A chain that could not have been assembled in place

The seven planets of TRAPPIST-1 have periods of 1.51, 2.42, 4.05, 6.10, 9.21, 12.35 and 18.77 days. Take the ratios of neighbours: 1.603, 1.672, 1.506, 1.509, 1.342, 1.519. Every one is within about two per cent of the ratio of two small integers — 8:5, 5:3, 3:2, 3:2, 4:3, 3:2.

Six consecutive near-commensurabilities is not something that happens by chance. It is a record of how the system was put together, and it says the planets arrived where they are by moving.

A resonance is a pendulum #

The useful way to think about a mean-motion resonance is not “the periods have a ratio of 3:2” but “a particular angle oscillates instead of circulating”.

For a $j:(j-1)$ resonance the angle is

$$\phi = j\lambda_{\text{out}} - (j-1)\lambda_{\text{in}} - \varpi,$$

built from the two mean longitudes and the longitude of periapsis. Away from resonance, $\phi$ increases steadily; the geometry of the pair drifts through every configuration and the mutual perturbations average away. In resonance, $\phi$ oscillates about a fixed value — it librates — and the pair repeatedly returns to the same relative configuration.

That is the whole difference, and it has an enormous consequence. A conjunction that happens at a random place each time contributes a random kick. A conjunction that happens at the same place every time contributes the same kick, over and over, and the effect accumulates.

The Kirkwood gaps are the destructive version of this: asteroids at a resonance with Jupiter receive repeated kicks in a consistent direction and their eccentricities grow until they cross a planet’s orbit. The Galilean moons are the constructive version: Io, Europa and Ganymede have periods in a 1:2:4 ratio and have kept them for billions of years.

Capture requires slow, convergent, damped approach #

Here is the part that turns a pattern into a history.

A pair of planets drifting apart passes through a resonance and out the other side; the resonant angle circulates faster and faster and there is no chance to lock. A pair drifting together approaches the resonance from outside, and the libration slows to zero as the separatrix is crossed — and if the crossing is slow enough, the pair is captured. The mathematics is an adiabatic invariant: the area enclosed by the trajectory in phase space is conserved when the parameters change slowly, and that conservation is what traps the system inside the librating region.

The requirements are specific:

• Convergent. The orbits must approach each other, not separate. Two planets migrating independently in a disc converge if the inner one moves inward more slowly than the outer, which is the generic case for planets of comparable mass.
• Slow. The drift must be slow compared with the libration period, or the pair sails through. “Slow” here means the migration timescale must exceed the resonant libration timescale, which for these systems is tens to hundreds of orbits.
• Damped. Something must remove the eccentricity that resonance pumps up, or the pair is driven back out of the resonance almost immediately.

Only one environment supplies all three: a gas disc. The gas drives migration, sets its rate, and damps eccentricity — all three, from one agent.

So a resonant chain is a fossil of the disc phase. It is not merely evidence that migration happened; it is evidence that migration happened slowly, convergently, and while gas was still present.

What holds a chain together once the gas has gone #

A resonance is not merely a configuration a system passes through; it is a restoring force. That is the second thing the pendulum picture supplies, and it is why the survivors survive.

Inside the librating region, a small displacement of the resonant angle produces a torque pushing it back. So a chain resists being pulled apart: a perturbation that would disrupt a non-resonant system merely sets the angle oscillating with a larger amplitude. The system is protected by the same mechanism that trapped it, and the protection extends to close approaches — planets in resonance can be packed more tightly than non-resonant ones without going unstable, because the resonance guarantees that conjunctions never happen at the dangerous place.

The clearest demonstration is Pluto, which crosses Neptune’s orbit and has done so for billions of years without incident. The 3:2 resonance ensures that Pluto is always near aphelion when Neptune is closest in longitude, so the two are never within 17 AU of each other. A geometry that looks lethal on a plan view is stable precisely because the timing is locked.

The same protection is what makes TRAPPIST-1 possible. Its seven planets are separated by only a few per cent in orbital distance — packed tightly enough that a non-resonant system of the same masses would go unstable in millions of years. The chain is not an ornament; it is what the system is standing on.

The chain is measured by the timing, not by the periods #

The periods above are known to many decimal places, but the resonance is not established by taking their ratios. Two planets can have a period ratio of exactly 1.5 and not be in resonance at all — what matters is whether the angle librates, and that is a statement about the dynamics rather than about the arithmetic.

The evidence comes from the transit times. In a resonant chain the mutual perturbations are large and coherent, so the timing variations are correspondingly large: minutes to tens of minutes, with periods of months to years.

The model being fitted is a direct integration of all eight bodies, because the many-body problem has no closed solution and a chain is about as far from the two-body case as a planetary system gets. Fitting it to those times gives the masses, the eccentricities and the libration amplitudes together. For TRAPPIST-1 the fit uses hundreds of transit times across seven planets and delivers all seven masses to a few per cent — which is the only way they could have been obtained, the star being far too faint for precision velocities.

What the chain paid for #

A resonant chain is worth something beyond its history, and TRAPPIST-1 is the demonstration.

Because the seven planets perturb each other strongly and all seven transit, one N-body fit to the transit times delivers seven masses. Because all seven transit, the same photometry delivers seven radii. So the system yields seven densities — the measurement that requires two independent methods everywhere else — around a star far too faint for a spectrograph to have weighed anything at all. The uniformity is itself an argument. Seven planets of nearly identical composition, spanning a factor of twelve in orbital distance, are not seven independent accidents; they are one batch of material distributed by a process that did not sort it. That is what migration through a disc would do, and what in-place formation from locally available solids would not.

Chains are common, and mostly broken #

The most informative fact about resonant chains is not that they exist but how many have come apart.

Only a few per cent of compact multi-planet systems are in resonance today. But the period-ratio distribution of all such systems has a distinctive shape: a modest excess just wide of the 3:2 and 2:1 ratios, and a deficit just narrow of them. That asymmetry is what a population of chains looks like after they have been disrupted and allowed to spread slightly outward — by tides, by leftover planetesimals, or by the resonance itself becoming unstable once the damping gas is gone.

So the current reading is that most compact systems formed in chains and most chains broke. The surviving ones — TRAPPIST-1, Kepler-223, TOI-178, Kepler-60 — are the minority that stayed intact, and they are correspondingly precious, because a broken chain records that migration happened while an intact one records how.

There is one more thing the broken chains say, and it is about timing rather than about dynamics. A chain that dissolves does so on a timescale set by how close to the separatrix it was left, and the systems observed today have had billions of years to do it. So the surviving chains are not a random sample of the ones that formed: they are the ones captured deepest, which by the argument above are the ones whose discs dispersed most slowly. The population of intact chains is a biased record of disc lifetimes, and the bias runs in a direction that can be computed.

That is a satisfying inversion of the usual situation. The rare configuration is the informative one, and the common one is what the rare configuration becomes.

It also sharpens what the eccentricity distribution of hot Jupiters is saying. Both are arguments from a population to a process, and both rely on catching systems in a transient state — mid-circularisation there, still-intact here. The difference is which way the erasure runs: tides destroy the evidence of violent migration, while the disruption of chains destroys the evidence of gentle migration. Neither population would look like this if the relevant process had finished everywhere.

What was actually measured #

Kepler-223, 2016. Four planets in a chain of 3:4, 2:3 and 3:4 resonances — the first system for which the resonant angles were shown to librate rather than merely to have suggestive ratios. The libration was established from transit timing, and the paper’s argument is explicitly that the configuration is reachable only by slow convergent migration.

TRAPPIST-1, 2017–2021. Seven planets, six near-commensurabilities, and a three-body Laplace-like relation linking each consecutive triple: the generalised angles

$$\phi_{i} = p\,\lambda_{i} - (p+q)\,\lambda_{i+1} + q\,\lambda_{i+2}$$

librate with small amplitudes for every triple in the chain. That is a far stronger statement than six pairwise ratios, and it is what makes the system the strongest migration evidence known.

TOI-178, 2021. Six planets in a 2:4:6:9:12:18 chain of period ratios, found with CHEOPS. The system’s innermost planet is not in the chain, and the densities do not vary monotonically with distance — a system that migrated in an orderly way and is not otherwise orderly at all.

The Galilean moons, since 1676. Io, Europa and Ganymede satisfy $n_{\text{Io}} - 3n_{\text{Eu}} + 2n_{\text{Ga}} = 0$ to about one part in $10^{9}$, and the corresponding angle librates about 180° with an amplitude of 0.064 degrees. The system was assembled the same way, in a circumplanetary disc rather than a circumstellar one, and it is the calibration case for everything above — the one chain whose members can be visited.

Which angle is the chain #

The generalised angles above are not a refinement of the pairwise ones. They are a different claim, and in TRAPPIST-1 they are the claim that holds.

Take any three consecutive planets. Each neighbouring pair has its own two-body resonant angle, and for several pairs in this system those angles circulate rather than librate — the pair sits near a commensurability without being locked to it. The three-body angle built from the same three longitudes nevertheless librates, with an amplitude of a few degrees, for every consecutive triple in the chain.

There is no contradiction in that. Two adjacent pairwise angles circulating slowly at nearly the same rate combine into a difference that does not circulate at all, in the way two clocks each running fast by the same amount keep time with one another. The three-body relation says that the drift rates are locked even where the individual configurations are not.

Which changes what the chain is evidence of. A single pairwise resonance can be produced by tides acting on one pair long after the gas has gone; six linked three-body relations cannot, because they require the whole system to have been assembled against itself rather than each pair against its own neighbour. The chain is one dynamical object, and the argument for slow convergent migration is carried by the links rather than by the ratios.

It also disposes of an objection the opening paragraph invites. Every period ratio quoted there is a per cent or two wide of exact commensurability, which looks like a chain that has already begun to come apart. Exactness was never the requirement: what is required is that a particular combination of angles stay bounded, and a set of orbits displaced slightly from the nominal ratios can satisfy that while no single pair sits at the bottom of its own resonance. Being near a resonance and being in one are different conditions, and the second is the one the timing measures.

Where the picture stops #

A near-commensurability is not a resonance. Period ratios cluster near small integers partly because of resonance and partly because compact systems are packed to a similar dynamical spacing, which produces ratios in the same range without any angle librating. Distinguishing the two needs the timing variations, and for most systems those are unmeasured.

Capture is probabilistic. Whether a pair is caught depends on the drift rate, the eccentricities at the time and the phase at the encounter. Simulations produce capture in most, but not all, cases with plausible parameters — so a system that is not in a chain is not evidence that it did not migrate.

Chains break, and the mechanism is not settled. Tidal dissipation on the innermost planet, interaction with a residual planetesimal disc, and instability at the moment the gas disperses are all candidates, and the observed period-ratio distribution does not clearly select one.

Libration amplitudes are hard to measure and are the real quantity of interest. A small amplitude means deep capture and a quiet history; a large one means the system has been knocked about since. The amplitudes come out of the same N-body fits that give the masses, and they are the least well constrained parameters in them — so the statements about how quietly a chain formed are weaker than the statements about whether it formed at all.

And a chain constrains the disc, not the whole history. It records the epoch during which gas was present and the planets were converging. What happened before — how the cores were assembled, and where — is untouched by this evidence.

Seven transits, and why they all happen #

One further thing the chain explains is why the system is visible at all in this much detail.

Seven planets transit the same star, which requires seven orbital planes aligned with each other to within about a tenth of a degree as seen from here. A random set of inclinations would show one or two transiting planets at most; a flat system either transits nearly as a unit or not at all, and the orientation of an orbit in space takes three angles of which only one is set by the transit geometry.

Flatness is exactly what disc migration produces, and for the same reason as the resonances: the gas that damped the eccentricities damped the inclinations too. So the chain and the coplanarity are two consequences of one history, and the system’s extraordinary observational richness — seven transits, seven sets of timing variations, seven densities — is a direct dividend of it having formed quietly.

The generalisation #

Resonant capture as a fossil of slow drift appears wherever a system has been swept through a commensurability.

The Galilean moons were captured as tides raised on Jupiter pushed them outward at different rates. Neptune’s migration outward through the Kuiper belt captured Pluto and the plutinos into the 3:2, and the number of objects caught measures how slowly Neptune moved — one of the strongest constraints on the outer solar system’s history. The Cassini division in Saturn’s rings is a 2:1 with Mimas.

The common structure is that a resonance is a trap with a narrow entrance: it can only be entered slowly and from one direction, so finding something inside it is a statement about the approach rather than about the destination. That is why an object in a resonance is worth so much more than an object merely near one.

One more sweep rate sits between the two the essay has drawn.

Where this goes next #

Everything so far has been dynamics — masses, orbits, histories. What none of it touches is what these planets are actually like, and that requires a completely different measurement: the light that passes through an atmosphere on its way past.

Later rungs on this anchor: resonant capture and the adiabatic invariant. The Laplace resonance derived. Three-body resonant angles. Chain breaking, and the period-ratio distribution. Resonant chains as constraints on disc lifetimes. Secular resonances, which involve precession rates rather than periods. The plutinos and Neptune’s migration. Ring gaps and shepherd moons. And whether the solar system was ever in a chain, which the Nice model says it was.