Planckian locus

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https://en.wikipedia.org/wiki/Planckian_locus 

 

Planckian locus in the CIE 1931 chromaticity diagram

In physics and color science, 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 as the blackbody temperature changes. It goes from deep red at low temperatures through orange, yellowish white, white, and finally bluish white at very high temperatures.

A color space is a three-dimensional space; 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, colorfulness, and luminance) which specify the color and brightness of a particular homogeneous visual stimulus. A chromaticity is a color projected into a two-dimensional space that ignores brightness. For example, the standard CIE XYZ color space 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.

The Planckian locus in the XYZ color space

CIE 1931 Standard Colorimetric Observer functions used to map blackbody spectra to XYZ coordinates

In the CIE XYZ color space, the three coordinates defining a color are given by X, Y, and Z:

${\displaystyle X_T={\int }_0^{\mathrm{\infty }}X(\lambda )M(\lambda ,T)d\lambda }$

${\displaystyle Y_T={\int }_0^{\mathrm{\infty }}Y(\lambda )M(\lambda ,T)d\lambda }$

${\displaystyle Z_T={\int }_0^{\mathrm{\infty }}Z(\lambda )M(\lambda ,T)d\lambda }$

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 functions of the CIE standard colorimetric observer, 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:

${\displaystyle M(\lambda ,T)=\frac{c_1}{{\lambda }^5}\frac{1}{\exp \left(\frac{c_2}{\lambda T}\right)-1}}$

where:

c₁ = 2πhc² is the first radiation constant

c₂ = hc/k 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 of the black body

h is Planck's constant

c is the speed of light

k is Boltzmann's constant

This will give the Planckian locus in CIE XYZ color space. If these coordinates are X_T, Y_T, Z_T where T is the temperature, then the CIE chromaticity coordinates will be

${\displaystyle x_T=\frac{X_T}{X_T+Y_T+Z_T}}$

${\displaystyle y_T=\frac{Y_T}{X_T+Y_T+Z_T}}$

Note that in the above formula for Planck’s Law, you might as well use c_1L = 2hc² (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 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 X_T, Y_T and Z_T, not the values relative to each other. Since X_T, Y_T and Z_T are usually normalized to Y_T = 1 (or Y_T = 100) and are normalized when x_T and y_T are calculated, the absolute values of X_T, Y_T and Z_T do not matter. For practical reasons, c₁ 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 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. uses a cubic spline:

${\displaystyle x_c=\{\begin{array}{ll}-0.2661239\frac{{10}^9}{T^3}-0.2343589\frac{{10}^6}{T^2}+0.8776956\frac{{10}^3}{T}+0.179910& 1667\text{K}\le T\le 4000\text{K}\\ -3.0258469\frac{{10}^9}{T^3}+2.1070379\frac{{10}^6}{T^2}+0.2226347\frac{{10}^3}{T}+0.240390& 4000\text{K}\le T\le 25000\text{K}\end{array}}$

${\displaystyle y_c=\{\begin{array}{ll}-1.1063814x_c^3-1.34811020x_c^2+2.18555832x_c-0.20219683& 1667\text{K}\le T\le 2222\text{K}\\ -0.9549476x_c^3-1.37418593x_c^2+2.09137015x_c-0.16748867& 2222\text{K}\le T\le 4000\text{K}\\ +3.0817580x_c^3-5.87338670x_c^2+3.75112997x_c-0.37001483& 4000\text{K}\le T\le 25000\text{K}\end{array}}$

Kim et al.'s approximation to the Planckian locus (shown in red). The notches demarcate the three splines (shown in blue).

 

Animation showing an approximation of the color of the Planckian Locus through the visible spectrum

The Planckian locus can also be approximated in the CIE 1960 color space, which is used to compute CCT and CRI, using the following expressions:

${\displaystyle \overline{u}(T)=\frac{0.860117757+1.54118254\times {10}^{-4}T+1.28641212\times {10}^{-7}{T}^2}{1+8.42420235\times {10}^{-4}T+7.08145163\times {10}^{-7}{T}^2}}$

${\displaystyle \overline{v}(T)=\frac{0.317398726+4.22806245\times {10}^{-5}T+4.20481691\times {10}^{-8}{T}^2}{1-2.89741816\times {10}^{-5}T+1.61456053\times {10}^{-7}{T}^2}}$

This approximation is accurate to within ${\displaystyle \left|u-\overline{u}\right|<8\times {10}^{-5}}$ and ${\displaystyle \left|v-\overline{v}\right|<9\times {10}^{-5}}$ for ${\displaystyle 1000K<T<15,000K}$. 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 Color temperature § Approximation.

Correlated color temperature

The correlated color temperature (T_cp) is the temperature of the Planckian radiator whose perceived colour most closely resembles that of a given stimulus at the same brightness and under specified viewing conditions

— CIE/IEC 17.4:1987, International Lighting Vocabulary (ISBN 3900734070)

The mathematical procedure for determining the correlated color temperature involves finding the closest point to the light source's white point on the Planckian locus. Since the CIE's 1959 meeting in Brussels, the Planckian locus has been computed using the CIE 1960 color space, also known as MacAdam's (u,v) diagram. Today, the CIE 1960 color space is deprecated for other purposes:

The 1960 UCS diagram and 1964 Uniform Space are declared obsolete recommendation in CIE 15.2 (1986), but have been retained for the time being for calculating colour rendering indices and correlated colour temperature.

— CIE 13.3 (1995), Method of Measuring and Specifying Colour Rendering Properties of Light Sources

Owing to the perceptual inaccuracy inherent to the concept, it suffices to calculate to within 2K at lower CCTs and 10K at higher CCTs to reach the threshold of imperceptibility.

Close up of the CIE 1960 UCS. The isotherms are perpendicular to the Planckian locus, and are drawn to indicate the maximum distance from the locus that the CIE considers the correlated color temperature to be meaningful: ${\displaystyle {\mathrm{\Delta }}_{uv}=\pm 0.05}$

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 (SPD) of a Planckian radiator follows Planck's law, and depends on the second radiation constant, ${\displaystyle c_2=hc/k}$. As measuring techniques have improved, the General Conference on Weights and Measures has revised its estimate of this constant, with the International Temperature Scale (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 illuminants, 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:

• ${\displaystyle c_2=1.432\times {10}^{-2}\text{m·K}}$ (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, 1.435 × 10⁻²
• ${\displaystyle c_2=1.4380\times {10}^{-2}\text{m·K}}$ (IPTS-48). In effect for Illuminant series D (formalized in 1967).
• ${\displaystyle c_2=1.4388\times {10}^{-2}\text{m·K}}$ (ITS-68), (ITS-90). Often used in recent papers.
• ${\displaystyle c_2=1.4387770(13)\times {10}^{-2}\text{m·K}}$ (CODATA, 2010)
• ${\displaystyle c_2=1.43877736(83)\times {10}^{-2}\text{m·K}}$ (CODATA, 2014)
• ${\displaystyle c_2=\mathrm{1.438776877...}\times {10}^{-2}\text{m·K}}$ (CODATA, 2018). Current value, as of 2020. The 2019 redefinition of the SI base units 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 c₂ 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.