Planckian Locus
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In
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
and
color science Color science is the science, scientific study of color including lighting and optics; Photometry (optics), measurement of light and colorimetry, color; the physiology, psychophysics, and color model, modeling of color vision; and color reproductio ...
, the Planckian locus or black body locus is the path or ''locus'' that the color of an incandescent black body would take in a particular
chromaticity space Chromaticity is an objective specification of the quality of a color regardless of its luminance. Chromaticity consists of two independent parameters, often specified as ''hue'' (''h'') and ''colorfulness'' (''s''), where the latter is alter ...
as the blackbody
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
changes. It goes from deep
red Red is the color at the long wavelength end of the visible spectrum of light, next to orange and opposite violet. It has a dominant wavelength of approximately 625–750 nanometres. It is a primary color in the RGB color model and a seconda ...
at low temperatures through
orange Orange most often refers to: *Orange (fruit), the fruit of the tree species '' Citrus'' × ''sinensis'' ** Orange blossom, its fragrant flower ** Orange juice *Orange (colour), the color of an orange fruit, occurs between red and yellow in the vi ...
,
yellow Yellow is the color between green and orange on the spectrum of light. It is evoked by light with a dominant wavelength of roughly 575585 nm. It is a primary color in subtractive color systems, used in painting or color printing. In t ...
ish,
white White is the lightest color and is achromatic (having no chroma). It is the color of objects such as snow, chalk, and milk, and is the opposite of black. White objects fully (or almost fully) reflect and scatter all the visible wa ...
, and finally bluish white at very high temperatures. A
color space A color space is a specific organization of colors. In combination with color profiling supported by various physical devices, it supports reproducible representations of colorwhether such representation entails an analog or a digital represe ...
is a
three-dimensional space In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values ('' coordinates'') are required to determine the position of a point. Most commonly, it is the three- ...
; that is, a color is specified by a set of three numbers (the CIE coordinates ''X'', ''Y'', and ''Z'', for example, or other values such as
hue In color theory, hue is one of the properties (called color appearance parameters) of a color, defined in the CIECAM02 model as "the degree to which a stimulus can be described as similar to or different from stimuli that are described as ...
,
colorfulness Colorfulness, chroma and saturation are attributes of perceived color relating to chromatic intensity. As defined formally by the International Commission on Illumination (CIE) they respectively describe three different aspects of chromatic ...
, and
luminance Luminance is a photometric measure of the luminous intensity per unit area of light travelling in a given direction. It describes the amount of light that passes through, is emitted from, or is reflected from a particular area, and falls wit ...
) which specify the perceived attributes of a particular homogeneous visual stimulus. A chromaticity is a color projected into a
two-dimensional space A two-dimensional space is a mathematical space with two dimensions, meaning points have two degrees of freedom: their locations can be locally described with two coordinates or they can move in two independent directions. Common two-dimensiona ...
that ignores brightness. For example, the standard
CIE XYZ color space In 1931, the International Commission on Illumination (CIE) published the CIE 1931 color spaces which define the relationship between the visible spectrum and human color vision. The CIE color spaces are mathematical models that comprise a "stan ...
projects directly to the corresponding chromaticity space specified by the two chromaticity coordinates known as ''x'' and ''y'', making the familiar chromaticity diagram shown in the figure. The Planckian locus, the path that the color of a black body takes as the blackbody temperature changes, is often shown in this standard chromaticity space.


Planckian locus in the XYZ color space

In the
CIE XYZ color space In 1931, the International Commission on Illumination (CIE) published the CIE 1931 color spaces which define the relationship between the visible spectrum and human color vision. The CIE color spaces are mathematical models that comprise a "stan ...
, the three coordinates defining a color are given by ''X'', ''Y'', and ''Z'': : X_T = \int_\lambda X(\lambda)M(\lambda,T)\,d\lambda : Y_T = \int_\lambda Y(\lambda)M(\lambda,T)\,d\lambda : Z_T = \int_\lambda Z(\lambda)M(\lambda,T)\,d\lambda where ''M''(''λ'',''T'') is the spectral radiant exitance of the light being viewed, and ''X''(''λ''), ''Y''(''λ'') and ''Z''(''λ'') are the
color matching function In 1931, the International Commission on Illumination (CIE) published the CIE 1931 color spaces which define the relationship between the visible spectrum and human color vision. The CIE color spaces are mathematical models that comprise a "sta ...
s of the CIE
standard colorimetric observer In 1931, the International Commission on Illumination (CIE) published the CIE 1931 color spaces which define the relationship between the visible spectrum and human color vision. The CIE color spaces are mathematical models that comprise a "stan ...
, shown in the diagram on the right, and ''λ'' is the wavelength. The Planckian locus is determined by substituting into the above equations the black body spectral radiant exitance, which is given by
Planck's law In physics, Planck's law (also Planck radiation law) describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature , when there is no net flow of matter or energy between the ...
: : M(\lambda,T) =\frac\frac where: : ''c1'' = 2\pi is the first radiation constant : ''c2'' = \frac is the second radiation constant and : ''M'' is the black body spectral radiant exitance (power per unit area per unit wavelength: watt per square meter per meter (W/m3)) : ''T'' is the
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
of the black body : ''h'' is the
Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
: ''c'' is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
: ''k'' is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
This will give the Planckian locus in CIE XYZ color space. If these coordinates are ''XT'', ''YT'', ''ZT'' where ''T'' is the temperature, then the CIE chromaticity coordinates will be : x_T = \frac : y_T = \frac Note that in the above formula for Planck's Law, you might as well use ''c''1L = 2''hc''2 (the first radiation constant ''for spectral radiance'') instead of ''c1'' (the “regular” first radiation constant), in which case the formula would give the
spectral radiance In radiometry, spectral radiance or specific intensity is the radiance of a surface per unit frequency or wavelength, depending on whether the Spectral radiometric quantity, spectrum is taken as a function of frequency or of wavelength. The Interna ...
''L''(''λ'',''T'') of the black body instead of the spectral radiant exitance ''M''(''λ'',''T''). However, this change only affects the ''absolute'' values of ''XT'', ''YT'' and ''ZT'', not the values ''relative to each other''. Since ''XT'', ''YT'' and ''ZT'' are usually normalized to ''YT'' = 1 (or ''YT'' = 100) and are normalized when ''xT'' and ''yT'' are calculated, the absolute values of ''XT'', ''YT'' and ''ZT'' do not matter. For practical reasons, ''c''1 might therefore simply be replaced by 1.


Approximation

The Planckian locus in ''xy'' space is depicted as a curve in the chromaticity diagram above. While it is possible to compute the CIE ''xy'' co-ordinates exactly given the above formulas, it is faster to use approximations. Since the
mired Contracted from the term micro reciprocal degree, the mired () is a unit of measurement used to express color temperature. Values in mireds are calculated by the formula: : M = \frac, where ''T'' is the colour temperature in units of kelvins an ...
scale changes more evenly along the locus than the temperature itself, it is common for such approximations to be functions of the reciprocal temperature. Kim et al. use a
cubic spline In numerical analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline (mathematics), spline where each piece is a third-degree polynomial specified in Hermite interpolation, Hermite form, that is, by its values and first derivat ...
: : x_\text=\begin -0.2661239 \frac - 0.2343589 \frac + 0.8776956 \frac + 0.179910 & 1667\,\text \leq T \leq 4000\,\text \\ -3.0258469 \frac + 2.1070379 \frac + 0.2226347 \frac + 0.240390 & 4000\,\text \leq T \leq 25000\,\text \end : y_\text=\begin -1.1063814 x_\text^3 - 1.34811020 x_\text^2 + 2.18555832 x_\text - 0.20219683 & 1667\,\text \leq T \leq 2222\,\text \\ -0.9549476 x_\text^3 - 1.37418593 x_\text^2 + 2.09137015 x_\text - 0.16748867 & 2222\,\text \leq T \leq 4000\,\text \\ +3.0817580 x_\text^3 - 5.87338670 x_\text^2 + 3.75112997 x_\text - 0.37001483 & 4000\,\text \leq T \leq 25000\,\text \end The Planckian locus can also be approximated in the
CIE 1960 color space The CIE 1960 color space ("CIE 1960 UCS", variously expanded ''Uniform Color Space'', ''Uniform Color Scale'', ''Uniform Chromaticity Scale'', ''Uniform Chromaticity Space'') is another name for the chromaticity space devised by David MacAdam. ...
, which is used to compute CCT and CRI, using the following expressions: : \bar(T)=\frac : \bar(T)=\frac This approximation is accurate to within \left, u-\bar \ < 8\times10^ and \left, v-\bar\<9\times10^ for 1000\,K. Alternatively, one can use the chromaticity (''x'', ''y'') coordinates estimated from above to derive the corresponding (''u'', ''v''), if a larger range of temperatures is required. The inverse calculation, from chromaticity co-ordinates (''x'', ''y'') on or near the Planckian locus to correlated color temperature, is discussed in '.


Correlated color temperature

The mathematical procedure for determining the
correlated color temperature Correlated color temperature (CCT, Tcp) refers to the "temperature of a Planckian radiator whose perceived color most closely resembles that of a given stimulus at the same brightness and under specified viewing conditions". The SI unit is the ...
involves finding the closest point to the light source's
white point A white point (often referred to as reference white or target white in technical documents) is a set of tristimulus values or chromaticity coordinates that serve to define the color "white" in image capture, encoding, or reproduction. Depending o ...
on the Planckian locus. Since the CIE's 1959 meeting in Brussels, the Planckian locus has been computed using the
CIE 1960 color space The CIE 1960 color space ("CIE 1960 UCS", variously expanded ''Uniform Color Space'', ''Uniform Color Scale'', ''Uniform Chromaticity Scale'', ''Uniform Chromaticity Space'') is another name for the chromaticity space devised by David MacAdam. ...
, also known as MacAdam's (u,v) diagram. Today, the CIE 1960 color space is deprecated for other purposes: Owing to the perceptual inaccuracy inherent to the concept, it suffices to calculate to within 2 K at lower CCTs and 10 K at higher CCTs to reach the threshold of imperceptibility.


International Temperature Scale

The Planckian locus is derived by the determining the chromaticity values of a Planckian radiator using the standard colorimetric observer. The relative
spectral power distribution In radiometry, photometry (optics), photometry, and color science, a spectral power distribution (SPD) measurement describes the Power (physics), power per unit area per unit wavelength of an illumination (lighting), illumination (radiant exitan ...
(SPD) of a Planckian radiator follows Planck's law, and depends on the second radiation constant, c_2=hc/k. As measuring techniques have improved, the
General Conference on Weights and Measures The General Conference on Weights and Measures (abbreviated CGPM from the ) is the supreme authority of the International Bureau of Weights and Measures (BIPM), the intergovernmental organization established in 1875 under the terms of the Metre C ...
has revised its estimate of this constant, with the
International Temperature Scale The International Temperature Scale of 1990 (ITS-90) is an equipment calibration standard specified by the International Committee of Weights and Measures (CIPM) for making measurements on the Kelvin and Celsius temperature scales. It is an appro ...
(and briefly, the ''International Practical Temperature Scale''). These successive revisions caused a shift in the Planckian locus and, as a result, the correlated color temperature scale. Before ceasing publication of
standard illuminant A standard illuminant is a theoretical source of visible light with a spectral power distribution that is published. Standard illuminants provide a basis for comparing images or colors recorded under different lighting. CIE illuminants The Inte ...
s, the CIE worked around this problem by explicitly specifying the form of the SPD, rather than making references to black bodies and a color temperature. Nevertheless, it is useful to be aware of previous revisions in order to be able to verify calculations made in older texts: * c_2 = (ITS-27). Note: Was in effect during the standardization of Illuminants A, B, C (1931), however the CIE used the value recommended by the U.S.
National Bureau of Standards The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sc ...
, 1.435 × 10−2 * c_2 = (IPTS-48). In effect for Illuminant series D (formalized in 1967). * c_2 = (ITS-68), (ITS-90). Often used in recent papers. * c_2 = (
CODATA The Committee on Data of the International Science Council (CODATA) was established in 1966 as the Committee on Data for Science and Technology, originally part of the International Council of Scientific Unions, now part of the International ...
2010) * c_2 = (
CODATA The Committee on Data of the International Science Council (CODATA) was established in 1966 as the Committee on Data for Science and Technology, originally part of the International Council of Scientific Unions, now part of the International ...
2014) * c_2 = (
CODATA The Committee on Data of the International Science Council (CODATA) was established in 1966 as the Committee on Data for Science and Technology, originally part of the International Council of Scientific Unions, now part of the International ...
2018). Current value, as of 2020. The
2019 revision of the SI In 2019, four of the seven SI base units specified in the International System of Quantities were redefined in terms of natural physical constants, rather than human artefacts such as the standard kilogram. Effective 20 May 2019, the 144th ...
fixed the Boltzmann constant to an exact value. Since the Planck constant and the speed of light were already fixed to exact values, that means that ''c''2 is now an exact value as well. Note that ... doesn't indicate a repeating fraction; it merely means that of this exact value only the first ten digits are shown.


See also

*
Ultraviolet catastrophe The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century and early 20th century classical physics that an ideal black body at thermal equilibrium would emit an unbounded quantity of en ...


References

{{reflist


External links


Numerical table of color temperature and the corresponding xy and sRGB coordinates for both the 1931 and 1964 CMFs
by Mitchell Charity. Color space