The flash a circling charge sends once a turn

An electron held in a circle by a magnet is accelerating all the time, and a charge that turns must glow has already established what that costs: Larmor’s formula, a power proportional to the square of the acceleration, radiated in the doughnut-shaped pattern of a dipole. That essay was about how much. This one is about what colour.

The obvious answer is the frequency the electron goes round at. A charge circling at $f_0$ turns its dipole at $f_0$, and a rotating dipole broadcasts at the frequency of its rotation, in the way a radio antenna broadcasts at the frequency of the current in it. For a slow electron that answer is right. In the 5.3-tesla field of a large fusion experiment, electrons with ten kiloelectronvolts of energy go round 146 thousand million times a second, and the plasma glows at 146 gigahertz — a microwave line so clean that its brightness is used as a thermometer for the plasma.

The same electron at a few gigaelectronvolts, bent on an arc ten metres in radius, would complete that circle about five million times a second. By the obvious answer it should broadcast at five megahertz, in the band that carries shortwave radio. It produces X-rays instead, at frequencies $10^{12}$ times higher, and whole laboratories are built to use them. The answer has changed by twelve orders of magnitude and the mechanism has not changed at all. Everything that follows is the arithmetic of that one fact.

A spike instead of a sine wave #

The field a distant observer receives from a charge is set by the charge’s acceleration at an earlier moment, the moment at which the signal now arriving left. The field that points where the charge is now derives the part of the field that falls off as $1/r$ and carries energy away; for a charge on a circle of angular frequency $\omega_0$, seen from far away in the plane of the orbit, it reduces to one line. Write $\tau$ for the orbital phase at the moment of emission, measured from the point where the charge is heading straight at the observer. Then the arriving field is

$$E \propto \frac{\beta(\cos\tau - \beta)}{(1-\beta\cos\tau)^3},$$

and it arrives at a phase $\phi = \tau - \beta\sin\tau$ of the observer’s clock. The second expression is where the story is. The charge is on a circle of radius $R$, so its distance from the observer changes by $R\sin\tau$ over the orbit, and a signal leaving from nearer arrives earlier by that distance divided by the speed of light. In units of the orbit, that correction is $\beta\sin\tau$.

At a fifth of the speed of light the arriving field is almost the rotating dipole of the obvious answer: a smooth oscillation once per turn, a little sharper at its crest than at its trough. At six tenths of the speed of light it has become a peak with long shallow wings. At nine tenths it is a spike — seven thousandths of a turn wide at half height, with the rest of the orbit contributing a trough so shallow that it barely leaves the axis. The drawing scales each curve to its own peak, which hides the other change: divided by the speed, the peak of the fastest is $1/(1-\beta)^2 = 100$ times a slow charge’s, because the denominator $(1-\beta\cos\tau)^3$ comes close to zero when the charge is heading at the observer.

The areas are the check that the drawing is honest. A radiation field carries no steady part — whatever pushes a distant charge one way during the flash must pull it back over the rest of the turn — so each curve must enclose as much area below the axis as above it. They do, to a part in a million of the peak. The spike at 0.9c is paid for by a trough that is tiny and lasts almost the whole orbit.

The width numbers are the argument in miniature. The fastest charge emitted the flash over 6.3 per cent of its turn, and the observer receives it in 0.7 per cent. The emission is not short. The arrival is, and the difference is not something that happens to the light on the way; it is a fact about when each part of it set off.

One power from the beam, two from the chase #

Two separate effects shorten the flash, and they scale differently with speed.

The first is the searchlight. The sky that crowds into a cone shows that a moving source’s radiation, isotropic or dipole-shaped in the source’s own frame, is thrown forward into a cone of half-angle about $1/\gamma$ in the laboratory. A charge on a circle carries that cone round with it, like a lighthouse, and the cone points at a given observer only while the charge’s direction of motion is within about $1/\gamma$ of the line of sight. That is a fraction of order $1/\gamma$ of the orbit, so the charge emits the flash over a time that falls as one power of $\gamma$.

The second is the chase. During exactly that stretch the charge is moving towards the observer at nearly the speed of light, so each part of the flash sets off from a point closer to the observer than the part before it. The light emitted at the end of the stretch has almost caught up with the light emitted at the start. The ratio of arrival time to emission time is $d\phi/d\tau = 1-\beta\cos\tau$, which at $\tau = 0$ is $1-\beta$, and for a fast charge that is $1/2\gamma^2$. Two more powers of $\gamma$.

On logarithmic axes the two effects separate cleanly. The width of the emission falls with slope −1, as the cone requires. The width of the arrival falls with slope −3, and the gap between the two lines is the chase. At $\gamma = 100$ the arrival is 18,870 times shorter than the emission — close to $2\gamma^2 = 20{,}000$ and slightly below it, because at the edges of the flash the charge is already turning away and following its light a little less closely. Near $\gamma = 1$ both widths are a sizeable fraction of the orbit and the lines converge: there is no searchlight and no chase, only a turning dipole.

The same compression appears elsewhere in a different disguise. The motion that measures faster than light watches a blob in a jet appear to cross the sky at seven times the speed of light, because it is moving almost straight at the observer and the light from the later positions has less distance to travel. The apparent speed of the blob and the brevity of this flash are the same factor, $1/(1-\beta\cos\theta)$, read one way as a speed and the other as a time. So is the Doppler factor of everything from an exchange of pulses, in which a stream of flashes sent by a moving source is received at a rate multiplied by the same kind of factor. Synchrotron radiation is the Doppler effect applied to a source that is on its way somewhere else by the time its signal is complete.

From one note to thousands #

A signal that repeats once per turn contains only frequencies that are whole multiples of the orbital frequency, and how much of each depends on its shape. A sine wave is all fundamental. A spike $1/\gamma^3$ of a period wide must contain harmonics up to roughly $\gamma^3$, because the only way to build a short event out of waves is to add many of them, and the shortest wave needed is about as long as the event. That is the relation sharpness has to be paid for establishes for any signal, and it is the whole reason the colour changes.

The transform is taken directly from the arriving field of the first figure, one harmonic at a time. It is also known in closed form — the strength of the $m$-th harmonic is $m\beta J'_m(m\beta)$, a derivative of a Bessel function, which Schott derived in 1912 without any figure — and the computed transform agrees with it harmonic by harmonic to a part in a million, which is the check that the spike in the first figure is the real field and not an artefact of how it was sampled.

At half the speed of light the spectrum is almost a single line. The second harmonic carries a fraction of the fundamental’s power and the ninth carries less than one per cent. That is the regime of the fusion plasma: a few weak harmonics, used in practice to separate hot electrons from cold ones, around one dominant frequency.

As the speed rises the spectrum is not simply stretched. Its strongest harmonic moves off the fundamental entirely, to about the 14th at 0.9c and the 453rd at 0.99c — close to $\gamma^3$ in each case, where the dashed lines sit — and the fall to one per cent moves out ten times further. Below the peak the power rises slowly and steadily with the harmonic number. The fundamental, which is all a slow charge sends, becomes one of the weakest things a fast charge sends.

In a real ring the harmonics are not resolved. Electrons in a storage ring are spread in energy and slightly in orbit, and the neighbouring harmonics, spaced by the orbital frequency of a few megahertz, blur into a continuum long before $m$ reaches the millions. What is left is the envelope.

One curve for every machine #

For $\gamma$ large, the envelope takes a shape that no longer depends on the speed, the radius or the charge. Measured in units of the critical frequency

$$\omega_c = \tfrac{3}{2}\gamma^3\omega_0,$$

every synchrotron spectrum is the same curve. The curve is a Bessel-function integral, $x\int_x^\infty K_{5/3}$, and like the harmonics it can be computed without a table — the integral over frequency can be done first, by hand, on the integral representation of the Bessel function.

The shape has two ends worth reading. At low frequency it rises gently, as the four-thirds power of the frequency when counted per logarithmic interval, so a synchrotron source is never dark at long wavelengths: the whole radio and optical band is lit by the slow edges of the flash. At high frequency it falls exponentially, because no amount of Fourier arithmetic can make a signal of fixed duration contain much at frequencies well above one over that duration. The critical frequency is defined so that the division between the two ends is exact: half the power is below it and half above, which the shaded area confirms to a tenth of a per cent.

The peak depends on how the spectrum is counted. Per unit frequency it sits at 0.288 of the critical frequency; per logarithmic interval, which is what a detector covering a factor of two in photon energy collects, it sits at 1.32. Neither is wrong. They answer different questions, and a table of “the” peak photon energy of a source has to say which one it means.

Eleven decades from one formula #

Only one number is left to supply for any machine: where the critical frequency falls. With the orbital frequency written as $c/2\pi\rho$ for a circle of radius $\rho$, the critical photon energy is $\tfrac{3}{2}\hbar c\gamma^3/\rho$ — a length set by the magnet’s geometry, divided into Planck’s constant times the speed of light, multiplied by the cube of the Lorentz factor.

The five rows cover eleven decades of photon energy, and the whole spread is the cube. The two electron rings between them are the light sources of the synchrotron era. Their radii differ by a factor of five and their energies by a factor of three, and the cube of the energy wins: the larger ring reaches 20.9 kiloelectronvolts, hard X-rays that pass through centimetres of tissue, while the smaller one delivers 3.11, soft X-rays suited to surfaces. LEP, the large electron–positron collider, was a particle-physics machine and not a light source, and at 104.5 gigaelectronvolts its critical energy was 818 kiloelectronvolts — gamma rays. Each electron lost about 3.4 gigaelectronvolts per turn to them, more than three per cent of its energy, replaced turn by turn by radio-frequency cavities. That loss, rising as the fourth power of the energy, is what ended the idea of circular electron colliders at higher energies.

The protons that later ran in the same tunnel make the point from the other side. They carried 67 times LEP’s energy, but a proton is 1,836 times heavier than an electron, so each had a Lorentz factor 27.4 times smaller, and the cube of that ratio outweighs the difference in radius and everything else. Their critical energy is 43.8 electronvolts, in the ultraviolet, 18,652 times below LEP’s. Mass enters the reach as its cube, which is why every machine built to produce synchrotron light accelerates electrons, and why the most energetic charges ever brought round a circle radiate in the ultraviolet while far less energetic electrons in the same tunnel reached gamma rays.

The tokamak electrons sit at the far left, off the formula. At a Lorentz factor of 1.02 there is no cube worth taking, and what is drawn is simply the fundamental — the one-note case, the rotating dipole of a charge that turns must glow. The magnetic force that holds them in their circle, as the force that does no work explains, sets a period that at low speed does not depend on the speed at all; it is the relativistic mass in $\gamma$ that finally puts the speed back in, and with it everything this essay is about.

Where the flash model stops #

The charge is classical. Everything here is Larmor’s field of a point charge, with the difficulties of a point charge’s own field set aside as the force a charge exerts on itself describes them. It fails when a single emitted photon carries a noticeable fraction of the electron’s energy — when $\hbar\omega_c$ approaches $\gamma mc^2$. For LEP the ratio was about $10^{-5}$ and the classical spectrum was excellent. Recoil against quantum emission matters in the strongest astrophysical magnetic fields and in laser–electron collisions, where the spectrum is cut off below the classical curve.

The charges radiate independently. A storage ring holds perhaps $10^{11}$ electrons in a bunch millimetres long, and at X-ray wavelengths their fields add with random phases, so the power is $N$ times one electron’s. Where a bunch is shorter than the wavelength, or has been organised into slices a wavelength apart, the fields add in phase and the power goes as $N^2$ — the free-electron laser, which is this essay’s flash made coherent and is a different subject.

The observer is in the plane of the orbit and far away. Out of the plane the flash is weaker, differently polarised and slightly longer; the in-plane view is the brightest and the one the numbers describe. Near the orbit, within a distance comparable to its radius, the far-field expression does not apply at all, for the reason the distance where a field changes its mind gives for any source: close in, the parts of the field that fall off faster than $1/r$ are not yet negligible.

The orbit is a circle and stays one. The charge loses energy to the radiation it sends, as the bill that arrives when the pushing stops accounts for, and on a circle that energy is drawn from its motion as it is sent. In a storage ring it is replaced every turn. A charge left to spiral in a magnetic field of astrophysical strength — the regime in which a radio source’s brightness is used to infer its electrons and its field — is another matter, and a question of measurement rather than of mechanism.

The searchlight and its polarisation #

Every figure here is a time series or a spectrum along one line of sight. What none of them draws is the searchlight itself — the narrow cone sweeping round the orbit that an observer anywhere in the plane catches once per turn — or the polarisation, which is linear in the plane of the orbit and turns elliptical above and below it. The harmonics are drawn as though they were resolved; in every real machine they are not, and the continuum a detector records is the envelope, broadened by the spread of energies in the beam and by the finite length of the magnets. The last figure places one number per machine and cannot show that a real light source is designed by choosing where the whole curve falls relative to the experiment’s needs.

Nor do the figures show where the flash physically is. The spike arrives in 0.007 per cent of a turn at $\gamma = 10$, but it was emitted over a stretch of the orbit a hundred times longer, and at no moment is the radiation a compact object travelling towards the observer; it becomes short only by arriving.

Still open: how short a flash can be made #

The flash from one bend is as short as the cube of the Lorentz factor allows, and at a light source that is about a femtosecond per electron. The pulses experimenters actually receive are far longer, tens of picoseconds, because an electron bunch is millimetres long and its electrons flash at different moments. Making the bunch itself shorter runs into the radiation: a very short bunch radiates coherently at long wavelengths, the coherent field acts back on the bunch, and the bunch lengthens or breaks up into microstructure. Where the limit set by the bunch’s own field lies for a given ring, and how far schemes that slice a short piece out of a long bunch with a laser can be pushed, are questions of accelerator design under active work rather than questions of principle.

The habit worth carrying away is to ask what an observer’s clock is doing before reading off a duration. A signal from a moving source is compressed or stretched by the factor $1-\beta\cos\theta$ between the time it is sent and the time it is received, and every “unexpectedly high” frequency, speed or brightness from something moving towards the observer is that factor before it is anything else. The electron in a storage ring is doing exactly what the slow electron in a fusion plasma does. It is simply doing it while running after its own light.