Astronomy & Celestial Mechanics
the sky, computed
Orbits

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.

Assumes Kozai–Lidov, Secular theory and Chaos.

15 min read 6 figures Small perturbations, large consequencesTwo bodies, and no more

A body on an inner orbit, disturbed by a distant companion on an outer one, feels a torque that averages over both orbits into a slow and systematic effect. For an inner orbit inclined by more than 39 degrees to the outer one, that torque turns inclination into eccentricity: the orbit becomes more elongated as it becomes less inclined, and then gives the eccentricity back, in cycles that repeat indefinitely. It is called the Kozai–Lidov mechanism, after the two who found it independently, and it explains why many asteroids, satellites, binary stars and planets have the orbits they do.

The cycles are bounded because something is conserved. The inner orbit’s angular momentum has a component along the outer orbit’s axis, proportional to 1−e2cos⁡i\sqrt{1-e^2}\cos i, and in the simplest version of the calculation that component never changes. Eccentricity and inclination trade along a curve that keeps it fixed; the curve puts a ceiling on the eccentricity, and since the component cannot change sign, the inclination can never pass through ninety degrees. A prograde orbit stays prograde.

That conservation is a property of the approximation, not of gravity. It holds when the outer body’s orbit is a circle and the averaged force from it is symmetric about its axis. When the outer orbit is eccentric, the next term in the expansion of the force — the octupole — breaks the symmetry, and the conserved quantity starts to drift.

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.
Fig. 1 The inclination of an inner orbit to the outer orbit’s plane against time, starting at 65 degrees and nearly circular, with and without the octupole. Without it, the orbit cycles between 65 degrees and about 40 and never approaches 90. With an octupole strength of 0.03 the cycles wander, and after about a hundred and twenty timescales the inclination crosses ninety degrees: the orbit has turned over, and it stays retrograde for sixty timescales before turning back.

The parameter that measures the asymmetry

The strength of the octupole relative to the quadrupole is a single number,

ε=aaout eout1−eout2,\varepsilon = \frac{a}{a_{\rm out}}\,\frac{e_{\rm out}}{1 - e_{\rm out}^2},

the ratio of the orbits’ sizes times a measure of the outer orbit’s eccentricity. It is zero when the outer orbit is circular, whatever the separation, and it grows when the outer orbit is elongated or the inner one is large relative to it. For a planet at one astronomical unit perturbed by a companion star at a hundred with an outer eccentricity of a half, ε is about 0.007; for a stellar binary with an inner separation a tenth of the outer and an outer eccentricity of 0.6, it is about 0.09.

The octupole enters the averaged energy with ε in front of it, so for small ε it is a small correction to each cycle. But it acts on the conserved quantity that the quadrupole leaves alone, and a small effect on a conserved quantity accumulates. Over many cycles, the angular-momentum component drifts, and the envelope of the cycles changes with it. How far it can drift, and whether it reaches zero, is decided by ε and by the starting inclination.

The quantity the quadrupole conserves, and the octupole does not. The inner orbit's angular momentum along the outer orbit's axis, √(1 − e²) cos i, against time, for the same two runs. At quadrupole order it is exactly constant, at 0.423: eccentricity and inclination trade against each other along a curve that keeps it fixed, and since it cannot change sign, the inclination cannot cross ninety degrees. With the octupole it wanders, slowly and then abruptly, and crosses zero at each flip — always at a moment of very high eccentricity, when √(1 − e²) is small and a small torque is enough to reverse it.
Fig. 2 The inner orbit’s angular momentum along the outer axis, 1−e2cos⁡i\sqrt{1-e^2}\cos i, against time for the same two runs. Without the octupole it is exactly constant. With it, it wanders gently for a hundred timescales, then falls abruptly through zero — the flip — and later rises back through zero again. Each crossing happens at a moment when the eccentricity is extreme and 1−e2\sqrt{1-e^2} is small.

The timing of the crossings is the key to how the flip happens. The angular-momentum component is the product of 1−e2\sqrt{1-e^2}, which measures how circular the orbit is, and cos⁡i\cos i. When the eccentricity approaches one, 1−e2\sqrt{1-e^2} approaches zero, and the whole component becomes small regardless of the inclination. A small torque can then carry it through zero. The octupole’s torque is weak, but at the peak of an extreme eccentricity cycle it has almost nothing to overcome. The flip is not a slow rotation of the orbit through ninety degrees; it is a quick reversal of a nearly radial orbit, whose orientation is barely defined, at the moment when the orbit is closest to being a straight line through the central body.

It is worth noticing what does not change. The inner orbit’s semi-major axis, and so its orbital period and energy, are untouched by any of this: the averaged torques exchange angular momentum between the orbits and redistribute it within the inner orbit, but they do no work on it. The octupole reshapes and reorients an orbit of fixed size. Everything dramatic in the figures — the flips, the plunging pericentres — happens along a curve of constant energy, and it is the small angular momentum near radial orbits that makes large reorientations cheap.

The price of turning over: a pericentre a thousand times closer

The same geometry makes each flip a moment of extreme eccentricity, far beyond the quadrupole’s ceiling.

How close to radial the inner orbit gets. One minus the inner orbit's eccentricity, on a logarithmic scale, against time, for the same two runs. At quadrupole order each cycle reaches the same maximum, 1 − e = 0.16, set by the conserved quantity. With the octupole, cycles differ from one another, and at each flip through ninety degrees the eccentricity spikes: the closest approach reaches 1 − e = 3.6·10⁻⁶, more than ten thousand times smaller. The pericentre shrinks by that factor at the moment of the flip, which is when tides, relativity or a collision with the central body can take over.
Fig. 3 One minus the eccentricity — the pericentre distance in units of the semi-major axis — against time, for the same runs. The quadrupole cycles all reach the same closest approach, 1 − e of about 0.16. With the octupole, the cycles vary, and at each flip the pericentre plunges by more than four orders of magnitude, to the floor of the figure.

That spike is the reason the octupole matters for the fate of real systems. In the quadrupole limit, the closest an inner orbit can come to its central body is fixed by its starting inclination; for a start at 65 degrees, a sixth of its semi-major axis. With the octupole, the same orbit can come ten thousand times closer. A planet at five astronomical units, whose regular cycles would never bring it inside nearly one, could be driven through a pericentre of a hundred thousand kilometres — inside its star.

In practice something else intervenes first. As the pericentre shrinks, tides raised on the planet by the star, and the relativistic precession of the orbit, grow steeply and begin to compete. The general-relativistic precession can switch the cycle off by detuning the resonance that drives it; tides can dissipate the orbital energy at each close pericentre and shrink the orbit permanently. Which happens depends on the system, and the octupole’s contribution is to make the extreme approach possible at all.

Which orbits turn over

At quadrupole order the maximum eccentricity has a closed form, emax⁡=1−53cos⁡2i0e_{\max} = \sqrt{1 - \tfrac53\cos^2 i_0}, and depends only on the starting inclination. The octupole breaks that too.

The closest approach reached, against the starting inclination. The smallest value of 1 − e reached within 300 quadrupole timescales, on a logarithmic scale, against the inner orbit's starting inclination, for octupole strengths ε = 0, 0.01, 0.03, starting nearly circular. At quadrupole order the answer follows the closed form eₘₐₓ = √(1 − (5/3) cos² i₀), reaching 1 − e ≈ 0.006 at 85 degrees, and no growth at all below 39. With the octupole, the orbits starting at high inclination reach far more extreme eccentricities — many reach the floor of the plot, 1 − e = 10⁻⁵, where the orbit is held from going further and where any real central body would long since have been struck — and at ε = 0.01, 2 of the 10 starting inclinations flip; at ε = 0.03, 5 of the 10 starting inclinations flip.
Fig. 4 The closest approach reached within three hundred timescales against the starting inclination, for octupole strengths of 0, 0.01 and 0.03. Without the octupole the result follows the closed form, shown faintly behind. With it, orbits starting above a threshold inclination reach extreme eccentricities and flip; the threshold falls from about eighty degrees at ε = 0.01 to about sixty-five at 0.03.

The pattern can be drawn as a window.

Which starting inclinations flip. For each octupole strength, the starting inclinations whose orbits flipped through ninety degrees at least once within 300 quadrupole timescales, starting nearly circular, marked as filled cells. At ε = 0 none flip. At ε = 0.01 the flipping orbits start between 80 and 85 degrees; At ε = 0.03 the flipping orbits start between 65 and 85 degrees. The window widens and moves to lower inclinations as the outer orbit becomes more eccentric or the inner orbit larger relative to it. Within a window, whether and when a given orbit flips depends sensitively on its starting phases.
Fig. 5 Starting inclinations whose orbits flipped at least once within three hundred timescales, for three octupole strengths, as filled cells. None flip without the octupole. At ε = 0.01 the flippers start between eighty and eighty-five degrees; at ε = 0.03, between sixty-five and eighty-five. The window widens toward lower inclinations as ε grows.

The published criterion for the test-particle case, derived from the averaged equations by Boaz Katz and collaborators and by Yoram Lithwick and Smadar Naoz in 2011, agrees with the drawn window: flips require ε above about a hundredth, and the range of inclinations that flip widens toward the Kozai critical angle as ε grows. What the window does not show is that it is ragged at its edges. Two orbits starting at the same inclination but different phases of their pericentres and nodes can behave quite differently, one flipping early and the other not at all within the time shown.

Near the top of the window, where the quadrupole cycle already carries the orbit close to radial, even a weak octupole is enough, and flips come quickly and repeatedly.

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 82 degrees with a nearly circular orbit, for octupole strengths ε = 0 and 0.01. 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 ≈ 22, 8 times in all within the time shown. The integration conserves the secular energy to 4·10⁻⁵.
Fig. 6 The inclination of an orbit starting at 82 degrees, with no octupole and with a weak one, ε = 0.01. Without it, the orbit cycles between 82 degrees and about 40. With it, the orbit first turns over after about twenty timescales and flips back and forth eight times in four hundred, spending long stretches retrograde. A system this close to perpendicular needs only a slightly eccentric outer orbit to lose its sense of rotation.

At high starting inclination the quadrupole cycle itself drives the orbit to 1−e2\sqrt{1-e^2} of about a tenth, so the conserved component is small from the start and the octupole has little to overcome. Such orbits flip readily and repeatedly, and they spend comparable times prograde and retrograde. That makes the direction of a highly inclined orbit in a triple a statement about recent history rather than initial conditions: an observer seeing it retrograde learns only that it is currently in one of its retrograde spells.

A secular problem that is chaotic

That raggedness is not a numerical artefact; it is the dynamics. The quadrupole problem has as many conserved quantities as degrees of freedom and is integrable: its orbits lie on smooth surfaces in phase space and never wander. The octupole removes one conserved quantity, and the problem becomes non-integrable. Regions of phase space near the separatrix — the boundary between the librating and circulating Kozai cycles — become chaotic, and an orbit in them wanders unpredictably between prograde and retrograde.

The integrations behind these figures show it directly. The same run, integrated with a step half as large, flips first at a different time: once at about 120 timescales, once at about 180. Both integrations conserve the averaged energy to a few parts in a million, so neither is wrong in the ordinary sense. The difference is the sensitivity to small changes that defines chaos, and a prediction with an expiry date is the right way to read any single run: the statistics of flips — whether an orbit in a given region flips, how often, how extreme the eccentricities — are robust; the timing of an individual flip is not.

That is an unusual kind of chaos, because it arises in averaged equations that have already removed the fast motion of both orbits. It is chaos in the slow evolution of orbital shapes, on timescales of thousands to millions of orbits, and it would be invisible to any calculation that kept only the leading term of the expansion.

A symmetry that was an approximation

The conservation that forbids flips was, for a long time, believed to be more general than it is. Lidov’s analysis of 1961 treated a satellite perturbed by the Moon, and Kozai’s of 1962 an asteroid perturbed by Jupiter; both took the perturber’s orbit to be a circle, for which the conservation is exact. When the calculation was extended to three bodies of comparable mass, a standard simplification — eliminating the nodes, by using the fact that the two orbits’ lines of nodes are always opposite in the plane perpendicular to the total angular momentum — seemed to preserve the same conserved quantity. In 2011 Smadar Naoz and collaborators showed that the simplification had been applied incorrectly, and that the inner orbit’s angular-momentum component along the total angular momentum is not conserved even at quadrupole order when the masses are comparable. Flips had been hiding in the equations for fifty years. The correction mattered most for exactly the systems where flips are most consequential: stellar triples and planets with massive companions, where the inner orbit’s angular momentum is not negligible against the outer one’s.

The episode is a small version of a large lesson. A three-body problem has no general solution, and every tractable version of it is an approximation that keeps some terms and drops others. The quadrupole Kozai problem is integrable and beautiful because it drops the terms that break its symmetry, and the conclusions drawn from it — the ceiling on eccentricity, the impossibility of flipping — were properties of what had been dropped. Adding the next term back did not refine those conclusions; it removed them.

Retrograde planets

The most direct evidence that orbits flip comes from hot Jupiters — giant planets orbiting their stars in a few days. They cannot have formed so close, and one route inward is exactly this one: a planet formed at several astronomical units, driven to extreme eccentricity by a distant companion, captured by tides at a close pericentre and circularised there. A planet circularised at twice the distance that would destroy it ends up in a tight circular orbit, but with whatever inclination to its star’s equator the Kozai cycling left it.

The Rossiter–McLaughlin effect, which measures the angle between a planet’s orbit and its star’s spin as the planet transits, has found many hot Jupiters on highly inclined orbits and some that are retrograde — orbiting opposite to their star’s rotation. What is measured is the sky-projected angle, and the true angle is often larger. At quadrupole order, starting from an orbit aligned with the star’s equator and inclined to a companion, a retrograde final orbit is hard to produce. With the octupole, a flip at the moment of extreme eccentricity delivers the planet to its star retrograde, and the distribution of observed obliquities, with its tail of retrograde planets, is one of the things the octupole was first invoked to explain.

Mergers, collisions and destroyed stars

The same mechanism applies to compact objects. A pair of black holes in a wide orbit, with a third body further out, is driven through cycles of eccentricity; at a flip the pericentre can become small enough for gravitational-wave emission to take over, and the pair merges within a few orbits — a merger whose signal carries the pair’s masses and, if caught early enough, its residual eccentricity, which is the fingerprint of the third body. Merging black holes with measurable eccentricity would point to this channel, since isolated binaries circularise long before merging.

For stars, the extreme pericentres can lead to collisions in stellar triples, to mass transfer and mergers that make blue stragglers, and to white dwarfs in triples colliding head-on, which has been proposed as one route to type Ia supernovae. In the centres of galaxies, a star orbiting a massive black hole with another black hole or a massive disc further out can be driven into the hole’s tidal radius and destroyed, raising the rate of tidal disruptions in galaxies whose nuclei host binaries. In each case the octupole’s role is the same: to make reachable a pericentre the quadrupole forbids.

Reading ε off a real system

For any particular system the octupole’s importance can be estimated from quantities that are often measured. The ratio of semi-major axes comes from the two orbital periods through Kepler’s third law; the outer eccentricity from the outer orbit’s radial-velocity curve or astrometry; and together they give ε directly. For planets in binary star systems, where the companion star’s orbit is known from decades of astrometry, ε ranges from below a thousandth, for planets close to their stars in wide binaries, to a few hundredths for planets far from their stars in eccentric binaries. The first are firmly in the quadrupole regime; the second are in the regime where flips are expected.

What is usually not measured is the mutual inclination between the two orbits, because it requires knowing both orbits’ orientations in space, not just on the sky. For transiting planets the inner orbit’s inclination to the sky is known and its node is not; for visual binaries the outer orbit’s full orientation is known. Combining the two constrains the mutual inclination only statistically. That is the gap between the theory, which is sharp about which inclinations flip, and the observations, which can usually say only that a given system might.

What the figures leave out

The integrations use the test-particle limit, in which the inner body’s mass is too small to affect the outer orbit. For comparable masses — stellar triples, or a pair of black holes with a third — the outer orbit’s angular momentum is not a fixed reference, the octupole’s form changes, and flips occur even when the inner orbits are less inclined. The equations are doubly averaged, over both orbits, and they fail when the inner orbit’s eccentricity is so extreme that its pericentre passage is fast compared with the outer orbit’s motion: at the deepest spikes the averaging itself breaks down, and direct integrations of the three bodies find the eccentricity can go higher still. The integrations also omit every short-range force — tides, relativity, the central body’s size — and so the plunges to the floor of the figures are what the gravity of three points allows, not what a real planet or star would survive.

The time unit is the quadrupole timescale up to a constant, which for a planet at five astronomical units and a solar-mass companion at five hundred is some millions of years; the octupole acts over tens to hundreds of those. The mechanism is therefore slow, and it competes with every other process that changes orbits over millions of years, including the passage of other stars. A star passing within a few thousand astronomical units of a wide triple can change the outer orbit’s eccentricity, and with it ε, abruptly; in the dense centre of a star cluster such encounters are frequent enough that the octupole’s slow drift is repeatedly reset, which can either hasten a flip or prevent one.

Still open: how often an orbit turns over in a real population

The window drawn is for one set of conditions. A population of triples, planets with companions or black holes in clusters spans a wide distribution of ε, of inclinations and of masses, and the fraction that flip within the system’s lifetime depends on all of them and on the short-range forces that can suppress the cycle before a flip. The measured distribution of hot-Jupiter obliquities, the eccentricities of merging black holes detected by gravitational waves and the rate of tidal disruptions are three independent samples of that fraction. Whether they agree with each other and with the flips drawn here — whether the octupole is the dominant route to extreme eccentricity or one of several — is still being assembled, one class of system at a time. Astrometry from the Gaia satellite, which measures the orbits of both planets and companion stars in three dimensions for thousands of nearby systems, is beginning to supply the missing mutual inclinations directly, and with them the first test of whether the systems predicted to flip are the ones whose planets are found misaligned.

Where this goes next

An eccentricity and an inclination trading, at 65° of mutual tilt. The secular equations integrated from a nearly circular orbit (e = 0.02) inclined at 65° to a distant perturber's plane, over three oscillations. Above, the eccentricity; below, the inclination, with the constant √(1−e²)cos i drawn as the flat line it is. The eccentricity climbs to 0.8380 and the inclination falls to 39.25° at the same instant, and neither is a coincidence: the product is fixed, so one can only rise as the other falls. That floor is the same for every starting tilt — at maximum eccentricity j = √(5/3)Θ, so cos i = √(3/5) and the inclination arrives at 39.23° whether the orbit began at 50° or at 89°. The closed form for a circular start is e_max = √(1 − (5/3)cos²i₀) = 0.8380, which contains nothing about the perturber — not its mass, not its distance. Those set the clock and not the amplitude, and the period here is 4.83 Kozai times. What the figure cannot show is what happens at the top of the cycle in a real system: at e = 0.838 the pericentre is 0.1620 of the semi-major axis, where tides, general relativity or a stellar surface all intervene, and the quadrupole picture ends. An inclination that turns into an eccentricity A distant companion cannot change an orbit's size or its energy. It can take a circular orbit tilted past 39.23 degrees and drive it to an eccentricity near one, and back, over and over — and the companion's mass and distance set only the clock. A ceiling the outer orbit imposes. The greatest eccentricity a cycle can drive an inner orbit to, against the mutual inclination it started at, for four ratios of the inner orbit's angular momentum to the outer one's. The bottom curve is the test-particle case and it is the familiar closed form, √(1 − (5/3)cos²i₀), reaching one at ninety degrees — recovered here by a solver that knows nothing about it, which is the check. Every other curve is lower. The reason is conservation: the cycle works by moving angular momentum out of the inner orbit, and it has nowhere to put it but the outer one. A test particle can give up all of its angular momentum because the outer orbit is infinite by comparison; an inner binary carrying 1 times as much as its companion cannot, because tilting the outer orbit by the required amount changes the mutual inclination and shuts the cycle down. At ninety degrees the ceiling falls from 1.000 to 0.866. The pericentre a mechanism can deliver is bounded by what the perturber can absorb, and the test-particle formula is an upper bound rather than a prediction. A ceiling the outer orbit imposes The closed form for how far a cycle can drive an eccentricity assumes the inner orbit is a test particle, so the angular momentum it gives up goes nowhere. A real inner binary of comparable mass has to be paid for: the outer orbit absorbs what the inner one loses, tilts, and changes the very inclination the cycle depends on. The true obliquity a projected angle allows, against the tilt of the star. The true angle ψ between a transiting planet's orbit and its star's spin axis, against the inclination of that spin axis to the line of sight, for projected angles λ of 0°, 30°, 60°, 90°, 130° — from cos ψ = sin i★ cos λ for an orbit seen edge-on. At i★ = 90°, a star seen equator-on, the projection loses nothing and ψ = λ. As the star tips towards pole-on the true angle moves towards 90° from whichever side it started, so every curve ends at 90° when i★ = 0. Two things follow exactly. The projected angle is a lower bound for an aligned-looking system and an upper bound for a retrograde-looking one — ψ always lies between λ and 90°. And λ = 90° needs no inclination at all: a planet whose projected orbit crosses the stellar equator at right angles is on a polar orbit whatever the tilt, which is why the pile-up of polar orbits could be seen in projected angles alone. An angle hidden by the tilt of the star The line profile of a transited star gives the angle between orbit and spin as projected on the sky. The true angle needs the tilt of the star's spin axis, and every way of measuring that tilt is blind in the same place — a star seen equator-on, which is exactly where an aligned system sits.

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ChaosConserved quantityHierarchical tripleKozai–LidovOctupoleOrbital flipRetrograde orbitSecular perturbation