Planckian locus

Planckian locus in the XYZ color space

In the CIE XYZ color space, 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 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: : M(\lambda,T) =\frac{c_{1}}{\lambda^5}\frac{1}{\exp\left(\frac{c_2}{{\lambda}T}\right)-1} where: : c1 = 2\pi{hc^2} is the first radiation constant : c2 = \frac{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 the Planck constant : c is the speed of light : k is the Boltzmann constant

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{X_T}{X_T+Y_T+Z_T} : 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 c1L = 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 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, 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. use a cubic spline: : x_\text{c}=\begin{cases} -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} \leq T \leq 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} \leq T \leq 25000,\text{K} \end{cases} : y_\text{c}=\begin{cases} -1.1063814 x_\text{c}^3 - 1.34811020 x_\text{c}^2 + 2.18555832 x_\text{c} - 0.20219683 & 1667,\text{K} \leq T \leq 2222,\text{K} \ -0.9549476 x_\text{c}^3 - 1.37418593 x_\text{c}^2 + 2.09137015 x_\text{c} - 0.16748867 & 2222,\text{K} \leq T \leq 4000,\text{K} \ +3.0817580 x_\text{c}^3 - 5.87338670 x_\text{c}^2 + 3.75112997 x_\text{c} - 0.37001483 & 4000,\text{K} \leq T \leq 25000,\text{K} \end{cases}

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: : \bar{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} : \bar{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 \left| u-\bar{u} \right| and \left|v-\bar{v}\right| 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

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 ()}}

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:

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 (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 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:

• 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, 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 2010)
• c_2 = (CODATA 2014)
• c_2 = (CODATA 2018). Current value, as of 2020. The 2019 revision of the SI 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.

References

References

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