Resonance clears a gap in one place and locks a moon in another

Jupiter tugs on every asteroid in the belt, and the tug is minute — a few parts in ten thousand of the Sun’s pull, at most. Over one orbit it very nearly cancels: the asteroid is pulled forward for half the encounter and back for the other half, and what remains is a residue that points in a direction determined by where Jupiter happened to be.

If the asteroid’s period bears no simple relation to Jupiter’s, that direction is different on every orbit, and over thousands of revolutions the residues point every which way and sum to nothing.

If the two periods are in a simple ratio, they do not. The asteroid meets Jupiter at the same place in its own orbit, over and over, and the residues line up and add. A perturbation too small to matter becomes, given enough repetitions, the thing that decides whether a body can be there at all.

Where a resonance is, and why #

The location of a resonance is not an observation. It is arithmetic, and the arithmetic is the harmonic law.

A $p{:}q$ mean-motion resonance means the inner body completes $p$ orbits while the outer completes $q$. Since $P^2 \propto a^3$, a body whose period is $q/p$ of Jupiter’s sits at

$$a = a_J\left(\frac{q}{p}\right)^{2/3}.$$

Putting Jupiter at 5.2028 AU and running the ratios gives 2.06 AU for 4:1, 2.50 for 3:1, 2.82 for 5:2, 2.96 for 7:3 and 3.28 for 2:1. Every one of those radii is a gap in the asteroid belt, and the gaps were noticed by Daniel Kirkwood in 1866 — from a catalogue of fewer than a hundred asteroids, which is a striking piece of pattern recognition.

The gaps are not narrow. The 3:1 gap is a few hundredths of an AU wide, and the 2:1 gap wider still. What sets the width is how far from exact commensurability the accumulation still beats the natural drift of the orbit, and that is a calculation about libration rather than about geometry.

Locked and unlocked, in one variable #

The right object to watch is not the semi-major axis but a combination of angles called the resonant angle — for a $p{:}q$ resonance, something of the form $\sigma = p\lambda_1 - q\lambda_2 - (p-q)\varpi$, built from the two bodies’ positions and the orientation of one orbit.

Its value says where in its own orbit the inner body is when the two meet. And its behaviour divides sharply into two cases.

If $\sigma$ librates — oscillates about a fixed value without ever running through a full turn — the two bodies always meet in the same configuration, and the resonance holds. If it circulates, running steadily through all values, the meeting place drifts and the resonance does not hold.

That is a pendulum, exactly. Near commensurability, $\sigma$ obeys $\ddot\sigma = -k\sin\sigma$, the equation of a pendulum under gravity. A pendulum swinging back and forth is a librating resonance; a pendulum with enough energy to go over the top is a circulating one; and the boundary between them, the separatrix, is the pendulum that arrives at the top with exactly zero speed.

The pendulum analogy is not a teaching device. It is the actual reduced equation, and everything about resonances — the width, the libration period, the fact that the boundary is a place where the period goes to infinity — is a property of pendulums.

Two outcomes from one mechanism #

The same accumulation empties the asteroid belt and holds Jupiter’s moons in place, and the difference between the two is worth stating carefully because it is not obvious.

In the belt, the resonance pumps the eccentricity. An asteroid at the 3:1 gap has its eccentricity driven up over a few hundred thousand years until the orbit crosses Mars’s, and then a close encounter removes it — thrown into the Sun, ejected on a hyperbola, or dropped into a planet-crossing orbit. Numerical integrations by Jack Wisdom in the early 1980s showed exactly this: an asteroid placed at 2.50 AU wanders quietly for a hundred thousand years and then, with no warning and no change of forcing, its eccentricity jumps. The gaps are not places asteroids avoided. They are places asteroids were removed from.

Around Jupiter, the resonance is a stable configuration. Io, Europa and Ganymede have periods in the ratio 1:2:4, so exactly, that the resonant angle librates by less than a degree. The arrangement is not a coincidence and was not there at formation: tidal forces from Jupiter pushed the moons outward at different rates, Io reached the commensurability with Europa first, and was captured into it. The capture is one-way — a system drifting slowly toward a resonance can fall in, and cannot drift back out.

The difference between the two cases is which quantity the resonance controls and whether that quantity has somewhere to go. In the belt it raises the eccentricity until a planet is in reach; around Jupiter it maintains a small eccentricity against tidal circularisation. In both cases the resonance is doing the same thing — it is the consequence that differs.

What the lock actually does #

The Laplace resonance has a consequence far larger than the orbital bookkeeping, and it is the reason the outer solar system is not uniformly frozen.

Being locked keeps Io’s eccentricity at 0.0041, which is small and would be zero without the resonance — tidal dissipation circularises an orbit, and if Io were alone it would have been circular long ago. Because the eccentricity is maintained, Io’s distance from Jupiter varies over each orbit, so the tidal bulge raised on it rises and falls, so the body is flexed, so it is heated.

The power involved is about $10^{14}$ watts, some forty times the Earth’s entire internal heat flow, into a body a quarter of the Earth’s diameter. Io has more than four hundred active volcanoes and is the most geologically active object in the solar system, and the energy comes from the orbits of two other moons.

The same chain, weaker, gives Europa a subsurface ocean and — through a different resonance with Dione — gives Enceladus its south-polar water jets. Every place in the outer solar system where liquid water is suspected is a place where a resonance is maintaining an eccentricity that tides would otherwise have removed.

The gap that is a pile, and the reason for the difference #

The mechanism produces gaps in one belt and concentrations in another, and the contrast is the sharpest test the whole account faces.

Neptune’s 3:2 mean-motion resonance sits at 39.4 AU, and instead of being empty it holds Pluto and several hundred other objects, collectively the plutinos. Neptune’s 2:1 at 47.8 AU holds another population. Nothing about the resonance mechanism has changed; the sign of the outcome has.

What differs is whether being locked puts a body in reach of the perturber or keeps it away. Pluto’s orbit crosses Neptune’s — its perihelion is inside Neptune’s orbit — so on a naive reading it should have been ejected long ago. It has not been, because the resonance guarantees that whenever Pluto is at perihelion, Neptune is nowhere near: the resonant angle librates about 180°, which is precisely the condition that the two are on opposite sides of the Sun at closest approach. The two never come within 17 AU of each other, despite crossing orbits.

In the asteroid belt the geometry runs the other way. A body at the 3:1 resonance has its eccentricity pumped, and there is a planet — Mars — sitting where the growing orbit reaches. The resonance does not deliver the asteroid to Jupiter; it delivers it to a planet the resonance says nothing about.

So the rule is not “resonances clear” or “resonances protect”. It is that a resonance fixes the phase of encounters, and whether that is protective depends on what else is in the neighbourhood. The same mechanism that keeps Pluto alive for the age of the solar system empties a band of the asteroid belt, and the difference is entirely in the arrangement of the other bodies.

What was actually measured #

The resonance in the Jovian system is verified to a precision that leaves no room for interpretation, and the measurement is old.

The ratios are checked by timing eclipses of the moons as they pass into Jupiter’s shadow — events sharp to a few seconds and observable since 1610. Modern determinations give the Laplace relation

$$n_{\text{Io}} - 3n_{\text{Europa}} + 2n_{\text{Ganymede}} = 0$$

to better than one part in $10^{5}$ of the mean motions involved, with the residual consistent with a libration of amplitude about 0.064 degrees. The eclipse timings that establish it are the same ones Rømer used to measure the speed of light. That is not “approximately 1:2:4”. It is a lock.

The belt is measured differently and the evidence is statistical. Roughly 1.3 million asteroids are catalogued with orbits, and the histogram of their semi-major axes shows the gaps at the computed radii with no fitting of any kind. The depletion is not total — a few objects sit in each gap at any moment, on their way out — and the residence times inferred from integrations match the observed sparse population.

There is a check that discriminates between explanations, and it is worth having. If the gaps were carved by collisions or by some property of the primordial disc, they would appear in the distribution of asteroids’ proper elements — the long-term averaged values — in the same way as in the osculating ones. They do not appear the same way, and more tellingly, the same resonance locations produce concentrations rather than gaps in the Kuiper belt, where Neptune’s 3:2 resonance holds Pluto and some four hundred other objects. One mechanism, two signs of effect, depending on whether the resonance protects a body from close encounters or delivers it into them.

The generalisation: it is not about gravity #

The resonance mechanism requires nothing astronomical. It requires a system with two frequencies and a weak coupling between them, and that description covers a great deal.

The mathematical statement is that a perturbation series in the coupling strength develops small denominators: terms containing $1/(p\omega_1 - q\omega_2)$, which blow up when the frequencies are commensurable. Poincaré identified this as the central obstruction to perturbation theory in celestial mechanics, and the whole of KAM theory — Kolmogorov, Arnold and Moser, in the 1950s and 60s — is about which orbits survive despite it. The answer is that orbits whose frequency ratio is badly approximable by rationals survive, and the most badly approximable number of all is the golden ratio, which therefore turns up in stability arguments about systems with no geometry in them.

The same small denominators appear in the design of particle accelerators, where the transverse oscillation frequencies must be kept off resonance or the beam is lost; in the rotation of asteroids, where spin–orbit resonances hold Mercury — the planet with the most eccentric orbit of the eight — at exactly 3:2 rotations per orbit; and in the locking of a laser to a cavity. The astronomical cases are simply the ones where the coupling is weak enough and the number of repetitions large enough that the accumulation is total.

The count is the point. A perturbation of one part in $10^4$, repeated $10^8$ times with the same sign, is not a perturbation any more. The solar system has had time for $10^{8}$ to $10^{9}$ orbits, and that is the number that converts a negligible force into a structural feature.

The longest chain known #

The mechanism’s most striking instance is not in the solar system, and it was found in 2017.

TRAPPIST-1 has seven planets, and every adjacent pair is close to a mean-motion resonance: 8:5, 5:3, 3:2, 3:2, 4:3 and 3:2 outward from the innermost. The whole system is a single resonant chain, in which three-body combinations of the mean motions are commensurable as well as the pairs, and the chain has held together for what is presumably billions of years.

A chain of that length cannot form in place. It is the fossil of migration: the planets formed further out, drifted inward through the gas disc at rates depending on their masses, caught each other in resonance one by one, and thereafter migrated as a locked unit. The configuration is a record of a process that finished before the disc dispersed.

It is also useful. The resonances mean the planets perturb each other measurably, so the transit times vary by minutes in a pattern that depends on the masses — and fitting those variations gives all seven masses without a single radial-velocity measurement.

It is worth restating the division the two outcomes rest on, because it is the whole content of the mechanism and it is not about strength. A resonance that a body drifts into while the trapped region is growing holds it; one that a body drifts through while the region is shrinking releases it; and one that a body sits in with no drift at all does neither. Whether a given commensurability clears a gap or locks a pair is therefore a question about what else is acting on the orbits, not about the resonance itself.

The same statement has a converse worth having. If a commensurability is observed to be occupied, something must have driven the orbits together slowly enough for the trapped region to grow around them — so an occupied resonance is evidence about a system’s history and an empty one is evidence about nothing in particular, since a gap can be cleared by several mechanisms and can also simply never have been filled.

Where the model stops #

Two-body resonances only. Real systems have three-body resonances, where a combination of three mean motions is commensurable. These are weaker, far more numerous, and are the dominant source of chaos in the asteroid belt away from the Kirkwood gaps.

The pendulum approximation. It holds near an isolated resonance. Where two resonances overlap, the pendulum picture fails completely and the motion is chaotic — Chirikov’s criterion says overlap is the mechanism, and it is what makes the boundaries of the gaps fuzzy rather than sharp — a second route to the sensitive dependence the three-body problem is famous for.

No dissipation. Capture into resonance requires slow drift, which requires tides or drag or mass loss. A conservative system cannot be captured; it can only have started there.

Fixed orbits for the perturber. Jupiter itself moves, and during the solar system’s early history it moved a great deal — the resonance locations swept across the belt as the giant planets migrated, and much of the belt’s structure is a fossil of that sweep rather than of the present configuration.

The figures have a specific limitation. The gap histogram is a distribution of present semi-major axes, and the resonance’s action is on the eccentricity — so the picture shows the outcome of a process whose actual variable is not on either axis. The libration plots show the variable that matters and cannot show where the body is. No single figure holds both, and the mechanism only makes sense when the two are read together.

One more width shows how much of the gap structure survives a coarser reading.

The ladder from here #

Around other stars the constructive case is commoner still, and a chain of successive resonances is a fossil of how the system was assembled — capture requires slow convergent drift, which requires gas.

Later rungs on this anchor: the resonant angle derived, and the pendulum reduction. Resonance width, and the libration period. Capture into resonance, and why it is one-way. The Kirkwood gaps computed, and Wisdom’s integrations. The Laplace resonance and the tidal heating it maintains. Spin–orbit resonances, and Mercury’s 3:2. Secular resonances, where it is the precession rates rather than the periods that lock. Resonance overlap and the Chirikov criterion. Neptune’s resonances and the plutinos. Resonant chains in exoplanet systems, of which TRAPPIST-1 is the longest known. And planetary migration, which is how the resonances were populated in the first place.

Kirkwood found the gaps in a table of 87 asteroids and correctly attributed them to Jupiter. The dynamical explanation of how Jupiter empties them took another 115 years and a computer.