Kirkwood gaps

Why is the asteroid belt swept empty at exactly the orbits whose timing is most perfectly regular?

In 1866 Daniel Kirkwood sorted the known asteroids by the size of their orbits and found the distribution was not smooth. There were narrow bands almost nobody occupied. What made them peculiar was not their distance from the Sun but their timing: an asteroid in one of those bands takes exactly one third, or two fifths, or one half of Jupiter's year to go round.

That ought to feel backwards. Regularity is usually what keeps things going — a metronome, a clock escapement, a well-tuned engine. Why should the orbits with the tidiest arithmetic be the ones swept clean?

Start with what Jupiter can do at all. It is a distant object; its pull on a belt asteroid is tiny compared with the Sun's. In one orbit it nudges the asteroid a little one way and a little the other, and almost everywhere those nudges land at scattered points in the asteroid's cycle and cancel out to nearly nothing.

Now ask what changes if the timing is exact. If the asteroid completes precisely three orbits while Jupiter completes one, then the geometry repeats: the closest approach happens at the same point of the asteroid's orbit, in the same direction, every single cycle. The nudge is the same size it was before — but now it always arrives in phase, so instead of cancelling it accumulates. That is the whole idea of a mean-motion resonance, and it is the same reason a child on a swing gains height from small pushes that a random shove would not deliver.

Check the arithmetic against the sky, because it is checkable. Kepler's third law says the orbital period goes as the three-halves power of the orbit size, so a period ratio fixes a distance. Jupiter orbits at about 5.20 astronomical units. Take one third of its period and the asteroid sits at 5.20 times one third to the two-thirds power — about 2.50 AU. Two fifths gives 2.82 AU, three sevenths gives 2.96 AU, one half gives 3.28 AU. Those are exactly where the observed gaps lie, which is strong evidence that the mechanism is timing and not, say, some local shortage of raw material.

So the pushes accumulate. Into what? Not into speed, mostly, but into shape. What the resonance pumps up is eccentricity — the orbit gets progressively more stretched while its average distance stays roughly put. Follow that far enough and the asteroid's perihelion drops inward until the orbit crosses those of Mars, then Earth. At that point the asteroid is no longer a belt object; it is a planet-crosser, and its remaining lifetime is short. Some are flung onto new orbits or out of the system entirely by a close encounter; a surprisingly large fraction end by spiralling into the Sun.

For a long time this was believed rather than demonstrated, because the available calculations could not follow an orbit long enough. The demonstration came in the 1980s, when Jack Wisdom showed that motion in the three-to-one resonance is genuinely chaotic: eccentricity does not creep up steadily but sits quietly for long intervals, then jumps to planet-crossing values, on timescales of tens of thousands to a million years. That intermittency is why earlier short integrations found nothing alarming.

Now the sting in the tail, and the reason the naive story is incomplete. If resonance destroyed, the one-to-one resonance — sharing Jupiter's own period — would be the emptiest place of all. It is one of the most crowded: thousands of Trojan asteroids sit there permanently. The three-to-two resonance holds the Hilda family, also stable and populated. Resonance, then, is not intrinsically destructive.

What separates them? Whether the resonance protects the object from close approach or drives it into one. In the Trojan and Hilda cases the repeating geometry is arranged so the asteroid is always far from Jupiter when the pattern comes round — the same clockwork that concentrates the kicks also guarantees they are delivered from a distance. In the three-to-one and two-to-one cases, the accumulated effect drives eccentricity instead, and eccentricity is what kills.

Two clarifications keep this honest.

The first is what a "gap" actually is. Nothing is empty in space. The gaps are holes in the distribution of one orbital parameter — the average distance from the Sun — not voids you could fly through. Asteroids on eccentric orbits pass through the region at 2.5 AU constantly; they simply do not average 2.5 AU. Saturn's ring gaps look superficially similar and also involve resonances, but work through a different balance, since ring particles collide with each other and asteroids essentially never do.

The second is that the gaps have a downstream consequence. The three-to-one resonance, together with a secular resonance near the belt's inner edge, is one of the main hatches by which belt material reaches the inner Solar System — so the mechanism that empties those bands also supplies a large share of near-Earth asteroids and of the meteorites that land here. The gaps are not a wastebasket but a conveyor. How much of today's near-Earth population came through which route is still argued; the resonance-driven eccentricity mechanism itself is not.

• Why the Trojan points are stable at all, and what the two-body Lagrange picture leaves out.
• How chaotic orbits are distinguished from merely complicated ones, and what a Lyapunov time actually measures.

Every term the collection defines is gathered in the glossary.