A temperature that is a speed

A synchrotron spectrum has no temperature in it. The electrons that make it follow a power law in energy, not a thermal distribution, and the spectrum they emit is a power law too, with no peak to read a temperature from. But every source of radio emission has a brightness temperature — the temperature a blackbody would need to have to produce the same surface brightness at that frequency — and it is a perfectly good number to compute from a flux, a frequency and an angular size. For a source that is not a blackbody, it is usually treated as a convenience.

For a synchrotron source it is more than that. There is a maximum brightness temperature that a steady synchrotron source can have, and it is set not by the source’s composition or history but by a process that turns the source’s own light against it. That ceiling makes the brightness temperature a physical quantity, and it makes a measurement that exceeds the ceiling into a measurement of something else.

A number made from three measurements #

The definition is worth making concrete, because the whole argument turns on how large the numbers get. A source of flux density $S$ at frequency $\nu$ that subtends a solid angle $\Omega$ has a brightness temperature $T_b = S c^2/(2k\nu^2\Omega)$, from the Rayleigh–Jeans form of the Planck law. Put in a one-jansky source at 15 gigahertz — a typical bright quasar core — with a Gaussian angular size of a tenth of a milliarcsecond, and the answer is about $5\times10^{11}$ K. Halve the angular size and the temperature quadruples.

Nothing in that calculation assumes the source is thermal; it simply asks how hot a blackbody would have to be to look as bright. For a thermal source the answer is the physical temperature. For a synchrotron source the answer is instead the mean energy of the electrons radiating at that frequency, expressed as a temperature — which is why it can be enormous, and why there is any reason at all for it to have a ceiling. An electron can only radiate as brightly as its own energy allows, and the electrons radiating at centimetre wavelengths in the fields of a tenth of a gauss or less found in compact cores have Lorentz factors of a few hundred, energies equivalent to $10^{11}$ to $10^{12}$ K. The ceiling that the next section derives from scattering sits at the same place for a reason: it is where the electrons that make the emission are about to lose their energy to the emission they made.

Electrons scattering their own light #

The same relativistic electrons that radiate synchrotron photons in a magnetic field also scatter photons that pass through them. An electron of Lorentz factor $\gamma$ that scatters a low-energy photon boosts its energy by a factor of about $\gamma^2$, so radio photons are turned into X-rays, and the electron loses the difference. The scattering of the microwave background by such electrons is a known and useful process: it measures the electron density independently of the field. But the microwave background is a weak photon field. In a compact source, the densest photon field the electrons meet is the synchrotron radiation they have just made.

The ratio of the two losses is the ratio of the energy density in photons to the energy density in the magnetic field. For a compact, self-absorbed source, the photon energy density is set by the brightness temperature, and the field is set, through the frequency at which the source turns over from optically thick to thin, by the same quantities. Kellermann and Pauliny-Toth worked the ratio out in 1969, and the answer is steep: the inverse-Compton power divided by the synchrotron power goes as the fifth power of the brightness temperature, divided by a constant that comes out close to $10^{12}$ K.

At $10^{11}$ K the ratio is $10^{-5}$, and the scattering is irrelevant. At $10^{12}$ K it is one. And then the process runs away, because the scattered photons form a photon field of their own, which the electrons scatter again, and each order of scattering multiplies the losses by the same factor. At twice the critical temperature, first and second order together make the losses a thousand times the synchrotron power. The electrons radiate away their energy in hours, as X-rays and gamma rays, and the source’s radio brightness collapses back below the ceiling.

This is the inverse-Compton catastrophe, and its consequence is sharp: a steady, incoherent synchrotron source at rest cannot be measured with a brightness temperature above about $10^{12}$ K. The number depends weakly on the upper cut-off of the electron spectrum and on the spectral index, but not by more than a factor of a few.

A softer ceiling underneath #

There is a second, less dramatic limit below the first, and real sources sit near it rather than near the catastrophe.

The field inferred from a self-absorbed source falls steeply as its brightness temperature rises, and the density of electrons needed to produce the observed flux in that weaker field rises. The ratio of particle energy to field energy therefore climbs as a high power of the brightness temperature — the power $p + 13/2$, which is 8.5 for the electron index of 2 typical of these sources, as Readhead showed in 1994. The two energies are equal near $5\times10^{10}$ K. A factor of two above that, the particles outweigh the field by a factor of 360. At the Compton ceiling they outweigh it by $10^{11}$.

A source so far from equipartition is not impossible, but it is odd. Its energy is almost all in particles, and a magnetic field that weak cannot confine them for long. Readhead’s argument was that sources should be found near the equipartition value if they are anywhere near balance, and the compact radio cores measured with the longest ground baselines, once corrected for the motions discussed below, do cluster there. The equipartition brightness temperature is a soft ceiling — a statement about where sources prefer to be — and the Compton limit is the hard one above it.

A limit set by the length of the instrument #

Testing either ceiling requires measuring an angular size, and the sizes involved are far smaller than any single telescope resolves. That is the domain of interferometry, and of very long baseline interferometry in particular, where telescopes on different continents record the same signal and correlate it later, and where the closure phase survives the atmosphere over each station. There is a surprising feature of what such an instrument can establish.

The finest angle an interferometer resolves is the wavelength divided by the baseline. The brightness temperature of a source of fixed flux, measured at that resolution, is proportional to the flux times the wavelength squared divided by the angle squared — and the wavelength squared in the numerator exactly cancels the one in the denominator. The largest brightness temperature an interferometer can measure depends only on the length of its longest baseline and the flux of the source, not on the wavelength at which it observes. Lobanov’s estimate is about $3.09\,(B/{\rm km})^2\,(S/{\rm mJy})$ kelvin.

For an array the size of the Earth, a one-jansky source can be tested only up to about $5\times10^{11}$ K — a factor of two below the Compton ceiling. That is not a coincidence of engineering. It means ground-based interferometry, however high its frequency, could never have demonstrated that a source exceeds the ceiling by direct imaging: any compact source brighter than the ceiling would simply appear unresolved, with a lower limit on its temperature just above the limit of the instrument. For decades the brightest radio cores were reported with brightness temperatures “above $10^{12}$ K”, and the phrasing was exact — that was as far as the Earth could see.

The way past it is to make the Earth bigger. A radio telescope in an elliptical orbit reaching 350,000 kilometres from the Earth, correlated with ground telescopes, gives baselines thirty times longer and brightness-temperature limits a thousand times higher. The measurements made that way, in the 2010s, found cores in quasars such as 3C 273 with brightness temperatures of $10^{13}$ K and more — directly, from resolved sizes, with no inference about variability. The ceiling was exceeded by a factor of ten or more.

Brighter because it is coming towards the observer #

A source measured above the ceiling is not a steady source at rest. The resolution of the paradox is that the emitting plasma is moving towards the observer at nearly the speed of light, and its brightness is boosted by relativity.

A blob of plasma moving with Lorentz factor $\Gamma$ at an angle $\theta$ to the line of sight has a Doppler factor $\delta = 1/[\Gamma(1 - \beta\cos\theta)]$. Its frequencies are multiplied by $\delta$, and its intensity by a higher power, because intensity divided by frequency cubed is invariant. The upshot for a brightness temperature measured from a resolved size is simple: the observed value is the rest-frame value multiplied by $\delta$. A source at the Compton ceiling in its own frame, moving with $\delta = 10$, is measured at $10^{13}$ K.

That would be a hypothesis rather than a result if $\delta$ could not be measured independently. It can, because the same motion produces a second, purely geometric effect. A blob moving at nearly the speed of light towards the observer, at a small angle, almost keeps pace with its own light, so the time between two observations of it is compressed and its sideways motion appears faster than light. The apparent transverse speed is $\beta\sin\theta/(1 - \beta\cos\theta)$, and it reaches $\sqrt{\Gamma^2-1}$ — almost ten times the speed of light for $\Gamma = 10$ — at the angle where the Doppler factor equals $\Gamma$. Measuring apparent speed from the motion of knots along a jet over years is routine, and the fastest jets show apparent speeds of 10 to 40 times the speed of light.

The two measurements agree. The Doppler factors needed to bring the brightest cores down to the equipartition or Compton value are the ones their apparent speeds imply, to within the scatter expected from the unknown viewing angle of each source. The brightness temperature is a speedometer. A radio core’s excess over $10^{12}$ K, or better over $5\times10^{10}$ K, measures how relativistic and how well aligned its jet is — and it does so for sources in which no knot has been followed at all.

A size inferred from a flicker #

There is a second route to a brightness temperature, and it gives much larger numbers. A source whose flux varies in a time $\Delta t$ cannot be larger than light travels in $\Delta t$, because its different parts could not otherwise vary together. That size, with the flux and the distance, gives a brightness temperature without resolving the source at all.

For a source moving towards the observer, the variability route is boosted more steeply than the direct one. The variability time is compressed by $\delta$, which shrinks the inferred size and raises the inferred temperature by $\delta^2$, and the flux is boosted too; the total factor is about $\delta^3$. So the variability temperatures of blazars, which reach $10^{13}$ to $10^{15}$ K for variations over weeks, need Doppler factors of only a few to ten to be consistent with the ceiling — well within what the apparent speeds allow.

Then there were the intraday variables: compact radio sources whose flux changed by tens of per cent within hours, implying brightness temperatures up to $10^{21}$ K. To bring those down to the ceiling needs a Doppler factor of a thousand, and no jet has ever shown a speed that implies one. For a decade this was a genuine crisis — either the physics of synchrotron sources was wrong, or something else was making the sources vary.

It was something else. The variations are twinkling. Radio waves from a very compact source pass through the ionised gas of the Milky Way’s interstellar medium, whose turbulent density fluctuations act as a moving screen of weak lenses — the same way air turbulence makes a star twinkle while a planet, being larger, does not. Scintillation of this kind affects a source only if its angular size is smaller than a critical angle of about ten microarcseconds, so the variability measured an angle, not a light-crossing time. The proof was geometric: the variations of one source arrived at two telescopes on opposite sides of the Earth minutes apart, as the scintillation pattern swept across the planet, and their timing changed over the year as the Earth’s velocity relative to the screen changed. A source that varies intrinsically cannot do either. The highest brightness temperatures ever inferred were an effect of the Milky Way’s gas, not of the quasars.

The X-rays a slow jet would have made #

The Compton argument can also be run backwards. If a compact core were at rest, and its brightness temperature were near the ceiling, the scattered photons would appear as X-rays with a flux that follows from the radio measurement alone — the size, the turnover frequency and the flux fix the field, the electron density and hence the scattered power. For many bright cores that predicted X-ray flux is orders of magnitude above what X-ray telescopes measure from the same objects.

The discrepancy is the same motion in another guise. If the source is moving with Doppler factor $\delta$, the rest-frame brightness temperature is lower, the rest-frame photon density is lower, and the self-Compton X-rays fall steeply — as a high power of $\delta$. Requiring the predicted X-rays to be no brighter than those observed therefore gives a lower limit on the Doppler factor, with no reference to any knot, apparent speed or variability. These inverse-Compton Doppler factors, computed for hundreds of sources, agree with the ones from apparent speeds and from variability in the population as a whole, with the scatter expected from three methods that make different assumptions about the geometry. Three independent routes — a missing X-ray flux, a knot moving faster than light, a temperature above a ceiling — all say that the plasma in the brightest radio cores is moving at Lorentz factors of ten or more, towards the observer.

What the ceiling does not cover #

The argument assumes incoherent emission — each electron radiating independently, so that the brightness temperature cannot exceed the electrons’ own energy expressed as a temperature, and the Compton catastrophe applies. Coherent emission, in which many particles radiate in phase like the elements of a radio antenna, is not bound by it. Pulsars reach brightness temperatures of $10^{25}$ to $10^{30}$ K, and the fast radio bursts higher still, and neither is a violation: their emission is coherent, and the numbers are evidence of it. The ceiling identifies incoherent sources as much as it limits them.

The Compton limit also assumes the source has had time to reach its steady state. A source that has just been injected with fresh electrons can briefly exceed it, before the scattering catches up, and some of the fastest flares may be caught in that transient. And the relation between observed and rest-frame temperatures depends on whether the emitting region is a moving blob or a steady stream of plasma, which changes the power of $\delta$ by one. None of these moves the conclusion by a factor of ten, but each moves the Doppler factor inferred from a given temperature by a factor of two, which is the level at which the comparison with apparent speeds is made.

Finally, the direct measurements with the longest baselines found brightness temperatures that are uncomfortably high even allowing for Doppler boosting. Some cores measured from space baselines need Doppler factors of fifty or more to come down to equipartition — at the upper end of what the kinematics of the same jets allow. Whether that means the cores are far from equipartition, that the plasma is being re-energised faster than it cools, or that the size estimates are biased by the limited coverage of a spacecraft’s orbit, is not yet settled.

Still open: how far above the ceiling a core really is #

The ceiling has been exceeded, directly, and the excess is explained by motion — but the size of the excess is at the edge of what the motion can supply. The next measurements that would settle it are denser: space baselines with better sampling, so that sizes of ten microarcseconds are measured rather than bounded, and simultaneous measurements of apparent speed, brightness and polarisation in the same jets, so that the Doppler factor from each can be compared source by source rather than across a population. The brightness temperature began as a way of describing how bright a source looks. It has become, in turn, a test of synchrotron physics, a speedometer for jets, and a probe of the Milky Way’s turbulent gas, and the one question it still asks is whether the fastest jets are moving as fast as their brightness says they must.