures (above 10¹⁰ K; such temperatures existed in the very early universe), where the thermal motion separates protons and neutrons in spite of the strong nuclear forces, electron-positron pairs appear and disappear spontaneously and are in thermal equilibrium with electromagnetic radiation. These particles form a part of the black body spectrum, in addition to the electromagnetic radiation.^[53]

Doppler effect for a moving black body

The relativistic Doppler effect causes a shift in the frequency f of light originating from a source that is moving in relation to the observer, so that the wave is observed to have frequency f':

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

where v is the velocity of the source in the observer's rest frame, θ is the angle between the velocity vector and the observer-source direction measured in the reference frame of the source, and c is the speed of light.^[54] This can be simplified for the special cases of objects moving directly towards (θ = π) or away (θ = 0) from the observer, and for speeds much less than c.

Through Planck's law the temperature spectrum of a black body is proportionally related to the frequency of light and one may substitute the temperature (T) for the frequency in this equation.

For the case of a source moving directly towards or away from the observer, this reduces to

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

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

This is an important effect in astronomy, where the velocities of stars and galaxies can reach significant fractions of c. An example is found in the cosmic microwave background radiation, which exhibits a dipole anisotropy from the Earth's motion relative to this black-body radiation field.

History

Balfour Stewart

In 1858, Balfour Stewart described his experiments on the thermal radiative emissive and absorptive powers of polished plates of various substances, compared with the powers of lamp-black surfaces, at the same temperature.^[23] Stewart chose lamp-black surfaces as his reference because of various previous experimental findings, especially those of Pierre Prevost and of John Leslie. He wrote "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." More an experimenter than a logician, Stewart failed to point out that his statement presupposed an abstract general principle, that there exist either ideally in theory or really in nature bodies or surfaces that respectively have one and the same unique universal greatest possible absorbing power, likewise for radiating power, for every wavelength and equilibrium temperature.

Stewart measured radiated power with a thermo-pile and sensitive galvanometer read with a microscope. He was concerned with selective thermal radiation, which he investigated with plates of substances that radiated and absorbed selectively for different qualities of radiation rather than maximally for all qualities of radiation. He discussed the experiments in terms of rays which could be reflected and refracted, and which obeyed the Stokes-Helmholtz reciprocity principle (though he did not use an eponym for it). He did not in this paper mention that the qualities of the rays might be described by their wavelengths, nor did he use spectrally resolving apparatus such as prisms or diffraction gratings. His work was quantitative within these constraints. He made his measurements in a room temperature environment, and quickly so as to catch his bodies in a condition near the thermal equilibrium in which they had been prepared by heating to equilibrium with boiling water. His measurements confirmed that substances that emit and absorb selectively respect the principle of selective equality of emission and absorption at thermal equilibrium.

Stewart offered a theoretical proof that this should be the case separately for every selected quality of thermal radiation, but his mathematics was not rigorously valid.^[55] He made no mention of thermodynamics in this paper, though he did refer to conservation of vis viva. He proposed that his measurements implied that radiation was both absorbed and emitted by particles of matter throughout depths of the media in which it propagated. He applied the Helmholtz reciprocity principle to account for the material interface processes as distinct from the processes in the interior material. He did not postulate unrealizable perfectly black surfaces. He concluded that his experiments showed that in a cavity in thermal equilibrium, the heat radiated from any part of the interior bounding surface, no matter of what material it might be composed, was the same as would have been emitted from a surface of the same shape and position that would have been composed of lamp-black. He did not state explicitly that the lamp-black-coated bodies that he used as reference must have had a unique common spectral emittance function that depended on temperature in a unique way.

Gustav Kirchhoff

In 1859, not knowing of Stewart's work, Gustav Robert Kirchhoff reported the coincidence of the wavelengths of spectrally resolved lines of absorption and of emission of visible light. Importantly for thermal physics, he also observed that bright lines or dark lines were apparent depending on the temperature difference between emitter and absorber.^[56]

Kirchhoff then went on to consider bodies that emit and absorb heat radiation, in an opaque enclosure or cavity, in equilibrium at temperature T.

Here is used a notation different from Kirchhoff's. Here, the emitting power E(T, i) denotes a dimensioned quantity, the total radiation emitted by a body labeled by index i at temperature T. The total absorption ratio a(T, i) of that body is dimensionless, the ratio of absorbed to incident radiation in the cavity at temperature T . (In contrast with Balfour Stewart's, Kirchhoff's definition of his absorption ratio did not refer in particular to a lamp-black surface as the source of the incident radiation.) Thus the ratio E(T, i) / a(T, i) of emitting power to absorption ratio is a dimensioned quantity, with the dimensions of emitting power, because a(T, i) is dimensionless. Also here the wavelength-specific emitting power of the body at temperature T is denoted by E(λ, T, i) and the wavelength-specific absorption ratio by a(λ, T, i) . Again, the ratio E(λ, T, i) / a(λ, T, i) of emitting power to absorption ratio is a dimensioned quantity, with the dimensions of emitting power.

In a second report made in 1859, Kirchhoff announced a new general principle or law for which he offered a theoretical and mathematical proof, though he did not offer quantitative measurements of radiation powers.^[57] His theoretical proof was and still is considered by some writers to be invalid.^[55][58] His principle, however, has endured: it was that for heat rays of the same wavelength, in equilibrium at a given temperature, the wavelength-specific ratio of emitting power to absorption ratio has one and the same common value for all bodies that emit and absorb at that wavelength. In symbols, the law stated that the wavelength-specific ratio E(λ, T, i) / a(λ, T, i) has one and the same value for all bodies, that is for all values of index i . In this report there was no mention of black bodies.

In 1860, still not knowing of Stewart's measurements for selected qualities of radiation, Kirchhoff pointed out that it was long established experimentally that for total heat radiation, of unselected quality, emitted and absorbed by a body in equilibrium, the dimensioned total radiation ratio E(T, i) / a(T, i), has one and the same value common to all bodies, that is, for every value of the material index i.^[59] Again without measurements of radiative powers or other new experimental data, Kirchhoff then offered a fresh theoretical proof of his new principle of the universality of the value of the wavelength-specific ratio E(λ, T, i) / a(λ, T, i) at thermal equilibrium. His fresh theoretical proof was and still is considered by some writers to be invalid.^[55][58]

But more importantly, it relied on a new theoretical postulate of "perfectly black bodies," which is the reason why one speaks of Kirchhoff's law. Such black bodies showed complete absorption in their infinitely thin most superficial surface. They correspond to Balfour Stewart's reference bodies, with internal radiation, coated with lamp-black. They were not the more realistic perfectly black bodies later considered by Planck. Planck's black bodies radiated and absorbed only by the material in their interiors; their interfaces with contiguous media were only mathematical surfaces, capable neither of absorption nor emission, but only of reflecting and transmitting with refraction.^[60]

Kirchhoff's proof considered an arbitrary non-ideal body labeled i as well as various perfect black bodies labeled BB . It required that the bodies be kept in a cavity in thermal equilibrium at temperature T . His proof intended to show that the ratio E(λ, T, i) / a(λ, T, i) was independent of the nature i of the non-ideal body, however partly transparent or partly reflective it was.

His proof first argued that for wavelength λ and at temperature T, at thermal equilibrium, all perfectly black bodies of the same size and shape have the one and the same common value of emissive power E(λ, T, BB), with the dimensions of power. His proof noted that the dimensionless wavelength-specific absorption ratio a(λ, T, BB) of a perfectly black body is by definition exactly 1. Then for a perfectly black body, the wavelength-specific ratio of emissive power to absorption ratio E(λ, T, BB) / a(λ, T, BB) is again just E(λ, T, BB), with the dimensions of power. Kirchhoff considered, successively, thermal equilibrium with the arbitrary non-ideal body, and with a perfectly black body of the same size and shape, in place in his cavity in equilibrium at temperature T . He argued that the flows of heat radiation must be the same in each case. Thus he argued that at thermal equilibrium the ratio E(λ, T, i) / a(λ, T, i) was equal to E(λ, T, BB), which may now be denoted B_λ (λ, T), a continuous function, dependent only on λ at fixed temperature T, and an increasing function of T at fixed wavelength λ, at low temperatures vanishing for visible but not for longer wavelengths, with positive values for visible wavelengths at higher temperatures, which does not depend on the nature i of the arbitrary non-ideal body. (Geometrical factors, taken into detailed account by Kirchhoff, have been ignored in the foregoing.)

Thus Kirchhoff's law of thermal radiation can be stated: For any material at all, radiating and absorbing in thermodynamic equilibrium at any given temperature T, for every wavelength λ, the ratio of emissive power to absorptive ratio has one universal value, which is characteristic of a perfect black body, and is an emissive power which we here represent by B_λ (λ, T) . (For our notation B_λ (λ, T), Kirchhoff's original notation was simply e.)^{[59][61][62][63][64][65]}

Kirchhoff announced that the determination of the function B_λ (λ, T) was a problem of the highest importance, though he recognized that there would be experimental difficulties to be overcome. He supposed that like other functions that do not depend on the properties of individual bodies, it would be a simple function. Occasionally by historians that function B_λ (λ, T) has been called "Kirchhoff's (emission, universal) function,"^{[66][67][68][69]} though its precise mathematical form would not be known for another forty years, till it was discovered by Planck in 1900. The theoretical proof for Kirchhoff's universality principle was worked on and debated by various physicists over the same time, and later.^[58] Kirchhoff stated later in 1860 that his theoretical proof was better than Balfour Stewart's, and in some respects it was so.^[55] Kirchhoff's 1860 paper did not mention the second law of thermodynamics, and of course did not mention the concept of entropy which had not at that time been established. In a more considered account in a book in 1862, Kirchhoff mentioned the connection of his law with Carnot's principle, which is a form of the second law.^[70]

According to Helge Kragh, "Quantum theory owes its origin to the study of thermal radiation, in particular to the "black-body" radiation that Robert Kirchhoff had first defined in 1859–1860."^[71]

A Startling Revelation on Climate Sensitivity

I have stumbled upon a post from the American Chemical Society, dated December 20, 2017, which appears to make a startling revelation on why climate sensitivity was determined to be so high as to predict catastrophic global warming by anthropogenic emissions of greenhouse gases.  The entire post is appended below, but I will quote the portion that contains the revelation:

"The increase in CO₂ from about 185 to about 265 ppm [from the last glacial period to the pre-industrial modern interglacial] gives a radiative forcing of

ΔF_CO2 = (5.35 W·m^–2) ln(265/185) = 1.9 W·m^–2

The radiative forcing for CH₄ is determined in a way analogous to that for CO₂. For the increase of CH₄ from about 375 to about 675 ppb, ΔF_CH4 ≈ 0.3 W·m^–2. Thus, the total radiative forcing, ΔF, due to these two greenhouse gases is about 2.2 W·m^–2. The predicted change in the average planetary surface temperature is

ΔT ≈ [0.3 K·(W·m^–2)^–1] (2.2 W·m^–2) ≈ 0.7 K

 

Analyses from multiple sites based on several different temperature proxies indicate that Earth’s average surface temperature increased between 3 and 4 K during the change from the last glacial period to the present era.

Our calculated temperature change, that includes only the radiative forcing from increases in greenhouse gas concentrations, accounts for 20-25% of this observed temperature increase. This result implies climate sensitivity factor perhaps four to five times greater, ∼1.3 K·(W·m^–2)^–1, than obtained by simply balancing the radiative forcing of the greenhouse gases. [Italics and bold mine] The analysis based only on greenhouse gas forcing has not accounted for feedbacks in the planetary system triggered by increasing temperature, including changes in the structure of the atmosphere."

Unless I am seriously misreading this, the author(s) are assuming that the entire temperature rise from the last glacial to current interglacial (which they take to be 3-4K instead of the correct value of ~8K) is due entirely to increases in atmospheric greenhouse gases (which account for only 0.7K) and associated feedbacks (accounting for an additional 2.3-3.3K).  As this is approximately what the IPCC claims, it is reasonable to suppose this assumption is the "empirical" basis for such high climate sensitivity.  Note also that no justification is offered for this assumption; it is simply asserted.

The increase in atmospheric water vapor, due to rising temperatures, is often cited as the main cause for this large feedback.  Curiously however, although water vapor is a greenhouse gas, and its increase should induce an additional temperature rise, I have never encountered an attempt to calculate how large it should be.  Curious, for the same equations used to calculate the effect of CO2 on temperature can be applied to water vapor; only the constant in the Arrhenius relation needs to be modified, and that can be done by noting that although there is (conservatively) ten times as much of the latter as the former, it is estimated to cause only three times as much greenhouse warming, thus implying a constant only about one-third, or 5.35/3 = 1.78.  As for the increase in atmospheric water vapor due to a temperature rise 0.7K, we can apply the rule of thumb that for every 10K increase there is about a doubling of water vapor pressure; thus, for 0.7K, this yields an increase of 5%, or a ratio of 1.05.  Using the same Arrhenius relationship as in the post below gives

ΔF_H2O = (1.78 W·m^–2) ln(1.05) = 0.087 W·m^–2

and applying the same approximation for calculating the temperature increase

ΔT ≈ [0.3 K·(W·m^–2)^–1] (0.087 W·m^–2) ≈ 0.026 K

which is a < 4% addition to the 0.7K temperature rise, or a total of 0.73 K.

Of course, this addition 0.026 K warming raises water vapor even more, but by an amount to small to be significant.  It is certainly nowhere close to the 2.3K-3.3K allegedly needed to account for the glacial-interglacial temperature difference.

Another strange feature of the post's analysis is the failure to mention albedo change due to the reduction of ice and snow covering the Earth's surface when a glacial age yields to an interglacial.  If I make the estimate that, during the last glacial, ice and snow covered approximately 25% more of the Earth's surface than now, its albedo would have been around .36 -- some 20% greater than the current value.  Employing the Stephan-Boltzmann relation estimates a temperature some 6K colder then, or some 75% of the actual temperature change (about 8K, remember).

Milankovitch cycles are also conspicuously absent, though it is difficult to calculate their effect on temperature with certainty.  What is certain is that relying on greenhouse gas changes alone is woefully insufficient to the task employed here, that they only amount to about 10% of the glacial-interglacial temperature rise.

Without further ado, the ACS post.
 

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ACS Climate Science Toolkit | How Atmospheric Warming Works

From link https://www.acs.org/content/acs/en/climatescience/atmosphericwarming/climatsensitivity.html

The concept of “climate sensitivity” is deceptively simple. How much would the average surface temperature of the Earth increase (decrease) for a given positive (negative) radiative forcing? The simplest approach to estimating climate sensitivity is to combine the energy balance for the incoming and outgoing energies and a simple atmospheric model to calculate how to counterbalance a given radiative forcing. If ΔF is the difference between incoming and outgoing energy flux (the equivalent of radiative forcing), we have

ΔF = (1 – α)S_ave – εσT_P^{4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .} (1)

In this equation, α is the Earth’s albedo, S_ave is the average solar energy flux, 342 W·m^–2, ε is the effective emissivity of the planetary system, σ is the Stefan-Boltzmann constant, and T_P is the average planetary surface temperature. If ΔF is zero, the energies are balanced. That is,

(1 – α)S_ave = εσT_P^{4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .} (2)

In the absence of greenhouse gases in the atmosphere, ε would be unity, and T_P would be 255 K. The greenhouse gases in the atmosphere give a lower effective emissivity that requires an increase of T_P to about 288 K to maintain energy balance.

For ΔF > 0, a positive radiative forcing, the incoming energy is higher than the outgoing. To counterbalance this forcing, the surface temperature has to increase by ΔT to produce a planetary radiative flux that is ΔF larger than the incoming flux. The required counterbalance, assuming no changes in other factors affecting the climate, is represented by this equation.

ΔF = εσ[T_P + ΔT]⁴ – (1 – α)S_ave^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .} (3)

The algebraic manipulation shown below in small printgives this relationship between radiative forcing and the counterbalancing temperature change that would be required to return the planet to energy balance.

ΔF = εσ[T_P + ΔT]⁴ – (1 – α)S_ave

ΔF = εσ[T_P (1 + ΔT/ T_P)]⁴ – (1 – α)S_ave

ΔF = εσT_P⁴(1 + ΔT/ T_P)⁴ – (1 – α)S_ave

Substituting εσT_P⁴ = (1 – α)S_ave, gives

ΔF = [(1 – α)S_ave ](1 + ΔT/ T_P)⁴ – (1 – α)S_ave

The factor (1 + ΔT/ T_P)⁴ can be expanded and approximated:

(1 + ΔT/ T_P)⁴ = 1 + 4(ΔT/ T_P) + 6(ΔT/ T_P)² + 4(ΔT/ T_P)³ + (ΔT/ T_P)⁴

(1 + ΔT/ T_P)⁴ ≈ 1 + 4(ΔT/ T_P)

Because ΔT is small, ΔT/ T_P << 1, and the higher order terms in the expansion are negligible. Substituting in the expression for ΔF, gives

ΔF ≈ [(1 – α)S_ave ] [1 + 4(ΔT/ T_P)] – (1 – α)S_ave

ΔF ≈ (1 – α)S_ave + 4(1 – α)S_ave (ΔT/ T_P) – (1 – α)S_ave

ΔF ≈ 4(1 – α)S_ave (ΔT/ T_P)]

Solving for ΔT, gives the climate sensitivity based on this simple approach.

ΔT ≈ Tp ΔF/[4(1 – α)S_ave]

ΔT ≈ Tp ΔF/[4(1 – α)S_ave] ≈ [0.3 K·(W·m^–2)^–1] ΔF (for Tp ≈ 288 Κ)^{. . . . . . .} (4)

To apply this approximation for climate sensitivity due to CO₂ and CH₄, we can examine a case for which the change in concentration of greenhouse gases is reasonably well known and whose temperature change from an initial constant temperature state to a higher constant temperature is also known. The figure shows Antarctic ice core data that span the time from the end of the last glacial period to the beginning of the present era. For our purposes, we need the initial and final concentrations of CO₂ and CH₄, and the average global temperature change. For this test, we assume, that radiative forcing by these gases is the only external forcing on the climate system. (The detailed time course of the changes is interesting and can be correlated with changes that are evident in other geological records from this time span, but are not relevant for our calculation.)

The figure is based on a figure from the NOAA Paleoclimatology Program website. The original reference is Eric Monnin, Andreas Indermühle, André Dällenbach, Jacqueline Flückiger, Bernhard Stauffer, Thomas F. Stocker, Dominique Raynaud, Jean-Marc Barnola, “Atmospheric CO2 Concentrations Over the Last Glacial Termination,” Science, 2001, 291, 112-114.

The increase in CO₂ from about 185 to about 265 ppm gives a radiative forcing of

ΔF_CO2 = (5.35 W·m^–2) ln(265/185) = 1.9 W·m^–2

The radiative forcing for CH₄ is determined in a way analogous to that for CO₂. For the increase of CH₄ from about 375 to about 675 ppb, ΔF_CH4 ≈ 0.3 W·m^–2. Thus, the total radiative forcing, ΔF, due to these two greenhouse gases is about 2.2 W·m^–2. The predicted change in the average planetary surface temperature is

ΔT ≈ [0.3 K·(W·m^–2)^–1] (2.2 W·m^–2) ≈ 0.7 K

 

Analyses from multiple sites based on several different temperature proxies indicate that Earth’s average surface temperature increased between 3 and 4 K during the change from the last glacial period to the present era.

Our calculated temperature change, that includes only the radiative forcing from increases in greenhouse gas concentrations, accounts for 20-25% of this observed temperature increase. This result implies climate sensitivity factor perhaps four to five times greater, ∼1.3 K·(W·m^–2)^–1, than obtained by simply balancing the radiative forcing of the greenhouse gases. The analysis based only on greenhouse gas forcing has not accounted for feedbacks in the planetary system triggered by increasing temperature, including changes in the structure of the atmosphere.

Water Vapor and Clouds

One of the most important sources of feedback in the planetary system, shown graphically below, is the increase in the vapor pressure of water as the ocean’s temperature increases. The vapor pressure increases by about 7% per degree kelvin. Warming oceans evaporate more water and a warmer atmosphere can accommodate more water vapor, the most important greenhouse gas. This feedback amplifies the warming effect of the non-condensable greenhouse gases and is responsible for a good part of the multiplier effect on climate sensitivity noted in the previous paragraph.

For the first calculation of atmospheric warming by increased CO₂, Arrhenius chose to consider a doubling of its concentration and climate science has stuck with this standard. Thus, most values for climate sensitivity are given today as the temperature change predicted for doubling the CO₂ concentration, ΔT_2xCO2, or the equivalent of its doubling, taking all the greenhouse gas radiative forcing into account. The IPCC’s analysis gives a very likely (> 90% probability) value of 3 K with a likely (> 66% probability) range from 2 to 4.5 K. Our radiative forcing for doubling CO₂ from 280 to 560 ppt was 3.53 W·m^–2, which gives ΔT_2xCO2 = 4.6 K (= [1.3 K·(W·m^–2)^–1][ 3.53 W·m^–2]). Although on the high side, this first level approximation is not wildly amiss and provides some insight into the factors that affect climate sensitivity.

Credit: Jerry Bell

An increase in atmospheric water vapor also affects cloud formation. The effect of clouds on the energy balance between incoming solar radiation and outgoing thermal IR radiation depends on the kinds of clouds and can result in either positive or negative feedback for planetary warming. Clouds are composed of tiny water droplets or ice crystals, which makes them very good black bodies for absorption and re-emission of thermal IR radiation. Unlike greenhouse gases that absorb and emit only at discrete wavelengths, clouds absorb and emit like black bodies throughout the thermal IR. The higher the top of the cloud, the lower the temperature from which emission takes place and the lower the energy emitted. Thus, the higher the cloud, the greater its positive feedback effect on planetary warming. Thin, wispy cirrus clouds very high in the troposphere near the stratosphere have the strongest warming effect while low-lying layers of stratus clouds have a weaker warming effect.

Cumulous and stratus clouds in the lower troposphere are opaque—we can’t see through them. The tiny water droplets or ice crystals in these clouds scatter visible light in all directions, including back into space, so they reduce the amount of solar energy that reaches the surface. That is, they increase the Earth’s albedo and therefore have a negative feedback effect on planetary warming. The very high cirrus clouds contain very little water (as ice) and are not opaque—we can see the sky through them. They do not scatter very much solar radiation and have only a weak negative feedback effect.

Studies of changes in ocean surface salinity since 1950 indicate that salinity is increasing where evaporation occurs and decreasing where rainfall is high. The implication is that increasing sea surface temperature intensifies the evaporation-condensation-precipitation water cycle.

Since clouds have both positive and negative feedback effects, which predominates and how will changing global temperature affect this balance? These are very uncertain aspects of climate science. The factors that control where clouds form, what kinds are formed, and how increased temperature and atmospheric water vapor affect their formation are complex. Computer modeling of the turbulence, condensation, and growth of water droplets in cloud formation requires large amounts of computer time and capacity. At present, even the fastest computers, running general circulation models (GCM) of the climate require so much computer capacity that the models cannot incorporate the further complexity of cloud formation. Thus, the GCMs incorporate algorithms that relate cloud formation to other parameters, such as relative humidity, to estimate their formation and effects. These lead to great variation in the model predictions, depending on the algorithms and parameters used, but generally suggest that, as the planet warms, clouds will be a positive feedback, although perhaps relatively weak.

Aerosol Radiative Forcing

Aerosol particulate matter, tiny particles or liquid droplets suspended in the atmosphere, generally scatter and absorb incoming solar radiation, thus contributing to the Earth’s albedo. Naturally occurring aerosol particles are mainly picked up by the wind as dust and water spray or produced by occasional volcanic eruptions. Poor land-use practices by humans can make dust storms worse and intensify the natural effects.

Human activities do, however, add significantly to aerosol sulfate particles as well as producing black carbon (soot) particles. Burning fossil fuels containing sulfur produces SO₂ that is oxidized in the atmosphere, ultimately forming hygroscopic sulfuric acid molecules and salts that act as nucleation sites for tiny water droplets. These aerosol sulfate particles tend to be quite small, so, for a given amount of emission, the number of particles is large and scatters a good deal of solar radiation. This scattering increases the albedo and produces negative radiative forcing by reducing the amount of solar radiation reaching the surface.

The atmospheric models used in this Atmospheric Warming module of the ACS Climate Science Toolkit have been one-dimensional. They have focused on the properties of an atmospheric column only as a function of altitude. Two-dimensional models can aggregate the properties of one-dimensional models over many different locations to get a more realistic average over the planet. However, a two-dimensional model still lacks an essential characteristic of the climate system—continuous exchange of matter and energy among the one-dimensional atmospheric cells. Atmospheric circulation, the winds, must be accounted for to make any attempt to capture the observable characteristics of climate changes—past, present, or future. Similarly, the vast ocean currents, albeit much slower than atmospheric circulation, must be accounted for over long modeling periods. Developing, testing, refining, and comparing general circulation models of the climate are vitally important to further our understanding of the climate and make predictions of its future more reliable. Exploring the GCMs is beyond the scope of this Toolkit, but, if you are interested, References and Resources has some leading references where you can begin to explore.

Secondarily, the high concentration of these tiny aerosol sulfate particles leads to the formation of clouds with high concentrations of tiny water droplets that scatter more solar radiation than a lower concentration of larger droplets. This change in the composition of clouds also increases the albedo and produces further negative radiative forcing, sometimes called the indirect aerosol effect. Also, these clouds are more stable against formation of precipitation, so can have a longer lifetime to reflect sunlight. The uncertainties surrounding the modeling of both the direct and indirect effects of aerosol particulate matter, especially those involving cloud formation, are large and add further to the uncertainty in predicting the climate sensitivity resulting from human activities. The uncertainty is captured in the error bars associated with aerosols in this IPCC graphic and is also the impetus for increased research to better understand aerosols and clouds.

Credit: Figure 2, FAQ 2.1, from the IPCC Fourth Assessment Report (2007), Chapter 2,
 

Changes in Atmospheric Constituents and in Radiative Forcing

Black carbon (soot) is released from incomplete fuel combustion. Burning biomass in inefficient cook stoves or to clear land and incomplete combustion of diesel fuel are large sources of black carbon over much of southeast Asia and other developing areas of the world. In the atmosphere, the particles absorb and scatter incoming solar radiation. As winds carry them across globe, some end up on snow and ice-covered ground where they reduce the albedo and produce positive radiative forcing. A new initiative to develop and distribute efficient cook stoves to replace those now in use, could greatly reduce black carbon emissions, reducing radiative forcing a bit and, as a bonus, improving the health of the populations using them.

Modelling Planetary Temperature

There is an equation being circulated among climate skeptics being touted, very oddly, as a "model" for planetary temperatures:  there are different forms of this equation, with one I've seen most being

T = 1 / ((R * p(rho)) / P) / M, where

T = temperature at planetary surface in Kelvins (absolute temperature),
R = Ideal Gas Law constant (explained below),
p(rho) = density of atmosphere at planetary surface,
P = pressure of atmosphere at planetary surface,
M = mean molecular weight of atmosphere at planetary surface.

The reason I call this equation odd when presented as a model for planetary temperature should be obvious from the complete absence of any inclusion of solar radiance.  As we all learned in elementary school, or should have learned, the sun is the primary source of heat for all bodies in the solar system.  So how can equation which completely omits this factor possibly be regarded as a temperature model?

Another oddity is that the equation is restricted to bodies possessing atmospheres of 10^5 pascals pressure or higher.  This is the surface pressure of Earth's atmosphere, so the restriction not includes all solar system bodies lacking substantial atmospheres (e.g., the Moon, Mercury, Ganymede, Pluto, etc.) but also the planet Mars, the surface pressure of which is only about 7 * 10^2 pascals.  One should wonder what makes Mars different?  There is, in fact, a straightforward answer to this question, which is quickly revealed once the true meaning of the equation is exposed.

Skeptical science should not be confused with, or used as justification for, bad science.  Most of you are probably familiar with the Ideal Gas Law, which relates the pressure, volume, number of moles, and temperature of an "ideal gas", and serves as a model of gas behavior under specified conditions; if not, a full explanation can be found at https://en.wikipedia.org/wiki/Ideal_gas_law.  There you will discover the so-called Ideal Gas Law equation:

PV = nRT, where

P = the pressure of the gas,
V = the volume of the gas,
n = the number of moles of the gas (a mole being an Avagadro's number number of anything, in this
      case molecules of gas, approximately 6.022×10²³.
R = the constant of proportionality, having the value of 8.314 J/(K·mol),
T = the temperature (in Kelvins) of the gas.

The temperature of an ideal monatomic gas is proportional to the average kinetic energy of its atoms. The size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure. The atoms have a certain, average speed, slowed down here two trillion fold from room temperature.
(From https://en.wikipedia.org/wiki/Kinetic_theory_of_gases)

I raise the Ideal Gas Law, and its associated equation, because, as it turns out, the "model" equation for planetary temperature given above is not a model at all but merely the gas law equation algebraically rearranged:

PV = nRT
T = (P * V) / (n * R)
T = (P * V * M) / ( n * M * R)

as (n * M) / V = gas density

T = (P * M) / (p(rho) * R)
T = 1 / ((R * p(rho)) / P) / M

The equation does not work well for Mars because its atmosphere is almost pure CO2 at cold temperatures; CO2 deviates significantly from an ideal gas, as do all gasses at low enough temperatures.

For a better model, though simple, of solar system body atmosphere temperatures, refer to the link https://en.wikipedia.org/wiki/Climate_model#Zero-dimensional_models, which I have reproduced below:

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A very simple model of the radiative equilibrium of the Earth

${\displaystyle (1-a)S\pi r^{2}=4\pi r^{2}\epsilon \sigma T^{4}}$

where

• the left hand side represents the incoming energy from the Sun
• the right hand side represents the outgoing energy from the Earth, calculated from the Stefan-Boltzmann law assuming a model-fictive temperature, T, sometimes called the 'equilibrium temperature of the Earth', that is to be found,

and

• S is the solar constant – the incoming solar radiation per unit area—about 1367 W·m⁻²
• ${\displaystyle a}$ is the Earth's average albedo, measured to be 0.3.^[2][3]
• r is Earth's radius—approximately 6.371×10⁶m
• π is the mathematical constant (3.141...)
• ${\displaystyle \sigma }$ is the Stefan-Boltzmann constant—approximately 5.67×10⁻⁸ J·K⁻⁴·m⁻²·s⁻¹
• ${\displaystyle \epsilon }$ is the effective emissivity of earth, about 0.612

The constant πr² can be factored out, giving

${\displaystyle (1-a)S=4\epsilon \sigma T^{4}}$

Solving for the temperature,

${\displaystyle T={\sqrt[{4}]{\frac {(1-a)S}{4\epsilon \sigma }}}}$

This yields an apparent effective average earth temperature of 288 K (15 °C; 59 °F). This is because the above equation represents the effective radiative temperature of the Earth (including the clouds and atmosphere). The use of effective emissivity and albedo account for the greenhouse effect.

This very simple model is quite instructive, and the only model that could fit on a page. For example, it easily determines the effect on average earth temperature of changes in solar constant or change of albedo or effective earth emissivity.

The average emissivity of the earth is readily estimated from available data. The emissivities of terrestrial surfaces are all in the range of 0.96 to 0.99 (except for some small desert areas which may be as low as 0.7). Clouds, however, which cover about half of the earth’s surface, have an average emissivity of about 0.5 (which must be reduced by the fourth power of the ratio of cloud absolute temperature to average earth absolute temperature) and an average cloud temperature of about 258 K (−15 °C; 5 °F). Taking all this properly into account results in an effective earth emissivity of about 0.64 (earth average temperature 285 K (12 °C; 53 °F)).

This simple model readily determines the effect of changes in solar output or change of earth albedo or effective earth emissivity on average earth temperature. It says nothing, however about what might cause these things to change. Zero-dimensional models do not address the temperature distribution on the earth or the factors that move energy about the earth.

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I must add here that a common argument used to refute or at least minimize the CO2 greenhouse effect is the claim, that at "only" 0.04% of the Earth's atmosphere, CO2 exists in too small quantity to account for significant warming.  This is misleading, however, for the pertinent number is the total mass of CO2 per square area.  Given that the total mass of the atmosphere is 5 * 10^15 tons (10 tons / meter^2), 0.04% of this is 3 trillion tons of CO2 (remember the molecular mass of CO2 is ~50% higher than nitrogen and oxygen).  This amounts to 6 kg CO2 / meter^2.  Compressed down to the density of water, atmospheric CO2 would make a layer 0.6 cm (~ quarter inch) thick over the Earth's surface.

I assume the nobody has a problem with imagining a layer of 1/4" of suitably tinted water (or other substance) absorbing a significant amount of the visible light impinging on it.  So it should not be difficult to imagine the same layer of CO2 absorbing a significant amount of infra-red radiation as well.  In fact, the IR spectrum of CO2 shows it would do so.  Thus, the argument that CO2 is a mere 0.04% of the atmosphere is not relevant.

For a detailed discussion of the greenhouse effect, see Dr. Judith Curry's blog, which I have reproduced at http://amedleyofpotpourri.blogspot.com/2017/12/best-of-greenhouse.html.

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From a skeptical point of view, the issue is not whether CO2 is a greenhouse gas, or that rising levels shouldn't cause rising temperatures (or that human activities haven't caused the bulk of the rise of the last 150 years), but by how much, and whether this represents a serious threat to civilization, or even life on our planet.  In this vein, an interesting paper published recently (http://newscenter.lbl.gov/2015/02/25/co2-greenhouse-effect-increase/) indicates that warming from CO2 is only about 2/3rds the commonly accepted value, leading to perhaps 0.5-1.0C warming by the end of the century.  The catastrophic predictions of some climate models will not come to pass if this is accurate, especially if CO2 levels are kept in the range of 500 ppm by 2100, which the paper also (indirectly) indicates.

Climate model

From Wikipedia, the free encyclopedia

 

Climate models are systems of differential equations based on the basic laws of physics, fluid motion, and chemistry. To “run” a model, scientists divide the planet into a 3-dimensional grid, apply the basic equations, and evaluate the results. Atmospheric models calculate winds, heat transfer, radiation, relative humidity, and surface hydrology within each grid and evaluate interactions with neighboring points.

Climate models use quantitative methods to simulate the interactions of the important drivers of climate, including atmosphere, oceans, land surface and ice. They are used for a variety of purposes from study of the dynamics of the climate system to projections of future climate.

All climate models take account of incoming energy from the sun as short wave electromagnetic radiation, chiefly visible and short-wave (near) infrared, as well as outgoing long wave (far) infrared electromagnetic. Any imbalance results in a change in temperature.

Models vary in complexity:

• A simple radiant heat transfer model treats the earth as a single point and averages outgoing energy
• This can be expanded vertically (radiative-convective models) and/or horizontally
• Finally, (coupled) atmosphere–ocean–sea ice global climate models solve the full equations for mass and energy transfer and radiant exchange.
• Box models can treat flows across and within ocean basins.
• Other types of modelling can be interlinked, such as land use, allowing researchers to predict the interaction between climate and ecosystems.

Box models

Box models are simplified versions of complex systems, reducing them to boxes (or reservoirs) linked by fluxes. The boxes are assumed to be mixed homogeneously. Within a given box, the concentration of any chemical species is therefore uniform. However, the abundance of a species within a given box may vary as a function of time due to the input to (or loss from) the box or due to the production, consumption or decay of this species within the box.
Simple box models, i.e. box model with a small number of boxes whose properties (e.g. their volume) do not change with time, are often useful to derive analytical formulas describing the dynamics and steady-state abundance of a species. More complex box models are usually solved using numerical techniques.

Box models are used extensively to model environmental systems or ecosystems and in studies of ocean circulation and the carbon cycle.^[1]

Zero-dimensional models

A very simple model of the radiative equilibrium of the Earth is

${\displaystyle (1-a)S\pi r^{2}=4\pi r^{2}\epsilon \sigma T^{4}}$

where

• the left hand side represents the incoming energy from the Sun
• the right hand side represents the outgoing energy from the Earth, calculated from the Stefan-Boltzmann law assuming a model-fictive temperature, T, sometimes called the 'equilibrium temperature of the Earth', that is to be found,

and

• S is the solar constant – the incoming solar radiation per unit area—about 1367 W·m⁻²
• ${\displaystyle a}$ is the Earth's average albedo, measured to be 0.3.^[2][3]
• r is Earth's radius—approximately 6.371×10⁶m
• π is the mathematical constant (3.141...)
• ${\displaystyle \sigma }$ is the Stefan-Boltzmann constant—approximately 5.67×10⁻⁸ J·K⁻⁴·m⁻²·s⁻¹
• ${\displaystyle \epsilon }$ is the effective emissivity of earth, about 0.612

The constant πr² can be factored out, giving

${\displaystyle (1-a)S=4\epsilon \sigma T^{4}}$

Solving for the temperature,

${\displaystyle T={\sqrt[{4}]{\frac {(1-a)S}{4\epsilon \sigma }}}}$

This yields an apparent effective average earth temperature of 288 K (15 °C; 59 °F).^[4] This is because the above equation represents the effective radiative temperature of the Earth (including the clouds and atmosphere). The use of effective emissivity and albedo account for the greenhouse effect.

This very simple model is quite instructive, and the only model that could fit on a page. For example, it easily determines the effect on average earth temperature of changes in solar constant or change of albedo or effective earth emissivity.

The average emissivity of the earth is readily estimated from available data. The emissivities of terrestrial surfaces are all in the range of 0.96 to 0.99^[5][6] (except for some small desert areas which may be as low as 0.7). Clouds, however, which cover about half of the earth’s surface, have an average emissivity of about 0.5^[7] (which must be reduced by the fourth power of the ratio of cloud absolute temperature to average earth absolute temperature) and an average cloud temperature of about 258 K (−15 °C; 5 °F).^[8] Taking all this properly into account results in an effective earth emissivity of about 0.64 (earth average temperature 285 K (12 °C; 53 °F)).

This simple model readily determines the effect of changes in solar output or change of earth albedo or effective earth emissivity on average earth temperature. It says nothing, however about what might cause these things to change. Zero-dimensional models do not address the temperature distribution on the earth or the factors that move energy about the earth.

Radiative-convective models

The zero-dimensional model above, using the solar constant and given average earth temperature, determines the effective earth emissivity of long wave radiation emitted to space. This can be refined in the vertical to a one-dimensional radiative-convective model, which considers two processes of energy transport:

• upwelling and downwelling radiative transfer through atmospheric layers that both absorb and emit infrared radiation
• upward transport of heat by convection (especially important in the lower troposphere).

The radiative-convective models have advantages over the simple model: they can determine the effects of varying greenhouse gas concentrations on effective emissivity and therefore the surface temperature. But added parameters are needed to determine local emissivity and albedo and address the factors that move energy about the earth.

Effect of ice-albedo feedback on global sensitivity in a one-dimensional radiative-convective climate model.^[9][10][11]

Higher-dimension models

The zero-dimensional model may be expanded to consider the energy transported horizontally in the atmosphere. This kind of model may well be zonally averaged. This model has the advantage of allowing a rational dependence of local albedo and emissivity on temperature – the poles can be allowed to be icy and the equator warm – but the lack of true dynamics means that horizontal transports have to be specified.^[12]

EMICs (Earth-system models of intermediate complexity)

Depending on the nature of questions asked and the pertinent time scales, there are, on the one extreme, conceptual, more inductive models, and, on the other extreme, general circulation models operating at the highest spatial and temporal resolution currently feasible. Models of intermediate complexity bridge the gap. One example is the Climber-3 model. Its atmosphere is a 2.5-dimensional statistical-dynamical model with 7.5° × 22.5° resolution and time step of half a day; the ocean is MOM-3 (Modular Ocean Model) with a 3.75° × 3.75° grid and 24 vertical levels.^[13]

GCMs (global climate models or general circulation models)

General Circulation Models (GCMs) discretise the equations for fluid motion and energy transfer and integrate these over time. Unlike simpler models, GCMs divide the atmosphere and/or oceans into grids of discrete "cells", which represent computational units. Unlike simpler models which make mixing assumptions, processes internal to a cell—such as convection—that occur on scales too small to be resolved directly are parameterised at the cell level, while other functions govern the interface between cells.
Atmospheric GCMs (AGCMs) model the atmosphere and impose sea surface temperatures as boundary conditions. Coupled atmosphere-ocean GCMs (AOGCMs, e.g. HadCM3, EdGCM, GFDL CM2.X, ARPEGE-Climat)^[14] combine the two models. The first general circulation climate model that combined both oceanic and atmospheric processes was developed in the late 1960s at the NOAA Geophysical Fluid Dynamics Laboratory^[15] AOGCMs represent the pinnacle of complexity in climate models and internalise as many processes as possible. However, they are still under development and uncertainties remain. They may be coupled to models of other processes, such as the carbon cycle, so as to better model feedback effects. Such integrated multi-system models are sometimes referred to as either "earth system models" or "global climate models."

Research and development

There are three major types of institution where climate models are developed, implemented and used:

• National meteorological services. Most national weather services have a climatology section.
• Universities. Relevant departments include atmospheric sciences, meteorology, climatology, and geography.
• National and international research laboratories. Examples include the National Center for Atmospheric Research (NCAR, in Boulder, Colorado, USA), the Geophysical Fluid Dynamics Laboratory (GFDL, in Princeton, New Jersey, USA), the Hadley Centre for Climate Prediction and Research (in Exeter, UK), the Max Planck Institute for Meteorology in Hamburg, Germany, or the Laboratoire des Sciences du Climat et de l'Environnement (LSCE), France, to name but a few.

The World Climate Research Programme (WCRP), hosted by the World Meteorological Organization (WMO), coordinates research activities on climate modelling worldwide.

A 2012 U.S. National Research Council report discussed how the large and diverse U.S. climate modeling enterprise could evolve to become more unified.^[16] Efficiencies could be gained by developing a common software infrastructure shared by all U.S. climate researchers, and holding an annual climate modeling forum, the report found.^[17]