The glow that says nothing about the surface

Point a thermal camera at a beaker of water just off the boil, in a room at twenty degrees, and it reads a hundred. Point it at a block of polished aluminium sitting in the same water bath and it reads about twenty-five.

Nothing has gone wrong. Both surfaces are at the same temperature, and the camera is doing exactly what it was built to do: measuring how much infrared arrives from each and reporting the temperature a black body would need to send that much.

The law, and where it comes from #

The reason the two ends of that curve meet the way they do is a statement about matter that contains no matter.

Put two bodies inside a sealed cavity, insulated from everything, and wait. Everything reaches the same temperature and stays there. Now consider one small patch of one body, and one narrow band of wavelengths, and one direction. In equilibrium the energy that patch radiates into that band and that direction must equal the energy it absorbs from the same band and direction, because otherwise it would warm or cool relative to its surroundings — and two bodies at different temperatures in an isolated enclosure is a heat flow with no temperature difference behind it, which is the thing the second law forbids.

$$\varepsilon(\lambda, \theta) = \alpha(\lambda, \theta).$$

No arrangement can move heat from a cold body to a hot one for nothing, and that prohibition is what the whole law rests on. A body that emitted more than it absorbed at some wavelength, placed in a cavity with another that did the opposite, would drive heat one way round with no work supplied. So emissivity must equal absorptivity at every wavelength and every angle — not as an approximation, but on pain of a perpetual motion machine.

The argument is worth admiring for what it leaves out. It says nothing about what the body is made of, whether the emission is from electrons or from vibrating molecules, or what mechanism the absorption uses. It applies to a lump of soot, a polished mirror, a sheet of glass and a cloud of gas equally, and it was published in 1859 — before anybody knew what light was made of and forty years before Planck.

Why a hole is black #

The law explains why the ideal case is available in a laboratory rather than only on paper.

What repeated chances do to a survival probability is why a hole is black. A ray entering a small opening strikes the wall, loses a fraction, strikes again, and the survival falls exponentially in the number of bounces — so even a wall reflecting 90 per cent each time returns almost nothing after twenty. The hole is black because the geometry gives the light no way out, not because the wall is.

So a hole in a box absorbs everything that enters it, whatever the box is made of, and by Kirchhoff’s law it must therefore emit as a black body. That is what every black-body source ever built is: a heated cavity with a hole in it. The material of the cavity is chosen for its melting point and not for its optical properties, which is the strongest possible statement of the law’s independence from matter.

The spectral form is the one that matters #

The version of the law usually quoted — a good absorber is a good emitter — is the wavelength-integrated shadow of the real statement, and the difference is what a whole industry rests on.

Kirchhoff’s law binds $\varepsilon$ to $\alpha$ at each wavelength separately. A surface may therefore be black at half a micrometre, where the Sun’s light arrives, and shiny at ten, where its own thermal glow leaves. It breaks no law, because those are different wavelengths, and nothing in the argument connects them.

That last case is the one worth pausing on. A surface with low solar absorptance and high thermal emittance radiates more than it takes in and cools below ambient in full sunlight, with no power supplied. It is not a violation of anything: the surface is exchanging radiation with the sky, which through the atmosphere’s transparent window at eight to thirteen micrometres is effectively at a few tens of kelvin, and the sky is where the heat is going.

A stack of quarter-wave layers reflects a band and passes the rest, and the band can be placed anywhere. That is how a selective surface is built: reflect where the sun is, emit where the atmosphere is transparent, and the same object that stays cool in sunlight would be an ordinary absorber if the two bands were swapped. Nothing about the material decides it — the layer thicknesses do.

The same argument in three other subjects #

The shape of Kirchhoff’s reasoning — put the thing in a box, wait, and demand that nothing happen — recurs wherever emission and absorption are two names for one coupling, and each instance was discovered separately.

An antenna transmits with the pattern it receives with. An aerial’s gain in a given direction is the same whether it is sending or listening, which is why a radio telescope’s beam can be measured by transmitting through it. The proof is the same: put two antennas in a cavity, and any asymmetry runs an engine.

A good absorber of sound is a good radiator of it. A porous panel that swallows a frequency in a room will, if heated, radiate acoustic energy at that frequency into a cold room. The effect is far too small to matter and the reciprocity behind it is not: it is what makes an anechoic chamber’s wedges work in both directions and what makes a loudspeaker usable as a microphone.

And an atom’s emission and absorption rates are locked together. Einstein’s 1917 argument put atoms in a cavity full of thermal radiation and required the populations to stay put; that forced a fixed ratio between the rate at which an excited atom drops on its own and the rates at which radiation drives it either way. The stimulated emission that makes a laser possible was deduced from the requirement that a box of atoms in equilibrium stay in equilibrium — before anybody had a use for it, and by the same argument Kirchhoff used sixty years earlier.

The common form is worth naming, since it is the most productive single move in the subject: assume equilibrium, forbid anything from happening, and read off a relation between rates that then holds whether or not anything is in equilibrium. It is called detailed balance, and it delivers relations no direct calculation of either rate would give.

Two people, two years, one law #

Kirchhoff published the ratio law in 1859 and the cavity argument in 1860. Balfour Stewart had published a version of the same result in Edinburgh in 1858, from experiments with plates of rock salt, and a dispute about priority ran for some years.

The dispute is instructive rather than merely unfortunate, because the two arguments were not the same. Stewart’s was a careful experimental generalisation supported by a heuristic about internal reflections; Kirchhoff’s was a proof from the impossibility of a certain kind of engine, and it delivered the universal function of wavelength and temperature that the cavity emits — a function whose existence Kirchhoff could establish and whose form he could not.

That is the sentence the next forty years were spent on. Kirchhoff had shown that a single curve, depending on nothing but temperature, was waiting to be measured; Stefan found its integral in 1879, Wien found where its peak sits in 1893 and its shape at short wavelengths in 1896, and Planck found the curve itself in 1900 by assuming something nobody wanted to assume. The whole of that programme was set going by an argument about a hot box that could not be allowed to develop a temperature difference on its own.

The measurement problem this creates #

Because emissivity and temperature enter the measured radiance together, thermal imaging is an inference and not a reading, and every practical use of it has a way of getting round the ambiguity.

Assume it. Most cameras have a setting, and most work is done at 0.95 because paint, skin, fabric, soil and vegetation are all near it. The error from getting it wrong is small when the surface is nearly black and enormous when it is not.

Cover it. A patch of matt tape on a shiny pipe reaches the pipe’s temperature and has a known emissivity, so the camera reads the tape. This is the standard field method and it is an admission that the measurement is of the tape.

Look at a cavity instead of a surface. A crack, a deep hole, a bundle of tubes, the gap between two bricks: any geometry in which the light has to reflect several times before it escapes has an effective emissivity far above the material’s own. This is why a furnace’s interior can be pyrometered accurately while its outside cannot, and why a rough surface reads hotter than a polished one of the same material at the same temperature.

Measure at two wavelengths. If the emissivity is the same at both — a grey body — the ratio of the two radiances gives the temperature with the emissivity cancelling out. That is a two-colour pyrometer, and it fails on exactly the surfaces the spectral form of the law permits.

A concentrator cannot raise anything above the temperature of the source, and the reason is the same second law. So no arrangement of mirrors makes a solar furnace hotter than the Sun’s surface, and no arrangement of optics makes a radiative cooler colder than the sky it is exchanging with. Both bounds come from the prohibition above rather than from any limitation of the optics.

An atmosphere is a selective surface #

The selective coating above was engineered: absorb in one band, emit in another, and settle at a temperature the grey calculation forbids. A planetary atmosphere is the same device, arrived at without design, and Kirchhoff’s law is what makes its behaviour intelligible.

Air is nearly transparent from 0.3 to 3 micrometres, which is where essentially all of the Sun’s energy arrives. It is strongly absorbing over most of the range from 4 to 100 micrometres, which is where a surface at 288 kelvin radiates — with one important gap, the window from about 8 to 13 micrometres where the air is again fairly clear.

Kirchhoff’s law then supplies the half of the story that gets left out. The gases absorbing in the infrared must emit in exactly the same bands, at exactly the same strength, at whatever temperature they happen to be. This is not an additional fact about carbon dioxide or water vapour; it is forced, band by band, by the argument at the top of this essay, and it would hold for any gas with those absorption bands whatever its chemistry.

So the atmosphere is not a blanket that traps heat, which is the usual metaphor and is misleading in a specific way. It is a radiator. It absorbs infrared from below and it radiates infrared in both directions — down toward the surface and up toward space — and it does so at its own temperature, which falls with altitude.

What that arrangement does to the surface follows from where the emission to space comes from. Radiation leaving the top of the atmosphere in an absorbing band originates from an altitude high enough that there is little absorber above it, and that altitude is cold. The planet as a whole must radiate away what it receives, so the outgoing flux is fixed; if it is being emitted from a cold layer, and the temperature falls with height at a rate set by convection, then the surface underneath must be warmer than the emitting layer by however much that lapse rate demands.

Two things about this are worth extracting because they are general rather than atmospheric. The first is that a selective surface’s temperature is set by the ratio of its absorptance where the energy arrives to its emittance where the energy leaves — the same fourth-root expression drawn in the figure above, with the same structure. The second is that the window matters enormously: the 8-to-13 micrometre gap is the channel through which the surface radiates directly to space, and it is the same window that makes passive radiative cooling possible at all. A surface, a coating and an atmosphere are the same problem.

Why a clean flame is invisible to a thermal camera #

A gas is the case where the spectral form of Kirchhoff’s law is not a refinement but the whole story, and the practical consequence is one that catches out anybody who points an infrared camera at a burner.

A solid emits a continuum, because its atoms are packed together closely enough that their energy levels have smeared into bands. A gas does not. Its molecules are far apart, their levels are sharp, and it can absorb only at the wavelengths its transitions allow — so by Kirchhoff it can emit only there too. Carbon dioxide has strong bands near 2.7 and 4.3 micrometres, water vapour near 2.7 and 6.3, and between the bands the gas is essentially transparent and therefore essentially non-emitting.

A camera operating in the 8-to-14 micrometre band is therefore looking through a clean flame rather than at it. A premixed natural-gas flame, hot enough to melt glass, reads cool or shows almost nothing, because in the wavelengths the instrument samples the gas has an emissivity of a few per cent. What the camera usually shows is whatever is behind the flame.

Add soot and everything changes. A diffusion flame that cracks its fuel produces carbon particles, each of which is a small solid body with a continuous spectrum, and the flame becomes an emitter across the whole infrared. That is why a yellow candle flame is bright and a blue Bunsen flame is not, despite the blue one being hotter: the yellow is incandescent soot, radiating as a near-black body, and the blue is gas radiating in bands plus a little chemiluminescence.

Two consequences follow, and both are engineering rather than curiosity. A sooty flame transfers heat by radiation and a clean one does not, so a boiler burning heavy oil heats its tubes mostly by radiation while one burning hydrogen must do it by convection — which changes the whole design. And a gas’s emissivity depends on how much of it there is: the absorption in a band is proportional to the number of molecules along the line of sight, so the emissivity of a furnace gas depends on the furnace’s dimensions and on the partial pressures, and is tabulated as a function of both rather than quoted as a property.

The last point is the one that most sharply separates a gas from a surface. A surface’s emissivity is a number belonging to the material. A gas’s is a number belonging to the geometry, and the same gas at the same temperature is nearly black across ten metres and nearly transparent across ten millimetres.

What it costs #

Equilibrium is assumed. The derivation puts the body inside an isothermal cavity and waits. Its conclusion is a property of the material and survives outside the cavity, but the reasoning that establishes it does not: a body far from equilibrium — a laser medium, a fluorescent screen, an aurora — emits at wavelengths it does not absorb, and none of it violates Kirchhoff because none of it is in equilibrium.

The body has been assumed opaque. A sheet of glass transmits, and what balances is then absorptivity plus reflectivity plus transmissivity, with the emissivity equal to the absorptivity and both small.

Grey has been assumed wherever a total was taken. The camera figure uses a single emissivity across the band, which is what a camera does; a real surface has a different one at each wavelength, and the correct calculation weights it by the Planck spectrum at the relevant temperature. For most surfaces the correction is small; for glass, plastics and thin films it is not.

Light carries momentum as well as energy, and the same balance argument applies to it: in an isothermal cavity the pressure on every wall is the same, whatever the walls are made of. That is the mechanical counterpart of the spectral statement, and it is where the photon gas’s equation of state comes from.

Where the model stops #

Nothing here predicts an emissivity. Kirchhoff’s law says the two numbers are equal and is silent about what either of them is. Computing an emissivity from first principles needs the optical constants of the material, which is a whole function of frequency and not a property.

Very small bodies break the geometry. An object smaller than the wavelength it is radiating does not have an emissivity in the ordinary sense; its emission and absorption are cross-sections, they can exceed its geometric area, and the ratio is still bound by the same argument while the language of a surface fraction stops applying.

Angle has been suppressed. The law holds direction by direction, and most surfaces are not Lambertian: a metal’s emissivity rises steeply toward grazing angles and a dielectric’s falls. A camera reading a curved pipe therefore sees a different emissivity at the centre and at the edge, and the resulting apparent temperature gradient across a uniformly hot cylinder is a standard trap.

And the near field is another world entirely. Two surfaces closer together than a thermal wavelength exchange heat by evanescent coupling at rates orders of magnitude above the Stefan–Boltzmann limit. Nothing is violated — the limit is derived for propagating radiation — but every intuition trained on the far field is wrong there.

A body’s temperature is what its energy content and its heat capacity say, and a radiation measurement infers that temperature from a spectrum. The inference assumes the body is in equilibrium and that its emissivity is known — and where either fails, a thermal camera reports a brightness temperature which is a property of the measurement rather than of the object.

The sky’s own spectrum is what a downward-looking radiative cooler exchanges with, and its blueness is a scattering exponent rather than a temperature. That is where this model stops: the sky is not a blackbody at any temperature, it has windows and bands, and a surface designed against it is designed against a spectrum rather than against a number.

What the pictures cannot show #

The camera figure draws a single emissivity on its horizontal axis, and no surface has one. It has a spectrum, and the number the axis shows is that spectrum averaged against the Planck curve at whichever temperature is being considered — so strictly the axis is a different quantity for each of the three curves drawn on it.

Nor can any of these figures show the balance itself. What Kirchhoff’s argument is about is two streams of energy, one leaving a patch of surface and one arriving at it, cancelling wavelength by wavelength and direction by direction in a cavity where nothing is happening. A picture of an equilibrium is a picture of nothing changing, and the content of it — that the cancellation holds separately in every one of an infinite number of channels — is not something a drawing can carry.

Where this ladder goes next #

Two rungs stand on blackbody. The first found that counting modes and giving each a kT produces a divergent spectrum, and that closing the gap required energy to come in lumps. This one goes the other way: it finds a statement about thermal radiation that needed no quantum mechanics, no model of matter and no spectrum — only the refusal of heat to flow from cold to hot.

The habit worth carrying away concerns which measurements are direct. An instrument reports a quantity it can reach, and the quantity wanted is usually one inference away, with an unmeasured parameter in the inference. A thermal camera reaches radiance and wants temperature, and emissivity sits between them; a spectrometer reaches a line strength and wants a column density, with an oscillator strength between them. The discipline is to name the parameter aloud, because a reading with a hidden assumption in it looks exactly like a reading.

What is left on this ladder is the spectrum’s other consequence: that a body’s colour, and not merely its brightness, is a thermometer. The peak wavelength and the total power depend on temperature alone for a black body, and turning either into a measurement requires exactly the emissivity correction this rung is about.