Blackbody and the colour of temperature

Almost everything in colour science is derived from measurements of human observers. Blackbody radiation is the exception: it comes from thermodynamics, it depends on one parameter, and the same formula describes a candle flame, a tungsten filament and a star.

The law #

The spectral radiant exitance of a blackbody at temperature $T$ is

$$M(\lambda, T) = \frac{2\pi h c^2}{\lambda^5}\cdot\frac{1}{e^{hc/\lambda k T} - 1}$$

with $h$ Planck’s constant, $c$ the speed of light and $k$ Boltzmann’s constant. All three are now exact by definition in SI, so the only input with any uncertainty is the temperature.

Every curve on this page is computed from that expression. None is tabulated, which matters for a reason beyond tidiness: it means the Planckian locus, the colour of a tungsten lamp and the whole apparatus of colour temperature follow from one physical formula rather than from a table someone else prepared.

Wien’s displacement law, used as a check #

Differentiating Planck’s law and setting the result to zero gives the wavelength of the peak:

$$\lambda_{\max} T = 2.898 \times 10^{-3}\ \mathrm{m\,K}$$

This is Wien’s displacement law and it is the reason hotter things look bluer. It is also the natural independent check on an implementation of Planck’s law: the peak of the computed curve must land where Wien says, and a transcription error in the exponent moves it immediately.

That check has a precondition which is easy to miss, and missing it produced the one interesting failure in building this page. The visible band runs 380 to 780 nm, so Wien’s peak falls inside it only for temperatures between about 3715 K and 7626 K. Outside that window the computed curve has no peak in the sampled range at all — its maximum is simply whichever end of the graph is brighter.

An early version asserted Wien’s law at every temperature. At 2000 K it reported a peak at 780 nm against Wien’s 1449 nm and failed, correctly but uselessly: the assertion was comparing the edge of the window to something outside the window. The fix was to make the assertion know its own precondition. Inside the band it checks the peak position; outside it checks that the curve is monotonic in the right direction, which is the correct statement about the visible range for a radiator whose peak is elsewhere.

The Planckian locus #

Integrating each blackbody spectrum against the matching functions gives a chromaticity, and sweeping the temperature traces a curve through the diagram — the Planckian locus. It runs from deep orange at low temperatures, through white in the region of daylight, to a pale blue at high ones.

Two things about that curve are worth noticing.

It sits well inside the display gamut over its whole useful range, which is why colour temperature is one of the few things about colour that reproduces easily. And it is a curve, not a region — so the vast majority of chromaticities are not on it, and correspond to no temperature at all.

Correlated colour temperature, and its limits #

Lamps are sold with a colour temperature on the box. Most of them are not blackbodies.

A fluorescent tube emits a few narrow mercury lines plus phosphor emission; an LED emits a blue peak plus a broad phosphor hump. Neither spectrum resembles a thermal curve in the slightest, and neither chromaticity lies on the Planckian locus. What the number on the box means is the correlated colour temperature: the temperature of the blackbody whose chromaticity is nearest, measured perpendicular to the locus in a chromaticity space chosen to make “nearest” perceptually sensible.

That is a projection, and projections discard information. Two lamps with the same correlated colour temperature can sit on opposite sides of the locus, one greenish and one pinkish, and look plainly different while carrying the same number. The industry has a second quantity for this — the distance from the locus, called Duv or tint — which is much less often quoted and is frequently what makes one 4000 K lamp look wrong beside another.

More seriously, correlated colour temperature says nothing about how a lamp will render surfaces. Two lamps with identical chromaticity can have very different spectra, and surfaces that match under one will not match under the other. This is what colour rendering indices attempt to quantify, and it is a separate axis from colour temperature entirely.

The naming problem #

Colour temperature runs backwards relative to ordinary usage. A 2700 K lamp is called warm and a 6500 K lamp is called cool, while the physics says the second is nearly two and a half times hotter.

The words come from the appearance rather than the thermodynamics, and the appearance in turn comes from association — firelight and candles at the low end, overcast sky at the high end. It is worth stating clearly once, because the reversal catches people who know the physics and expect the naming to follow it.

Why tungsten wastes most of its energy #

The hero figure carries a practical fact that the normalisation partly hides. Each curve is scaled to its own peak, so all five look comparably bright. In absolute terms they are not: total radiated power goes as $T^4$, and a 2000 K source emits vastly less than a 6500 K one.

More to the point, a tungsten filament at 2856 K peaks at about 1015 nm, well into the infrared. The visible range catches only the rising tail of the curve. This is why incandescent lamps convert only a few per cent of their input into visible light and the rest into heat — not a manufacturing defect but a direct consequence of Planck’s law at a temperature the filament can survive. Running it hotter would improve the efficiency and shorten the life sharply, and the trade-off between the two set the design of every incandescent bulb ever made.

Daylight is not a blackbody #

The sun is very nearly one, at about 5800 K at its surface. What arrives at the ground is not.

Sunlight passes through an atmosphere that absorbs at specific wavelengths, and the sun’s own outer layers imprint Fraunhofer absorption lines before the light leaves. Scattering removes short wavelengths preferentially, which is why the sky is blue and why light from the sky alone is much bluer than light from the sun’s disc. Daylight at the ground is a mixture of the two in a ratio that changes with the weather and the time of day.

The CIE handles this by not pretending. The D-series illuminants are reconstructed from three tabulated basis functions — a mean daylight spectrum plus two components capturing how real daylight varies — combined in proportions set by the desired correlated colour temperature. D65 is therefore a measurement-derived object, and the difference between it and a 6504 K blackbody is precisely the structure a thermal source cannot have.

This is a good example of the general shape of the subject. Where physics supplies an answer, use it: illuminant A is a formula. Where it does not, measure and tabulate, and say which was done.

Why the naming runs backwards #

Colour temperature is reported in a way that inverts ordinary usage. A 2700 K lamp is sold as warm and a 6500 K lamp as cool, while the thermodynamics says the second is more than twice as hot.

The words follow the appearance and the appearance follows association — firelight, candles and low sun at one end, overcast sky and shade at the other. Worth stating once plainly, because it reliably catches people who know the physics and expect the vocabulary to agree with it.

There is a second reversal hiding underneath. A lamp at 2700 K looks warm when compared side by side with a 6500 K one, but a room lit entirely by either looks approximately white to somebody sitting in it, because the visual system adapts to whatever it is given. The warm/cool distinction is a comparison, not an appearance, and it largely evaporates when there is nothing to compare against.

What the number on the box leaves out #

Two lamps sharing a correlated colour temperature can differ in ways the number does not express, and both differences are common enough to matter.

The first is tint: the distance from the Planckian locus, perpendicular to it. Two 4000 K lamps can sit on opposite sides, one faintly green and one faintly pink, and look plainly different while carrying identical labels. The quantity exists and is called Duv, and it is very rarely printed.

The second is spectral shape, which is invisible in any chromaticity-based number at all. Two lamps with identical white points can render surfaces quite differently, because the surfaces multiply the spectrum before the eye integrates it. Colour rendering indices attempt to capture this by measuring how a set of standard samples shift between the test lamp and a reference of the same correlated temperature — which is a real attempt at a hard problem, and is widely criticised for the choice of samples and for reducing a many-dimensional failure to one number.

Why an incandescent lamp is mostly a heater #

The hero figure normalises each curve to its own peak, which makes them comparable in shape and hides something important about magnitude.

Total radiated power goes as the fourth power of temperature, so a 6500 K source radiates roughly a hundred times more than a 2000 K one. More to the point for lighting: a tungsten filament runs at about 2856 K, and Wien’s law puts its peak at roughly 1015 nm — well into the infrared, past the right-hand edge of every graph on this page.

The visible band therefore catches only the rising tail of the curve. A few per cent of the energy emerges as light and the rest as heat, and this is not a manufacturing failure but a direct consequence of Planck’s law at a temperature the filament can survive. Running it hotter shifts the peak toward the visible and improves the efficiency; it also evaporates the tungsten far faster, and the trade-off between the two set the design of every incandescent bulb ever manufactured.

This also explains a piece of everyday experience. Dimming an incandescent lamp lowers the filament temperature, which moves the peak further into the infrared and makes the light both dimmer and redder. Dimming an LED does not, because an LED is not a thermal source — its spectrum is fixed by a band gap and a phosphor. Lamps that mimic the reddening on dimming do so deliberately, in software, because people expect it.

What a star’s colour reports #

The same law applied at much higher temperatures is how stellar surface temperatures are measured, and it is worth noticing that this is the one place in the subject where colour is used as an instrument rather than studied as a phenomenon.

A star’s spectrum is approximately Planckian, so its colour fixes its surface temperature — blue-white stars run above 10,000 K, the sun sits near 5800 K, and the red giants are down near 3000 K. Astronomers measure this as a colour index: the difference between the brightness measured through two standard filters, which is a two-channel version of the same collapse that three cone classes perform.

The colours are also subject to the same limitation as everything else here. A photograph of a blue-white star is showing a chromaticity inside the display gamut, and the star’s actual chromaticity, being close to the Planckian locus, genuinely is reachable — which makes stars one of the few astronomical subjects whose colours can be reproduced honestly.

What was computed here #

Planck’s law is evaluated directly from the exact SI constants at each of the eighty-one sampled wavelengths. Illuminant A is that computation at 2856 K, normalised to 100 at 560 nm in the CIE convention, and its chromaticity comes out at $(0.4475, 0.4074)$ against the published $(0.4476, 0.4074)$.

The Wien check runs on every temperature drawn. Inside the visible band it compares the computed peak against $2.898\times10^{-3}/T$ with a twelve-nanometre tolerance — at 5000 K the curve peaks at 580 nm against Wien’s 579.6. Outside the band it verifies monotonicity in the direction the peak position implies, so a 2000 K curve must rise across the whole visible range and a 10000 K curve must fall.

Daylight is deliberately not computed this way, because daylight is not a blackbody. D65 is reconstructed from the CIE basis functions, and the difference between that reconstruction and a 6504 K blackbody is exactly the atmospheric structure that makes daylight what it is.

The one law in the subject that is not about people #

Worth saying plainly, because it is unusual here. Almost everything in colour science descends from measurements of human observers — the matching functions, the discrimination ellipses, the difference formulae. All of it is conventional in the sense that a different sample of people would have produced slightly different numbers.

Planck’s law is not. It follows from thermodynamics and quantum statistics, its three constants are exact by definition, and it would be the same law for any observer with any set of receptors. The Planckian locus is the one curve on the chromaticity diagram that comes from physics rather than from psychophysics — the diagram’s axes are conventional, and the curve drawn on them is not.

That makes colour temperature unusually well founded as colour quantities go, and it is worth keeping separate from the correlated colour temperature printed on lamp boxes, which is a projection onto that curve from somewhere off it.

What the pictures cannot show #

The normalisation is a real distortion and the caption states it. Five curves at their true relative magnitudes would show the 2000 K one as a nearly invisible line along the bottom, which is honest about the energy and useless about the shape.

The swatches beside each temperature are also less meaningful than they look. They show the chromaticity of each radiator at a common luminance, which is exactly what an adapted observer does not see — anyone in a room lit by a 2700 K lamp adapts to it and perceives it as roughly white. The figure shows the stimuli side by side, which is a comparison no observer makes in practice.

Who found it, and when #

Wien derived his displacement law in 1893 and a spectral distribution law that worked at short wavelengths and failed at long ones. Rayleigh and Jeans produced a law that worked at long wavelengths and diverged at short ones — the ultraviolet catastrophe.

Planck resolved it in 1900 by assuming energy was exchanged in discrete quanta, a step he described as an act of desperation and regarded for years as a mathematical device rather than a physical claim. It was the first appearance of the quantum, and it arrived from a curve-fitting problem about the colour of hot objects.

Where this goes next #

The reason a lamp’s spectrum matters beyond its chromaticity is the illuminant is half the answer. The reason a 2700 K room still looks white to somebody sitting in it is constancy is the default. And the general framing is a spectrum is not a colour.