
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'':
:
:
:
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 ...
:
:
where:
: ''c
1'' =
is the
first radiation constant
: ''c
2'' =
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/m
3))
: ''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 ''X
T'', ''Y
T'', ''Z
T'' where ''T'' is the temperature, then the CIE chromaticity coordinates will be
:
:
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 ''c
1'' (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 ''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, ''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 ...
:
:
:

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:
:
:
This approximation is accurate to within
and
for