Concept

Solar system stability — where it appears

The question of whether the planets' orbits remain orderly over billions of years or can be driven into collisions or ejection. Integrations find the system chaotic, with Mercury's orbit having a small chance of destabilising before the Sun ages.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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

Which pairs of planets the angular momentum deficit can bring together. For each adjacent pair of planets, the system's angular momentum deficit divided by the critical deficit at which that pair's orbits could touch, on a logarithmic scale; below one the pair cannot collide under secular evolution, whatever happens. Blue: using only the deficit of the pair's own subsystem — the four terrestrial planets (7.8 × 10⁻⁹ in units of a solar mass times √AU) or the four giants (4.3 × 10⁻⁶). Orange: using the whole Solar System's deficit, which the giants hold almost entirely. On their own budget the inner pairs sit at 0.21, 0.12, 0.34 — safe. On the whole budget they sit at 112, 65, 187. The giants are safe either way, at 0.05, 0.18, 0.36. The terrestrial planets are stable because almost none of the giants' deficit reaches them, not because there is too little of it.

A deficit the giants hold and Mercury could borrow

The angular momentum the planets' orbits lack of circular motion is conserved by every secular interaction, to every order, so it is a budget no planet can overdraw. Compared with what it would take to bring a pair of orbits into contact, it proves without an integration that the terrestrial planets cannot collide among themselves — and shows that they are safe only because the giants, who hold five hundred times as much, lend them almost none. A loan of half a per cent of Jupiter's would be enough to unsettle Mercury.

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

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

Named alongside it

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

Binomial statistics Chaos Ensemble Lyapunov exponent Numerical integration Standard map Angular momentum deficit Compact planetary systems Eccentricity Laplace–Lagrange theory Rare event sampling Roundoff error

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