A ceiling on the light and not on the mass

The Eddington limit is the luminosity at which the push of radiation on the electrons in a gas equals the pull of gravity on the protons they drag with them. It is one line of algebra, and it depends on the mass and nothing else. It was derived for a sphere, and the first thing everyone learns about it afterwards is that almost nothing in the universe accretes spherically: gas falling onto a compact object has angular momentum, settles into a disc, and a disc is flat. The radiation leaves through the faces and the gas arrives through the edge, so neither is directly in the other’s way.

That observation, which is correct, is usually taken to mean that the limit does not apply to discs. It does apply. It applies locally, at each radius, and what it limits is not the rate at which gas can arrive but the rate at which the energy of that gas can leave as light. Above the Eddington rate the two part company, and the whole of the physics is in where the surplus goes.

Why the light grows as a logarithm #

Measure the supply in its natural unit. A thin disc converts about a tenth of the rest-mass energy of the gas it accretes into light, so there is a rate of supply, $\dot M_E = L_E/(0.1\,c^2)$, at which it would radiate exactly its Eddington luminosity. Call the actual supply $\dot m$ in units of that rate. Below $\dot m = 1$ nothing unusual happens: the luminosity is $\dot m$ times the limit, and the gas radiates its tenth on the way in.

Above $\dot m = 1$ the disc cannot radiate everything, but it does not fail everywhere at once. The energy released per unit area of a disc rises steeply towards its centre, so at large radii the local flux is far below the local limit, and the disc is thin and unremarkable there. Moving inward, there is a radius at which the flux emerging from the disc’s surface reaches the value that would lift the surface gas off — the local Eddington flux — and inside that radius the disc can radiate no more than that. Shakura and Sunyaev worked out in 1973 where this happens. The radius is proportional to the supply: about $\tfrac{27}{4}\dot m$ Schwarzschild radii, the spherisation radius, because inside it the disc puffs up until it is roughly as thick as it is wide.

Outside the spherisation radius the disc radiates normally, and it releases about the Eddington luminosity in total, because the spherisation radius is by construction the place where the disc outside it has released exactly that. Inside it, each annulus radiates at its own limit. The local limit on a flat annulus is proportional to the vertical component of gravity, which falls off in a way that makes each factor of two in radius contribute the same amount of light. Integrated from the inner edge out to the spherisation radius, that gives a luminosity proportional to the logarithm of the ratio of the two radii — and since the outer one grows as $\dot m$, the total is

$$L \approx L_E\,(1 + \ln \dot m).$$

Ten times the Eddington supply gives 3.3 times the limit. A hundred times gives 5.6. A thousand times gives 7.9. The limit bends rather than breaks, and it bends so hard that no supply anyone has proposed gets a disc much past ten times its limit in real, isotropic luminosity.

An efficiency that falls as the supply rises #

The logarithm has a direct consequence for how much of the supplied energy comes out as light at all.

A thin disc’s efficiency is set by how deep in the potential its inner edge sits: a tenth for a non-rotating black hole, up to about four tenths for one spinning as fast as possible. Above the Eddington rate that number stops describing the light. The supplied energy is still released — the gas still falls through the same potential — but only $(1+\ln\dot m)/\dot m$ of the thin-disc fraction comes out as radiation. At a hundred times the Eddington supply the disc radiates about half a per cent of the rest-mass energy; at a thousand, less than a tenth of a per cent.

The rest has to go somewhere, and there are exactly two places it can go. It can leave the system as the kinetic energy of an outflow — gas lifted off the puffed-up inner disc by the very radiation pressure that limits the luminosity. Or it can stay with the gas, as photons that are made inside the flow and cannot escape before the flow carries them into the hole. Both happen in every simulation of the regime, in proportions that depend on things nobody has yet pinned down — the stress that moves angular momentum outward, the magnetic field, the black hole’s spin. The two limiting cases are worth drawing separately, because they lead to opposite conclusions about the one thing astronomers most want to know: how fast the object gains mass.

A disc that throws away what it cannot radiate #

In the first limiting case the surplus is expelled. Each annulus inside the spherisation radius radiates at its local limit and blows off whatever gas it cannot keep, so the rate at which gas flows inward decreases towards the centre.

The shape of the result is the whole point. Inside the spherisation radius the inflow falls in proportion to the radius, because each annulus can only accept as much gas as it can radiate the energy of. Three different supplies — ten, a hundred and a thousand times the Eddington rate — produce three different spherisation radii and three different winds, and exactly the same rate of arrival at the inner edge: in this idealised form, a little under half the Eddington rate. The limit on the light has turned into a limit on the growth. The hole receives what a thin disc at its limit would have given it, and the surplus leaves as a wind carrying up to a few tenths of the speed of light.

That wind is observed. In the ultraluminous X-ray sources — point sources in nearby galaxies brighter than any stellar-mass object could be at its limit — high-resolution X-ray spectra show absorption lines of highly ionised iron, oxygen and neon blueshifted by 0.1 to 0.3 of the speed of light — a speed read off a line’s edge, as it is for the winds of hot stars. The outflow is real, and it carries a mechanical power comparable to the radiated luminosity. The same kind of wind, driven by the same radiation pressure, is seen in some quasars, where it is the leading candidate for the mechanism that couples a black hole’s growth to the gas supply of its whole galaxy.

A disc that swallows its own light #

In the second limiting case the surplus is kept. The gas in a thick, super-Eddington flow is optically thick, and a photon made inside it has to diffuse out through many scatterings; if the gas is falling inward faster than the photon can diffuse to the surface, the photon is carried in with it. Inside a trapping radius — which also grows in proportion to the supply — the photons are advected into the hole with the gas.

In that case the luminosity is still about $L_E(1+\ln\dot m)$, because the light that does escape comes from outside the trapping radius, where the disc behaves much as it does in the wind picture. But the mass does not leave. Nearly all of the supplied gas arrives at the hole, and the object grows at $\dot m$ times the Eddington rate while shining at only a few times its limit.

The difference between the two pictures is a factor of the supply in the growth rate, and at a supply of ten it is a factor of twenty-two in the time to reach a billion solar masses. At the Eddington rate a black hole e-folds every fifty million years — the Salpeter time for an efficiency of a tenth — and growing a hundred-solar-mass seed to a billion takes 16 e-folds, 806 million years. Quasars are seen at redshift seven, less than 800 million years after the Big Bang, with masses of a billion suns inferred from the widths of their emission lines, and their seeds could not have formed before the first stars did, a hundred million years or more into that interval. At the Eddington rate the arithmetic does not close. With trapping, a supply of ten times the Eddington rate closes it in less than a tenth of the time available, and the problem disappears — provided the supply was there.

In the wind picture it becomes worse. A hole fed above its limit that blows the surplus away grows more slowly than one fed gently at the limit, because the thin disc at least delivers its whole supply. So the question of how the first quasars grew is, in large part, the question of which of these two limiting cases the flows around their seeds were closer to — and the luminosity, which is the only thing observed, is the same in both.

A modest excess seen down a funnel #

The ultraluminous X-ray sources raised the opposite question. They are too bright rather than too heavy: point sources off the nuclei of nearby galaxies, with luminosities of $10^{39}$ to $10^{41}$ erg per second if they radiate equally in every direction, which is ten to a thousand times the limit of a ten-solar-mass black hole. For fifteen years the favoured explanation was that they were black holes of a thousand solar masses or more — the missing intermediate-mass population. Then pulsations were found in several of them: they are neutron stars, of 1.4 solar masses, with a limit of $1.8\times10^{38}$ erg per second, radiating hundreds of times more than that.

The $1+\ln\dot m$ law cannot deliver a factor of a hundred from any plausible supply. What it can do is combine with geometry. The same wind that carries off the surplus is optically thick, and it surrounds the inner disc with a funnel through which the escaping light is channelled. An observer looking down the funnel sees the light concentrated into a fraction of the sky, and infers — by assuming the source radiates equally in every direction — a luminosity larger by the inverse of that fraction. King proposed in 2009, from the statistics of the ultraluminous sources’ soft components, that the funnel fills a fraction of about $73/\dot m^2$ of the sky once the supply exceeds about 8.5 times the Eddington rate.

On this reading the most extreme sources are unremarkable. M82 X-2, the first pulsating one found, has an apparent luminosity about a hundred times its limit; with the collimation law that needs a supply of about forty times the Eddington rate and a beam filling 4.6 per cent of the sky, and the true luminosity is under five times the limit. NGC 5907 ULX1, apparently 570 times its limit, needs a supply of 87 and a beam filling one per cent. Nothing in the physics is extreme except the viewing angle.

There is a catch, and it is not small. A beam filling one per cent of the sky is narrow, and a pulsar seen through it is a rotating magnetised star whose emission pattern sweeps round with the spin. Light channelled up a narrow funnel is scattered many times on the way out, and scattering washes out rapid variability. Yet the pulsations are seen, with amplitudes of ten or twenty per cent. Either the beam is much wider than the collimation law says, and the true luminosity is correspondingly larger, or something other than the funnel is making room for the light — and the leading alternative is a magnetic field strong enough to reduce the electron-scattering opacity itself. In a field above about $10^{13}$ gauss, photons well below the electrons’ cyclotron energy scatter far less than free electrons would let them, and the local limit rises by a large factor. The first sources of this kind would then be magnetars fed above their nominal limit, rather than ordinary pulsars seen down a funnel.

A spin-up that counts the mass #

Everything so far has been inferred from light, and light is exactly what the super-Eddington regime makes ambiguous. The pulsating sources offer a measurement of something else.

A neutron star accreting from a disc is caught by its own magnetic field at some radius, where the field’s pressure matches the inflow’s, and the gas then flows along field lines onto the poles. The angular momentum that gas carried at the radius where it was caught is delivered to the star, and the star spins up — the same torque that turns old pulsars into millisecond ones. That spin-up is observed: M82 X-2’s period of 1.37 seconds shortens by $2\times10^{-10}$ seconds every second. A torque is angular momentum per unit time, and gas cannot deliver more angular momentum than it carries — which at the capture radius is its Keplerian value. So the observed spin-up fixes a minimum rate of mass arrival, with no reference to the X-rays at all.

The floor depends on where the gas is caught, since gas caught further out carries more angular momentum per gram and less of it is needed. The capture radius cannot be beyond the corotation radius, about 2,000 kilometres for a period of 1.37 seconds, or accretion would spin the star down rather than up; and it cannot be inside the star. Across that whole range the floor sits between three and thirty-nine times the Eddington rate. The supply is super-Eddington, measured as a torque rather than as a luminosity. Whatever the geometry of the light, the mass is really arriving that fast.

It also bounds the beaming from the other side. If the X-rays were isotropic, the supply would have to be about a hundred times the Eddington rate, which is above the floor everywhere — so isotropy is not excluded by the torque alone, but it would require the gas to deliver only a fraction of the angular momentum it carries, which is what a field threading the disc beyond the capture radius does. A floor of three, if the gas is caught near corotation, leaves room for a beaming factor of thirty; a floor of forty, if it is caught near the surface, leaves room for very little beaming at all. The capture radius depends on the dipole field strength, which is what the magnetar interpretation and the beaming interpretation disagree about. The torque turns a question about geometry into a question about one radius.

What the two limiting cases leave out #

The figures idealise twice. The wind picture assumes that every annulus inside the spherisation radius radiates at exactly its own limit and expels exactly the rest; the trapping picture assumes that all the gas arrives and the photons made inside the trapping radius are all swallowed. Radiation-magnetohydrodynamic simulations, which follow the gas, the field and the photons together, find something between: outflows carrying a substantial fraction of the supply, trapped photons carrying another, and a black hole that grows at several times the Eddington rate while radiating at a few times its limit. The fractions depend on the hole’s spin and the magnetic flux threading it, and the most efficient flows, around rapidly spinning holes with strong fields, can even radiate more than the $1+\ln\dot m$ law allows by extracting energy from the spin.

Both pictures also assume the supply is there to be accreted. A seed black hole in the early universe sits in a halo whose gas it heats with its own radiation, and a supply of ten times the Eddington rate has to be delivered through that heated gas, for a sustained hundred million years, to the region a few thousand Schwarzschild radii across where the disc is. That is a question about how gas reaches a galaxy’s centre, not about the disc, and it is at least as hard.

And the beaming law is an empirical calibration, not a derivation. The factor 73 comes from fitting the soft thermal components of ultraluminous sources under the assumption that the funnel’s walls are what emits them. A different reading of the same spectra gives a different law, and the tension with the observed pulsations is evidence that it is at best part of the story.

The same regime, briefly, in other places #

Super-Eddington flows are not confined to exotic sources. A star torn apart by a massive black hole returns its debris at a rate that, for a hole of a million solar masses, exceeds the Eddington rate by a factor of a hundred for weeks, and the flares are observed to be near their limit rather than a hundred times above it — the surplus is presumably blown away or swallowed, and the optical emission of many such events is interpreted as reprocessing in exactly the thick outflow the wind picture predicts. A neutron star in a binary transferring mass on its thermal timescale receives far more than it can accept, and the system SS 433, a black hole or neutron star fed at thousands of times its Eddington rate, blows off the surplus in two precessing jets moving at a quarter of the speed of light. The supply in both cases is set by something unrelated to the accretor, and in both the object’s own response to an excess is to limit its light and let the mass go wherever the geometry allows.

Still open: which fate the first black holes had #

The Eddington limit on a disc turns out to be two statements that are usually made as one. It is a limit on the luminosity, which it enforces, up to a logarithm, by any means available. It is not a limit on the rate at which mass arrives, and where the arriving mass goes above the limit is set by the balance between outflow and trapping — which the luminosity does not reveal. In the nearby ultraluminous sources, the balance can be approached through winds seen in absorption and through pulsar torques that count the mass directly. In the first quasars, where the question matters most, neither is available: the objects are too distant for their winds to be resolved and have no pulsars in them. Whether the black holes seen at redshift seven were fed gently from massive seeds or fed far above their limits from small ones, and whether the gas that fed them stayed or was blown away, is still being decided — by simulations of the flows, by searches for the fainter, earlier population that the second picture predicts should exist, and by gravitational-wave detectors in space that would hear the seeds merge.