A limit written for free electrons

A brightness that would blow the star apart is set by one balance: the outward push of light on the gas against the inward pull of gravity. Both act on the same gas, so the gas’s mass cancels, and the limit depends only on the star’s mass and on how strongly the gas scatters light — its opacity. For ionised hydrogen the opacity is that of free electrons scattering photons, the Thomson cross-section, and the limit for a neutron star of 1.4 solar masses is about $1.8\times10^{38}$ erg a second.

Past the limit, the light and the mass part company: gas poured onto a compact object faster than the limit allows is either blown away or swallowed, and the light that escapes rises only slowly above the limit — a few times, perhaps ten. That account explains most of the ultraluminous X-ray sources in nearby galaxies as stellar-mass black holes accreting at high rates with their light beamed by a funnel. It struggles with the sources that pulse.

The cross-section in the limit #

A pulsation with a period of a second is a rotating neutron star: nothing larger could spin that fast, and a black hole has no surface structure to make a regular pulse. So the pulsating ultraluminous sources are neutron stars, and their apparent luminosities are fifty to six hundred times the limit for a neutron star’s mass. Beaming — the light confined to a narrow cone that happens to point at the observer — can make up part of the factor, and the funnel geometry that collimates a super-Eddington disc’s light has been fitted to them. But a pulse is a feature of the star’s own rotating emission, and light confined to a narrow scattering funnel loses its pulses; the observed pulse amplitudes of ten or twenty per cent argue against beaming by hundreds.

The other way out is in the limit’s own formula. The Eddington luminosity is

$$L_{\rm E} = \frac{4\pi G M m_p c}{\sigma},$$

and $\sigma$, written as the Thomson cross-section, is a statement about electrons that are free — that respond to the oscillating electric field of a light wave by oscillating in step and reradiating. An electron in a magnetic field of $10^{14}$ gauss is not free in that sense.

An electron bound to a field #

In a strong magnetic field an electron’s motion across the field is quantised into Landau levels, and in the classical picture it gyrates around the field lines at the cyclotron frequency. Its energy quantum, the cyclotron energy, is

$$E_{\rm cyc} = 11.6\ \mathrm{keV}\times\frac{B}{10^{12}\ \mathrm{G}}.$$

For an ordinary pulsar’s field of $10^{12}$ gauss it is about twelve kilo-electronvolts, in the range of the X-rays such a star emits. For a magnetar’s $10^{14}$ gauss it is over a mega-electronvolt, far above them.

A photon whose energy is well below the cyclotron energy and whose electric field oscillates across the magnetic field — the extraordinary mode — tries to push the electron sideways, and the electron, held to its gyration, barely moves. It scatters the photon with a cross-section reduced by the square of the ratio of the photon’s energy to the cyclotron energy. A photon polarised along the field — the ordinary mode — can push the electron along the field line, where it is free, and is scattered almost normally except when travelling nearly along the field.

The reduction is enormous for the extraordinary mode. At ten kilo-electronvolts in a field of $10^{14}$ gauss the cross-section is about a ten-thousandth of Thomson’s, so the force light exerts on the gas in that mode is ten thousand times smaller and the luminosity it can carry before blowing the gas away ten thousand times larger. The limit is an opacity, and the opacity is a property of how free the electrons are.

How far it can go #

A neutron star’s accreted gas is threaded by the field, and the radiation diffusing out through it is a mixture of the two modes. The flux is carried preferentially by the mode the gas scatters least, as heat flows preferentially through the better conductor of two in parallel, so the extraordinary mode’s low opacity dominates the diffusion — but the ordinary mode carries half the photons at a near-normal opacity, and photons change mode as they scatter.

The figure is a bound, not a calculation of any real source. Detailed models of the column of gas falling onto a neutron star’s magnetic pole — where the gas stops in a shock, and the radiation it releases escapes sideways through the column’s walls — find maximum luminosities of order $10^{40}$ erg a second for surface fields near $10^{14}$ gauss, a factor of fifty to a hundred above the free-electron limit. That covers M82 X-2 and NGC 7793 P13 without beaming. NGC 5907 ULX1, at nearly six hundred times the limit, needs either the most favourable field or some beaming on top.

Stars that have such fields #

Fields of $10^{14}$ gauss are not hypothetical. The magnetars — a few dozen isolated neutron stars that emit X-ray bursts and occasional giant flares — have dipole fields of $10^{14}$ to $10^{15}$ gauss, measured the same way every pulsar’s field is measured: from how fast the star spins down. A rotating magnet loses energy as radiation, at a rate set by its dipole moment, so the period and its rate of increase together give the dipole field. Magnetars spin slowly, with periods of seconds, and slow down fast, and the combination says their dipoles are a hundred to a thousand times stronger than an ordinary pulsar’s.

Their X-ray emission is powered not by rotation, which supplies too little energy, but by the decay of the field itself, which stresses and cracks the crust. Some magnetars show signs that their surface fields are stronger still than their dipoles — absorption features that vary through the rotation and are best explained as proton resonances in small, intense loops of field near the surface. The multipolar configuration the ultraluminous pulsars would need is thus observed, in a different class of neutron star, by the same kind of line that is ambiguous in the ultraluminous sources.

What is not known is whether a neutron star born with a magnetar’s field can end up in a close binary accreting at an enormous rate. Magnetars are young — their spin-down ages are thousands of years — and their fields decay over tens of thousands; a neutron star in a binary that has had time to begin transferring mass is usually older. A star whose dipole has decayed while its small-scale surface field persists would be one route. Another is that the strong field is not primordial but generated or buried by the accretion itself.

A field the spin will not allow #

The magnetic explanation needs a field of $10^{13}$ to $10^{14}$ gauss where the radiation escapes — at the surface, near the accreting pole. The spin of each pulsar constrains the field from a different direction, and the two constraints do not agree unless the field has structure.

A neutron star accretes through its magnetosphere: the disc is stopped where the field’s pressure matches the inflowing gas’s, at the magnetospheric radius, and the gas then flows along field lines to the poles. For that to happen, the magnetospheric radius must lie inside the corotation radius, where an orbit keeps pace with the star’s spin. If the magnetosphere extends beyond corotation, its field lines sweep faster than the gas orbits, and the spinning magnet flings the gas away instead of accepting it. The star would spin down, not up. The angular momentum the accreted gas delivers is what makes the pulsars spin up, and it requires accretion.

The magnetospheric radius depends on the dipole — the part of the field that falls off slowest with distance and dominates thousands of kilometres out, where the disc is stopped. The cap the spin sets is therefore a cap on the dipole: about $10^{13}$ gauss for M82 X-2, comparable for the others. A dipole of $10^{14}$ gauss, a magnetar’s, is excluded.

The way out is a field that is strong only near the surface. A neutron star’s field need not be a pure dipole. Higher multipoles — quadrupoles, octupoles, small-scale loops — fall off with distance as higher powers and are negligible at the magnetospheric radius, but at the surface, near the accreting pole, they can be much stronger than the dipole. A star with a dipole of $10^{12}$ to $10^{13}$ gauss and multipolar fields of $10^{14}$ gauss near its surface would accrete normally, spin up as observed, and radiate through a column where the opacity is suppressed a hundredfold.

The same distinction — the field that sets the lever arm against the field that acts locally — governs how ordinary stars lose their spin. A star’s wind is levered by its dipole and not by the total field, because only the large-scale field reaches far enough out to hold the wind in corotation. A neutron star’s accretion is regulated by the same dipole for the same reason, and the light it emits is governed by whatever field is at the base. Which field matters depends on the distance at which the question is asked.

One line, two fields #

There is one way to measure a neutron star’s field directly rather than infer it, and it has been tried on these sources. Electrons scattering photons at the cyclotron energy resonate, and the resonance shows as an absorption feature in the spectrum at that energy. Cyclotron lines measure the fields of ordinary accreting pulsars to a few per cent, and they are where the canonical $10^{12}$ gauss comes from.

A proton gyrates too, and its cyclotron energy at the same field is smaller by the ratio of the masses. In a field of $10^{15}$ gauss the proton’s resonance falls at a few kilo-electronvolts, in the same range as the electron’s resonance in an ordinary field. An absorption line in the X-ray spectrum therefore measures the field only once the particle making it is known, and the two possibilities differ by a factor of 1,836.

Absorption features have been reported in the spectra of two ultraluminous pulsars or candidates. In NGC 300 ULX1 a feature near 13 kilo-electronvolts was read as an electron line and a field of about $10^{12}$ gauss, consistent with an ordinary dipole. In a source in the galaxy M51, a feature near 4.5 kilo-electronvolts was read as a proton line and a field of about $10^{15}$ gauss — a magnetar’s — or alternatively as an electron line from a field of a few times $10^{11}$. The proton reading requires the field to be confined close to the surface, where the protons are, which is exactly the multipolar configuration the spin argument requires. The readings are interpretations, and the ambiguity is not resolved by the line’s energy alone.

What would resolve it is the line’s behaviour. An electron line’s energy changes with the height in the accretion column at which it forms, which changes with the luminosity, in a direction that has been observed in ordinary pulsars. A proton line’s width and strength depend differently on the gas’s temperature. And the two have different dependences on the angle at which the column is seen, which varies through each pulse. Following the line through the pulse and with the luminosity is the test.

One in the Milky Way #

All three sources in the first figure are in other galaxies, millions of parsecs away, and their luminosities rest on distances to those galaxies and on the assumption that the X-rays are isotropic. In 2017 a transient in the Milky Way, Swift J0243.6+6124, became the first Galactic pulsar to cross into the ultraluminous range, peaking at a few times $10^{39}$ erg a second — more than ten times the Eddington limit — during an outburst lasting months, at a distance of about five kiloparsecs fixed by the parallax of its companion star.

A Galactic source can be studied in a way the extragalactic ones cannot. Its X-rays are bright enough for detailed spectra and timing at every luminosity through the outburst, and its distance does not depend on a galaxy’s. Through the outburst its pulse profile changed shape as it passed a luminosity of about $10^{38}$ erg a second, which column models identify as the point where a radiation-dominated shock forms above the magnetic pole and the column begins to radiate sideways — the transition the magnetic account predicts, at the luminosity it predicts for a field of a few times $10^{13}$ gauss near the surface. It is one source, and a modest excess compared with NGC 5907 ULX1, but it is the case where the column’s behaviour can be watched rather than inferred.

The surprising part: the same equation twice #

The Eddington limit has now appeared twice among these stars, and its two appearances are mirror images. For the heaviest stars, the limit is approached from below: the luminosity grows with mass faster than the limit does, and the most massive stars sit close to it, losing mass in winds as they press against it. The opacity there is that of free electrons plus the lines of heavy elements, and a star poor in heavy elements has a lower opacity and a higher limit.

For a magnetised neutron star — a star held up by the degeneracy of its neutrons, whose mass is known to be about 1.4 solar masses — the limit is exceeded, and the reason is the same quantity moved in the same direction: the opacity is lowered, not by removing the heavy elements but by binding the electrons. In both cases the limit is not a property of mass and luminosity alone. It is a property of how strongly the gas that the light must pass through can be pushed, and the gas can be made harder to push.

What the figures assume #

The cross-section is the extraordinary mode’s low-energy limit. The full magnetic scattering cross-section has resonances at the cyclotron energy and its harmonics, depends on the angle between the photon and the field, and includes vacuum polarisation effects at fields above about $4\times10^{13}$ gauss, where the vacuum itself becomes birefringent. The square law drawn here is the leading behaviour below resonance.

The gain is a bound. The figure treats the two modes as independent channels carrying flux in parallel; in reality photons convert between modes when they scatter, and the column’s geometry matters. Published column models give smaller gains than the bound.

The spin constraint uses the simplest magnetospheric radius. The radius at which a disc is stopped is uncertain by a factor of two, depending on the disc’s structure and how the field threads it, and at super-Eddington rates the inner disc is thick and radiation-pressure dominated. The cap on the dipole moves by a factor of a few with those assumptions; it does not move by the factor of ten that would admit a magnetar dipole for M82 X-2.

The luminosities are peak values. These sources vary by factors of ten or more, and the ratios to the limit drawn in the first figure are for their brightest states.

Still open: whether the brightest pulsars are magnetars in disguise #

Two accounts of the pulsating ultraluminous sources remain in contention. In one, they are ordinary neutron stars with ordinary dipoles, accreting at hundreds of times the Eddington rate through thick discs whose winds collimate the light, so that the true luminosity is only modestly above the limit and the apparent excess is mostly geometry. In the other, they are neutron stars with strong multipolar fields near their surfaces, whose suppressed opacity lets them radiate a large fraction of the excess genuinely, with only modest beaming.

The two predict different things. Beaming predicts that the pulsed fraction should fall as the luminosity rises and the funnel narrows; a magnetic column predicts that the pulse profile should change with luminosity as the column grows. Beaming predicts many more such systems seen off-axis as fainter sources; magnetic suppression predicts that the brightest should be the most strongly magnetised. There is also a calorimeter in the sky around some of these sources. Several ultraluminous X-ray sources sit inside large ionised nebulae, hundreds of parsecs across, whose emission lines record how much ionising radiation the source has emitted in all directions over tens of thousands of years — a measure of the luminosity that no beaming can inflate, because the nebula surrounds the source. Where such a nebula exists, it fixes the true luminosity and the beaming factor follows by division; for most of the pulsating sources no such nebula has been found or measured. A confirmed cyclotron line, followed through a pulse and through a change of luminosity, would settle the field at the surface and with it how much of the light is real. For now the limit written for free electrons is known to be the wrong limit for these stars, and how wrong is the open question.