A disc that throws its spin upward

An accretion disc has to get rid of angular momentum to let its gas fall inward, and the standard answer to how it does so is turbulence: the magnetorotational instability turns any weak field into a tangle whose stresses carry angular momentum outward through the disc, gram by gram, from the inner gas to the outer. The mechanism is exact in its linear stage and robust in simulations, and it produces the small effective viscosity that observed discs require. But it has two weaknesses. It needs the gas to be ionised enough to hold the field, which the cold, dense midplanes of protoplanetary discs are not; and it moves angular momentum outward within the disc, which only postpones the question of where it finally goes.

There is another route, and it removes the angular momentum from the disc altogether. A magnetic field that threads the disc — with a net vertical component, rather than a tangle — can lift gas off the disc’s surface and fling it away along the field lines, and gas leaving that way carries off far more angular momentum per gram than it had in the disc. The disc accretes by throwing its spin upward. The mechanism needs no instability and no turbulence, and in the regions where turbulence fails it may be the only way a disc accretes at all.

A bead on a whirled wire #

Close to the disc, where the field is strong compared with the gas’s inertia, a field line behaves like a rigid wire. Its footpoint is anchored in the disc and rotates with it at the local orbital angular speed $\Omega_0$, and the whole line, as far as it stays rigid, rotates at that speed too. Gas on the line cannot cross it — the gas is frozen to the field — but it can slide along it, and in the frame rotating with the footpoint the only forces on it are gravity and the centrifugal force. Its motion is that of a bead on a wire whirled around a vertical axis.

The figure draws the effective potential along such a wire, in units of $GM/r_0$ where $r_0$ is the footpoint radius, for three inclinations. Along a vertical wire, moving up means moving away from the star at constant distance from the axis: gravity pulls the bead back and the centrifugal force does not help, so the footpoint is the bottom of a potential well and gas stays in the disc. Along a wire tilted outward, moving along it also moves the bead further from the axis, where the centrifugal potential is lower. For a small tilt gravity still wins; for a large one the centrifugal term wins, the footpoint becomes the top of a hill, and gas placed there slides away outward and upward.

The dividing angle is exactly 30 degrees from the vertical for a Keplerian disc, a result derived by Blandford and Payne in 1982. It follows from the curvature of the effective potential at the footpoint.

Expanding the potential to second order in the distance along the line gives a curvature of $1 - 4\sin^2\theta$ in units of $\Omega_0^2$. The first term is the vertical pull of gravity, which holds gas to the disc midplane; the second is the combination of gravity’s radial pull and the centrifugal force, which cancel exactly at the footpoint for a Keplerian orbit and push outward away from it. The curvature changes sign where $\sin\theta = 1/2$. A field configuration that bends outward by more than 30 degrees at the disc surface — as the field lines of a disc threaded by a large-scale field naturally do, since they spread apart as they rise — is a set of launching ramps.

The energy for the launch comes from the disc’s rotation. The field line forces the gas on it to corotate with the footpoint, so as the gas slides outward to larger radius it is being spun up, and the torque that spins it up is transmitted back along the field line to the footpoint, where it acts on the disc as a brake. The gas gains angular momentum and the disc loses it. That exchange is the whole point.

A gram of wind for several grams of accretion #

The corotation cannot last. As the gas moves outward, its inertia grows and the field weakens, and at some distance — the Alfvén radius, where the gas’s speed reaches the speed of magnetic waves along the field — the field can no longer force it to corotate. Beyond that point the gas moves off on its own, carrying whatever angular momentum it has been given. By the Alfvén radius, which is at $\lambda r_0$, the gas has been held at the footpoint’s angular speed while its lever arm grew by a factor $\lambda$, so it leaves with specific angular momentum $\lambda^2\Omega_0 r_0^2$ — $\lambda^2$ times what it had in the disc.

The same arithmetic describes the wind that spins down a sun-like star: a magnetised wind that corotates out to its Alfvén radius removes angular momentum out of all proportion to its mass. In a star it is how rotation is lost; in a disc it is how accretion is gained.

Each gram of wind removes the angular momentum of $\lambda^2 - 1$ grams of disc gas, which is what those grams must lose to fall inward. Balancing the angular momentum over the whole region from which the wind is launched gives a mass-loss rate in the wind that is only a fraction of the accretion rate through the disc: the logarithm of the ratio of the outer to the inner launching radius, divided by $2(\lambda^2 - 1)$. For a lever arm of 3 and a wind launched over two decades of radius, the wind carries 29 per cent of the inflow; for a lever arm of 6, 6 per cent. The rest of the gas accretes, with no turbulence involved.

Young stars show exactly such outflows. Their jets, collimated and fast, are measured to carry about a tenth of the mass being accreted onto the star, which is what lever arms of a few imply — and the jets rotate, in the sense of the disc, at speeds that let their angular momentum be estimated from the spectra of the gas on either side of the jet’s axis. The angular momentum carried by the jets of the best-observed young stars is comparable to what the disc must lose at the jets’ launching radii, a check made with line-of-sight velocity gradients across a few tens of astronomical units.

A speed set by the same number #

The same lever arm fixes how fast the wind leaves.

Conservation of energy and angular momentum along the field line, for a wind cold enough that its thermal pressure does not matter, gives a terminal speed of $\Omega_0 r_0\sqrt{2\lambda^2 - 3}$. A lever arm of 3 gives 3.9 times the Keplerian speed at the footpoint; a lever arm of 10, fourteen times. For a young star whose disc launches the wind from a tenth of an astronomical unit, where the orbital speed is about ninety kilometres a second, a lever arm of 3 gives a jet at a few hundred kilometres a second — the speeds measured, from the Doppler shifts of the forbidden lines of oxygen, sulphur and iron in the jets of T Tauri stars.

The speed and the mass-loss fraction are two measurements of one number. A fast jet with a large lever arm is efficient: it carries away much angular momentum per gram and needs little mass. A slow outflow with a small lever arm is wasteful: it must carry off a large fraction of the inflow to remove the same angular momentum. The observed jets, fast and light, sit on the efficient side, which is part of the evidence that they are magnetocentrifugal rather than driven by pressure.

Why a weak field wins #

The striking thing about disc winds is how weak a field can drive them effectively, and the reason is geometry.

The turbulent stress in a disc transports angular momentum radially, across the disc’s edge: the torque it exerts between one ring and the next acts over the area of the cylinder separating them, $2\pi r \times 2H$, where $H$ is the disc’s thickness. The wind’s torque acts on the disc’s upper and lower faces, over an area $2\pi r \times 2\,dr$ for each ring — so per unit radius, the wind’s lever is larger than the turbulence’s by a factor of order $r/H$, which for a thin disc is ten to thirty. A magnetic stress far weaker than the turbulent one, acting on the faces, can exert the larger torque.

Written in terms of the plasma β — the ratio of gas pressure to the magnetic pressure of the threading field — and with the toroidal field wound up by the rotation to be comparable to the vertical one, the ratio of wind torque to turbulent torque is $2(r/H)/(\alpha\beta)$. For a turbulent α of 0.01 and $r/H$ of 10, the two are equal at β of 2,000: a field whose pressure is one two-thousandth of the gas’s. Stronger than that, and the disc accretes mainly through its wind. Such fields are not exotic; they are about what the interstellar field would give if a fraction of it were dragged into the disc when the star formed, and simulations of discs threaded by a net vertical field routinely launch winds that dominate the accretion at β of ten thousand or less.

The dead zone that still accretes #

The consequence matters most in protoplanetary discs. Their midplanes between a tenth of an astronomical unit and several are too weakly ionised for the magnetorotational instability to operate: the field diffuses through the neutral gas faster than the instability can grow. A disc that relied on turbulence would pile up gas there, in a dead zone that could not accrete. Observed protoplanetary discs accrete at rates of $10^{-8}$ solar masses a year or so throughout their lives, which raised the question of how.

Winds answer it. The upper layers of the disc, exposed to the star’s X-rays and ultraviolet, are ionised enough to couple to the field even where the midplane is not, and a wind launched from those layers removes angular momentum from the disc column below. Simulations that include the non-ideal effects of weak ionisation — ohmic diffusion, the Hall effect, and ambipolar diffusion, where the ions and neutrals drift apart — find that the magnetorotational turbulence is suppressed almost everywhere in the inner few astronomical units of such a disc, and that a laminar, wind-driven flow carries the accretion instead. The accretion then happens in a thin layer near the surface where the field’s torque acts, not throughout the column, which changes where the gas flows and where the dust it carries piles up.

The Hall effect adds a twist that no other transport mechanism has. Its sign depends on whether the field threading the disc is parallel or antiparallel to the disc’s rotation axis, and in one orientation it strengthens the wound-up field and the wind’s torque, while in the other it weakens them. Two otherwise identical discs, differing only in which way their field points, would accrete differently — a bimodality that has been proposed as one reason protoplanetary discs of similar stars differ so much in structure.

What a wind-driven disc does to its planets #

The difference between turbulent and wind-driven accretion is not only a question of how fast gas reaches the star; it changes the structure of the disc in which planets form and move. A turbulent disc carries its accretion flow through its whole thickness and diffuses its gas and small dust smoothly. A wind-driven disc can carry its accretion in a thin, fast layer near the surface while the midplane, where the dust settles and the planets sit, is almost still. The midplane is then far less turbulent than the standard models assume, which lets dust settle into a thin layer and grow, and which leaves any change in the disc’s structure — a ring, a gap, a pressure maximum — to persist rather than being smoothed away.

Planets migrate through a disc by exchanging angular momentum with the gas around them, and the torques involved are a delicate balance of contributions that nearly cancel; the part exerted by gas turning round in the planet’s orbit depends on how fast the disc’s gas flows past the planet radially. In a wind-driven disc that radial flow is concentrated in the surface layer, or can even reverse in the midplane, and the co-orbital torque changes accordingly. Migration rates computed for turbulent discs can be wrong by factors of several, in either direction, for a disc that accretes through its wind. The rings and gaps seen in the dust of young discs at tens of astronomical units, often read as the marks of forming planets, are equally consistent in some models with a wind-driven disc concentrating its flux into bands with no planets at all.

The same lever, other engines #

The magnetocentrifugal mechanism is not specific to young stars. Around black holes, winds launched by the disc’s field carry off part of the accretion flow in the X-ray binaries and in active galactic nuclei, and absorption lines of highly ionised gas outflowing at a few hundred to tens of thousands of kilometres a second are seen in both. Some of those winds are driven by radiation — the pressure of the light the disc cannot radiate away fast enough — some by the disc’s heating of its own surface, and some by the magnetic lever; the three predict different relations between the wind’s speed, its ionisation and its launching radius, and distinguishing them is a large part of what high-resolution X-ray spectroscopy of those winds is for. The jets of radio galaxies, collimated over hundreds of thousands of light years, are thought to be launched by a related mechanism in which the field is anchored in the black hole’s spinning spacetime rather than in the disc.

What the bead picture leaves out #

The rigid-wire picture is exact only near the footpoint, where the field dominates the gas’s inertia; the full solution requires solving the magnetohydrodynamic equations for the shape of the field lines and the flow along them together, and Blandford and Payne’s original solutions were self-similar — the same at every radius up to a scale — which real discs are not. The lever arm is an output of that full solution, not a free number, and it depends on how much mass the disc loads onto each field line: a heavily loaded line bends more and has a smaller lever arm. The mass loading in turn depends on the disc’s surface heating and ionisation, which the bead picture ignores.

The torque comparison assumes the toroidal field is comparable to the vertical one at the disc surface, and that a single α describes the turbulence; real discs have both mechanisms operating at once in different layers, with the field’s geometry set by how the flux is transported through the disc over its life. And a wind needs a net vertical flux to thread the disc in the first place. Whether discs keep the flux they are born with, lose it by diffusion, or concentrate it by the accretion flow itself is unsettled, and it decides whether the wind picture applies to a disc at all.

Still open: where the disc’s flux comes from and where it goes #

The magnetocentrifugal wind turns a question about turbulence into a question about magnetic flux. A disc threaded by enough vertical field accretes through its wind with a small mass loss and no turbulence; a disc threaded by too little must rely on the instability, and where the instability is suppressed it cannot accrete efficiently at all. How much flux a disc inherits from the collapsing cloud that formed it, and whether that flux is dragged inward by the accretion or diffuses outward through the disc, determines which kind of disc it is — and the answer is not known for any observed disc. The rotation of young stellar jets, measured across their axes, and the magnetic fields in discs, mapped from the polarisation of their dust emission, are beginning to give the numbers; the dead zones, the planet-forming regions of the discs, are where the difference between the two pictures is largest.