The circling that comes in quanta

A charge that turns must glow found that an electron going round in a circle radiates, and the flash a circling charge sends once a turn followed the same electron in a magnetic field from slow to fast: at low speed it broadcasts one frequency, the frequency at which the force that does no work turns it, $\omega_c = eB/m$; at high speed the chase after its own light crushes that radiation into a brief flash once a turn, rich in high harmonics. In both cases the circling is continuous. The electron can go round with any energy, and it radiates that energy away as smoothly as it likes.

On the surface of a neutron star the field is ten thousand times stronger than any built on Earth, and the argument breaks in a new place. The electron’s speed is not the problem. The problem is that the circling itself comes in steps, the steps are as large as hard X-rays, and the radiation that should drain them is so fast that no electron is ever found anywhere but on the bottom step. What a neutron star’s field does to light is then no longer emission at all. It is a resonance, and the resonance writes the strength of the field into the X-ray spectrum as a gap.

The quantum of going round #

A charge in a uniform magnetic field moves freely along the field and circles across it at $\omega_c = eB/m$, whatever the radius of the circle. Quantum mechanics allows the circling only certain energies, the Landau levels, spaced by $\hbar\omega_c$ like the levels of a harmonic oscillator — which is what the circling, projected onto any line across the field, is. Lev Landau worked them out in 1930, and they underlie the quantum Hall effect, the oscillations of a metal’s magnetisation with field, and the way the potentials that are not unique still give unique energies.

For an electron the step is $\hbar\omega_c = 11.58$ keV for every $10^8$ tesla. Across the fields that exist in nature that one proportionality covers the electromagnetic spectrum.

In a laboratory magnet of a few tesla the step is a fraction of a millielectronvolt, a microwave, far smaller than the thermal energy of anything warmer than a few kelvin. Fusion plasmas are heated by beaming in microwaves at exactly this frequency: at six tesla the electrons circle 168 thousand million times a second and absorb power from a beam tuned to it. On white dwarfs with fields of thousands of tesla the cyclotron radiation is visible and infrared light, appearing as broad humps in the spectra of the binaries called polars and making their light strongly circularly polarised. On neutron stars feeding on a companion, with fields of a few hundred million tesla, the step is tens of kilo-electronvolts, a hard X-ray. And near $4.41 \times 10^9$ tesla the step becomes as large as the electron’s rest energy, $mc^2$, and the electron’s circling is relativistic even in its lowest excited state. Magnetars, with fields of $10^{10}$ to $10^{11}$ tesla, are past that point, and their electron lines would lie in gamma rays; the proton, 1,836 times heavier, has its line there in X-rays instead.

Ten thousand times any magnet #

Fields of $10^8$ tesla are not made by any current anyone could drive. They are inherited. A massive star’s core, before it collapses, is a ball of iron about the size of the Earth that may carry a field of a thousand tesla or so inside it — magnetic white dwarfs, which are cores that did not collapse, show fields up to a hundred times that — and the core’s plasma conducts so well that the field cannot get out: the flux through the star is carried inward with the collapsing matter. The flux is field times area, the area falls as the square of the radius, and a core shrinking from six thousand kilometres to twelve concentrates its field by a factor of a quarter of a million. Fields of a few hundred million tesla follow without any dynamo at all, and a dynamo in the hot, rapidly spinning young star can push some of them much further.

Such a field dominates every other force on an electron near the surface. Gravity there costs a proton about fourteen mega-electronvolts for every kilometre it climbs, and still the field is the stronger constraint: it confines an electron’s sideways motion to a few picometres. Atoms are squeezed into needles along the field, and even the structure of matter at the surface — whether it is a gas, a liquid or a solid of chains of atoms — depends on the field more than on the temperature.

The proton circles too, and its quantum is smaller by the mass ratio, 6.3 electronvolts at $10^8$ tesla, deep in the ultraviolet and far below the thermal energy of the plasma. Its radiative decay, which goes as the inverse cube of the mass for a given field, is slower by a factor of six thousand million. Protons in an accreting pulsar’s field therefore circle classically, as electrons do in a laboratory, and the X-ray line is the electrons’ alone. The same physics, a factor of a hundred million weaker, is what the image made of frequencies uses inside a hospital scanner, where a proton’s spin turns about the field at a radio frequency proportional to it and the frequency is read as position. In both the field is measured by a frequency that it alone sets; on a neutron star the frequency happens to be an X-ray, and the reading is done by the gap it leaves rather than by the signal it sends.

Steps that crowd together #

The levels are equally spaced only while the circling is slow. In general the $n$-th level lies at $mc^2(\sqrt{1 + 2nB/B_c} - 1)$ above the ground, with $B_c = m^2c^2/e\hbar = 4.41 \times 10^9$ tesla, and when $B$ is a sizeable fraction of $B_c$ each step up adds a little less energy than the one before, because the electron’s mass is rising with its speed. At an accreting pulsar’s tenth of the critical field the second level sits at 1.92 times the first rather than two. Far above it the energies grow only as the square root of the level number.

This is the flash a circling charge sends once a turn seen from the other end. There, a relativistic electron’s harmonics were integer multiples of a fundamental that itself fell as $1/\gamma$; here the levels are fixed by the field and the departure from integer spacing is the quantum statement of the same relativistic slowing. In a measured spectrum the effect is a way of checking that a line and its supposed harmonic belong together: the harmonic should sit a little below twice the fundamental, not at it.

A step nothing stays on #

An electron in the first Landau level is a circling charge carrying one quantum of circling, and a circling charge radiates. The rate at which it falls back to the ground level is, to lowest order, exactly the classical Larmor loss of a charge circling with energy $\hbar\omega_c$:

$$\Gamma = \frac{4}{3}\,\alpha\,\frac{\hbar\omega_c^2}{mc^2},$$

which grows as the square of the field. The circling period shrinks only as its inverse.

In a one-tesla magnet the excited electron goes round seventy thousand million times, for 2.6 seconds, before it radiates its quantum. That is why electrons in a laboratory field can be treated as circling continuously: collisions and thermal kicks move them between levels thousands of times before radiation has any say. In a neutron star’s $10^8$ tesla the first level lasts a quarter of a femtosecond. The electron makes about seven hundred turns and falls back. In the hot, dense plasma above the star, at a temperature of a few kilo-electronvolts, collisions come thousands of times more slowly than that.

The consequence is drastic. Whatever lifts an electron to the first level — a collision, or a photon of the right energy — it is back on the ground within a femtosecond. The whole electron population sits in the lowest Landau level. Across the field the electrons are frozen into the smallest circles quantum mechanics allows, a few picometres across; along it they move freely with whatever thermal speed they have. The plasma is, for practical purposes, a gas of beads sliding on wires.

The spring a field supplies #

A photon arriving with exactly the energy of the first step is absorbed, lifting an electron, and within a femtosecond the electron radiates a photon of the same energy back out — in a different direction. Nothing is converted to heat; the photon is redirected. That is scattering, and at the step’s energy it is enormous.

The cross-section that forgets the colour treated an electron bound to an atom as a mass on a spring, shaken by light: weak far below the spring’s resonance, enormous at it, and above it falling to the flat Thomson value. In a magnetic field the spring is literal. An electron circling across the field is an oscillator with natural frequency $\omega_c$, a light wave of the right polarisation drives it, and its cross-section near $\omega_c$ has exactly the resonant shape of the bound electron, with the field in place of the atom. At the peak, for the lifetime computed above, the cross-section exceeds Thomson’s by a factor of order $(\omega_c/\Gamma)^2$ — about twenty million at $10^8$ tesla — before the thermal motion of the electrons smears it. A layer of plasma that is nearly transparent to X-rays on either side of the resonance is opaque within it.

A gap in the X-rays #

The X-rays from an accreting neutron star come from gas falling onto its magnetic poles at half the speed of light, stopped in a column or a mound a kilometre or so across and heated to tens of millions of kelvin. The emerging spectrum is a hard continuum, bending over at a few tens of kilo-electronvolts, and every photon of it has to cross the magnetised plasma near the pole to get out.

Photons near the cyclotron energy are scattered again and again and many of them are turned back into the star or out of the beam, so the spectrum seen from outside carries a notch. Its width comes from the electrons’ motion along the field, which Doppler-shifts the resonance as seen by an incoming photon — a few kilo-electronvolts for electrons at several keV — and its depth from how much plasma the photons cross. A weaker notch appears near the second level, and in a few pulsars up to four harmonics have been seen.

The first was found in 1976, in the spectrum of the pulsar Hercules X-1, by a balloon-borne telescope flown by Joachim Trümper’s group. It was at first read as an emission line and is now understood as absorption near forty kilo-electronvolts. It was the first direct measurement of a neutron star’s magnetic field — a field until then inferred only from the rate at which pulsars slow down, a calculation that assumes the star is a spinning dipole in empty space. A spectral line needs no such assumption about how the star loses its spin. More than thirty accreting pulsars now have measured lines, from about ten to about a hundred kilo-electronvolts, and the fields they imply sit between about $10^8$ and $10^9$ tesla, about a tenth of the critical field, where the slowing of young radio pulsars had already put neutron-star fields.

What the line actually measures #

Converting a line energy to a field requires three corrections, and each moves the answer by more than the line’s own uncertainty.

The first is the relativistic spacing just described, small at these fields. The second is gravity. The line is formed near the surface of a star of about 1.4 solar masses and twelve kilometres, and the redshift that weighs a dead star found that the surface of such a star reddens its light by about z = 0.24. A line seen at 38 keV left the surface at 47. Read without the redshift, the field comes out too low by the same factor.

The third is position, and it is the one that makes the line interesting rather than merely useful.

A dipole field falls as the cube of the distance from the star’s centre. At the surface of a twelve-kilometre star, a kilometre of height reduces the field by a fifth, while the gravitational redshift changes far more slowly. A line formed a kilometre up the accretion column appears at 31 kilo-electronvolts instead of 38, and one formed three kilometres up at 21. The line measures the field where the scattering is, and the scattering is wherever the column’s structure puts it.

That is why a cyclotron line is not a constant of a star. In several pulsars the line energy moves with the X-ray brightness — upward as the star brightens in some, downward in others — and the two senses are read as two regimes of the accretion column: at low accretion rates the infalling gas is stopped close to the surface by its own collisions, and brighter means a slightly deeper, higher-field stopping layer; at high rates the radiation itself decelerates the gas in a shock that rises with luminosity, carrying the scattering layer up into weaker field. In Hercules X-1 the line has also drifted downward by a few kilo-electronvolts over two decades, a slow change in the column or in the field near the pole that is not understood. A line that moves with the light is a measurement of where in the column the light is made.

Where the bead-on-a-wire picture stops #

The figures treat the field as a perfect dipole, the star as a static sphere, and the electrons as a thermal gas in the lowest level, scattering one photon at a time. The real line-forming region is a column threaded by field lines that bend, with gas falling through it at a large fraction of the speed of light, so the resonance seen by a photon depends on the direction it travels relative to both the field and the flow, and the line’s shape is computed by following millions of photons through a model column. The figures’ line widths and depths are therefore illustrative; the energies, the lifetimes and the scaling with field are not.

The lifetime is the lowest-order rate. Near the critical field it acquires relativistic corrections, and the electron’s spin, which flips with the circling in the full treatment, shifts the higher levels slightly; at the fields of accreting pulsars these corrections are a few per cent. The spectrum figure does not show the polarisation of the scattered light, which differs strongly between the two modes a magnetised plasma supports and which X-ray polarimeters have only recently begun to measure. And the picture says nothing about magnetars, where the field exceeds the critical value, the vacuum itself becomes birefringent — a consequence of the field nobody can transform away becoming nonlinear at about the same scale — and the electron line moves out of the X-ray band entirely. Variable absorption features seen in a few magnetars at a few kilo-electronvolts have been interpreted as proton lines, which put the field in loops near the surface at $10^{10}$ to $10^{11}$ tesla; that reading is not settled.

The domain of the argument is a magnetised plasma in which the cyclotron quantum exceeds the thermal energy and the radiative decay outruns collisions: fields above about $10^7$ tesla for the plasmas near accreting neutron stars. Below that the circling is classical, and the radiation is the continuous glow of the earlier essays. Above it the circling is quantised and the radiation is a resonance.

Still open: where on the star the field is #

The line says what the field is where the line forms. The torque the infalling gas exerts on the star, which spins it up or down, says what a dipole would have to be to couple the star to its disc at the observed rate. For several pulsars the two agree to within the uncertainties; for a few the line implies a field several times stronger than the torque suggests, which may mean that the field near the pole has strong components on scales smaller than the star — tangled, multipolar field that dominates near the surface and has died away by the distance at which the disc is held off. How much of a neutron star’s field is in such small-scale structure, how long it survives, and whether accretion buries it under a layer of fresh conducting matter, as has been proposed to explain why old accreting stars have weaker fields than young ones, are not settled.

The line itself is a consequence of a field strong enough to make the circling discrete. On a neutron star the cyclotron quantum is 11.6 keV per 10⁸ tesla, an excited electron radiates its quantum in 2.6 × 10⁻¹⁶ s after about seven hundred turns, and so every electron sits in the lowest Landau level and the plasma scatters only at the step: a resonance that carves a line whose energy, corrected for the star’s redshift and for the height at which it forms, is the field. The classical circling charge glows at any energy; the quantised one is silent except at one, and the silence is a measurement.