Astronomy & Celestial Mechanics
the sky, computed
Concept

Laplace resonance — where it appears

The three-body commensurability holding Io, Europa and Ganymede at period ratios of one to two to four. It keeps the three from ever being in conjunction at the same longitude, and it maintains the eccentricities that drive Io's tidal heating.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

The Kirkwood gaps. Asteroid numbers against semi-major axis, with the resonant radii marked. Each gap sits where the orbital period is a simple fraction of Jupiter's, and each of those radii is computed from the harmonic law rather than placed by eye.

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

When two orbital periods are in a simple ratio, small tugs stop averaging away and start accumulating. Sometimes that empties a region entirely. Sometimes it holds three moons together for the age of the solar system.

gravitation · Resonance Io's measured heat needs k₂/Q = 0.016, and Enceladus's needs 0.011. Tidal surface heat flux against orbital eccentricity, from Ė = (21/2)(k₂/Q)GM_p²R⁵ne²/a⁶ evaluated at each satellite's own orbit, drawn at a common k₂/Q of 0.015. Every line has slope 2 because the dissipation is quadratic in e and nothing else on this axis varies. The filled marks are each body at its actual eccentricity; the two ringed ones are the bodies with a measured surface heat flux, and they are the only points here that are observations. Solving each of those for k₂/Q gives 0.016 for Io and 0.011 for Enceladus — within a factor of 1.5 of one another, for a warm silicate body and a 500-kilometre ball of ice, which ought to be a coincidence and is instead the sharpest problem in the subject: nothing about Enceladus's ice can plausibly dissipate at 0.011, and the number is what the heat requires all the same. The dashed line is the Earth's measured surface heat flux, 0.087 W/m², which Io exceeds by a factor of 28 — the most volcanically active body in the solar system is the fourth largest moon of the fifth planet, and the reason is entirely in the orbit.

A moon heated by not being allowed to relax

Tidal dissipation goes as the square of an eccentricity that tides themselves destroy, so a moon radiating tidal heat is spending something it cannot have saved. Io's would be gone in a hundred and forty thousand years, and the resonance that keeps putting it back is the reason there are volcanoes.

gravitation · Tidal heating 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 Dissipation against viscosity for Io: a peak at 10^13.5 Pa s, and two solutions at the observed rate. The imaginary part of the Love number — the part that turns tidal work into heat — against the viscosity of the body's interior, for a Maxwell rheology at Io's size, density and forcing period. Both axes are logarithmic and the curve is not monotonic, which is the whole content of the figure. At high viscosity the body is elastic: it stores the energy the tide puts in and gives it back, and dissipates nothing. At low viscosity it is fluid: it deforms all the way and does so in phase with the forcing, and dissipates nothing again. Everything happens in between, at viscosities for which the Maxwell time — viscosity divided by rigidity — is comparable to the orbital period, and the peak here is 0.735 at 10^13.5 pascal seconds. Two consequences follow, and they pull in opposite directions. The peak is an upper limit: a homogeneous body of this size cannot dissipate more than that however its viscosity is chosen, so a measured heat flow above it would refute the model rather than constrain it. And below the peak the observed value is met twice — at 10^11.5 and at 10^15.5 pascal seconds — so a heat flow alone does not say which side of the peak the interior is on. Breaking that degeneracy needs a second observable, and the usual one is the phase of the response rather than its size. The picture treats the body as one homogeneous Maxwell solid, which is certainly wrong for a moon with a molten layer; a partial melt concentrates the dissipation and shifts the peak.

One heat flow, and two viscosities

Io radiates a hundred thousand gigawatts of tidal heat, and that number is supposed to say something about the rock inside it. It does, and not what one would expect — because dissipation vanishes at both extremes of viscosity, the measured heat is produced by two different interiors and cannot choose between them.

gravitation · Tidal heating One measurement, one line, and every point on it is an interior. The Love number against the quality factor, both logarithmic. An orbital measurement — a moon observed to be receding, a spin observed to be slowing — determines only the ratio of the two, so it picks out a diagonal band rather than a point, and every interior along that band reproduces the observation exactly. A body that deforms twice as easily and dissipates half as efficiently is indistinguishable from one that does the opposite. What breaks it is a measurement of the deformation on its own: a spacecraft tracking the body's gravity field through a tidal cycle measures k₂ directly, which is a vertical line here, and the intersection gives Q = 5.36·10⁴. The uncertainty on that answer is the two fractional errors added in quadrature, 20 per cent, and it is dominated by whichever of the two was worse — which for every body in the solar system is the orbital rate rather than the Love number.

A heat flow that depends on a number nobody can compute

Every tidal rate in astronomy — a moon receding, a spin slowing, an orbit circularising, a satellite melting — is proportional to one combination of two quantities that no orbital measurement can separate. One of them describes how much a body deforms and the other how badly it leaks, and only a spacecraft can tell them apart.

gravitation · Tidal heating

Named alongside it

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

LibrationForced eccentricityLove numberQuality factorTidal dissipationAdiabatic invariantConvergent migrationHeat flowMean motion resonanceResonance captureResonant angleResonant chain

All concepts