Capture is a direction, not a strength

A chain of planets in resonance cannot have been assembled where it sits, and the previous rung of this anchor made that argument from the improbability of the arrangement. This rung makes it from the mechanism, and the mechanism turns out to have a property that is stronger than a probability: capture is impossible in one direction.

The pendulum underneath #

A mean-motion resonance is a pendulum, and this collection has drawn the pendulum before. The resonant angle — a particular combination of the two bodies’ longitudes and pericentres, chosen so that it is slowly varying when the periods are commensurate — obeys

$$\ddot\varphi + K\sin\varphi = 0$$

to leading order, with $K$ set by the perturbing mass and by how close the period ratio is to the exact commensurability. Deriving that reduction is the business of averaging away everything that is not slowly varying, and what survives the averaging is one angle out of the six an orbit has. Below a critical energy the angle oscillates about zero and the body is locked; above it the angle runs and the body is not.

Area, and why it is conserved #

The key idea is not about resonances at all. A pendulum whose stiffness is changed slowly conserves the area its trajectory encloses in the plane of angle against angular velocity. That is adiabatic invariance, and “slowly” means slowly compared with the oscillation period rather than compared with anything else.

The area enclosed by the separatrix — the boundary between librating and circulating orbits — goes as the square root of the stiffness. So as the resonance strengthens, the separatrix’s area grows while the trajectory’s area does not, and the trajectory finds itself further inside the boundary than it was. As the resonance weakens, the separatrix shrinks, and eventually it shrinks past the trajectory.

The consequence has a sign built into it. A body already librating stays librating when the resonance strengthens and stops when it weakens. A body circulating just outside can be swept up when the separatrix grows past it and can never be swept up when it shrinks.

That is capture, and it is one-way.

Why the area is the right thing to track #

It is worth being explicit about why the argument is made in terms of an area rather than an energy, because the energy is not conserved and the area is.

As the resonance strengthens, the body’s oscillation energy rises: a stiffer pendulum swinging through the same angle has more energy. So an energy-based argument would have to account for where that energy came from, and the answer — from the slow change in the parameter — is exactly the thing being treated as external. The area sidesteps that. It is the action of the oscillation, and the theorem says the action does not change under a slow parameter change however much the energy does.

The theorem has a condition, and the condition is what makes capture probabilistic rather than certain in real systems. It holds when the fractional change in the parameter over one oscillation period is small. Near the separatrix the oscillation period diverges — a trajectory infinitesimally inside the boundary takes infinitely long to complete a circuit — so the condition always fails there, however slow the drift. That is why a trajectory being swept across the separatrix has an outcome that depends on its phase at the moment of crossing, and why capture probabilities are numbers between zero and one rather than certainties.

The eccentricity the capture leaves behind #

Capture does not merely lock the period ratio. It changes the eccentricity, and by an amount the same conservation argument fixes.

Inside a resonance the body’s angular momentum is exchanged with the perturber in a way that couples the semi-major axis to the eccentricity, so continued convergent migration after capture drives the eccentricity up. The pair is dragged closer together in period ratio, cannot get closer in ratio because the lock holds, and puts the difference into eccentricity instead.

That is why resonant pairs are eccentric, and it is a check on the mechanism that does not rely on the period ratio at all. It also sets a limit: at high enough eccentricity the resonance’s own stability fails, or another dissipative process damps it, and the system stops.

What makes a resonance strengthen #

Two bodies drift toward each other in period ratio when something removes energy or angular momentum from one relative to the other. In a planetary system the something is usually a gas disc.

A planet embedded in a disc exchanges angular momentum with it at every Lindblad resonance, and the sum of those exchanges is a net torque. The direction and magnitude depend on the local disc profile, but two planets in the same disc generally migrate at different rates because the rate depends on mass and position, and the difference is what matters. When the outer planet migrates inward faster than the inner one, the period ratio falls toward a commensurability and the resonance strengthens: convergent. When the inner one migrates outward faster, the ratio rises away and the resonance weakens: divergent. The first captures; the second cannot.

The chain, and what it records #

A system of several planets locked in a chain of resonances is therefore a record of convergent migration, and there is no other way to make one.

The argument is worth stating as a chain of impossibilities. The planets cannot have formed at those exact period ratios, because formation does not know about ratios. They cannot have arrived at them by divergent migration, because divergent crossing never captures. They cannot have been captured one at a time by chance, because capture requires the separatrix to grow past the trajectory and a stationary separatrix never does. What remains is that they moved toward one another slowly enough for the adiabatic condition to hold, in a medium that provided the drift — and the medium has to have been gas, because nothing else in a young system can absorb that much angular momentum.

Where the argument gets uncomfortable #

Two facts sit awkwardly with it and are worth stating rather than skipping.

The first is that most multi-planet systems are not in resonance. The period ratio distribution of the Kepler systems has a broad peak away from the commensurabilities, and a small excess just wide of the strong ones rather than at them — a distribution measured on a census whose selection function is understood better than almost any other in the subject. If migration through a disc were universal and capture were as reliable as the argument above suggests, resonant chains should be the norm.

The second is that capture is not certain even for convergent migration. The adiabatic argument requires the drift to be slow compared with the libration period; a fast sweep carries the trajectory across the resonance before the separatrix can grow around it, and the body passes through. The probability of capture is therefore a function of the migration rate and of the body’s eccentricity when it arrives — an eccentric body has a larger trajectory area and is harder to enclose.

The evidence, in other words, is entirely architectural, and the next section says what that means in practice.

What was actually measured #

Nothing in this essay has been watched happening. Migration takes a hundred thousand years at least; capture takes a few libration periods, which for a planetary resonance is decades to centuries. Predictions of orbital configurations have expiry dates measured in millions of years, so the history has to be reconstructed rather than integrated. The evidence is architectural.

The first piece is the existence of chains. Several systems have three or more planets in a chain of first-order resonances, with period ratios matching to a fraction of a per cent. That precision is measured directly, from transit ephemerides that span years.

The second is that the resonant angles librate rather than circulate. That is the actual test of a resonance, as opposed to a near-commensurability, and it requires knowing the pericentres — which comes from the transit timing variations rather than from the periods alone. The third is the solar system’s own case. The Laplace resonance among Io, Europa and Ganymede is a chain of exactly this kind, and its assembly by convergent migration driven by tidal expansion of Io’s orbit is the worked example everything else is compared against — with the difference that the drift there is tidal rather than disc-driven and is still going on.

Divergence, and the one thing it does produce #

The rule that divergent crossing never captures is not a rule that divergent crossing does nothing. It leaves a signature of its own, and the signature is the reason the observed period ratios sit where they do.

A pair crossing a resonance divergently receives a kick: the resonance acts for the time the pair takes to cross it, and what it leaves behind is an eccentricity jump rather than a lock. The size of the jump depends on the crossing speed, and the pair emerges wide of the commensurability with more eccentricity than it went in with.

That is the standard account of how the giant planets of the solar system reached their present configuration: a slow divergent crossing of the two-to-one resonance between Jupiter and Saturn, which captured nothing, excited eccentricities and inclinations throughout the system, and scattered small bodies everywhere. The evidence is not the planets’ present period ratio, which is unremarkable, but the eccentricities and inclinations of everything smaller. It is worth separating two things the word capture is doing, because they are different events with the same name. One is the moment a trajectory crosses the separatrix and becomes trapped, which is what the phase portrait shows and what the probability applies to. The other is everything that happens afterwards: the continued migration that deepens the lock, pumps the eccentricity and, if it goes on long enough, drives the pair into a regime where the resonance itself becomes unstable. The first is quick and probabilistic; the second is slow and deterministic. Observations see only the outcome of both together, and separating them requires knowing how long the disc lasted after capture — which is exactly the quantity the resonant systems are used to measure. That circularity is not vicious, because the two effects leave different signatures: the fraction of systems in resonance measures the first, and the eccentricity and libration amplitude of those that are measure the second.

One assumption underneath the whole argument deserves stating, because it is the one that fails in the systems where capture is most interesting. The adiabatic invariant belongs to a pendulum, and a pendulum is what a single isolated resonance reduces to. Two resonances close together in period ratio do not add: their separatrices overlap, the region between them stops being foliated by closed curves, and there is no area to conserve. That is the Chirikov criterion, and it is the same overlap that empties a Kirkwood gap rather than filling it. In a chain of three or more planets the two-body resonances are joined by three-body ones, which lie between them and are weaker but not absent, so a real chain crossing is adiabatic in the gaps and chaotic where the resonances meet. Capture into a chain is therefore probabilistic in a second and quite different way from the one this essay has described, and the probability is estimated from integrations rather than from an invariant.

The same rule with only one planet moving #

The argument so far has had two bodies drifting toward each other. There is a second geometry that captures just as effectively and looks quite different, and it is the one the outer solar system is the evidence for.

Suppose a single planet migrates outward through a disc of small bodies. Its resonances are at fixed period ratios, so they move outward with it — and as they sweep across the disc, each small body is overtaken by a resonance rather than drifting into one.

From the small body’s point of view, the resonance’s strength rises as it approaches and falls as it recedes, which is exactly the strengthening-then-weakening the separatrix argument needs. A body swept over slowly enough is enclosed as the separatrix grows past it, and then carried outward with the resonance because leaving it would require the lock to break.

That is how Neptune’s outward migration is thought to have populated its resonances. The three-to-two resonance holds a large population of Kuiper belt objects, Pluto among them, all with period ratios matching Neptune’s to a fraction of a per cent and none of them plausibly formed at those exact ratios. The two-to-one holds a smaller population, and several higher-order resonances hold smaller ones still.

The check that makes it convincing is not the existence of the populations but their eccentricities. A body carried outward inside a resonance has its eccentricity pumped, by the same coupling described above, and the amount depends on how far it was carried. So the resonant populations should be eccentric, they should be more eccentric the further they were transported, and the inclinations should be raised as well. All three are observed.

A single migrating planet leaves a record in the bodies it swept up, and the record is quantitative: the distance Neptune moved is read off the eccentricities of the objects it is carrying.

There is one further consequence of the sweeping geometry that the two-planet case does not have. A body the resonance passes over without capturing is not left unchanged: it receives a kick as the resonance goes by, and the kicks accumulate over the many resonances a migrating planet drags across the disc. So the population that was not captured is stirred, and the observed dynamical excitation of the non-resonant Kuiper belt is part of the same accounting as the resonant populations. A mechanism that captures selectively necessarily perturbs everything it fails to capture, and both halves of that are observable.

What a libration amplitude is worth #

The resonant angle librates, and the amplitude of that libration is a measured quantity that carries information nothing else does.

The adiabatic argument gives it directly. A body captured very slowly is enclosed by a separatrix that grows past it gently, and it ends with a small enclosed area — a narrow libration. A body captured quickly, near the limit where the adiabatic condition fails, ends with a large one. So the amplitude is a fossil of the capture rate.

That makes it, in principle, a measurement of how fast the migration was, which is a quantity nothing else in the observable record constrains. A system whose planets librate through a few degrees migrated slowly; one librating through tens of degrees did not.

Two complications stand between the principle and a number, and both are worth stating.

The first is that anything dissipative acting after capture damps the amplitude. Tides in the planets, or residual gas, remove libration energy and narrow the swing, so an observed small amplitude is consistent with slow capture followed by nothing and with fast capture followed by damping. Distinguishing them requires knowing the dissipation, which is the quantity this subject is least able to supply.

The second is that measuring the amplitude requires the resonant angle, which requires the pericentre directions as well as the periods. For a transiting system those come out of the timing variations, and they come out correlated with the masses — so the amplitude and the planet masses are determined together and their uncertainties are not independent.

An observable that is a fossil of a rate is rare enough to be worth a great deal, and the reason the resonant systems attract the attention they do is that they are the only places a migration rate leaves a mark at all.

None of this is available for the two-planet case in the same way, because there the disc that drove the migration is gone and left nothing to count.

The pendulum picture has an extreme worth drawing, because it shows how little separates a captured body from one that was never captured at all.

That middle curve is the whole difficulty of the subject in one line. Its period is arbitrarily long — logarithmically divergent as the energy approaches the separatrix — so a body near the boundary takes an arbitrarily long time to complete one libration and is exquisitely sensitive to anything that happens meanwhile. A perturbation of any size, applied at the right moment, moves it decisively to one side or the other.

So capture is not a threshold that a body crosses once. It is a boundary that a body may sit on for a long time, being pushed across and back by whatever else is in the system, and the outcome is settled only when the driving stops or the boundary moves away. That is why capture probabilities are computed rather than derived, and why they are quoted as probabilities at all.

It is also why the observed librations are informative. A body deep inside the resonance, oscillating with a small amplitude, has been there long enough for the amplitude to have been damped, and the damping mechanism and its timescale are then constrained. A body near the separatrix has either arrived recently or is on its way out, and either way its amplitude is a clock reading rather than a stable state. Distinguishing the two cases needs the damping timescale, and the damping timescale needs a mechanism — tides for a satellite, gas drag for a body in a disc, collisions for an asteroid family. So a libration amplitude is only a clock once something else has supplied the rate, and the systems where the rate is known are the ones the whole argument is calibrated on.

Where the ladder goes #

The next rung has to be what breaks a chain. Chains are common in the youngest systems and rare in the old ones, so something dismantles them: a phase of dynamical instability after the disc dissipates is the leading candidate, and it predicts a particular distribution of eccentricities and inclinations afterwards — a distribution that has to be read against the fact that no planet has an eccentricity of its own, since secular exchange keeps redistributing what an instability deposits.

The other direction is the capture probability itself, as a function of drift rate and eccentricity — a calculation that turns the sharp one-way rule of this essay into a smooth function, and which is the quantity a population synthesis actually needs.