Astronomy & Celestial Mechanics
the sky, computed
Exoplanets

The gas that turns round beside the planet

The two spiral wakes a planet raises nearly cancel, and what decides the direction it drifts is a third, smaller torque from the gas sharing its orbit. That gas makes a U-turn in front of the planet and behind it, and whether the turn pushes the planet outward depends on how the disc was heated and on a viscosity nobody has measured.

Assumes Planet migration, Lagrange points and Resonance.

16 min read 5 figures Small perturbations, large consequencesTwo bodies, and no more

A planet embedded in a gas disc raises two spiral wakes, one inside its orbit and one outside, and the torques from them nearly cancel. The inner wake pushes the planet forward, the outer one drags it back, and what survives the subtraction is a few per cent of either — negative in almost every disc anyone has written down, and large enough to carry an Earth-mass planet into its star in a few hundred thousand years.

That residual is called the Lindblad torque, after the resonances at which the wakes are launched. It is not the whole torque. There is gas that sits at the same distance from the star as the planet does and goes round at nearly the same rate, and that gas does not launch a wake at all. It makes U-turns. Its contribution to the torque is smaller than either wake, comparable to their difference, and of either sign — so it is the quantity that decides which way a planet actually moves.

Where a disc pushes a planet outward. The sign of the total torque on a low-mass planet, as a function of the power-law indices of the disc's surface density (Σ ∝ r^−α, horizontal) and temperature (T ∝ r^−β, vertical), from the fitting formulae of Paardekooper and collaborators for an adiabatic disc with γ = 1.4 and the horseshoe torque unsaturated. Above the line the total is positive and the planet migrates outward; below it the planet migrates inward. The line is straight because every part of the torque is linear in the two indices, and it is steep in α because a shallow density profile strengthens the vortensity part and weakens nothing. The minimum-mass nebula (α = 1.50, β = 0.50) sits at a total of −3.76 in units of Γ₀/γ; The viscously heated inner disc (α = 0.60, β = 0.90) sits at a total of +0.74 in units of Γ₀/γ; The irradiated outer disc (α = 1.07, β = 0.43) sits at a total of −2.65 in units of Γ₀/γ. The minimum-mass nebula every textbook starts from is on the inward side by a wide margin, and a disc heated by its own accretion is on the outward side: the same planet can migrate either way depending on how the gas is heated, which the Lindblad torque alone could never say.
Fig. 1 The sign of the total torque on a low-mass planet across the space of disc models, labelled by how steeply the gas’s surface density and temperature fall with distance from the star. Above the line the planet moves outward. The minimum-mass nebula — the disc reconstructed by smearing the planets back out into gas — lies well on the inward side; a disc heated by its own accretion lies on the outward side. Every part of the torque is linear in the two slopes, which is why the boundary is a straight line and why a modest change in how the gas is heated moves a disc across it.

Orbits that turn round instead of passing

Gas slightly inside a planet’s orbit goes round faster than the planet and gas slightly outside goes round slower, so in a frame rotating with the planet the inner gas overtakes it and the outer gas falls behind. Far from the orbit, that relative motion carries gas past the planet, and the passing is what raises the wakes.

Close to the orbit the relative motion is too slow to carry gas past. Gas just inside the orbit catches up with the planet, is pulled outward by it as it approaches, crosses onto an orbit slightly outside the planet’s, and — now going slower than the planet — drifts back the way it came. Half an orbit later it meets the planet from the other side, is pulled inward, crosses back, and starts catching up again. In the rotating frame its path is a horseshoe with its ends at the planet, the same shape a small body traces when it shares an orbit with a planet around the two triangular Lagrange points. Some of Saturn’s moons do exactly this with each other, swapping orbits every few years.

Each U-turn exchanges angular momentum with the planet. Gas crossing outward in front of the planet gains angular momentum and the planet loses it; gas crossing inward behind the planet loses angular momentum and the planet gains it. If the gas on the two legs of the horseshoe were identical, the two exchanges would cancel exactly. They are not identical when the disc has a gradient, because the gas arriving at the front of the horseshoe comes from a slightly different part of the disc from the gas arriving at the back. The net torque — the horseshoe drag — measures the difference between them.

Two quantities that the U-turn carries across

What matters is not the gas’s density alone but two conserved quantities that the gas carries with it around the horseshoe.

The first is the vortensity — the ratio of the gas’s vorticity to its surface density. In a flow with no viscosity it is conserved along each streamline, so a parcel making a U-turn carries its vortensity from one side of the planet’s orbit to the other. If the disc’s vortensity varies with radius, the horseshoe region becomes a place where gas of one vortensity has been swapped with gas of another, and the density perturbation that results exerts a torque on the planet. For a disc whose surface density falls as the three-halves power of radius, the vortensity is flat, and this part of the torque vanishes. That is exactly the slope of the minimum-mass nebula, which is why the nebula every textbook starts from carries no vortensity torque at all.

The second is the entropy. A parcel making a U-turn is compressed as it approaches the planet and, if it cannot radiate on the timescale of the turn, it keeps its entropy. Gas brought from a region of higher entropy into one of lower entropy is hotter and less dense than its new surroundings at the same pressure, and the density asymmetry this produces in front of and behind the planet exerts a second torque. Its sign is set by whether the entropy falls or rises with radius — which depends on whether the temperature falls faster than the density.

Three torques, and which one decides the sign. The torque on a low-mass planet in three model discs, split into the Lindblad torque and the two parts of the horseshoe torque, in units of Γ₀/γ, unsaturated, for γ = 1.4. minimum-mass nebula: Lindblad -3.20, vortensity +0.00, entropy −0.56, total −3.76; viscously heated inner disc: Lindblad -3.97, vortensity +0.99, entropy +3.72, total +0.74; irradiated outer disc: Lindblad -3.12, vortensity +0.47, entropy +0.00, total −2.65. The Lindblad torque is always negative and changes little between the models, since it depends on the gradients only weakly. The two horseshoe torques change a great deal: the vortensity part vanishes for Σ ∝ r^−3/2, which is why the minimum-mass nebula has none, and the entropy part is set by whether the entropy falls outward, which it does when the temperature falls faster than the density can compensate. The residual whose sign decides the direction of migration is therefore decided by the smaller, more fragile torques, not by the larger one.
Fig. 2 The three parts of the torque in the three model discs. The Lindblad torque is negative in every case and barely changes: it depends weakly on the gradients because the wakes are launched well away from the planet, where a few per cent of difference in the local slope is a small effect. The two horseshoe parts change a great deal. In the minimum-mass nebula the vortensity part is zero and the entropy part is slightly negative; in the viscously heated disc both are positive and the entropy part alone nearly cancels the Lindblad torque; in the irradiated disc the entropy gradient is flat and only a small vortensity torque is left.

The consequence is that the sign of migration is decided by the smaller, more fragile parts of the torque. The Lindblad torque sets a baseline that is always inward and of known size, and the horseshoe torque either adds to it or overwhelms it depending on thermodynamics that the wakes are almost blind to.

Why the minimum-mass nebula is the worst case

The minimum-mass nebula is a model of the Sun’s disc reconstructed by taking each planet’s heavy elements, adding enough hydrogen and helium to restore solar composition, and spreading the result in an annulus around the planet’s orbit. It has a surface density falling as the three-halves power of radius and a temperature falling as the square root — the temperature of a thin disc lit by the star at a grazing angle.

In that disc the vortensity is flat, so its horseshoe torque is zero, and the entropy falls only slowly outward, so its entropy torque is small and negative. Every part of the total points inward. That is why the first calculations of disc migration, which used this disc because it was the one available, found planets falling in so fast.

A disc that is accreting onto its star is different. Accretion releases orbital energy as heat, mostly close to the star, and a disc heated that way has a temperature falling steeply with radius and a surface density falling more gently — the product of viscosity and surface density is constant in a steady accreting disc, and viscosity rises with temperature. In the model drawn, the temperature falls as r−0.9r^{-0.9} and the surface density as r−0.6r^{-0.6}. The entropy then falls outward steeply, and gas carried inward across the planet’s orbit on the back leg of the horseshoe is hotter and thinner than its surroundings. The resulting torque is positive, large and outward, and it more than cancels the Lindblad torque.

So a planet can migrate outward in the inner disc of a young star, and the same planet migrates inward in the outer disc, where starlight rather than accretion sets the temperature. That is not a small correction to the picture of migration; it reverses its direction over a large part of the disc.

Whether the gas can cool during the turn

The entropy torque carries one more condition, and it is the one the previous account of this problem flagged as the pivot: a disc that cools efficiently gives one answer and one that does not gives another.

A parcel making a U-turn is compressed and then released on the timescale of the turn itself, which for gas close to the planet’s orbit is a fraction of the libration period — tens to hundreds of orbits. If the gas can radiate the heat of compression faster than that, it does not keep its entropy. It stays at the local temperature, the density asymmetry that the entropy gradient would have produced never forms, and the entropy torque disappears. The disc behaves isothermally and only the vortensity part of the horseshoe torque is left.

Whether a disc can cool on that timescale depends on its optical depth. Close to the star, where the surface density is high and the dust is opaque, radiation diffuses out of the disc slowly and the gas is effectively adiabatic during a turn: the entropy torque is at full strength. Far from the star, where the disc is thin and cold, radiation escapes almost immediately and the torque is isothermal. The transition happens where the cooling time equals the turn time, and it falls at a few to a few tens of astronomical units in a typical young disc — which is to say across the range where giant-planet cores are thought to form.

This is the same physics seen from the other side. The previous figures used the ratio of specific heats of an adiabatic gas, 1.4, and assumed the gas keeps its entropy through the turn. A disc that radiates efficiently has an effective ratio close to one and a much weaker entropy torque, so the outward region of the opening figure shrinks — and the inner disc, where the steep temperature gradient lives, is also where the adiabatic assumption is best. The outward migration is strongest precisely where the conditions for it are most nearly met, and that coincidence is part of why the mechanism has survived two decades of scrutiny.

A torque that has to be kept alive

There is a catch, and it is the most important physical point about the horseshoe torque. The horseshoe region is closed. Gas on horseshoe orbits goes round and round the same U-turns, and after a few turns it has mixed itself: the vortensity and entropy on the two legs have been swapped back and forth until they are the same everywhere in the region. Then there is no difference between the gas in front of the planet and the gas behind it, and the torque vanishes. This is called saturation.

The only thing that can prevent it is diffusion. Viscosity carries vortensity into the horseshoe region from the disc outside it, and thermal diffusion carries heat in, so if either process is fast enough to restore the gradients before the libration erases them, the torque survives. If diffusion is too slow, the torque saturates to zero. If it is too fast, the gas diffuses across the horseshoe before completing a turn, the horseshoe region never forms, and the torque reverts to a smaller value computed as though the gas passed by rather than turned.

Outward only inside a window of viscosity. The total torque on a planet in the viscously heated inner disc (Σ ∝ r^−0.6, T ∝ r^−0.9, aspect ratio h = 0.05) as the disc's viscosity parameter runs from 10⁻⁶ to 10⁻¹, for planets of 1 and 10 Earth masses, with thermal diffusion taken equal to viscous diffusion. The dashed line is the unsaturated value, +0.74. At low viscosity the gas on horseshoe orbits is mixed by its own libration faster than diffusion can restore the gradients, the horseshoe torque saturates towards zero and only the negative Lindblad torque is left. At high viscosity diffusion wins so completely that the horseshoe region never forms and the torque reverts to the smaller linear value. A 1-Earth-mass planet migrates outward between α ≈ 10^−4.4 and 10^−3.5; a 10-Earth-mass planet migrates outward between α ≈ 10^−2.9 and 10^−2.0: the window moves with mass because the horseshoe's width goes as the square root of the planet's mass and its libration time with it. The viscosity of real discs is not known to better than an order of magnitude, which is exactly the width of these windows.
Fig. 3 The total torque in the accretion-heated disc as the viscosity parameter runs over five decades, for planets of one and ten Earth masses. At low viscosity the horseshoe torque saturates and only the inward Lindblad torque is left; at high viscosity the horseshoe region is washed out and the torque falls back towards the smaller linear value. Between the two there is a window in which the total is positive. For an Earth mass the window sits near a viscosity parameter of 10−410^{-4}; for ten Earth masses it sits near 10−2.510^{-2.5}, because the horseshoe’s width grows as the square root of the planet’s mass and a wider horseshoe takes longer to refill.

The width of those windows is about an order of magnitude in viscosity, and an order of magnitude is roughly how well the viscosity of a protoplanetary disc is known. The number is inferred from the rate at which discs accrete onto their stars and from the time they take to disperse, and it has been revised downward repeatedly as observations of disc turbulence — from the widths of molecular lines and from how thin the dust layers of resolved discs are — have suggested discs are much less turbulent than was assumed. In a disc with a viscosity parameter of 10−410^{-4} an Earth-mass planet can be pushed outward and a ten-Earth-mass core cannot. In a disc with a viscosity parameter of a few times 10−310^{-3} it is the other way round.

Outward only inside a window of viscosity. The total torque on a planet in the minimum-mass nebula (Σ ∝ r^−1.5, T ∝ r^−0.5, aspect ratio h = 0.05) as the disc's viscosity parameter runs from 10⁻⁶ to 10⁻¹, for planets of 1 and 10 Earth masses, with thermal diffusion taken equal to viscous diffusion. The dashed line is the unsaturated value, −3.76. At low viscosity the gas on horseshoe orbits is mixed by its own libration faster than diffusion can restore the gradients, the horseshoe torque saturates towards zero and only the negative Lindblad torque is left. At high viscosity diffusion wins so completely that the horseshoe region never forms and the torque reverts to the smaller linear value. A 1-Earth-mass planet never migrates outward here; a 10-Earth-mass planet never migrates outward here: the window moves with mass because the horseshoe's width goes as the square root of the planet's mass and its libration time with it. The viscosity of real discs is not known to better than an order of magnitude, which is exactly the width of these windows.
Fig. 4 The same calculation in the minimum-mass nebula. There is no window at all: both horseshoe parts are zero or negative there when unsaturated, so saturation can only remove a contribution that was not helping, and at every viscosity both planets migrate inward. The whole outward window of the previous figure exists because of the steep temperature gradient of a disc heated by accretion, and a disc model that does not have one cannot produce it at any viscosity.

This is the part of the subject that makes the question of migration a question about disc physics rather than about planets. The same planet in the same place is pushed outward or inward depending on a turbulent viscosity that is not measured in any disc, and on a heating profile that is modelled rather than observed.

A radius the planets converge on

The torque’s dependence on the local gradients has a consequence that turns an embarrassment into a mechanism. The gradients change with radius. Close to the star a disc is heated mainly by accretion; further out it is heated mainly by starlight, and the transition between the two regimes is a place where the temperature slope changes. So is the ice line, where the opacity changes abruptly because the dust grains acquire ice mantles, and so is the edge of the region where the gas is ionised enough to be turbulent.

A radius the planets converge on. The unsaturated total torque on a low-mass planet against distance from the star, in a disc whose inner part is heated by its own accretion (Σ ∝ r^−0.6, T ∝ r^−0.9) and whose outer part is heated by starlight (Σ ∝ r^−1.07, T ∝ r^−0.43), with the transition centred at 5 AU. Inside, the torque is positive and planets move outward; outside, it is negative and they move inward; the sign changes at 4.3 AU. A zero that is positive on the inside and negative on the outside is convergent: planets arriving from either side stop there, so a change in how the disc is heated makes a trap. Where real discs put such transitions — the ice line, the edge of the region where the gas is ionised enough to be turbulent, the inner edge — and how sharp they are, sets where traps sit; the figure's point is that the trap is made by the temperature gradient's change of slope, not by anything that stops the planet.
Fig. 5 The unsaturated torque against distance from the star in a disc whose inner part is heated by accretion and whose outer part is heated by starlight, with the transition centred at five astronomical units. Inside, the torque is positive; outside, negative; the sign changes at 4.3 astronomical units. A planet inside moves outward to the zero and a planet outside moves inward to it. A zero with that orientation is stable, and it is where planets collect.

The transition drawn is between two heating regimes, but the same arithmetic applies to any place where a slope changes. The ice line is one, since the dust opacity drops by a large factor where the grains lose their ice and the temperature profile bends there; there are several such lines rather than one, each for a different volatile. The inner edge of the region where the gas is too weakly ionised for the magnetorotational instability to make it turbulent is another, and there the viscosity itself jumps, which changes the saturation of the horseshoe torque as well as the surface-density slope.

That makes a trap, and traps change the arithmetic of the problem. A core that forms anywhere within a few astronomical units of a trap drifts to it and stops, rather than drifting into the star. Several cores arriving at the same trap meet each other there, which is where they can capture into resonance — and a chain of planets in successive resonances is exactly what convergent migration into a common stopping point should produce. The chains that cannot have been assembled in place are the strongest evidence that planets migrated, and traps are one of the leading candidates for where they were assembled.

A trap also moves. As the disc accretes and thins, its heating by accretion weakens, the transition moves inward, and the planets sitting in the trap are carried inward with it on the disc’s own timescale rather than on the much faster timescale of free migration. That is a mechanism for delivering planets to short periods slowly, which is what the hot planets that cannot have formed where they are require of any explanation.

What the formulae are and are not

Every number in these figures comes from fitting formulae calibrated against two-dimensional hydrodynamical simulations of a planet in a disc, which give the three torque parts in terms of the two slopes, the ratio of specific heats and two saturation parameters. They are good to perhaps twenty per cent in the regime they were fitted in, and they are what every population-synthesis model of planet formation uses.

They leave out three things that matter. The first is the third dimension: a real disc has vertical structure, and the horseshoe region of a low-mass planet is narrower than the disc is thick, so the flow around the planet is not two-dimensional. Three-dimensional simulations find the torques broadly consistent but not identical. The second is the planet’s own heat. A growing planet accretes solids and releases their energy, and a hot planet heats the gas around it; simulations that include this find an additional positive torque — a thermal torque — that can dominate for planets below a few Earth masses and is not in the formulae drawn here. The third is non-linearity. The formulae assume the planet is small enough that the disc responds linearly, and above about ten Earth masses in a typical disc that fails as the planet starts to carve the gas away from its orbit, which changes the horseshoe region’s width and eventually removes it.

And they give a torque, not a history. Where a planet ends up depends on how the disc’s slopes change with radius and time, and those come from a disc model — in which the heating, the opacity and the viscosity are all inputs. The formulae are the most reliable part of a calculation whose other parts are much less reliable.

What the picture cannot show

The figures draw a single planet in a disc described by power laws. Real discs observed at high resolution are not power laws: they have rings, gaps and spirals on scales of a few astronomical units, and each ring edge is a place where the surface-density slope is locally steep and can change sign. A ring edge is a trap of the vortensity kind rather than the entropy kind, and it may be the more common kind; nothing in the smooth models above can show where they are.

The torque is also never measured. No planet in a disc has had its migration rate or direction observed; the torques are inferred from simulation, and the simulations are checked only against each other and against the statistics of planetary systems billions of years after the disc has gone. The trap drawn at 4.3 astronomical units is a consequence of the model’s heating profile, and a different profile with the same physics would put it somewhere else.

Still open: the viscosity the whole sign depends on

Every figure here has the viscosity in it, and every conclusion about direction turns on it. The saturation window moves by an order of magnitude in viscosity between an Earth mass and ten; the entropy torque needs thermal diffusion to be comparable to viscous diffusion; the trap’s existence depends on the horseshoe torque being unsaturated. Observations of disc turbulence have been driving the inferred viscosity down for a decade, and in a disc with very low viscosity the horseshoe torque saturates for almost every planet mass, the outward windows close, and migration is inward everywhere except where the disc itself is structured. Whether planets are trapped by their discs’ thermodynamics or only by their discs’ rings is the question the next generation of disc observations — resolving the gas rather than the dust — is set up to answer.

The other route bypasses the gas altogether: a planet delivered to its star on an eccentric orbit, long after the disc has gone, leaves an inner edge at a place set by its own radius rather than by anything a disc did.

Where this goes next

An edge where two torques balance, 21 kilometres from the shepherd. Two torques on the edge of a ring, against distance from a shepherding moon, both axes logarithmic and both scaled by the same combination of surface density, radius and orbital rate so that only their shapes are being compared. The flat line is the viscous torque, which comes from collisions between ring particles and does not care how far away anything is; it is drawn for a kinematic viscosity of 12 square centimetres a second, within the range ring seismology gives. The falling line is the moon's, summed over the first-order resonances that crowd together as the gap narrows, which makes it an inverse cube. A flat curve and an inverse cube cross once, and the crossing is where an edge can sit: closer in the moon wins and pushes the material back, further out viscosity wins and the ring spreads. For Daphnis and the Keeler gap the balance lands 20.8 kilometres out against a measured half-width of 21, which is agreement to well inside the uncertainty on the viscosity — and it is the only handle anybody has on that viscosity, since the quantity being inferred is the collision rate among particles a metre across, a billion kilometres away. A torque that nearly cancels A planet embedded in a gas disc pulls on the material inside its orbit and outside it, and the two torques are almost equal and opposite. What survives the subtraction is a per cent of either, and it is still enough to carry a planet from where it formed to its star in less time than the disc lasts. The five Lagrange points at mass fraction 0.12. The five points at which a small body can keep station with two larger ones. The three on the line of centres are roots of a quintic and are unstable; the two forming equilateral triangles are stable for a sufficiently lopsided mass ratio. Five places that keep station, in a problem with no solution Three bodies under gravity cannot be solved. Restrict the problem slightly and five exact answers fall out anyway — three of them roots of a quintic, two of them perfect equilateral triangles. Libration and circulation of a resonant angle. The critical angle of a resonance over time, integrated from the pendulum equation it obeys. Below the separatrix the angle oscillates about a fixed value — the body is locked; above it, the angle runs without bound and the body is not. A chain that could not have been assembled in place Seven planets whose successive period ratios are all close to small whole numbers. Capture into a resonance requires the orbits to converge slowly, and converging slowly is something planets can only do in a disc.

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Angular momentumCorotation torqueEntropyHorseshoe orbitLindblad resonanceMigration trapPlanet migrationProtoplanetary discSurface densityTorqueViscosity