A torque that nearly cancels

A planet exists where one cannot form, and the first rung of this anchor established the problem: hot Jupiters sit at a hundredth of an astronomical unit, where the disc is too hot for ice and too sparse for enough solids, so they must have arrived from further out. This rung is about the mechanism, and about why it is embarrassingly efficient.

The mechanism is a torque between the planet and the gas it is embedded in. What makes it interesting is that the torque is a difference between two much larger numbers, so its sign and size are decided by quantities that would otherwise be third-order details of a disc model. A calculation whose answer is a one per cent residual is a calculation in which everything matters.

Two torques, and the subtraction #

A planet on a circular orbit in a disc raises a spiral wake in the gas, and the wake exerts a torque back on the planet. The calculation is done resonance by resonance: at each radius where the gas’s orbital frequency and the planet’s are in a simple ratio, the perturbation is coherent and angular momentum is exchanged.

Material inside the planet’s orbit goes round faster, so the planet pulls it back and it pushes the planet forward: an outward torque. Material outside goes round more slowly, so the planet drags it forward and it pulls the planet back: an inward torque.

The two are almost equal. Each is enormous — enough to move a planet across the disc in a few hundred orbits — and their sum is a per cent or two of either, with a sign that depends on the disc’s density and temperature gradients. Because the residual is a small difference of large quantities, its sign and magnitude are sensitive to everything: the surface density profile, the temperature profile, whether the disc can radiate, and whether the gas immediately around the planet is co-rotating with it. That sensitivity is the whole difficulty of the subject.

Why the timescale is a problem #

Put numbers in. For an Earth-mass planet at five astronomical units in a disc of the mass the solar system is thought to have formed from, the residual torque moves it inward on a timescale of a few hundred thousand years. The disc lasts a few million.

So a planet that reaches an Earth mass early in the disc’s life should reach the star before the disc disperses. The migration rate scales with the planet’s mass, so a ten-Earth-mass core migrates ten times faster still.

That is not what is observed. Planetary systems exist, with planets at every separation, and the cores that would become giant planets have to survive long enough to accrete gas. The theory as stated predicts that they do not.

What stops it #

Several mechanisms have been proposed and the most robust is the one the hero figure draws: the planet stops migrating smoothly when it becomes massive enough to clear a gap around itself.

The criterion is the same balance as a shepherd moon’s. The planet’s torque tries to push gas away; the disc’s viscosity and pressure try to fill the gap in. Above a critical mass the torque wins, a gap opens — the same way a resonance clears a gap in the asteroid belt and locks a moon elsewhere — and the planet is no longer embedded in a smooth disc — it sits in a hole, with material on both sides that can only reach it slowly.

Once a gap is open the migration changes character. The planet is locked to the disc’s own viscous evolution rather than sliding through it, which is much slower, and the direction depends on which way the disc is spreading.

The critical mass depends on the disc’s viscosity and thickness, and for a typical protoplanetary disc it comes out around a Saturn mass. Below that a planet migrates fast; above it, slowly. That is a mechanism for stopping giant planets and not for stopping the cores that have to grow into them, which is the part still unresolved.

The other ways out #

Three further mechanisms are worth naming because each solves a different part of the problem.

The first is a trap. If the disc has a place where the density or temperature gradient reverses — at the inner edge, at the boundary between magnetically active and inactive regions, or at the ice line — the residual torque changes sign there, and a planet arriving from either side stops. The second is the disc’s dispersal. A disc does not fade uniformly; it is cleared from the inside out by photoevaporation, opening a hole that expands. A planet outside the hole stops migrating when the gas inside it is gone.

The third is that the planets stop each other. Two migrating planets converging in period ratio are convergently migrating, and convergent migration captures into resonance. A captured pair migrates as a unit, more slowly than either would alone, and a chain of three or four is slower still.

The regime that runs away #

There is a third mode of migration between the two described above, and it is the one that behaves worst.

A planet massive enough to perturb the disc substantially but not massive enough to clear a gap sits in an awkward intermediate state: it has carved a partial gap, so the surface density around it is no longer smooth, and the gas on horseshoe orbits in its co-orbital region is being processed at a rate comparable to the planet’s own migration.

In that situation the torque acquires a feedback. As the planet migrates, material passes across its orbit from one side to the other, and the mass flux through the co-orbital region exerts a torque proportional to how fast the planet is already moving. If the coefficient exceeds one, the migration accelerates itself.

The result is runaway: a planet can cross a substantial fraction of the disc in a few tens of orbits, in whichever direction it happened to be drifting when the feedback engaged. It is called Type III migration, it is fast enough to be an instability rather than a drift, and its direction is set by an initial condition rather than by the disc’s gradients.

Two things make it hard to assess. It requires the co-orbital mass deficit — the difference between what is in the partial gap and what would be there in a smooth disc — to exceed the planet’s own mass, which is a condition on the disc’s structure rather than on the planet. And it is a numerical result: the linear theory that gives the Type I torque cannot contain it, because the feedback is intrinsically nonlinear, so its existence and its rate come from simulations whose resolution near the planet is the limiting factor.

A migration mode whose speed depends on the speed it already has is not a correction to the others, and whether real planets pass through it decides whether the intermediate mass range is a place a planet can linger or a place it crosses in a hurry.

The number the whole thing turns on #

It is worth writing the residual out, because the smallness is the point and the smallness is not an accident.

The torque from a single Lindblad resonance carries the square of the planet’s mass, the local surface density, and a function of the disc’s aspect ratio. Summing the inner resonances gives a positive number; summing the outer gives a negative one of nearly the same size. The reason they nearly cancel is geometric: the resonances are placed symmetrically about the planet in frequency, and the disc’s properties vary slowly across the region they occupy.

What breaks the symmetry is that the outer resonances sit slightly closer to the planet in radius than the inner ones do, because of the way the epicyclic frequency runs, and that the surface density and temperature are not constant. Both effects are of order the disc’s aspect ratio, which is a few per cent — so the residual is a few per cent of either sum, and its sign depends on gradients rather than on values.

That is why the subject is so sensitive to disc models. A change in the temperature profile that would be invisible in any observation flips the sign of the residual, and with it the direction a planet moves.

What was actually measured #

Nothing about the torque. Protoplanetary discs are observed, and planets in them are being observed for the first time, but the exchange of angular momentum between them is inferred entirely from structure.

The strongest evidence is the rings and gaps in discs themselves. Submillimetre imaging of nearby young stars shows discs with concentric gaps, at radii of tens of astronomical units, in objects only a million years old. The natural interpretation is that planets have opened them, and the gap widths imply masses of tens of Earth masses.

That interpretation is not forced. Gaps can also be produced by condensation fronts, by magnetic effects, and by instabilities in the dust rather than the gas. The discriminant would be detecting the planet, and two have been detected so far, in one system. The third piece is the resonant chains, and their fragility is informative. Chains are common in young systems and rare in mature ones, which says migration happened and something later broke most of its products.

Reading a mass off a gap #

The resolved discs turned the subject from a theoretical one into an observational one, and it is worth setting out how a gap becomes a planet mass, because the chain is short and every link is contested.

A gap’s width is set by the balance the hero figure draws: the planet’s torque opening it against the disc’s viscosity and pressure closing it. Wider gap, more massive planet — with the width depending also on the disc’s aspect ratio and its viscosity, neither of which is measured directly.

A gap’s depth is set by the same balance and by how much material can flow through the gap region. Deeper gap, more massive planet, with the same two unmeasured quantities entering differently.

So width and depth give two equations, and with three unknowns — planet mass, viscosity, aspect ratio — the system does not close. What is done in practice is to assume a viscosity, which is the least well determined of the three, and read off a mass with an uncertainty that spans an order of magnitude.

The masses obtained that way for the rings in the best-studied discs come out between a few Earth masses and a Jupiter, depending on the assumption. That range is wide enough to include and exclude several of the interesting possibilities, which is why the direct detections matter so much: one planet imaged in a gap calibrates every gap in that disc.

There is a further complication that has emerged as the images improved. Some discs have gaps at radii where nothing plausible could have formed and where the migration timescale from anywhere else is longer than the disc’s age. Those are read either as evidence for a mechanism other than a planet — a condensation front, a magnetically dead zone — or as evidence that the discs are older than their apparent ages, and both readings have consequences for everything else in this essay.

There is a statistical version of the same question that avoids committing to any one disc. Count the gaps across a large sample and ask whether their radii cluster: planets should be distributed smoothly, while a condensation front sits at a temperature and therefore at a radius that scales with the star’s luminosity. The observed radii do not obviously cluster once the luminosity scaling is applied, which is weak evidence for planets and is a long way from a demonstration.

The comparison that would settle it is between the gap radii and the stellar luminosities in a sample large enough to have a range of both, and the samples are only now becoming large enough to attempt it.

The awkward summary #

The state of the subject is worth stating plainly rather than smoothing over, because it is unusual.

The mechanism is not in doubt: a planet in a gas disc exchanges angular momentum with it, the calculation of how much has been done many times by different methods, and the answers agree. The problem is that the answer is too large. A theory that predicts migration timescales shorter than disc lifetimes by an order of magnitude, in a universe full of planets that did not migrate all the way in, is a theory with something missing rather than a theory that is wrong.

What is missing is most likely in the co-orbital region — the horseshoe of gas that shares the planet’s orbit, whose contribution to the torque is not a small residual and whose behaviour depends on how efficiently that gas can radiate. Torques from that region can be positive, and calculating them requires radiation hydrodynamics rather than a resonance sum. There is a second reason to state the difficulty this way rather than as a failure. The migration calculation is one of the few places in planet formation where the physics is clean enough to give a number with no free parameters, and the number disagrees with the world. That is a far more useful situation than a calculation with three tunable constants that can be made to agree with anything.

What the disagreement has produced, over two decades, is a list of specific things that must be true: the disc must have structure, or the co-orbital region must contribute more than the resonances do, or planets must form later than assumed, or all three. Each of those is testable against observations that did not exist when the problem was posed — the resolved discs, the resonant chains, the radius distribution — and each of them is being tested now. A theory that was merely approximately right would have generated none of that, and the list would still be a list of adjustable choices rather than of consequences.

Worth adding, because it is the part most often left out of the summary: the difficulty is not that the residual is small but that it is small and the two large numbers are known well. Each Lindblad torque can be computed to a few per cent from linear theory, checked against numerical hydrodynamics, and it agrees. So the residual is not lost in the uncertainty of the pieces; it is a genuine, computable quantity whose value happens to depend on gradients that a disc model has to supply and observations cannot. That is a different situation from a calculation that is merely imprecise, and it points somewhere specific. The place it points is the corotation region, where the torque is not a sum over resonances at all but an average over material on horseshoe orbits, and where the answer depends on whether that material can radiate away the heat it gains as it turns. A disc that cools efficiently gives one answer and one that does not gives another of the opposite sign, and the transition between the two regimes happens at radii and surface densities that real discs occupy.

Where the ladder goes #

The next rung is the co-orbital torque itself, which is where the residual’s sign is actually decided and which requires the disc’s thermodynamics rather than its geometry.

The other direction is the alternative route entirely: high-eccentricity migration, in which a planet is delivered by a distant companion driving its eccentricity up and tides circularising it afterwards. That mechanism needs no gas, works long after the disc is gone, and predicts misaligned orbits — and it is switched off entirely by the inner orbit’s own relativistic precession in part of the parameter space, which is why the observed population contains both aligned and misaligned hot Jupiters.