The radio that only heavier electrons can make

In 1974 Donald Gurnett, sorting through the radio data of satellites far out in the Earth’s magnetosphere, found that the planet was a powerful radio source. Bursts of emission between about fifty and five hundred kilohertz, with total powers of up to a billion watts, came from above the auroral zones at the same times as bright aurorae. Nobody on the ground had noticed, because the ionosphere reflects every frequency that low back upwards — the frequency the sky returns only at a slant followed how the ionosphere’s electrons turn back waves below their plasma frequency, a few megahertz — so the Earth’s loudest broadcast is aimed at space. Jupiter had been known to do the same thing since 1955, at frequencies up to about forty megahertz, which do pass the Earth’s ionosphere and were picked up by accident on a radio telescope being used for something else.

The frequencies were the clue. They match the frequency at which electrons circle the magnetic field in the regions above the aurora, just as a charge that turns must glow would lead one to expect. The intensities did not match anything spontaneous: the radio is far too bright for the number of electrons there, and its brightness temperature, the temperature a black body would need to emit as much, runs to $10^{20}$ kelvin and more. Something was amplifying it, and the explanation, given by C. S. Wu and L. C. Lee in 1979, turned out to rest on an effect most treatments of kiloelectronvolt electrons throw away as negligible.

A frequency that depends on the energy #

An electron moving across a magnetic field $B$ circles it at the cyclotron frequency $\Omega = eB/m$, which the circling that comes in quanta followed down to the quantum steps of a neutron star’s field. Classically the frequency has a property that makes it the standard clock of plasma physics: it does not depend on how fast the electron is moving. A faster electron goes round a bigger circle in the same time.

That is true only while the electron is slow. Relativistically, the momentum is $\gamma m v$ and the frequency becomes $\Omega/\gamma$: a faster electron is effectively heavier, and goes round more slowly. For an electron of energy $E$ the shift is $E/mc^2$ — a part in five hundred at one kiloelectronvolt, two per cent at ten. It looks like a correction to be dropped, and at these energies it changes almost nothing about the spontaneous glow: the flash a circling charge sends once a turn found that an electron’s radiation breaks into many harmonics only when it circles near the speed of light, while a ten-kiloelectronvolt electron radiates almost entirely at its fundamental. What the correction changes is the electron’s response to a wave. The spring that becomes a light-clock met the same thing in a mechanical oscillator: an oscillator whose frequency depends on its energy behaves differently in kind, not merely in degree, from one whose frequency is fixed, because an exchange of energy with anything else changes its phase as it goes.

The circle of electrons a wave can reach #

Consider a wave travelling across the field at frequency $\omega$, with its electric field rotating in the same sense as the electrons. An electron exchanges energy with the wave steadily only if it keeps in step with it — if its own circling frequency equals the wave’s. With the relativistic frequency the condition is $\Omega/\gamma = \omega$, which picks out the electrons of one particular energy, $\gamma = \Omega/\omega$, whatever the direction of their motion. In the plane of velocities along and across the field, those electrons lie on a circle centred at zero, of radius $c\sqrt{1-\omega^2/\Omega^2}$.

Two consequences follow straight from the drawing. There is no resonance at all above the cyclotron frequency, because no electron circles faster than $\Omega$; the resonant frequencies lie entirely below it. And a wave just below $\Omega$ meets only the electrons of one speed — a few per cent below, the circle runs through the ten-kiloelectronvolt electrons and through nothing else. The wave can be tuned to pick out a shell in velocity space. Without the relativistic mass, there would be no circle. Every electron would circle at exactly $\Omega$, a wave at $\Omega$ would be resonant with all of them at once, and a wave at any other frequency with none.

More fast electrons than slow ones #

Whether a wave grows or dies depends on the balance between the resonant electrons that give it energy and those that take it. An electron slightly ahead of the wave in phase is pushed back and loses energy to it; one slightly behind is pushed forward and gains. If there are equal numbers of each, nothing happens. What tips the balance is how the number of electrons changes with speed across the resonance. If there are more electrons just below the resonant speed than just above it, the wave, by pushing electrons up and down in speed, moves more of them upward than downward, gains nothing, and loses energy. If there are more just above, it gains.

A gas in thermal equilibrium has fewer electrons at every higher speed. Its density in velocity space falls everywhere, and waves passing through it are absorbed: this is the collisionless damping that the sound a plasma carries on its electrons’ heat found draining an ion-acoustic wave, where the slope of the ions’ distribution at the wave’s speed decided the loss. To amplify, the distribution must have a region where it rises with speed — a population inversion in velocity, the counterpart of the inverted levels of a laser, which hotter than any temperature there is found to be a state with more members up than down. The electrons that produce aurorae supply exactly that. They are accelerated downwards along the field to several kiloelectronvolts, and as they descend into the strengthening field, the force that does no work turns their motion along the field into motion across it, conserving their energy and their magnetic moment. A beam becomes a shell with a hole in it: a horseshoe, filled at all angles except those of electrons coming back up, which have been absorbed by the atmosphere. Inside the horseshoe, the density rises with speed.

Amplification that relativity makes possible #

The growth rate of the wave is a sum over the resonant electrons of the slope of the distribution across the field, $\partial f/\partial v_\perp$, weighted by $v_\perp^2$, taken round the resonance circle. For a distribution that depends only on speed, the sum round the circle is simply proportional to the slope of $f$ at the circle’s radius, $\tfrac43 v_r^3 f'(v_r)$.

The shell amplifies in a narrow band just below the cyclotron frequency, where the resonance circle runs through the inner side of the shell, and absorbs a little further below, where the circle runs through its outer edge. The Maxwellian absorbs at every frequency. That much would be expected of any instability fed by an inversion. The striking part is what happens if the relativistic mass is removed. Then the resonance is the whole plane of velocities at once, and the sum of $v_\perp^2\,\partial f/\partial v_\perp$ over the whole plane can be integrated by parts: it is $-2\int v_\perp f$, which is negative for every distribution there is. Without relativity, a wave crossing a magnetic field at the cyclotron frequency is absorbed by every population of electrons, inverted or not, because the electrons with too little perpendicular speed always outweigh in the sum. The mass shift of two per cent turns an absorber into an amplifier by letting the wave choose which electrons to talk to.

The same physics in the language of individual electrons is a story about phase. An electron that gives energy to the wave becomes lighter and circles faster; one that takes energy becomes heavier and circles more slowly. Electrons that start spread evenly round their orbits drift in phase relative to the wave according to which way their energy is changing, and they gather into a bunch. If the wave’s frequency is set slightly below the electrons’ own circling frequency, the bunch forms at the phase where the electrons are losing energy to the wave, and the wave grows coherently. Richard Twiss proposed this relativistic bunching in 1958 for astrophysical radio sources, and it was found independently in the Soviet Union and in the United States as the principle of the gyrotron — a microwave tube in which a hollow beam of electrons, spiralling in a strong field, gives up its perpendicular energy to a cavity mode just below the cyclotron frequency. Gyrotrons delivering a megawatt each at 170 gigahertz are the heaters planned for the fusion plasma of ITER. The aurora runs the same machine without a cavity, the amplification happening in a single pass through the source.

A brightness no electron could have #

The first argument that the radio is amplified rather than merely emitted is a comparison of temperatures. Any body that emits by spontaneous processes — each electron radiating on its own — cannot be brighter, at a given frequency, than a black body at its own temperature, because emission and absorption are tied together by the same atoms. The brightness no lens can increase found the same ceiling on any passive optical system: radiance cannot be concentrated beyond that of its source. The ten-kiloelectronvolt electrons above the aurora have a temperature, in that sense, of about a hundred million kelvin. The radio, worked out from its measured power, the size of its source and its bandwidth, has a brightness temperature of $10^{20}$ kelvin or more — a trillion times too bright for any population of those electrons radiating independently.

There are only two ways to exceed the ceiling. One is a source with a negative temperature, in which the emission outweighs the absorption, and the other is coherence, in which many electrons radiate in phase so that their fields, rather than their powers, add. A maser is both at once: the inversion supplies the negative absorption and the stimulated emission keeps the electrons in step with the wave. The brightness temperature was therefore an argument for a maser before any mechanism had been found, in the same way as the brightness of the first astrophysical masers, the hydroxyl and water lines of star-forming clouds, had been.

A frequency that is a thermometer #

The peak of the growth sits below the cyclotron frequency by an amount set by the electrons’ energy, because the resonance circle has to reach the shell. The offset is close to the fraction by which the electrons are heavier.

For the few-kiloelectronvolt electrons of the terrestrial aurora, the radio comes out within about one per cent of the local cyclotron frequency. Satellites flying through the source regions — the FAST spacecraft in the late 1990s was the first to do it with enough resolution — measured the horseshoe distributions directly and found the radio at frequencies just below the cyclotron frequency of the field they were sitting in, by the amount the electrons’ energy predicts. Since the field is known from the spacecraft’s magnetometer, the frequency of the emission measures the electrons’ energy, independently of any particle detector. It is a rare case of a relativistic correction being used as an instrument at the energies of a television tube.

A spectrum that is a map of height #

The frequency at which the radio is made depends on where it is made, because the field does. Along a planet’s polar field line the field falls roughly as the cube of the distance from the centre, so the cyclotron frequency falls by a factor of eight between the surface and one planetary radius up.

So a spectrum of the emission, taken from far away, is a map of where along the field lines the fast electrons are. The terrestrial emission’s spread from fifty to five hundred kilohertz says that the source extends from about half an Earth radius to two and a half, which is where satellites find the accelerated electrons and the evacuated cavities they come from. Jupiter’s emission cuts off near forty megahertz because there is no field strong enough above the cloud tops to circle electrons faster. Saturn’s is weaker and lower, matching its weaker field. The same emission has been looked for, and found in some cases, from brown dwarfs and from low-mass stars, where its frequency gives the strength of a magnetic field that no other measurement reaches.

The source must also satisfy a condition the figures do not show. The wave can only escape if the plasma in the source is tenuous — its plasma frequency well below the cyclotron frequency — because otherwise the cold electrons modify the wave so much that it cannot exist just below $\Omega$. Above the aurora, the electric fields that accelerate the beam also empty the region of cold plasma, leaving the density cavities in which the instability can run. The emission therefore requires two things together: a strong field and a nearly empty space, both of which the auroral acceleration region supplies and the dense ionosphere below does not. The numbers make the contrast sharp. The plasma frequency is 8.98 kilohertz times the square root of the electron density per cubic centimetre. In the auroral cavities the density falls to about one electron per cubic centimetre, a plasma frequency of nine kilohertz against a cyclotron frequency of a few hundred: a ratio of a few per cent. In the ionosphere’s F layer, with a hundred thousand electrons per cubic centimetre, the plasma frequency is nearly three megahertz and the cyclotron frequency about one and a half: the ratio is two, and the maser cannot run however inverted the electrons are. In the solar corona the ratio is usually large for the same reason, and the cyclotron maser is invoked there only for the sudden, millisecond spikes of radio from flaring loops, where the field is strongest and the density lowest.

What the drawings leave out #

Every figure here treats a wave travelling exactly across the field. Waves at an angle have a resonance curve that is an ellipse rather than a circle, displaced along the field by the Doppler shift of the electrons’ parallel motion, and the loss-cone hole of the horseshoe, rather than the shell’s inner slope, can then drive growth too; which of the two dominates in a given source depends on its geometry and has been argued over from satellite data. The shell is idealised as depending only on speed, which makes the integral round the circle exact; the real horseshoe has structure in angle, and its missing cone contributes. The growth rate is drawn scaled, because its absolute value depends on the density of the fast electrons, which varies by orders of magnitude between sources.

The figures also cannot show the saturation. A maser grows until it has used up the inversion that feeds it, and the auroral radio, in a single pass through a source a few hundred kilometres wide, converts about one per cent of the energy of the precipitating electrons into radio. How it saturates — by flattening the inner slope of the horseshoe, by trapping electrons in the wave, or by the wave leaving the source before it can do either — is calculated rather than drawn, and the calculations disagree in detail. The domain of the drawings is a wave crossing the field, a shell or Maxwellian of electrons of one to a hundred kiloelectronvolts, and a tenuous source.

Still open: what limits the brightest of these sources #

The emission from Jupiter’s moon Io is one of the most regular radio signals in the solar system, switched on when Io’s position in its orbit puts the field line it drags across into the right orientation for the beam to reach the Earth, and it follows the same mechanism. Similar bursts are now seen from cool stars and from a few objects at the boundary between stars and planets, with brightness temperatures that require coherent emission and with frequencies that, read as cyclotron frequencies, imply fields of thousands of gauss. Whether the same maser runs in all of them, how the electrons are accelerated where there is no solar wind to drive them, and how the energy of a system’s rotation is turned into the beams that feed it are questions being answered one source at a time.

The mechanism itself is settled. An electron circling a magnetic field at Ω/γ is heavier by $E/mc^2$ and slower by the same fraction, so a wave crossing the field below Ω resonates only with electrons on a circle of radius $c\sqrt{1 - \omega^2/\Omega^2}$; it grows if more electrons lie just outside that circle than just inside, which a horseshoe of auroral electrons supplies and no thermal gas can, and without the mass shift it would be absorbed by every distribution there is. The Earth’s loudest radio signal exists because ten-kiloelectronvolt electrons weigh two per cent more than electrons at rest.