Astronomy & Celestial Mechanics
the sky, computed
Concept

Convergent migration — where it appears

Two orbits drifting toward each other in period ratio, so that a resonance between them strengthens with time. It is the only direction in which resonance capture is possible, which makes a chain of planets in resonance a record of how the system arrived at its spacing.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as resonant chain — the same set of essays touches all of them, so they are one junction rather than several.

Libration and circulation of a resonant angle. The critical angle of a resonance over time, integrated from the pendulum equation it obeys. Below the separatrix the angle oscillates about a fixed value — the body is locked; above it, the angle runs without bound and the body is not.

A chain that could not have been assembled in place

Seven planets whose successive period ratios are all close to small whole numbers. Capture into a resonance requires the orbits to converge slowly, and converging slowly is something planets can only do in a disc.

exoplanets · Resonance A resonance keeps what convergence brings it and releases what divergence takes away. The resonant angle of a body inside a resonance whose strength is changing, for the two signs of that change. The angle obeys a pendulum, and the strength of the pendulum is set by how close the two orbits are; migration changes it slowly compared with the swing, which is the condition under which the area a trajectory encloses is conserved. Convergent migration strengthens the resonance, so the separatrix grows around a trajectory of fixed area and the swing narrows — from 1.05 radians to 0.77 across the figure, the body ending more deeply locked than it began. Divergent migration weakens it, the separatrix shrinks, and it eventually passes inside the trajectory: the angle stops oscillating and begins to run, 37.5π in the last quarter of the drawing alone, and the lock is gone for good. Nothing here is dissipative and nothing is random. The whole asymmetry is the sign of one derivative, which is why a chain of planets in resonance is direct evidence that they migrated toward one another, and why a chain cannot survive a phase in which they moved apart.

Capture is a direction, not a strength

A resonance holds a body that drifts into it from one side and lets go of one that drifts out the other way, and the asymmetry is not about how strong the resonance is. It is the sign of a derivative — whether the trapped region is growing or shrinking — which is why a chain of planets in resonance is direct evidence that they migrated toward each other.

exoplanets · Resonance

Named alongside it

The objects these essays reach for when they reach for this one.

Adiabatic invariantLaplace resonanceLibrationResonance captureResonant chainDynamical stabilityMean motion resonanceMigrationPlanet migrationProtoplanetary discResonant angleSeparatrix

All concepts