Astronomy & Celestial Mechanics
the sky, computed
Concept

Chaos — where it appears

Sensitive dependence on initial conditions, in which nearby trajectories separate exponentially and a prediction acquires an expiry date. In the solar system it arises from the overlap of neighbouring resonances rather than from any added ingredient.

Named by 10 essays across 4 fields — each of them below, with the objects they name alongside it.

A separation of 10⁻¹⁰ becomes 0.0285. The distance between two copies of Burrau's problem, started 10⁻¹⁰ apart in one coordinate of one body and then stepped in lockstep — one loop, one step size, both states advanced by it — for 60 time units. The vertical axis is logarithmic, so the straight stretch is exponential growth, and its slope is the Lyapunov exponent: 0.3413 per time unit, fitted over the 604 samples lying between ten times the starting offset and a tenth of the system's own size. That is an e-folding every 2.93 time units, so the separation multiplies by ten every 6.75 and by 7.8·10⁸ across the whole run. It has not saturated within the drawn interval: the growth rate over the last tenth of the run is still 0.6002 per time unit, 176% of the fitted exponent, and the separation has reached only 0.90% of the system's own size. The exponent is what sets a prediction horizon, and this run is drawn short of it on purpose: the straight line is the measurement, and the flat part that follows is arithmetic about how far apart two bounded systems can get.

A prediction with an expiry date

The inner solar system's Lyapunov time is about five million years, so a centimetre of error becomes an orbit in a hundred million. The ephemeris dies while the system survives — because the elements stay bounded when the phase does not, and only one of those is what stability means.

gravitation · Chaos 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 An inner orbit's inclination, with and without the octupole. The inclination of an inner orbit to the plane of a distant perturber against time, in units of the quadrupole timescale, starting at 65 degrees with a nearly circular orbit, for octupole strengths ε = 0 and 0.03. At ε = 0 — a circular outer orbit — the inclination oscillates in the regular Kozai–Lidov cycle and never crosses ninety degrees, because the angular momentum along the outer orbit's axis is conserved and cannot change sign. With the outer orbit eccentric, the inclination drifts from cycle to cycle and crosses ninety degrees, the orbit turning from prograde to retrograde, first at t ≈ 120, 2 times in all within the time shown. The integration conserves the secular energy to 7·10⁻⁶.

An orbit that turns over

A distant third body makes an inner orbit trade inclination for eccentricity in slow, regular cycles, and one quantity — the inner orbit's angular momentum along the outer orbit's axis — keeps the trade bounded and forbids the orbit from ever tipping past ninety degrees. That conservation holds only if the outer orbit is a circle. Make it eccentric and the quantity drifts, the cycles stop repeating, and at a moment of extreme eccentricity the inner orbit turns over, from prograde to retrograde, with its pericentre pressed a thousand times closer than the regular cycle allowed.

orbits · Kozai–Lidov 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 Which of three bodies is thrown out. Of 60 integrations of three bodies of mass 1, 2, 3 released from rest in random triangles, 57 broke up within the time integrated, each leaving one body unbound and a binary behind. The escaper was the body of mass 1 in 46 runs (81 per cent), 2 in 6 runs (11 per cent), 3 in 5 runs (9 per cent). The bars beside them are the shares the statistical theory of break-up predicts to leading order, in proportion to the escaper's mass to the power −3: 86, 11, 3 per cent. Every run is deterministic and none of them could be predicted without integrating it, but the ensemble is: the lightest body leaves most often when the masses differ, because the energy of the binary left behind is shared between the escaper and the binary's recoil in inverse proportion to their masses, and the lightest body gets the most speed. Energy is conserved in every run to better than 3·10⁻⁴.

The lightest body leaves

Three stars close together, with no hierarchy between them, cannot stay together. One of them is thrown out, which one is unpredictable in any single case, and across many cases it is almost always the lightest — a statistical law emerging from a deterministic chaos that cannot be integrated far enough to forecast.

gravitation · The three-body problem How long a chaotic orbit stays away, with and without islands. The fraction of recurrence times to a small square in the chaotic sea of the standard map that exceed a given number of iterations, on logarithmic axes, counted along one orbit of 6·10⁷ iterations at K = 1.5 and K = 7 (772,165 recurrences, mean 78; 548,043 recurrences, mean 109). Dashed: exponentials with the same means. At K = 7, where the islands have all but vanished, the recurrences are exponential: the orbit forgets where it was in a few iterations. At K = 1.5 they part from the exponential after a few hundred iterations and fall instead as a power of time, with exponent 1.17 between a thousand and a hundred thousand iterations; the step at the left is the share of visits that simply stay in the square for a second iteration. A recurrence a thousand times longer than the mean is found once in 257,388 recurrences, where an exponential would give one in e to the thousand. Each mean is the area of the chaotic sea divided by the area of the square — 110 when the sea fills the torus, 78 at K = 1.5, where the islands take 29 per cent of it. The islands shorten the average absence and lengthen its tail.

A chaotic orbit clings to the edge of order

In a chaotic sea with no islands, an orbit forgets where it was within a few steps, and the time it stays away from any region falls off exponentially. Put islands of regular motion in the sea and the long absences multiply by factors of millions: the orbit reaches an island's edge, is caught in the hierarchy of smaller islands round it, and behaves as if it were regular for thousands of steps. The tail becomes a power law, and a rare few absences carry a large share of the orbit's whole history.

gravitation · Chaos Sixty futures that began a trillionth apart. The momentum of the standard map at K = 3, a kicked rotor whose phase space is almost entirely chaotic, for 60 members started 10⁻¹² apart in angle, against iteration. For 41 iterations they are one trajectory; then the separation, growing by a factor of about 2.0 each step, reaches the size of the system and they become independent random walks. Three of the 60 cross the threshold at |p| = 90 within 400 iterations (dark), the stand-in for an orbit that goes unstable. Which members cross is decided by differences far below anything that could be measured; how many cross is a property of the dynamics, and it is the only thing the ensemble can be asked for.

A probability for a system with one future

The Solar System will do one thing, and an integration of it is wrong about which thing within a hundred million years. What long integrations report instead is a probability — one per cent that Mercury's orbit goes unstable before the Sun dies — and a probability of a single event is a strange object. It is a count over an ensemble of futures that differ by less than anything measurable, which makes sense only because chaos makes those futures independent; and how many of them are needed is a binomial question with an unforgiving answer for rare events.

gravitation · Numerical integration A rare crossing reached in stages. Multilevel splitting on a kicked rotor (the standard map at K = 10), aimed at the event |p| > 450 within 400 iterations. 16 members are run until each reaches the first level or runs out of time; those that reach it are copied back up to 16, each copy restarted from its parent's crossing point with its angle nudged by a thousandth of a radian, and run on toward the next. The levels (dashed) are not fixed values of |p| but curves that rise as time runs out, so a member counts as closer to the event only if it has the time left to finish. 4 levels; the fractions passing them are 0.13, 0.06, 0.63, 0.88, and their product, 0.0043, is the estimate. Each stage is a common event; the rare one is never waited for.

A rare event reached by cloning the near misses

Counting how often an ensemble of integrations goes unstable works for an event of one in a hundred and becomes hopeless for one in a million, because a ten per cent answer needs a hundred events and so a hundred million futures. Splitting gets there by a different route: wait for the members that come closest, copy them, and run the copies on. Each stage measures a common event and the rare one is a product of common ones — provided the copies forget their parents and the stages know how much time is left.

gravitation · Numerical integration An arrival held without a burn, and released. A body approaching a planet from a heliocentric orbit just outside the planet's, drawn in the planet's rotating frame (the Sun far to the left, the planet's orbital motion upward), in units in which the Hill radius is 0.69. It drifts in along the shear of the differential rotation, passes through the neck at the Lagrange point L2 into the region the planet controls, loops round the planet 3.3 times and leaves through a neck again. No burn is drawn: the capture and the release are both the Sun's tide working against the planet's pull. For Mars one unit of distance is 1.56 million km and an orbit round the planet near the Hill radius takes about a year.

An arrival the Sun hands over

A spacecraft that reaches a planet with no speed to spare can be captured with no burn at all, passing through the neck between the planet's gravity and the Sun's into a loose, looping orbit. The capture is temporary and chaotic — a few revolutions, and arrivals a few hundred kilometres apart end differently — but while it lasts the spacecraft is already bound, and the burn that keeps it is taken from there.

spaceflight · Escape

Named alongside it

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

Numerical integrationStandard mapLyapunov timeResonance overlapThree-body problemBinomial statisticsCantorusEnsembleHierarchical tripleInvariant curveKAM theoryLyapunov exponent

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