Astronomy & Celestial Mechanics
the sky, computed
Concept

Resonance overlap — where it appears

The condition in which neighbouring resonances are wide enough to touch. At that point the invariant curves separating them break and trajectories wander chaotically, and it is the standard criterion for the onset of chaos in a nearly integrable system.

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

The last curve across the cylinder, before and after it breaks. Surfaces of section for the standard map at K = 0.5, 0.9716, 1.4, each 22 trajectories iterated 41 times from a column of starting values. Nothing here is placed: every dot is an iterate. At K = 0.5 the islands are separate and the space between them is filled with curves that run all the way round in θ — a trajectory cannot get from one island to the next, and one started beside the hyperbolic fixed point wanders 2.91 in p — 0.46 of a cylinder — and stops. At K = 0.9716, which is Greene's threshold to four figures, the last of those curves is on the point of going and the same trajectory still only reaches 4.84, which is 0.77 of a cylinder. At K = 1.4 it covers 6.1 cylinders: the barrier is gone and there is nothing left to stop it. That transition is what a chaotic zone in the asteroid belt is, drawn without any asteroids.

Where the chaos comes from

A Lyapunov time says how long a prediction lasts. It does not say what destroyed it. The mechanism is two resonances whose libration widths overlap, and the transition can be watched happening on a surface of section as one number is turned up.

gravitation · Chaos A wave whose crests count out 35 grams per square centimetre of ring. Optical depth across a spiral density wave in Saturn's A ring, drawn outwards from the 5:3 inner Lindblad resonance with Mimas at 131,988 km. The wave is launched at the resonance on the left and damps away to the right. Its wavelength is not constant: it starts at about 17.6 km and has shortened to 3.58 km by the last crest drawn, because the wavenumber grows in proportion to the distance from resonance and the ring's own self-gravity is the only restoring force in the dispersion relation. That makes the pattern a chirp with exactly one unknown in it. Fitting the 24 crest positions actually drawn here — the square of each one's distance from resonance against its number, which is a straight line — returns a surface density of 35.0 grams per square centimetre against the 35 the profile was built from. The rings have been weighed this way rather than by anything touching them: the mass per unit area follows from counting bright bands in a light curve as a star sets behind the ring.

A ring weighed by the wave crossing it

Saturn's rings are a few tens of metres thick and spread over an area larger than the Earth, made of pieces nobody can resolve, and nothing has ever landed on them. Their mass per unit area is nevertheless known to a few per cent — from the rate at which the crests of a wave crowd together as it travels outwards.

gravitation · Planetary rings Two equilibria become four, and three worlds sit near the join. The Cassini equilibria of a spin axis, drawn against the ratio of its own precession rate to the rate at which its orbit plane turns, for an orbit inclination of 1.5 degrees. Each column of dots is the full set of obliquities at which the two precessions keep step at that ratio, found by root-finding rather than by tracing a remembered curve. Below α cos ε/|g| = 1.135 there are two such obliquities and above it there are four, and the figure checks both counts on either side of the join. The three marked bodies are placed by their own measured precession constants: the Earth with the Moon at 2.67, safely on the four-state side; the Earth without it at 0.86; and Mars at 1.06. Two of the three sit within a few tenths of the bifurcation, which is the whole reason their obliquities are not constants: near the join the equilibria are close together, the libration around them is wide, and a body pushed between neighbouring resonances wanders. The Moon's contribution to the Earth's precession constant is what moves the first mark away from that region, and the second mark is the same planet with that contribution removed. This is a two-frequency model of a many-frequency system, and the real chaos comes from the overlap of resonances it does not contain.

A tilt that is not a constant

The Earth's axis leans by 23.4 degrees, and that lean is what makes the seasons. It is also a dynamical variable with its own equilibria, its own resonances and its own chaos — and on Mars the same variable has swung between nearly zero and sixty degrees without anything having to happen.

sky · Obliquity A chaotic orbit that waits, and then moves. The momentum of one trajectory of the standard map at K = 1.2, unwrapped and counted in whole turns of 2π, against the iteration number, over 39,892 iterations. The motion inside each band is chaotic from the first step — the trajectory's Lyapunov time is 3.7 iterations — but it lingers near one integer resonance for thousands of iterations before reaching the next: the waits were 3,098, 1,464, 9,461, 15,957, 6,058, 3,854 iterations. Each arrival at the next integer resonance, drawn dashed, is marked. Between the integer resonances lie the remnants of the broken invariant curves, which no longer block the trajectory but still hold it, and each passage is quick compared with the wait before it.

A barrier that leaks at a rate that can be computed

When the last invariant curve between two resonances breaks, it does not vanish. It becomes a cantorus — a curve full of gaps — and a chaotic orbit has to find the gaps to get through. The time that takes grows as the cube of how close the system is to the threshold, and it has nothing to do with the Lyapunov time, which is why an orbit can be hopelessly unpredictable and still stay put for the age of the solar system.

gravitation · Chaos Mars's tilt over fifty million years, and a steadier twin. The obliquity of a Mars-like planet against time over 50 million years, integrated from 25 degrees with its orbital plane wobbling in five secular modes. With Mars's precession constant, 8.26 arcseconds a year, the axis wanders between 1 and 70 degrees, lingering near some values and lurching between them. The same planet precessing 6 times faster — about the Earth's rate, which the Moon's pull makes possible — stays between 23.2 and 30.8 degrees, because its precession is too fast to resonate with any of the orbital-plane frequencies. The orbital perturbations are illustrative, of the size Mars's own are; the chaos is computed from them.

A past that can only be given as odds

Mars's axis is tilted by twenty-five degrees today, and nothing holds it there. Its slow precession falls among the frequencies at which its orbital plane wobbles, the resonances overlap, and the tilt wanders chaotically between nearly zero and sixty degrees. Two calculations that start a hair apart part company within a few tens of millions of years — so Mars's obliquity a billion years ago is not a number anybody will compute. What can be computed is how much of its history the planet spent at each tilt, and that distribution is a prediction the ice at its poles can test.

sky · Obliquity Which hierarchical triples survive, against the outer orbit's pericentre and eccentricity. Each cell is one integration of a triple: an equal-mass inner binary on a circular orbit of unit separation, and a third body of 1 times the binary's mass on an outer orbit of the given eccentricity with its pericentre at the given distance from the binary's centre of mass, all in one plane and going the same way round. Cells coloured as survivors kept the binary intact and ejected nothing for 30 outer orbits — 37 of 50; the others broke up, shaded darker the longer they lasted. The curve is the Mardling–Aarseth stability criterion, a pericentre of 2.8 inner separations times ((1 + q)(1 + e)/√(1 − e))^0.4, derived from where the resonances between the outer orbit and the inner one begin to overlap, and meant for survival over very many orbits. Every triple outside it survived. Inside it, 17 of 30 also survived this run; the outermost break-up is at a pericentre of 3.0 for a circular outer orbit and 2.6 at e = 0.6. The boundary drawn from a finite integration is therefore a lifetime rather than a line — triples between the break-ups and the curve survived a run of this length and are not thereby shown to be stable; the criterion is a statement about how the resonances are arranged, and a finite integration can only fail to refute it.

A third body kept at a safe distance

Most triple stars last for billions of years because they are hierarchical — a close pair and a distant third. How distant is distant enough has a formula, the formula comes from overlapping resonances, and an integration of any finite length can only fail to refute it.

gravitation · The three-body problem

Named alongside it

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

ChaosLyapunov timeInvariant curveKAM theoryObliquityPrecession constantSecular resonanceStandard mapSurface of sectionAction–angle variablesAdiabatic invariantCantorus

All concepts