Planckian locus

The 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 = ∫ 0 ∞ X ( λ ) M ( λ , T ) d λ Y T = ∫ 0 ∞ Y ( λ ) M ( λ , T ) d λ Z T = ∫ 0 ∞ Z ( λ ) M ( λ , T ) d λ

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 ( λ , T ) = c 1 λ 5 1 exp ⁡ ( c 2 λ T ) − 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

x T = X T X T + Y T + Z T y T = 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:

x c = { − 0.2661239 10 9 T 3 − 0.2343580 10 6 T 2 + 0.8776956 10 3 T + 0.179910 1667 K ≤ T ≤ 4000 K − 3.0258469 10 9 T 3 + 2.1070379 10 6 T 2 + 0.2226347 10 3 T + 0.240390 4000 K ≤ T ≤ 25000 K

y c = { − 1.1063814 x c 3 − 1.34811020 x c 2 + 2.18555832 x c − 0.20219683 1667 K ≤ T ≤ 2222 K − 0.9549476 x c 3 − 1.37418593 x c 2 + 2.09137015 x c − 0.16748867 2222 K ≤ T ≤ 4000 K + 3.0817580 x c 3 − 5.87338670 x c 2 + 3.75112997 x c − 0.37001483 4000 K ≤ T ≤ 25000 K

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.

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

u ¯ ( T ) = 0.860117757 + 1.54118254 × 10 − 4 T + 1.28641212 × 10 − 7 T 2 1 + 8.42420235 × 10 − 4 T + 7.08145163 × 10 − 7 T 2

v ¯ ( T ) = 0.317398726 + 4.22806245 × 10 − 5 T + 4.20481691 × 10 − 8 T 2 1 − 2.89741816 × 10 − 5 T + 1.61456053 × 10 − 7 T 2

This approximation is accurate to within | u − u ¯ | < 8 × 10 − 5 and | v − v ¯ | < 9 × 10 − 5 for 1000 K < T < 15 , 000 K

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

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.

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.

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 = h c / 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 = 1.432 × 10 − 2 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

c 2 = 1.4380 × 10 − 2 m·K (IPTS-48). In effect for Illuminant series D (formalized in 1967).

c 2 = 1.4388 × 10 − 2 m·K (ITS-68), (ITS-90). Often used in recent papers.

c 2 = 1.4387770 ( 13 ) × 10 − 2 m·K (CODATA, 2006). Current value, as of 2010.