Emissivity

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.

Some specific forms of emissivity are detailed below.

Hemispherical emissivity

Hemispherical emissivity of a surface, denoted ε, is defined as

${\displaystyle \epsilon =\frac{M_{\mathrm{e}}}{M_{\mathrm{e}}^{\circ }},}$

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

${\displaystyle \begin{array}{cc}{\epsilon }_{\nu }& =\frac{M_{\mathrm{e},\nu }}{M_{\mathrm{e},\nu }^{\circ }},\\ {\epsilon }_{\lambda }& =\frac{M_{\mathrm{e},\lambda }}{M_{\mathrm{e},\lambda }^{\circ }},\end{array}}$

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

${\displaystyle {\epsilon }_{\mathrm{\Omega }}=\frac{L_{\mathrm{e},\mathrm{\Omega }}}{L_{\mathrm{e},\mathrm{\Omega }}^{\circ }},}$

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

${\displaystyle \begin{array}{cc}{\epsilon }_{\nu ,\mathrm{\Omega }}& =\frac{L_{\mathrm{e},\mathrm{\Omega },\nu }}{L_{\mathrm{e},\mathrm{\Omega },\nu }^{\circ }},\\ {\epsilon }_{\lambda ,\mathrm{\Omega }}& =\frac{L_{\mathrm{e},\mathrm{\Omega },\lambda }}{L_{\mathrm{e},\mathrm{\Omega },\lambda }^{\circ }},\end{array}}$

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".

In practice, approximation models are often used to describe this dependence. One of the simplest is the cosine model:

${\displaystyle \epsilon (\alpha )={\epsilon }_0\cdot {\cos }^n(\alpha ),}$

where ${\displaystyle {\epsilon }_0}$ is the emissivity in the normal direction and the exponent n characterizes the deviation of the surface from ideally diffuse behavior. For thermographic applications, this angular dependence must be incorporated into the quantitative thermography equation, because the effective radiance detected by a thermal camera pixel depends on both the emissivity function ${\displaystyle \epsilon (\alpha )}$ and the contribution of reflected background radiation.

Examples of the cosine-power emissivity model The parameter n in the function ε(α) = ε₀·cosⁿ(α) controls how strongly the emissivity decreases with the angle α from the surface normal. Typical interpretations are:

• n = 1 – Emissivity decreases moderately with angle. Such a surface radiates less efficiently at oblique incidence, and reflected background radiation contributes more strongly to the apparent temperature.
• n = 2 – Stronger angular dependence. At larger viewing angles, the detected emissivity is significantly lower, leading to a greater share of reflected radiation and a larger error in apparent temperature measurement if not corrected.
• n = 3 – Steep angular falloff. At oblique incidence the emitted component can become small relative to the reflected component; the apparent temperature may be strongly biased toward the background unless ε(α) and the viewing geometry are accounted for.

In practical thermography, surfaces with higher n values require precise correction for reflected apparent temperature in order to avoid underestimation of the true object temperature.

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 meters (10 μm). Visible light has a wavelength range of about 0.4×10⁻⁶ to 0.7×10⁻⁶ meters (400 to 700 nm) 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.

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 300 K (27 °C; 80 °F).

Closely related properties

Absorptance

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. 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.

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 1,273 K (1,000 °C; 1,832 °F), will emit 97% of its energy at wavelengths below 14 μm.

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.

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). 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.

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.

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. 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 parametrizations 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. 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.

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. Nitrogen (N
₂) and oxygen (O
₂) – the primary atmospheric components – interact less significantly with thermal radiation in the infrared band. 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. 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). Carbon dioxide (CO
₂) and other greenhouse gases contribute about ε=0.2 to ε_a when atmospheric humidity is low. Researchers have also evaluated the contribution of differing cloud types to atmospheric absorptivity and emissivity.

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.

Effective emissivity due to atmosphere

The IPCC reports an outgoing thermal radiation flux (OLR) of 239 (237–242) W m^-2 and a surface thermal radiation flux (SLR) of 398 (395–400) W m^-2, 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 ${\displaystyle \mathrm{O}\mathrm{L}\mathrm{R}={ϵ}_{\mathrm{e}\mathrm{f}\mathrm{f}}\sigma T_{se}^4}$ where ${\displaystyle {ϵ}_{\mathrm{e}\mathrm{f}\mathrm{f}}\approx 0.6}$ is the effective emissivity of Earth as viewed from space and ${\displaystyle T_{\mathrm{s}\mathrm{e}}\equiv {\left[\mathrm{S}\mathrm{L}\mathrm{R}/\sigma \right]}^{1/4}\approx }$ 289 K (16 °C; 61 °F) is the effective temperature of the surface.

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. In 1860, Gustav Kirchhoff published a mathematical description of their relationship under conditions of thermal equilibrium (i.e. Kirchhoff's law of thermal radiation). 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. 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. By 1900 Max Planck empirically derived a generalized law of blackbody radiation, thus clarifying the emissivity and absorptivity concepts at individual wavelengths.

Other radiometric coefficients