Interstellar medium

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

 

The distribution of ionized hydrogen (known by astronomers as H II from old spectroscopic terminology) in the parts of the Galactic interstellar medium visible from the Earth's northern hemisphere as observed with the Wisconsin Hα Mapper (Haffner et al. 2003).

In astronomy, the interstellar medium (ISM) is the matter and radiation that exist in the space between the star systems in a galaxy. This matter includes gas in ionic, atomic, and molecular form, as well as dust and cosmic rays. It fills interstellar space and blends smoothly into the surrounding intergalactic space. The energy that occupies the same volume, in the form of electromagnetic radiation, is the interstellar radiation field.

The interstellar medium is composed of multiple phases distinguished by whether matter is ionic, atomic, or molecular, and the temperature and density of the matter. The interstellar medium is composed, primarily, of hydrogen, followed by helium with trace amounts of carbon, oxygen, and nitrogen. The thermal pressures of these phases are in rough equilibrium with one another. Magnetic fields and turbulent motions also provide pressure in the ISM, and are typically more important, dynamically, than the thermal pressure is. In the interstellar medium, matter is primarily in molecular form, and reaches number densities of 10⁶ molecules per cm³ (1 million molecules per cm³). In hot, diffuse regions of the ISM, matter is primarily ionized, and the density may be as low as 10⁻⁴ ions per cm³. Compare this with a number density of roughly 10¹⁹ molecules per cm³ for air at sea level, and 10¹⁰ molecules per cm³ (10 billion molecules per cm³) for a laboratory high-vacuum chamber. By mass, 99% of the ISM is gas in any form, and 1% is dust. Of the gas in the ISM, by number 91% of atoms are hydrogen and 8.9% are helium, with 0.1% being atoms of elements heavier than hydrogen or helium, known as "metals" in astronomical parlance. By mass this amounts to 70% hydrogen, 28% helium, and 1.5% heavier elements. The hydrogen and helium are primarily a result of primordial nucleosynthesis, while the heavier elements in the ISM are mostly a result of enrichment (due to stellar gravity and radiation pressure) in the process of stellar evolution.

The ISM plays a crucial role in astrophysics precisely because of its intermediate role between stellar and galactic scales. Stars form within the densest regions of the ISM, which ultimately contributes to molecular clouds and replenishes the ISM with matter and energy through planetary nebulae, stellar winds, and supernovae. This interplay between stars and the ISM helps determine the rate at which a galaxy depletes its gaseous content, and therefore its lifespan of active star formation.

Voyager 1 reached the ISM on August 25, 2012, making it the first artificial object from Earth to do so. Interstellar plasma and dust will be studied until the estimated mission end date of 2025. Its twin Voyager 2 entered the ISM on November 5, 2018.

Voyager 1 is the first artificial object to reach the interstellar medium.

Interstellar matter

Table 1 shows a breakdown of the properties of the components of the ISM of the Milky Way.

Table 1: Components of the interstellar medium

Component	Fractional volume	Scale height (pc)	Temperature (K)	Density (particles/cm3)	State of hydrogen	Primary observational techniques
Molecular clouds	< 1%	80	10–20	102–106	molecular	Radio and infrared molecular emission and absorption lines
Cold neutral medium (CNM)	1–5%	100–300	50–100	20–50	neutral atomic	H I 21 cm line absorption
Warm neutral medium (WNM)	10–20%	300–400	6000–10000	0.2–0.5	neutral atomic	H I 21 cm line emission
Warm ionized medium (WIM)	20–50%	1000	8000	0.2–0.5	ionized	Hα emission and pulsar dispersion
H II regions	< 1%	70	8000	102–104	ionized	Hα emission and pulsar dispersion
Coronal gas Hot ionized medium (HIM)	30–70%	1000–3000	106–107	10−4–10−2	ionized (metals also highly ionized)	X-ray emission; absorption lines of highly ionized metals, primarily in the ultraviolet

The three-phase model

Field, Goldsmith & Habing (1969) put forward the static two phase equilibrium model to explain the observed properties of the ISM. Their modeled ISM included of a cold dense phase (T < 300 K), consisting of clouds of neutral and molecular hydrogen, and a warm intercloud phase (T ~ 10⁴ K), consisting of rarefied neutral and ionized gas. McKee & Ostriker (1977) added a dynamic third phase that represented the very hot (T ~ 10⁶ K) gas that had been shock heated by supernovae and constituted most of the volume of the ISM. These phases are the temperatures where heating and cooling can reach a stable equilibrium. Their paper formed the basis for further study over the subsequent three decades. However, the relative proportions of the phases and their subdivisions are still not well understood.

The atomic hydrogen model

This model takes into account only atomic hydrogen: A temperature higher than 3000 K breaks molecules, while that lower than 50000 K leaves atoms in their ground state. It is assumed that the influence of other atoms (He ...) is negligible. The pressure is assumed to be very low, so the durations of the free paths of atoms are longer than the ~ 1 nanosecond duration of the light pulses that constitute ordinary, temporally incoherent light.

In this collisionless gas, Einstein's theory of coherent light-matter interactions applies: all the gas-light interactions are spatially coherent. Suppose that a monochromatic light is pulsed, then scattered by molecules with a quadrupole (Raman) resonance frequency. If the “length of light pulses is shorter than all involved time constants” (Lamb (1971)), an “impulsive stimulated Raman scattering (ISRS)” (Yan, Gamble & Nelson (1985)) applies: the light generated by incoherent Raman scattering at a shifted frequency has a phase independent of the phase of the exciting light, thus generating a new spectral line, and coherence between the incident and scattered light facilitates their interference into a single frequency, thus shifting the incident frequency. Assume that a star radiates a continuous light spectrum up to X-rays. Lyman frequencies are absorbed in this light and pump atoms mainly to the first excited state. In this state, the hyperfine periods are longer than 1 ns, so an ISRS “may” redshift the light frequency, populating high hyperfine levels. Another ISRS “may” transfer energy from hyperfine levels to thermal electromagnetic waves, so the redshift is permanent. The temperature of a light beam is defined by its frequency and spectral radiance with Planck's formula. As entropy must increase, “may” becomes “does”. However, where a previously absorbed line (first Lyman beta, ...) reaches the Lyman alpha frequency, the redshifting process stops, and all hydrogen lines are strongly absorbed. But this stop is not perfect if there is energy at the frequency shifted to Lyman beta frequency, which produces a slow redshift. Successive redshifts separated by Lyman absorptions generate many absorption lines, frequencies of which, deduced from absorption process, obey a law more dependable than Karlsson's formula.

The previous process excites more and more atoms because a de-excitation obeys Einstein's law of coherent interactions: Variation dI of radiance I of a light beam along a path dx is dI=BIdx, where B is Einstein amplification coefficient which depends on medium. I is the modulus of Poynting vector of field, absorption occurs for an opposed vector, which corresponds to a change of sign of B. Factor I in this formula shows that intense rays are more amplified than weak ones (competition of modes). Emission of a flare requires a sufficient radiance I provided by random zero point field. After emission of a flare, weak B increases by pumping while I remains close to zero: De-excitation by a coherent emission involves stochastic parameters of zero point field, as observed close to quasars (and in polar auroras).

Structures

Three-dimensional structure in Pillars of Creation.

 

Map showing the Sun located near the edge of the Local Interstellar Cloud and Alpha Centauri about 4 light-years away in the neighboring G-Cloud complex

The ISM is turbulent and therefore full of structure on all spatial scales. Stars are born deep inside large complexes of molecular clouds, typically a few parsecs in size. During their lives and deaths, stars interact physically with the ISM.

Stellar winds from young clusters of stars (often with giant or supergiant HII regions surrounding them) and shock waves created by supernovae inject enormous amounts of energy into their surroundings, which leads to hypersonic turbulence. The resultant structures – of varying sizes – can be observed, such as stellar wind bubbles and superbubbles of hot gas, seen by X-ray satellite telescopes or turbulent flows observed in radio telescope maps.

The Sun is currently traveling through the Local Interstellar Cloud, a denser region in the low-density Local Bubble.

In October 2020, astronomers reported a significant unexpected increase in density in the space beyond the Solar System as detected by the Voyager 1 and Voyager 2 space probes. According to the researchers, this implies that "the density gradient is a large-scale feature of the VLISM (very local interstellar medium) in the general direction of the heliospheric nose".

Interaction with interplanetary medium

The interstellar medium begins where the interplanetary medium of the Solar System ends. The solar wind slows to subsonic velocities at the termination shock, 90–100 astronomical units from the Sun. In the region beyond the termination shock, called the heliosheath, interstellar matter interacts with the solar wind. Voyager 1, the farthest human-made object from the Earth (after 1998), crossed the termination shock December 16, 2004 and later entered interstellar space when it crossed the heliopause on August 25, 2012, providing the first direct probe of conditions in the ISM (Stone et al. 2005).

Interstellar extinction

The ISM is also responsible for extinction and reddening, the decreasing light intensity and shift in the dominant observable wavelengths of light from a star. These effects are caused by scattering and absorption of photons and allow the ISM to be observed with the naked eye in a dark sky. The apparent rifts that can be seen in the band of the Milky Way – a uniform disk of stars – are caused by absorption of background starlight by molecular clouds within a few thousand light years from Earth.

Far ultraviolet light is absorbed effectively by the neutral components of the ISM. For example, a typical absorption wavelength of atomic hydrogen lies at about 121.5 nanometers, the Lyman-alpha transition. Therefore, it is nearly impossible to see light emitted at that wavelength from a star farther than a few hundred light years from Earth, because most of it is absorbed during the trip to Earth by intervening neutral hydrogen.

Heating and cooling

The ISM is usually far from thermodynamic equilibrium. Collisions establish a Maxwell–Boltzmann distribution of velocities, and the 'temperature' normally used to describe interstellar gas is the 'kinetic temperature', which describes the temperature at which the particles would have the observed Maxwell–Boltzmann velocity distribution in thermodynamic equilibrium. However, the interstellar radiation field is typically much weaker than a medium in thermodynamic equilibrium; it is most often roughly that of an A star (surface temperature of ~10,000 K) highly diluted. Therefore, bound levels within an atom or molecule in the ISM are rarely populated according to the Boltzmann formula (Spitzer 1978, § 2.4).

Depending on the temperature, density, and ionization state of a portion of the ISM, different heating and cooling mechanisms determine the temperature of the gas.

Heating mechanisms

Heating by low-energy cosmic rays

The first mechanism proposed for heating the ISM was heating by low-energy cosmic rays. Cosmic rays are an efficient heating source able to penetrate in the depths of molecular clouds. Cosmic rays transfer energy to gas through both ionization and excitation and to free electrons through Coulomb interactions. Low-energy cosmic rays (a few MeV) are more important because they are far more numerous than high-energy cosmic rays.

Photoelectric heating by grains

The ultraviolet radiation emitted by hot stars can remove electrons from dust grains. The photon is absorbed by the dust grain, and some of its energy is used to overcome the potential energy barrier and remove the electron from the grain. This potential barrier is due to the binding energy of the electron (the work function) and the charge of the grain. The remainder of the photon's energy gives the ejected electron kinetic energy which heats the gas through collisions with other particles. A typical size distribution of dust grains is n(r) ∝ r^−3.5, where r is the radius of the dust particle. Assuming this, the projected grain surface area distribution is πr²n(r) ∝ r^−1.5. This indicates that the smallest dust grains dominate this method of heating.

Photoionization

When an electron is freed from an atom (typically from absorption of a UV photon) it carries kinetic energy away of the order E_photon − E_ionization. This heating mechanism dominates in H II regions, but is negligible in the diffuse ISM due to the relative lack of neutral carbon atoms.

X-ray heating

X-rays remove electrons from atoms and ions, and those photoelectrons can provoke secondary ionizations. As the intensity is often low, this heating is only efficient in warm, less dense atomic medium (as the column density is small). For example, in molecular clouds only hard x-rays can penetrate and x-ray heating can be ignored. This is assuming the region is not near an x-ray source such as a supernova remnant.

Chemical heating

Molecular hydrogen (H₂) can be formed on the surface of dust grains when two H atoms (which can travel over the grain) meet. This process yields 4.48 eV of energy distributed over the rotational and vibrational modes, kinetic energy of the H₂ molecule, as well as heating the dust grain. This kinetic energy, as well as the energy transferred from de-excitation of the hydrogen molecule through collisions, heats the gas.

Grain-gas heating

Collisions at high densities between gas atoms and molecules with dust grains can transfer thermal energy. This is not important in HII regions because UV radiation is more important. It is also less important in diffuse ionized medium due to the low density. In the neutral diffuse medium grains are always colder, but do not effectively cool the gas due to the low densities.

Grain heating by thermal exchange is very important in supernova remnants where densities and temperatures are very high.

Gas heating via grain-gas collisions is dominant deep in giant molecular clouds (especially at high densities). Far infrared radiation penetrates deeply due to the low optical depth. Dust grains are heated via this radiation and can transfer thermal energy during collisions with the gas. A measure of efficiency in the heating is given by the accommodation coefficient:

$${\displaystyle \alpha =\frac{T_2-T}{T_d-T}}$$

where T is the gas temperature, T_d the dust temperature, and T₂ the post-collision temperature of the gas atom or molecule. This coefficient was measured by (Burke & Hollenbach 1983) as α = 0.35.

Other heating mechanisms

A variety of macroscopic heating mechanisms are present including:

• Gravitational collapse of a cloud
• Supernova explosions
• Stellar winds
• Expansion of H II regions
• Magnetohydrodynamic waves created by supernova remnants

Cooling mechanisms

Fine structure cooling

The process of fine structure cooling is dominant in most regions of the Interstellar Medium, except regions of hot gas and regions deep in molecular clouds. It occurs most efficiently with abundant atoms having fine structure levels close to the fundamental level such as: C II and O I in the neutral medium and O II, O III, N II, N III, Ne II and Ne III in H II regions. Collisions will excite these atoms to higher levels, and they will eventually de-excite through photon emission, which will carry the energy out of the region.

Cooling by permitted lines

At lower temperatures, more levels than fine structure levels can be populated via collisions. For example, collisional excitation of the n = 2 level of hydrogen will release a Ly-α photon upon de-excitation. In molecular clouds, excitation of rotational lines of CO is important. Once a molecule is excited, it eventually returns to a lower energy state, emitting a photon which can leave the region, cooling the cloud.

Radiowave propagation

Atmospheric attenuation in dB/km as a function of frequency over the EHF band. Peaks in absorption at specific frequencies are a problem, due to atmosphere constituents such as water vapor (H₂O) and carbon dioxide (CO₂).

Radio waves from ≈10 kHz (very low frequency) to ≈300 GHz (extremely high frequency) propagate differently in interstellar space than on the Earth's surface. There are many sources of interference and signal distortion that do not exist on Earth. A great deal of radio astronomy depends on compensating for the different propagation effects to uncover the desired signal.

Discoveries

The Potsdam Great Refractor, a double telescope with 80cm (31.5") and 50 cm (19.5") lenses inaugurated in 1899, used to discover interstellar calcium in 1904.

In 1864, William Huggins used spectroscopy to determine that a nebula is made of gas. Huggins had a private observatory with an 8-inch telescope, with a lens by Alvin Clark; but it was equipped for spectroscopy which enabled breakthrough observations.

In 1904, one of the discoveries made using the Potsdam Great Refractor telescope was of calcium in the interstellar medium. The astronomer Johannes Frank Hartmann determined from spectrograph observations of the binary star Mintaka in Orion, that there was the element calcium in the intervening space.

Interstellar gas was further confirmed by Slipher in 1909, and then by 1912 interstellar dust was confirmed by Slipher. In this way the overall nature of the interstellar medium was confirmed in a series of discoveries and postulizations of its nature.

In September 2020, evidence was presented of solid-state water in the interstellar medium, and particularly, of water ice mixed with silicate grains in cosmic dust grains.

History of knowledge of interstellar space

Herbig–Haro object HH 110 ejects gas through interstellar space.

The nature of the interstellar medium has received the attention of astronomers and scientists over the centuries and understanding of the ISM has developed. However, they first had to acknowledge the basic concept of "interstellar" space. The term appears to have been first used in print by Bacon (1626, § 354–455): "The Interstellar Skie.. hath .. so much Affinity with the Starre, that there is a Rotation of that, as well as of the Starre." Later, natural philosopher Robert Boyle (1674) discussed "The inter-stellar part of heaven, which several of the modern Epicureans would have to be empty."

Before modern electromagnetic theory, early physicists postulated that an invisible luminiferous aether existed as a medium to carry lightwaves. It was assumed that this aether extended into interstellar space, as Patterson (1862) wrote, "this efflux occasions a thrill, or vibratory motion, in the ether which fills the interstellar spaces."

The advent of deep photographic imaging allowed Edward Barnard to produce the first images of dark nebulae silhouetted against the background star field of the galaxy, while the first actual detection of cold diffuse matter in interstellar space was made by Johannes Hartmann in 1904 through the use of absorption line spectroscopy. In his historic study of the spectrum and orbit of Delta Orionis, Hartmann observed the light coming from this star and realized that some of this light was being absorbed before it reached the Earth. Hartmann reported that absorption from the "K" line of calcium appeared "extraordinarily weak, but almost perfectly sharp" and also reported the "quite surprising result that the calcium line at 393.4 nanometres does not share in the periodic displacements of the lines caused by the orbital motion of the spectroscopic binary star". The stationary nature of the line led Hartmann to conclude that the gas responsible for the absorption was not present in the atmosphere of Delta Orionis, but was instead located within an isolated cloud of matter residing somewhere along the line-of-sight to this star. This discovery launched the study of the Interstellar Medium.

In the series of investigations, Viktor Ambartsumian introduced the now commonly accepted notion that interstellar matter occurs in the form of clouds.

Following Hartmann's identification of interstellar calcium absorption, interstellar sodium was detected by Heger (1919) through the observation of stationary absorption from the atom's "D" lines at 589.0 and 589.6 nanometres towards Delta Orionis and Beta Scorpii.

Subsequent observations of the "H" and "K" lines of calcium by Beals (1936) revealed double and asymmetric profiles in the spectra of Epsilon and Zeta Orionis. These were the first steps in the study of the very complex interstellar sightline towards Orion. Asymmetric absorption line profiles are the result of the superposition of multiple absorption lines, each corresponding to the same atomic transition (for example the "K" line of calcium), but occurring in interstellar clouds with different radial velocities. Because each cloud has a different velocity (either towards or away from the observer/Earth) the absorption lines occurring within each cloud are either blue-shifted or red-shifted (respectively) from the lines' rest wavelength, through the Doppler Effect. These observations confirming that matter is not distributed homogeneously were the first evidence of multiple discrete clouds within the ISM.

This light-year-long knot of interstellar gas and dust resembles a caterpillar.

The growing evidence for interstellar material led Pickering (1912) to comment that "While the interstellar absorbing medium may be simply the ether, yet the character of its selective absorption, as indicated by Kapteyn, is characteristic of a gas, and free gaseous molecules are certainly there, since they are probably constantly being expelled by the Sun and stars."

The same year Victor Hess's discovery of cosmic rays, highly energetic charged particles that rain onto the Earth from space, led others to speculate whether they also pervaded interstellar space. The following year the Norwegian explorer and physicist Kristian Birkeland wrote: "It seems to be a natural consequence of our points of view to assume that the whole of space is filled with electrons and flying electric ions of all kinds. We have assumed that each stellar system in evolutions throws off electric corpuscles into space. It does not seem unreasonable therefore to think that the greater part of the material masses in the universe is found, not in the solar systems or nebulae, but in 'empty' space" (Birkeland 1913).

Thorndike (1930) noted that "it could scarcely have been believed that the enormous gaps between the stars are completely void. Terrestrial aurorae are not improbably excited by charged particles emitted by the Sun. If the millions of other stars are also ejecting ions, as is undoubtedly true, no absolute vacuum can exist within the galaxy."

In September 2012, NASA scientists reported that polycyclic aromatic hydrocarbons (PAHs), subjected to interstellar medium (ISM) conditions, are transformed, through hydrogenation, oxygenation and hydroxylation, to more complex organics – "a step along the path toward amino acids and nucleotides, the raw materials of proteins and DNA, respectively". Further, as a result of these transformations, the PAHs lose their spectroscopic signature which could be one of the reasons "for the lack of PAH detection in interstellar ice grains, particularly the outer regions of cold, dense clouds or the upper molecular layers of protoplanetary disks."

In February 2014, NASA announced a greatly upgraded database for tracking polycyclic aromatic hydrocarbons (PAHs) in the universe. According to scientists, more than 20% of the carbon in the universe may be associated with PAHs, possible starting materials for the formation of life. PAHs seem to have been formed shortly after the Big Bang, are widespread throughout the universe, and are associated with new stars and exoplanets.

In April 2019, scientists, working with the Hubble Space Telescope, reported the confirmed detection of the large and complex ionized molecules of buckminsterfullerene (C₆₀) (also known as "buckyballs") in the interstellar medium spaces between the stars.

Planckian locus

From Wikipedia, the free encyclopedia

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.