The temperature a gas takes from the light around it

The gas law relativity leaves exact heated a gas until its particles moved near the speed of light and found that most of what Maxwell and Boltzmann said about it changes while the gas law survives. It ended with a question about the hottest plasmas in the universe: what temperature a gas settles at when its particles hardly ever collide, and something else — light — is the only thing it can exchange energy with. For a gas of free electrons in a strong radiation field that question has a clean answer. It comes from one scattering, repeated, and it is set by the colour of the light, not its brightness.

Two effects in one scattering #

A photon with a momentum found that when a photon scatters off an electron at rest, the electron recoils and the photon leaves with less energy: Compton’s shift, which for a photon of energy ε takes away on average a fraction ε/mc² of it, mc² being the electron’s rest energy, 511 kiloelectronvolts. That is the whole story only if the electron starts at rest. In a hot gas the electrons are moving, in every direction at once, and a photon meeting a moving electron is Doppler-shifted twice, once into the electron’s frame and once back out. Averaged over directions the two shifts do not cancel: head-on encounters are more frequent than overtaking ones, because the electron runs into photons coming towards it, and they boost the photon. The average gain is $4kT/mc^2$ of its energy per scattering, for electrons at temperature T.

The two effects together give the rule the first figure draws:

$$\frac{\langle\Delta\varepsilon\rangle}{\varepsilon} = \frac{4kT - \varepsilon}{mc^2}.$$

A photon whose energy is less than four times the electrons’ thermal energy gains, on average, from each scattering; one whose energy is more loses. Off electrons at ten million kelvin the crossing is at 3.45 kiloelectronvolts, in the X-ray band: visible and ultraviolet photons are slowly heated by such a gas, hard X-rays slowly cooled. Each change is small, a few per cent at most, and the rule holds as long as both kT and ε are well below mc². Its two terms have the same origin, the conservation of momentum and energy in a collision between a photon and a moving electron, which the kick a fast particle can give an electron worked out exactly for a single encounter.

Where the four comes from #

The factor of four in the gain is worth unpacking, because it is relativity in a thermal disguise. A photon meeting an electron moving at speed βc, at an angle θ, has in the electron’s frame an energy multiplied by $\gamma(1-\beta\cos\theta)$. The electron scatters it, in its own frame, without much change of energy but into a new direction; back in the laboratory frame its energy is multiplied again, by $\gamma(1+\beta\cos\theta')$ for the new direction. Averaged over all the directions of the incoming photons and all the directions they leave in, the product of the two factors exceeds one by $\tfrac43\gamma^2\beta^2$: the photon gains, because the encounters that boost it are more numerous than those that lower it.

For a thermal gas of non-relativistic electrons the average of $\beta^2$ is $3kT/mc^2$, and $\tfrac43$ of that is $4kT/mc^2$. The same factor $\tfrac43\gamma^2\beta^2$ sets the power a single fast electron gives to the photons around it, the inverse Compton scattering by which a relativistic electron turns radio photons into X-rays. A hot gas heating soft photons and a fast electron upscattering them are one process: the gas is a crowd of electrons each doing, gently and in every direction, what the fast electron does violently in one.

The gas that followed the light after it went dark #

The rule has a large example in the history of the universe. For its first few hundred thousand years the universe was filled with blackbody radiation and an ionised gas, and the free electrons scattered the radiation so often that the gas could not be at any temperature but the light’s. When the gas became neutral, at about three thousand kelvin, almost all the free electrons were captured and the coupling weakened. But a small fraction of electrons, about one in ten thousand, remained free, and through them the light went on setting the gas’s temperature for a long time afterwards: matter and radiation kept the same temperature, both falling as the universe expanded, until the expansion had made the radiation too dilute to keep up. Only then, by most calculations around a redshift of a hundred and fifty, did the gas fall off the light’s temperature and begin to cool faster, as a gas expanding on its own does.

That this happened at all, with so few free electrons, is a measure of how strongly a blackbody’s light couples to an electron: the radiation’s energy density was then so high that a single free electron among ten thousand atoms was enough to hold the whole gas at the light’s temperature. It held it, as the blackbody figure says it must, at the light’s temperature exactly, because in a blackbody the induced scattering that the simple rule omits was doing its four per cent of the work.

The balance over a spectrum #

Turn the rule round and it says what happens to the electrons. Every photon that gains energy took it from the electrons; every photon that lost it gave it to them. A gas of electrons bathed in light, exchanging energy with it only by scattering, is heated by the hard photons and cooled by the soft ones, and settles where the two balance. Each photon scatters at the same rate in the Thomson limit, so the balance is

$$\int u(\varepsilon)\,(\varepsilon - 4kT)\,d\varepsilon = 0, \qquad kT_C = \tfrac14\,\langle\varepsilon\rangle_u,$$

a quarter of the photons’ mean energy, weighted by the radiation’s energy density. This is the Compton temperature. Two things about it are worth stating plainly. It depends on the shape of the spectrum, not on its intensity: brighter light gets the gas to $T_C$ faster, but does not change $T_C$. And it is independent of the gas’s density, as long as the gas is thin enough that collisions between its particles, and the emission they cause, are slower than the scattering. Near a bright source of hard radiation that condition holds over large volumes, and the gas there has a temperature set by the source’s colour.

The four per cent a blackbody is owed #

The rule makes a prediction that cannot be right, and the error is instructive. Put the gas in blackbody light at temperature $T_r$. The energy-weighted mean photon energy of a blackbody is $4\zeta(5)/\zeta(4)$ times $kT_r$, about 3.83, so the rule says the gas settles at 0.957 of the light’s temperature. But a gas and blackbody light left alone together must end at one temperature: if they did not, heat would flow forever between two things in equilibrium, and the second law would fail.

The missing four per cent is quantum statistics. Photons are bosons, and a photon is more likely to be scattered into a state that already holds photons than into an empty one — induced scattering, the scattering counterpart of the stimulated emission that makes a laser. In blackbody light, where the low-energy states hold several photons each, induced scattering pushes photons down in energy towards the crowded states more than the simple rule allows, which takes energy away from the photons and gives it to the electrons. Aleksandr Kompaneets derived the full equation for photons scattering off a thermal gas in the 1950s, induced terms included; with them, blackbody light leaves the gas at exactly its own temperature, as it must. The outcomes identical photons refuse found photons’ bosonic preference in two photons meeting at a beam-splitter. Here it adds four per cent to a temperature.

In the dilute light near real astrophysical sources, far fewer photons occupy each state than in a blackbody at the same colour, induced scattering is negligible, and the simple rule stands.

A quasar’s light, and the heat it sets #

The light from the centre of an active galaxy, where gas falling onto a black hole shines across the whole spectrum, is a natural place for the rule to matter. A rough model of a quasar’s spectrum has about the same power in every decade of photon energy, from the infrared to hard X-rays around a hundred kiloelectronvolts. The fourth figure computes the Compton temperature of gas bathed in such light as the cutoff is moved. With a cutoff at a hundred kiloelectronvolts the gas settles at seventeen million kelvin, ten times hotter than the Sun’s corona, heated by nothing but light. Calculations with measured quasar spectra give similar values, of order ten million kelvin, and the temperature sets which gas near a quasar can stay bound to its galaxy and which is driven off as a wind.

The rule says the temperature should keep rising in proportion to the cutoff. It does not, and the figure shows why: as photons approach mc² the Thomson rule fails.

Where the photon stops being a light wave on a charge #

The rule assumes the electron responds to the photon as a classical charge shaken by a wave, with the cross-section that the cross-section that forgets the colour found to be the same at every frequency. That holds while the photon’s energy is small compared with the electron’s rest energy. As it approaches mc², the Klein–Nishina formula takes over: the cross-section falls, because the electron recoils within the encounter and cannot follow the wave, and the fraction of energy the photon gives up stops growing in proportion to its energy. A photon of 511 keV, the electron’s own rest energy, scatters at less than half Thomson’s rate and gives up about a third of its energy; a fifty-MeV photon scatters at two per cent of Thomson’s rate.

The product of the two, the energy a photon hands over for each unit of incoming flux, peaks near mc² and then falls. Very hard photons are poor heaters, and the Compton temperature of a gas under light extending far into the gamma rays levels off. The computed curve, using the full Klein–Nishina cross-section integrated over every angle, rises with the cutoff until about a hundred kiloelectronvolts and then bends over, reaching seventy million kelvin at a ten-MeV cutoff where Thomson’s rule would have given nearly two thousand million. The same physics that the photon a fast electron throws back used for inverse Compton sources, read in the other direction, keeps light from heating electrons to the relativistic temperatures the simple rule would allow.

Which photons heat and which cool #

The last figure splits the balance by photon energy for the quasar-like spectrum at its Compton temperature. Every decade of photon energy below 4kT cools the gas by about the same amount: the light has equal energy per decade, and each soft photon’s gain is small, but they are many. All the heating comes from the decade or two of hard X-rays just below the cutoff, few photons each handing over a good share of its energy. The areas are equal, as they must be at the balance.

The figure also says what changes the temperature. Strip the hard X-rays from the light — gas shielded by an intervening cloud that absorbs them, or a source whose spectrum is soft — and the remaining light is purely a coolant: the gas is driven down until other processes, collisions and the emission they cause, take over. The Compton temperature is a property of the light’s hardest part as much as of its average.

Where the balance holds #

The argument assumes the electrons exchange energy with the light faster than with anything else. In a dense gas they also collide with ions and with each other, emit bremsstrahlung and line radiation, and those processes have temperatures of their own to drive towards; the gas settles at the Compton temperature only where scattering dominates, which near a bright source of hard radiation means large volumes of thin, ionised gas, and elsewhere means it is one term among several. It also assumes the ions share the electrons’ temperature, which requires collisions between them, and in the thinnest plasmas they do not have time: the electrons can sit at the Compton temperature while the ions are hotter or colder. And the figures treat the radiation field as given, where in a gas thick enough to scatter each photon many times the spectrum is itself reshaped by the scattering: the soft photons are pushed up towards 4kT and pile up there, which is how the X-ray spectra of many accreting black holes are made. In those spectra the rule runs the other way, with the gas setting the light rather than the light setting the gas. A corona of electrons at a hundred kiloelectronvolts or so above the disc of an accreting black hole scatters the disc’s soft photons again and again, each scattering raising them by a few per cent, until the spectrum is a power law that rolls over near a few times the electrons’ kT. X-ray astronomers read the corona’s temperature from where that rollover falls, and the values they find, tens to hundreds of kiloelectronvolts, are themselves held below the electrons’ rest energy by the same Klein–Nishina limit and by the pairs of electrons and positrons that photons above mc² create.

What the figures cannot show #

The figures show temperatures and rates, not time. How fast a gas reaches its Compton temperature depends on the radiation’s energy density: in the light a few light-days from a bright quasar it takes minutes, in the faint light far out in a galaxy it takes longer than the age of the universe, and only where the time is short does the Compton temperature describe the gas. Nor can the figures show the gas’s motion. The Compton temperature is a temperature of the electrons’ random motion; light can also push gas bodily, by radiation pressure, and drag moving gas, the radiation drag that the bath that pushes back found, and those effects can matter as much as the heating.

Still open: whether the electrons and ions share one temperature #

Near the black hole at the centre of the Galaxy and in other faint accretion flows, the gas is so thin that electrons and ions hardly exchange energy at all in the time the gas takes to fall in, and the radiation, the turbulence and the magnetic fields heat them differently. Models of these flows assume a ratio between the ions’ and electrons’ temperatures, and the images and spectra they predict depend on it: the brightness of the ring the Event Horizon Telescope imaged, for example, is set by the electrons, which may be far cooler than the ions. Whether the electrons are heated mainly by light, as here, by turbulence or by magnetic reconnection, and what ratio the two temperatures settle at, are argued from simulations and observations without a settled answer. The Compton temperature is the cleanest case of a gas whose temperature is set from outside, and the real flows are gases in which several such settings compete.

The habit worth carrying away is to ask what a system can exchange energy with. A free electron and a photon exchange energy in every scattering, (4kT − ε)/mc² of the photon’s energy on average, so gas bathed in light settles where 4kT equals the light’s energy-weighted mean photon energy — 0.957 of a blackbody’s own temperature until induced scattering makes it exactly one, 1.7 × 10⁷ K in a quasar’s light — independent of how bright the light is, and capped near 10⁸ K because photons approaching mc² scatter less and give up only part of their energy. Light does not only illuminate a thin gas. It sets its temperature.