Planckian locus

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 perceived attributes 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.

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:

${\displaystyle X_T={\int }_{\lambda }X(\lambda )M(\lambda ,T)d\lambda }$

${\displaystyle Y_T={\int }_{\lambda }Y(\lambda )M(\lambda ,T)d\lambda }$

${\displaystyle Z_T={\int }_{\lambda }Z(\lambda )M(\lambda ,T)d\lambda }$

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:

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

where:

c₁ = ${\displaystyle 2\pi h{c}^2}$ is the first radiation constant

c₂ = ${\displaystyle \frac{hc}{k}}$ 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 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. use a cubic spline:

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

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

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 ${\displaystyle 1000K<T<15000K}$. 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 § Approximation.

The color of an infinitely hot blackbody.
#94b1ff

Infinite temperature

The Planckian locus terminates at CIE xy coordinates (0.2399, 0.2340), or an sRGB value of (148, 177, 255) or #94b1ff, a light blue color known as perano. This is because of the Rayleigh-Jeans law, which states that at frequencies much lower than the peak frequency of a black-body radiator, spectral power is inversely proportional to the fourth power of the wavelength. As a black-body's temperature approaches infinity, its color converges to the aforementioned shade of blue.