A nebula that cannot be beamed at

A neutron star accreting gas cannot shine much more brightly than about two times ten to the thirty-eight ergs a second. Above that, the pressure of its own light on the infalling electrons would blow the gas away before it arrived. Yet X-ray sources in nearby galaxies have been found pulsing with the periods of spinning neutron stars at apparent luminosities of $10^{40}$ and even $10^{41}$ ergs a second — a hundred to five hundred times the limit. The essay on a limit written for free electrons followed one explanation: a strong magnetic field near the star suppresses the electrons’ opacity, so the limit written for free electrons does not apply to them. It ended on the other explanation, and on an instrument in the sky that could choose between them.

The other explanation is geometry. Gas fed to a compact object faster than it can radiate does not all fall in: the excess is blown off in a thick wind from the inner disc, and the wind leaves a narrow open funnel along the axis through which the light escapes. A source radiating at a few times its limit, all of it squeezed into a cone covering a few per cent of the sky, looks to an observer inside the cone like an isotropic source thirty times brighter. The apparent luminosity is the true luminosity divided by the beaming fraction, and an observer in the beam cannot measure either separately.

What can is something that sees the source from every direction at once. Several ultraluminous sources sit inside nebulae of ionised gas a hundred parsecs and more across, and inside expanding shells of swept-up gas larger still. This essay works out what those structures measure and how well.

Why a beam fools a single observer #

Every measurement of a source’s luminosity from one place assumes that the source shines equally in every direction: the flux at the Earth times the area of a sphere through the Earth gives the power. For a star that is very nearly true. For an accreting compact object it is not obviously true, because the geometry of the flow — a disc, a wind, a funnel — has preferred directions.

A source that beams its light into a fraction b of the sky appears, to an observer in the beam, brighter by a factor of one over b. No property of the light itself reveals that: its spectrum and its variability are those of the light that escaped through the funnel, and those can be made similar for a beamed and an unbeamed source by adjusting the model. Statistical arguments help a little. If ultraluminous sources were strongly beamed, many more of them would be pointing elsewhere, and the number of sources at lower apparent luminosities would be correspondingly larger. But the luminosity function of X-ray sources in galaxies has too many other contributors for that to be decisive.

The pulsing sources add constraints of their own. Their spin periods shorten measurably over months, and the rate at which a neutron star spins up measures the torque of the gas reaching its magnetosphere, and so the mass inflow there — a quantity that, like a clock that runs down and says what it is, comes from timing rather than brightness and is blind to beaming, though it depends on the field and on where the magnetosphere sits. A neutron star’s pulses come from its rotating magnetic poles, and if the light escaped through a narrow funnel aligned with the disc’s axis rather than the star’s spin, the pulses would be smeared out by scattering in the funnel’s walls. Strong pulsations argue against very narrow beams. They do not say how wide.

How the funnel forms #

The beaming is not an arbitrary assumption; it follows from what happens to gas supplied faster than the limit allows. Far from the compact object the disc is thin and radiates comfortably below its local limit. Closer in, the radiation from each ring of the disc rises until, inside a radius proportional to the supply rate — the spherisation radius — it would exceed the limit, and the disc puffs up and loses mass in a wind from every ring inside that radius, shedding just enough to keep each ring at its own limit. The luminosity grows only as the logarithm of the supply, and the rest of the mass leaves in the wind.

The wind is dense and optically thick. It surrounds the inner disc in every direction except along the axis, where centrifugal support leaves a hollow cone, and the radiation from the inner disc escapes mainly through that cone, scattering off its walls. The higher the supply rate, the thicker the wind and the narrower the cone, and simple versions of the picture make the beaming fraction fall as the inverse square of the supply in units of the limit. A source fed at a hundred times its limit would radiate at about five times it, beamed into about a hundredth of the sky, and appear five hundred times brighter than it should.

A nebula sees every direction #

Gas surrounding a source absorbs the ionising photons the source emits in all directions and re-emits their energy as recombination lines. For hydrogen, the number of Balmer photons emitted per second counts the hydrogen-ionising photons absorbed per second; for singly ionised helium, the He II line at 4686 Å counts the photons above 54.4 electronvolts, the energy needed to strip helium’s second electron. Very few stars emit enough such photons to make a strong He II nebula, so a bright 4686 Å line round an X-ray source is the source’s signature.

The nebula’s line is a calorimeter. It responds to the total number of He⁺-ionising photons emitted into every direction the nebula covers — which, for a nebula surrounding the source, is every direction — averaged over the time it takes the gas to recombine and the light to cross the nebula, thousands to tens of thousands of years. Beaming cannot inflate it. A source that sends its photons into a narrow cone lights only the part of the nebula in the cone, and the total line luminosity counts only the photons actually emitted.

The calorimeter needs a conversion. Only a fraction of a source’s luminosity is in photons above 54 electronvolts, and that fraction depends on the source’s spectrum in the far ultraviolet and soft X-rays — exactly the range the Galaxy’s own gas absorbs, so it is never observed directly and has to be modelled from the spectrum measured on either side.

The line a beamed source would light #

With a conversion in hand, the line luminosity follows from the source’s true output.

The nebula’s line falls in proportion to the beaming fraction, because the nebula sees the true luminosity — the apparent one multiplied by b. A source that appears at $10^{40}$ ergs a second and is not beamed would light its nebula’s 4686 Å line at a few times $10^{36}$; one that appears at $10^{41}$ but is beamed into a hundredth of the sky would light it to exactly the same level, like an unbeamed source at $10^{39}$.

The He II lines measured round ultraluminous sources lie between $10^{35}$ and $10^{37}$ ergs a second. The best-studied, in the dwarf galaxy Holmberg II, has a line luminosity that, with plausible conversions, requires the source to emit in all directions roughly what it appears to emit towards the Earth: an ultraluminous source that is not strongly beamed. That was one of the first arguments that at least some of these sources genuinely radiate at or above the Eddington limit of a stellar-mass black hole, and not merely appear to.

A ratio that ranks the sources #

The comparison reduces to one ratio: the nebula’s line luminosity divided by the source’s apparent X-ray luminosity.

An unbeamed source emitting a tenth of its light above 54 electronvolts gives a ratio of about five ten-thousandths; a ratio a hundred times smaller means a beaming fraction of about a hundredth. The figure’s three lines show the conversion’s uncertainty: a factor of ten in the fraction of the light that ionises He⁺ moves every inferred b by the same factor of ten.

That uncertainty sets what the method can do. It cannot fix one source’s beaming to better than a factor of a few, because the conversion is not known to better than that. It can rank sources reliably, because the conversion is similar for sources with similar X-ray spectra: a source whose ratio is a hundred times smaller than another’s, with a similar spectrum, is beamed a hundred times more strongly. And it can catch the extreme cases. A pulsing source at $10^{41}$ whose nebula shows a He II line of $10^{36}$ would need a beaming fraction of a few hundredths whatever the conversion, and one whose nebula shows $10^{38}$ would need none.

The bubble round the nebula #

Many ultraluminous sources sit inside larger structures: shells of shocked gas a hundred to three hundred parsecs across, expanding at around a hundred kilometres a second, whose emission lines show that they are heated by shocks rather than by photons. These are bubbles blown by the source’s wind — the same wind that, in the beaming picture, forms the funnel’s walls.

A wind blowing steadily into uniform gas sweeps it into a shell whose radius grows as the wind power to the one-fifth and the time to the three-fifths, a self-similar solution worked out for the winds of massive stars in the 1970s. Inverted, the shell’s radius and expansion speed give the wind’s power. A shell of a hundred parsecs expanding at a hundred kilometres a second needs about two times ten to the thirty-nine ergs a second of kinetic power — twelve times the Eddington luminosity of a neutron star, carried by the wind alone. The largest shells need ten times more.

The logic is the one by which a disc’s turbulence was shown to be more than supernovae can pay for: an energy demanded by an observed structure, set against the power its supposed source can supply. That is a second calorimeter, and an independent one. It measures mechanical power rather than radiation, it depends on the density of the surrounding gas rather than on a spectral conversion, and it is just as immune to beaming: the wind’s momentum is deposited in the shell in every direction it blows. Where both calorimeters exist round one source, they test each other.

An average over a million years #

The bubble records the wind over its whole age.

A shell of a hundred to three hundred parsecs expanding at a hundred kilometres a second is between half a million and two million years old. Its power is an average over that time. The source has to have been blowing at that rate for that long — which says the super-Eddington phase is not a brief flare but a sustained state of the binary, lasting a substantial fraction of the time the companion star takes to transfer its envelope.

The two calorimeters average over different times. The nebula’s recombination lines respond within thousands of years, so they measure the recent output; the bubble’s size and speed respond over its whole age. A source whose nebula is faint and whose bubble is large has recently weakened, or has recently turned its beam away from the gas the nebula is made of. A source whose nebula is bright and whose bubble is small has recently switched on. The pair is a clock as well as a calorimeter.

Sources that face away #

Beaming has a consequence that does not depend on any calorimeter.

If sources are beamed into a fraction b of the sky with random orientations, for every one seen as ultraluminous there are 1/b − 1 others pointing elsewhere. At a beaming fraction of a hundredth, ninety-nine are hidden for each one seen. Their bubbles and nebulae would still be there, blown and lit by sources that look ordinary from the Earth — or invisible, hidden behind their own optically thick winds.

That is a testable prediction. Surveys of nearby galaxies in the emission lines of shocked and photoionised gas find bubble nebulae with the sizes and line ratios of the ones round ultraluminous sources. Some contain no ultraluminous source at all. Whether there are as many of those as beaming requires — ninety-nine for each one seen, if the beaming is strong — is a census that can be made with existing telescopes, and its result would bound the beaming fraction from a direction entirely independent of the sources’ light.

One source counted three ways #

Put the three measurements of one source’s energy side by side.

For a neutron star radiating ten times its Eddington luminosity into five per cent of the sky, the observer in the beam infers three and a half times ten to the forty. The nebula measures the true radiative output, eighteen times ten to the thirty-eight, and the bubble the wind power, about the same. The two calorimeters agree with each other and with a modestly super-Eddington source; they disagree with the apparent luminosity by the beaming factor, which is how they measure it.

The figure also says something about the other explanation. A neutron star whose strong field lets it radiate genuinely at a hundred times its limit, unbeamed, would light its nebula and blow its bubble at a hundred times the limit too — ten to the forty — and the calorimeters would agree with the apparent luminosity. So a source with a nebula whose line and bubble match its apparent X-ray luminosity is radiating what it appears to, whatever makes that possible; a source whose calorimeters fall short is beamed. The two explanations the essay on the magnetic limit set against each other make different predictions here, and the measurement that separates them has been made for a handful of ultraluminous sources, though for few of the pulsing ones.

What the calorimeters assume #

The nebula surrounds the source. A calorimeter that covers only part of the sky measures only part of the output. Nebulae round ultraluminous sources are often lopsided, shaped by the density of the gas round them, and a beam pointed into a low-density gap would escape without lighting anything.

The conversion is a model. The fraction of the output above 54 electronvolts is inferred from the spectrum’s shape on either side of the unobservable range, and different models of the accretion flow’s soft spectrum differ by factors of several.

The bubble’s surroundings are uniform. The wind power scales with the density of the gas the bubble expands into, which is measured from the shell’s emission lines only roughly, and a bubble expanding into a density gradient grows faster on one side than the other.

The wind is steady. The self-similar solution assumes a constant wind. A source that has varied over the bubble’s age leaves a bubble whose size reflects the average and whose speed reflects the recent past, and inverting the two as if they came from a steady wind mixes the times.

Nothing else contributes. Massive stars and supernovae in the same region also ionise gas and blow bubbles, and the winds of massive stars carry power that is itself hard to measure — some may be too hot to see in the lines usually used. The He II line is a cleaner signature than the bubble, because few stars make it, but some Wolf–Rayet stars do, and a nebula’s line luminosity has to be checked against the stars round it.

Still open: how many of the brightest sources are beamed #

The calorimeters say that some ultraluminous sources radiate roughly what they appear to and others may be beamed by modest factors. For most of the pulsing neutron stars — the sources that most directly challenge the Eddington limit, because their masses are known to be small — no nebula has been measured well enough to decide. Several sit in regions where no nebula is detected at all, which could mean they are recent arrivals, that their gas is too thin to light, or that their light goes elsewhere.

The measurement needed is straightforward and slow: deep integral-field spectroscopy of the gas round each pulsing source, to find and measure any He II nebula and any shocked shell, combined with X-ray spectra good enough to pin the soft-photon fraction. A sample of a dozen, with two calorimeters each, would say whether the pulsing sources are mostly geometry or mostly physics — whether the limit written for free electrons is being evaded by a field that changes the electrons, or simply obeyed by sources that point their light at the Earth.