Emissivity Explained

The emissivity of the surface of a material is its effectiveness in emitting energy as thermal radiation. Thermal radiation is electromagnetic radiation that most commonly includes both visible radiation (light) and infrared radiation, which is not visible to human eyes. A portion of the thermal radiation from very hot objects (see photograph) is easily visible to the eye.

The emissivity of a surface depends on its chemical composition and geometrical structure. Quantitatively, it is the ratio of the thermal radiation from a surface to the radiation from an ideal black surface at the same temperature as given by the Stefan–Boltzmann law. (A comparison with Planck's law is used if one is concerned with particular wavelengths of thermal radiation.) The ratio varies from 0 to 1.

The surface of a perfect black body (with an emissivity of 1) emits thermal radiation at the rate of approximately 448 watts per square metre (W/m) at a room temperature of 25C.

Objects generally have emissivities less than 1.0, and emit radiation at correspondingly lower rates.^[1]

However, wavelength- and subwavelength-scale particles,^[2] metamaterials,^[3] and other nanostructures^[4] may have an emissivity greater than 1.

Practical applications

Emissivities are important in a variety of contexts:

Insulated windows: Warm surfaces are usually cooled directly by air, but they also cool themselves by emitting thermal radiation. This second cooling mechanism is important for simple glass windows, which have emissivities close to the maximum possible value of 1.0. "Low-E windows" with transparent low-emissivity coatings emit less thermal radiation than ordinary windows.^[5] In winter, these coatings can halve the rate at which a window loses heat compared to an uncoated glass window.^[6]

Solar heat collectors: Similarly, solar heat collectors lose heat by emitting thermal radiation. Advanced solar collectors incorporate selective surfaces that have very low emissivities. These collectors waste very little of the solar energy through emission of thermal radiation.^[7]

Thermal shielding: For the protection of structures from high surface temperatures, such as reusable spacecraft or hypersonic aircraft, high-emissivity coatings (HECs), with emissivity values near 0.9, are applied on the surface of insulating ceramics.^[8] This facilitates radiative cooling and protection of the underlying structure and is an alternative to ablative coatings, used in single-use reentry capsules.

Passive daytime radiative cooling: Daytime passive radiative coolers use the extremely cold temperature of outer space (~2.7 K) to emit heat and lower ambient temperatures while requiring zero energy input.^[9] These surfaces minimize the absorption of solar radiation to lessen heat gain in order to maximize the emission of LWIR thermal radiation.^[9] It has been proposed as a solution to global warming.^[10]

Planetary temperatures: The planets are solar thermal collectors on a large scale. The temperature of a planet's surface is determined by the balance between the heat absorbed by the planet from sunlight, heat emitted from its core, and thermal radiation emitted back into space. Emissivity of a planet is determined by the nature of its surface and atmosphere.^[11]

Temperature measurements: Pyrometers and infrared cameras are instruments used to measure the temperature of an object by using its thermal radiation; no actual contact with the object is needed. The calibration of these instruments involves the emissivity of the surface that's being measured.

Mathematical definitions

In its most general form, emissivity can be specified for a particular wavelength, direction, and polarization.

However, the most commonly used form of emissivity is the hemispherical total emissivity, which considers emissions as totaled over all wavelengths, directions, and polarizations, given a particular temperature.^[12]

Some specific forms of emissivity are detailed below.

Hemispherical emissivity

Hemispherical emissivity of a surface, denoted ε, is defined as^[13]

\varepsilon = (M _{e})/(M ^{\circ} _{e}) ,

where

• M_e is the radiant exitance of that surface;
• M_e° is the radiant exitance of a black body at the same temperature as that surface.

Spectral hemispherical emissivity

Spectral hemispherical emissivity in frequency and spectral hemispherical emissivity in wavelength of a surface, denoted ε_ν and ε_λ, respectively, are defined as^[13]

\begin{align} \varepsilon _{\nu} & = (M _{e,\nu})/(M ^{\circ} _{e ,\nu}) , \\ \varepsilon _{λ} & = (M _{e,λ})/(M ^{\circ} _{e , λ}) , \end{align}

where

• M_e,ν is the spectral radiant exitance in frequency of that surface;
• M_e,ν° is the spectral radiant exitance in frequency of a black body at the same temperature as that surface;
• M_e,λ is the spectral radiant exitance in wavelength of that surface;
• M_e,λ° is the spectral radiant exitance in wavelength of a black body at the same temperature as that surface.

Directional emissivity

Directional emissivity of a surface, denoted ε_Ω, is defined as^[13]

\varepsilon _{\Omega} = (L _{e,\Omega})/(L ^{\circ} _{e ,\Omega}) ,

where

• L_e,Ω is the radiance of that surface;
• L_e,Ω° is the radiance of a black body at the same temperature as that surface.

Spectral directional emissivity

Spectral directional emissivity in frequency and spectral directional emissivity in wavelength of a surface, denoted ε_ν,Ω and ε_λ,Ω, respectively, are defined as^[13]

\begin{align} \varepsilon _{\nu,\Omega} & = (L _{e,\Omega,\nu})/(L ^{\circ} _{e ,\Omega,\nu}) , \\ \varepsilon _{λ,\Omega} & = (L _{e,\Omega,λ})/(L ^{\circ} _{e ,\Omega, λ}) , \end{align}

where

• L_e,Ω,ν is the spectral radiance in frequency of that surface;
• L_e,Ω,ν° is the spectral radiance in frequency of a black body at the same temperature as that surface;
• L_e,Ω,λ is the spectral radiance in wavelength of that surface;
• L_e,Ω,λ° is the spectral radiance in wavelength of a black body at the same temperature as that surface.

Hemispherical emissivity can also be expressed as a weighted average of the directional spectral emissivities as described in textbooks on "radiative heat transfer".

Emissivities of common surfaces

Emissivities ε can be measured using simple devices such as Leslie's cube in conjunction with a thermal radiation detector such as a thermopile or a bolometer. The apparatus compares the thermal radiation from a surface to be tested with the thermal radiation from a nearly ideal, black sample. The detectors are essentially black absorbers with very sensitive thermometers that record the detector's temperature rise when exposed to thermal radiation. For measuring room temperature emissivities, the detectors must absorb thermal radiation completely at infrared wavelengths near 10×10⁻⁶ metre.^[14] Visible light has a wavelength range of about 0.4–0.7×10⁻⁶ metre from violet to deep red.

Emissivity measurements for many surfaces are compiled in many handbooks and texts. Some of these are listed in the following table.^{[15] [16]}

Material	data-sort-type="number"	Emissivity
Aluminium foil	0.03
Aluminium, anodized	0.9^[17]
Aluminium, smooth, polished	0.04
Aluminium, rough, oxidized	0.2
Asphalt	0.88
Brick	0.90
Concrete, rough	0.91
Copper, polished	0.04
Copper, oxidized	0.87
Glass, smooth uncoated	0.95
Ice	0.97-0.99
Iron, polished	0.06
Limestone	0.92
Marble, polished	0.89–0.92
Nitrogen or Oxygen gas layer, pure	~0^[18]
Paint, including white	0.9
Paper, roofing or white	0.88–0.86
Plaster, rough	0.89
Silver, polished	0.02
Silver, oxidized	0.04
Skin, human	0.97–0.999
Snow	0.8–0.9
Polytetrafluoroethylene (Teflon)	0.85
Transition metal disilicides (e.g. MoSi2 or WSi2)	0.86–0.93
Vegetation	0.92-0.96
Water, pure	0.96

Notes:

1. These emissivities are the total hemispherical emissivities from the surfaces.
2. The values of the emissivities apply to materials that are optically thick. This means that the absorptivity at the wavelengths typical of thermal radiation doesn't depend on the thickness of the material. Very thin materials emit less thermal radiation than thicker materials.
3. Most emissitivies in the chart above were recorded at room temperature, .

Closely related properties

Absorptance

See main article: Kirchhoff's law of thermal radiation. There is a fundamental relationship (Gustav Kirchhoff's 1859 law of thermal radiation) that equates the emissivity of a surface with its absorption of incident radiation (the "absorptivity" of a surface). Kirchhoff's law is rigorously applicable with regard to the spectral directional definitions of emissivity and absorptivity. The relationship explains why emissivities cannot exceed 1, since the largest absorptivity—corresponding to complete absorption of all incident light by a truly black object—is also 1.^[19] Mirror-like, metallic surfaces that reflect light will thus have low emissivities, since the reflected light isn't absorbed. A polished silver surface has an emissivity of about 0.02 near room temperature. Black soot absorbs thermal radiation very well; it has an emissivity as large as 0.97, and hence soot is a fair approximation to an ideal black body.^{[20] [21]}

With the exception of bare, polished metals, the appearance of a surface to the eye is not a good guide to emissivities near room temperature. For example, white paint absorbs very little visible light. However, at an infrared wavelength of 10×10⁻⁶ metre, paint absorbs light very well, and has a high emissivity. Similarly, pure water absorbs very little visible light, but water is nonetheless a strong infrared absorber and has a correspondingly high emissivity.

Emittance

Emittance (or emissive power) is the total amount of thermal energy emitted per unit area per unit time for all possible wavelengths. Emissivity of a body at a given temperature is the ratio of the total emissive power of a body to the total emissive power of a perfectly black body at that temperature. Following Planck's law, the total energy radiated increases with temperature while the peak of the emission spectrum shifts to shorter wavelengths. The energy emitted at shorter wavelengths increases more rapidly with temperature. For example, an ideal blackbody in thermal equilibrium at, will emit 97% of its energy at wavelengths below .^[8]

The term emissivity is generally used to describe a simple, homogeneous surface such as silver. Similar terms, emittance and thermal emittance, are used to describe thermal radiation measurements on complex surfaces such as insulation products.^{[22] [23] [24]}

Measurement of Emittance

Emittance of a surface can be measured directly or indirectly from the emitted energy from that surface. In the direct radiometric method, the emitted energy from the sample is measured directly using a spectroscope such as Fourier transform infrared spectroscopy (FTIR).^[24] In the indirect calorimetric method, the emitted energy from the sample is measured indirectly using a calorimeter. In addition to these two commonly applied methods, inexpensive emission measurement technique based on the principle of two-color pyrometry.^[24]

Emissivities of planet Earth

The emissivity of a planet or other astronomical body is determined by the composition and structure of its outer skin. In this context, the "skin" of a planet generally includes both its semi-transparent atmosphere and its non-gaseous surface. The resulting radiative emissions to space typically function as the primary cooling mechanism for these otherwise isolated bodies. The balance between all other incoming plus internal sources of energy versus the outgoing flow regulates planetary temperatures.^[25]

For Earth, equilibrium skin temperatures range near the freezing point of water, 260±50 K (-13±50 °C, 8±90 °F). The most energetic emissions are thus within a band spanning about 4-50 μm as governed by Planck's law.^[26] Emissivities for the atmosphere and surface components are often quantified separately, and validated against satellite- and terrestrial-based observations as well as laboratory measurements. These emissivities serve as input parameters within some simpler meteorlogic and climatologic models.

Surface

Earth's surface emissivities (ε_s) have been inferred with satellite-based instruments by directly observing surface thermal emissions at nadir through a less obstructed atmospheric window spanning 8-13 μm.^[27] Values range about ε_s=0.65-0.99, with lowest values typically limited to the most barren desert areas. Emissivities of most surface regions are above 0.9 due to the dominant influence of water; including oceans, land vegetation, and snow/ice. Globally averaged estimates for the hemispheric emissivity of Earth's surface are in the vicinity of ε_s=0.95.^[28]

Atmosphere

Water also dominates the planet's atmospheric emissivity and absorptivity in the form of water vapor. Clouds, carbon dioxide, and other components make substantial additional contributions, especially where there are gaps in the water vapor absorption spectrum.^[29] Nitrogen and oxygen - the primary atmospheric components - interact less significantly with thermal radiation in the infrared band.^[30] Direct measurement of Earths atmospheric emissivities (ε_a) are more challenging than for land surfaces due in part to the atmosphere's multi-layered and more dynamic structure.

Upper and lower limits have been measured and calculated for ε_a in accordance with extreme yet realistic local conditions. At the upper limit, dense low cloud structures (consisting of liquid/ice aerosols and saturated water vapor) close the infrared transmission windows, yielding near to black body conditions with ε_a≈1.^[31] At a lower limit, clear sky (cloud-free) conditions promote the largest opening of transmission windows. The more uniform concentration of long-lived trace greenhouse gases in combination with water vapor pressures of 0.25-20 mbar then yield minimum values in the range of ε_a=0.55-0.8 (with ε=0.35-0.75 for a simulated water-vapor-only atmosphere).^[32] Carbon dioxide and other greenhouse gases contribute about ε=0.2 to ε_a when atmospheric humidity is low.^[33] Researchers have also evaluated the contribution of differing cloud types to atmospheric absorptivity and emissivity.^{[34] [35] [36]}

These days, the detailed processes and complex properties of radiation transport through the atmosphere are evaluated by general circulation models using radiation transport codes and databases such as MODTRAN/HITRAN. Emission, absorption, and scattering are thereby simulated through both space and time.

For many practical applications it may not be possible, economical or necessary to know all emissivity values locally. "Effective" or "bulk" values for an atmosphere or an entire planet may be used. These can be based upon remote observations (from the ground or outer space) or defined according to the simplifications utilized by a particular model. For example, an effective global value of ε_a≈0.78 has been estimated from application of an idealized single-layer-atmosphere energy-balance model to Earth.^[37]

Effective emissivity due to atmosphere

The IPCC reports an outgoing thermal radiation flux (OLR) of 239 (237–242) W m and a surface thermal radiation flux (SLR) of 398 (395–400) W m, where the parenthesized amounts indicate the 5-95% confidence intervals as of 2015. These values indicate that the atmosphere (with clouds included) reduces Earth's overall emissivity, relative to its surface emissions, by a factor of 239/398 ≈ 0.60. In other words, emissions to space are given by

OLR = \epsilon _{eff\sigmaT} ^{4} ^{se} where \epsilon _{eff} ≈ 0 . 6 is the effective emissivity of Earth as viewed from space and T _{se} \equiv \left[ SLR /\sigma\right] ^{1/4} ≈ is the effective temperature of the surface.^[38]

History

The concepts of emissivity and absorptivity, as properties of matter and radiation, appeared in the late-eighteenth thru mid-nineteenth century writings of Pierre Prévost, John Leslie, Balfour Stewart and others.^{[39] [40] [41]} In 1860, Gustav Kirchhoff published a mathematical description of their relationship under conditions of thermal equilibrium (i.e. Kirchhoff's law of thermal radiation).^[42] By 1884 the emissive power of a perfect blackbody was inferred by Josef Stefan using John Tyndall's experimental measurements, and derived by Ludwig Boltzmann from fundamental statistical principles.^[43] Emissivity, defined as a further proportionality factor to the Stefan-Boltzmann law, was thus implied and utilized in subsequent evaluations of the radiative behavior of grey bodies. For example, Svante Arrhenius applied the recent theoretical developments to his 1896 investigation of Earth's surface temperatures as calculated from the planet's radiative equilibrium with all of space.^[44] By 1900 Max Planck empirically derived a generalized law of blackbody radiation, thus clarifying the emissivity and absorptivity concepts at individual wavelengths.^[45]

See also

• Albedo
• Black-body radiation
• Passive daytime radiative cooling
• Radiant barrier
• Reflectance
• Sakuma–Hattori equation
• Stefan–Boltzmann law
• View factor
• Wien's displacement law

Further reading

• Web site: Spectral emissivity and emittance . Temperatures.com, Inc. . Southampton, PA . https://web.archive.org/web/20170424062858/http://www.spectralemissivity.com/ . 24 April 2017. An open community-focused website & directory with resources related to spectral emissivity and emittance. On this site, the focus is on available data, references and links to resources related to spectral emissivity as it is measured & used in thermal radiation thermometry and thermography (thermal imaging).
• Web site: Emissivity Coefficients of some common Materials . engineeringtoolbox.com . Resources, Tools and Basic Information for Engineering and Design of Technical Applications. This site offers an extensive list of other material not covered above.

Notes and References

1. The Stefan–Boltzmann law is that the rate of emission of thermal radiation is σT⁴, where σ = 5.67×10⁻⁸ W/m²·K⁴, and the temperature T is in kelvins. See Book: Trefil , James S. . The Nature of Science: An A-Z Guide to the Laws and Principles Governing Our Universe . . 2003 . 9780618319381 . 377 . registration .
2. Book: Bohren . Craig F. . Huffman . Donald R. . 1998 . Absorption and scattering of light by small particles . Wiley . 978-0-471-29340-8 . 123–126.
3. Narimanov . Evgenii E. . Smolyaninov . Igor I. . 2012 . Beyond Stefan–Boltzmann Law: Thermal Hyper-Conductivity . 1109.5444 . Conference on Lasers and Electro-Optics 2012 . OSA Technical Digest . Optical Society of America . QM2E.1 . 10.1364/QELS.2012.QM2E.1 . 978-1-55752-943-5 . 10.1.1.764.846 . 36550833.
4. Golyk . V. A. . Krüger . M. . Kardar . M. . Heat radiation from long cylindrical objects . Phys. Rev. E . 2012 . 85 . 4 . 046603 . 10.1103/PhysRevE.85.046603 . 22680594 . 1109.1769 . 2012PhRvE..85d6603G . 1721.1/71630 . 27489038 . free .
5. Book: The Low-E Window R&D Success Story . Windows and Building Envelope Research and Development: Roadmap for Emerging Technologies . . February 2014 . 5 . http://energy.gov/sites/prod/files/2014/02/f8/BTO_windows_and_envelope_report_3.pdf#page=15 .
6. Book: Essentials of Energy Technology . Fricke . Jochen . Borst . Walter L. . . 2013 . 978-3527334162 . 37 .
7. Book: Essentials of Energy Technology . 9. Solar Space and Hot Water Heating . Fricke . Jochen . Borst . Walter L. . Wiley-VCH . 2013 . 978-3527334162 . 249 . https://books.google.com/books?id=zn1nAgAAQBAJ&pg=PA249 .
8. Improved oxidation resistance of high emissivity coatings on fibrous ceramic for reusable space systems . Shao. Gaofeng. etal. Corrosion Science . 2019 . 146. 233–246 . 10.1016/j.corsci.2018.11.006 . 1902.03943 . 2019Corro.146..233S. 118927116.
9. Aili . Ablimit . Yin . Xiaobo . Yang . Ronggui . October 2021 . Global Radiative Sky Cooling Potential Adjusted for Population Density and Cooling Demand . Atmosphere . 12 . 11 . 1379 . 10.3390/atmos12111379 . free . 2021Atmos..12.1379A .
10. Chen . Meijie . Pang . Dan . Chen . Xingyu . Yan . Hongjie . Yang . Yuan . Passive daytime radiative cooling: Fundamentals, material designs, and applications . EcoMat . 2022 . 4 . 10.1002/eom2.12153 . 240331557 . Passive daytime radiative cooling (PDRC) dissipates terrestrial heat to the extremely cold outer space without using any energy input or producing pollution. It has the potential to simultaneously alleviate the two major problems of energy crisis and global warming. . free .
11. Web site: Climate Sensitivity . 2014-07-21 . American Chemical Society.
12. Book: Siegel . Robert . Howell . John R. . Thermal Radiation Heat Transfer . 1992 . Taylor & Francis . 0-89116-271-2 . 3.
13. Web site: Thermal insulation — Heat transfer by radiation — Physical quantities and definitions. ISO 9288:2022. ISO catalogue. 1989. 2015-03-15.
14. For a truly black object, the spectrum of its thermal radiation peaks at the wavelength given by Wien's Law: λ_max = b/T, where the temperature T is in kelvins and the constant b ≈ 2.90×10⁻³ metre-kelvins. Room temperature is about 293 kelvins. Sunlight itself is thermal radiation originating from the hot surface of the Sun. The Sun's surface temperature of about 5800 kelvins corresponds well to the peak wavelength of sunlight, which is at the green wavelength of about 0.5×10⁻⁶ metres. See Book: Saha, Kshudiram . The Earth's Atmosphere: Its Physics and Dynamics . . 2008 . 9783540784272 . 84 .
15. Book: Brewster , M. Quinn . Thermal Radiative Transfer and Properties . . 1992 . 9780471539827 . 56 .
16. Book: 2009 ASHRAE Handbook: Fundamentals - IP Edition . American Society of Heating, Refrigerating and Air-Conditioning Engineers . 2009 . Atlanta . . 978-1-933742-56-4. "IP" refers to inch and pound units; a version of the handbook with metric units is also available. Emissivity is a simple number, and doesn't depend on the system of units.
17. The visible color of an anodized aluminum surface does not strongly affect its emissivity. See Web site: Emissivity of Materials . Electro Optical Industries, Inc. . https://web.archive.org/web/20120919150451/http://www.electro-optical.com/eoi_page.asp?h=Emissivity%20of%20Materials . 2012-09-19 . live.
18. Trogler . William C. . The Environmental Chemistry of Trace Atmospheric Gases . Journal of Chemical Education . 72 . 11 . 973 . 1995 . 10.1021/ed072p973 . 1995JChEd..72..973T .
19. Book: Siegel, Robert . Thermal Radiation Heat Transfer, Fourth Edition . . 2001 . 9781560328391 . 41 .
20. Web site: Table of Total Emissivity . dead . https://web.archive.org/web/20090711135115/http://www.monarchserver.com/TableofEmissivity.pdf . 2009-07-11 . Table of emissivities provided by a company; no source for these data is provided.
21. Web site: Influencing factors . evitherm Society - Virtual Institute for Thermal Metrology . 2014-07-19 . dead . https://web.archive.org/web/20140112002045/http://www.evitherm.org/default.asp?ID=216 . 2014-01-12 .
22. Web site: ASTM C835 - 06(2013)e1: Standard Test Method for Total Hemispherical Emittance of Surfaces up to 1400°C . 2014-08-09 . ASTM International.
23. Book: Green Building: Principles and Practices in Residential Construction . Abe . Kruger . Carl . Seville . . 2012 . 9781111135959 . 198 .
24. Saad . Abdullah A. . Martinez . Carlos . Trice . Rodney W. . 2023-02-13 . Radiation heat transfer during hypersonic flight: A review of emissivity measurement and enhancement approaches of ultra-high temperature ceramics . International Journal of Ceramic Engineering & Science . 5 . 2 . en . 10.1002/ces2.10171 . 2578-3270. free .
25. Web site: Climate and Earth's Energy Budget . 14 January 2009 . NASA Earth Observatory . 10 October 2022.
26. Book: Petty . Grant W. . A first course in atmospheric radiation . 2006 . Sundog Publ. . Madison, Wisc.. 978-0972903318 . 68 . 2 .
27. Web site: ASTER global emissivity database: 100 times more detailed than its predecessor . 17 November 2014 . NASA Earth Observatory . 10 October 2022.
28. Web site: Joint Emissivity Database Initiative . NASA Jet Propulsion Laboratory . 10 October 2022.
29. Web site: Remote Sensing: Absorption Bands and Atmospheric Windows . . 17 September 1999 . 28 October 2022.
30. Höpfner . M. . Milz . M. . Buehler . S. . Orphall . J. . Stiller . G. . 24 May 2012 . The natural greenhouse effect of atmospheric oxygen (O₂) and nitrogen (N₂) . Geophysical Research Letters . en . 39 . L10706 . 10.1029/2012GL051409 . 2012GeoRL..3910706H . 128823108 . 1944-8007.
31. Liu . Lei . Zhang . Ting . Wu . Yi . Niu . Zhencong . Wang . Qi . 2018 . Cloud Effective Emissivity Retrievals Using Combined Ground-Based Infrared Cloud Measuring Instrument and Ceilometer Observations . Remote Sensing . 10 . 2033 . 2033 . 10.3390/rs10122033 . 2018RemS...10.2033L . free .
32. Mendoza . Victor M.. . Vallanueva . Elba E. . Garduno . Rene . Sanchez-Meneses . Oscar . Atmospheric emissivity with clear sky computed by E-Trans/HITRAN . Atmospheric Environment . 31 January 2017 . 155 . 174–188 . 10.1016/j.atmosenv.2017.01.048 . 2017AtmEn.155..174M . 1352-2310 .
33. Staley . D.O. . Jurica . G.M. . Effective atmospheric emissivity under clear skies . Applied Meteorology and Climatology . 1 March 1972 . 11 . 2 . 349–356 . 10.1175/1520-0450(1972)011<0349:EAEUCS>2.0.CO;2 . 1972JApMe..11..349S . free.
34. Web site: Clouds and Radiation . . 1 March 1999 . Graham, Steve . 28 October 2022.
35. Cox . Stephen K. . Observations of cloud infrared effective emissivity . Atmospheric Sciences . 1 February 1976 . 33 . 2 . 287–289 . 10.1175/1520-0469(1976)033<0287:OOCIEE>2.0.CO;2 . 1976JAtS...33..287C . free.
36. Chylek . Petr . Ramaswamy . V. . Simple approximation of infrared emissivity of water clouds . Atmospheric Sciences . 1 January 1982 . 39 . 1 . 171–177 . 10.1175/1520-0469(1982)039<0171:SAFIEO>2.0.CO;2 . 1982JAtS...39..171C . free.
37. Web site: ACS Climate Science Toolkit - Atmospheric Warming - A Single-Layer Atmosphere Model . . 1 December 2022.
38. Book: . IPCC . IPCC . 2021 . Climate Change 2021: The Physical Science Basis . Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change . 4 . V. . Masson-Delmotte . P. . Zhai . A. . Pirani . S. L. . Connors . C. . Péan . S. . Berger . N. . Caud . Y. . Chen . L. . Goldfarb . M. I. . Gomis . Cambridge University Press (In Press) .
39. Prévost. Pierre. April 1791. Mémoire sur l'équilibre du feu. Observations Sur la Physique. fr. XXXVIII. 1. 314–323.
40. Book: Leslie, John . An Experimental Inquiry into the Nature and Propagation of Heat . 1804 . J. Mawman . Edinburgh .
41. Book: Stewart, Balfour . 1866 . An Elementary Treatise on Heat . Oxford, Clarendon Press .
42. Kirchhoff . Gustav . 1860 . Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme and Licht . Annalen der Physik und Chemie. 109 . 275–301 . 10.1002/andp.18601850205 . free . 1860AnP...185..275K . 2 .
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