Astronomy & Celestial Mechanics
the sky, computed
The observed sky

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.

Assumes Precession, Seasons and Secular theory.

16 min read 8 figures The sky as a clockSmall perturbations, large consequences

The seasons are a consequence of a number. The Earth’s rotation axis is tilted by 23.44 degrees from the perpendicular to its orbit, so each hemisphere is presented to the Sun for half the year and turned away for the other half, and the Sun’s path across the sky changes accordingly. Change the number and the climate changes with it: at zero there are no seasons anywhere, and at sixty degrees the poles receive more annual sunlight than the equator does.

The number is not a constant, and the interesting question is not whether it varies but what kind of variable it is.

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.
Fig. 1 The obliquities at which a spin axis can sit still, against the ratio of its own precession rate to the rate at which its orbit plane turns. Each column is the full set of equilibria at that ratio, found by root-finding rather than traced from memory. Below a threshold there are two of them and above it there are four, and the join is where the behaviour changes. Three worlds are marked by their own measured precession constants, and two of the three sit within a few tenths of that join.

Two precessions, and the angle between them

The dynamics involve two slow motions and the relationship between them is the whole subject.

The first is the precession of the spin axis. The Sun and Moon pull on the Earth’s equatorial bulge, and the resulting couple makes the axis sweep a cone about the perpendicular to the orbit, once in about 25,772 years. The rate is αcos⁡ε\alpha\cos\varepsilon, where α\alpha is a constant of the planet — its dynamical ellipticity, its rotation rate, and the masses and distances of whatever is pulling on it.

It is accompanied by a smaller oscillation with the Moon’s node in it, which does not accumulate. The second is the precession of the orbit plane itself. The other planets perturb the Earth’s orbit — no planet has an eccentricity or an inclination of its own — and its plane is not fixed: it turns slowly about the invariable plane of the solar system, with a small inclination, at a rate conventionally called gg or ss.

The obliquity is the angle between the spin axis and the orbit normal. Both are moving. If they precess at very different rates the angle between them is nearly constant and the obliquity is nearly a constant of the motion. If they precess at similar rates they can lock — and near the lock, the angle between them is free to swing.

The path of the celestial pole over 25,772 years. The circle the Earth's rotation axis traces among the stars, at a radius equal to the obliquity, with the bright stars that fall near it and the years at which each is closest. Polaris is the pole star for a few centuries either side of now, and nothing else on the circle is nearly as close.
Fig. 2 The first of the two motions, drawn among the stars it passes. The pole traces a circle of angular radius equal to the obliquity, once in 25,772 years, and the bright stars near that circle take turns being the pole star. What the picture holds fixed is the radius of the circle — the obliquity — and this essay is about the fact that the radius is itself a variable on a longer timescale.

Where an axis can sit still

The clean way to see the structure is to work in the frame that turns with the orbit’s node. In that frame the problem is a single degree of freedom with a Hamiltonian, and the equilibria have a name.

H(X,ψ)=α2X2+g[Xcos⁡I−1−X2 sin⁡Icos⁡ψ],X=cos⁡ε,H(X,\psi) = \frac{\alpha}{2}X^2 + g\left[X\cos I - \sqrt{1-X^2}\,\sin I \cos\psi\right],\qquad X=\cos\varepsilon,

with ψ\psi the longitude of the spin axis measured from the node and II the orbit’s inclination to the reference plane. The equilibria of this — the places where both the obliquity and the longitude hold still, so that the whole geometry rotates rigidly — are the Cassini states, named for the seventeenth-century description of the Moon’s rotation that turned out to be a special case of them.

Where a spin axis can sit still. Level curves of the Colombo Hamiltonian for a spin axis precessing at α cos ε about an orbit normal that is itself precessing at rate g about the invariable pole, with an orbit inclination of 1.5 degrees and α/g = 2. The horizontal axis is the longitude of the spin axis measured from the orbit's node and the vertical axis is the obliquity; the curves are traced by finding where the Hamiltonian crosses each level on a grid rather than drawn as ellipses that look right. The 4 marked points are the Cassini states, the obliquities at which the two precessions keep step so that the axis holds a fixed geometry — 1.5°, 59.5°, 60.5°, 179.5° — and each one is checked to satisfy α sin ε cos ε + g sin(ε − I) = 0 to a part in a billion. How many there are is not a matter of degree: four exist when α/g exceeds the three-halves power of the sum of sin I and cos I each raised to two thirds, which here is 1.135, and two when it does not, so a planet whose spin slowly changes can find two of its equilibria annihilate. Closed curves around a state are libration, and an axis on one of them oscillates in obliquity for ever without drifting; the curves that run the full width are circulation. What this cannot show is the real Solar System, which has not one orbital precession frequency but a dozen, and it is their overlap rather than any one of them that makes an obliquity chaotic.
Fig. 3 Level curves of that Hamiltonian at a spin precession twice the orbital one. The marked points are the equilibria, and the heavier curve through the unstable one is the separatrix. Inside it an axis librates — its obliquity oscillates around an equilibrium for ever and never drifts; outside it, the longitude circulates and the obliquity varies with it by a bounded amount. Each state satisfies the equilibrium condition to a part in a billion, which is what the figure checks rather than the labels it prints.

The number of equilibria is not a matter of degree. Four exist when

αcos⁡ε∣g∣>(sin⁡2/3I+cos⁡2/3I)3/2,\frac{\alpha\cos\varepsilon}{|g|} > \left(\sin^{2/3}I + \cos^{2/3}I\right)^{3/2},

and two exist when it does not. At the boundary two of them merge and annihilate. A planet whose spin precession rate changes slowly — because it is being tidally despun, or because its moon is receding — crosses that boundary, and when it does, an axis that had been librating in a state that has just ceased to exist is released.

The Earth, with and without the Moon

The Earth’s precession constant is about 54.9 arcseconds a year, which at the present obliquity gives a spin precession of 50.4. The dominant orbital frequency is around 18.9. The ratio is 2.7, comfortably on the four-state side of the boundary and not near any single resonance.

Remove the Moon and the picture changes. The Moon contributes about two-thirds of the lunisolar couple, so without it α\alpha falls to something like a third of its value, the ratio falls to near 0.9, and the Earth sits essentially on the boundary — in the region where the equilibria are close together, the libration is wide, and neighbouring resonances overlap.

That is the real content of the claim that the Moon stabilises the Earth’s climate, and it is worth stating precisely because the popular version overstates it. The Moon does not hold the axis rigid. It moves the Earth’s spin precession rate away from the band where the orbital frequencies live, so that the obliquity librates by about a degree and a half instead of wandering over tens of degrees. The stabilisation is a matter of being off resonance, not of being held.

A wobble that should have stopped seventy years ago. Left, the path of the Earth's rotation pole across its own crust over 13 years, as the sum of two circular motions: the 433-day Chandler wobble at 150 milliarcseconds and the annual wobble at 90. The spiral is a beat, and its period measured off the drawn path is 6.39 years against the 6.39 the two frequencies require. Right, the same path's radius against time. Two numbers in this figure are the argument. The first is the Chandler period itself: a rigid Earth of dynamical ellipticity 0.0032737 would wobble freely at 305 days, and the observed 433 is 42 per cent longer because the Earth deforms under its own wobble and the oceans move with it — the period is a measurement of the planet's elasticity, made by watching a free motion rather than by forcing anything. The second is the damping: at a quality factor of about 100 the wobble should decay in 38 years, and it has been running for as long as anyone has watched. Something is exciting it continuously, and the excitation is fluctuating pressure at the bottom of the ocean and in the atmosphere. What the figure cannot show is the excitation itself, which is not periodic and is only visible statistically.
Fig. 4 The fastest of the axis’s motions, and the one with no external cause at all. The Chandler wobble is a free oscillation of the Earth about its own figure axis with a period of 433 days — not the 305 a rigid Earth would have, the difference being elasticity and the oceans — and it beats against the annual forced wobble to produce the six-year envelope drawn here. It damps in a few decades and has not stopped, so something is exciting it continuously. Precession moves where the axis points over millennia; this moves it by metres over a year, and both are the same axis.

Mars

Mars has no large moon and a precession constant of about 8.3 arcseconds a year against an orbital frequency near 7.1. The ratio is close to one, which is close to the boundary, and it is close to it in a spectrum where several modes lie nearby.

The consequence, established by long numerical integrations, is that the Martian obliquity is chaotic. It has spent time near zero and time above forty-five degrees, and the transitions are not driven by any external event — they are the system’s own dynamics, exploring the region where resonances overlap.

A criterion that is out by 2.5× and still used. Chirikov's overlap criterion, drawn as the quantity it compares. Each primary resonance of the standard map is a pendulum of half-width 2√K in the momentum, and neighbouring resonances are 2π apart; the criterion says the last curve between them goes when the two half-widths add to the separation, which happens at K = π²/4 = 2.467. The threshold that is actually measured is 0.9716, marked with the second rule, and the transport test confirms it on the map itself: at 0.9 of it a trajectory wanders 4.18 in p and at 1.15 of it 1.8 whole cylinders. The gap is a factor of 2.54, and it is not a defect in the criterion so much as an admission of what it leaves out: between any two primary resonances lies an infinite family of higher-order ones, and they have eaten the space long before the primaries reach each other.
Fig. 5 The mechanism, in the setting where it is easiest to draw. Two resonances, each with its own separatrix, and a control parameter that widens both. While they are separated the trajectories between them are regular and each resonance traps whatever is inside it. When the widths grow enough to touch, a trajectory can pass from one to the other, and from there to a third — and the motion becomes a wander through a connected region rather than an oscillation in a closed one. Nothing has been added to the system to make this happen, and it is where the chaos in the solar system comes from as well.

The observational consequences are visible. Mars has layered deposits at both poles, with a stratigraphy that records repeated cycles of ice deposition and removal, and mid-latitude ice deposits that cannot be in equilibrium with the present climate. At high obliquity the poles are the warmest part of the planet in summer and the polar caps sublimate, redistributing water to the mid-latitudes; at low obliquity the reverse. The layers are a record of the axis’s excursions, read the way a tree ring is read.

What sets the constant

Everything above depends on α\alpha, which is a property of the planet:

α∝32 C−AC n2ω,\alpha \propto \frac{3}{2}\,\frac{C-A}{C}\,\frac{n^2}{\omega},

where (C−A)/C(C-A)/C is the dynamical ellipticity — the flattening of the body, expressed as a difference of moments of inertia — nn is the orbital mean motion and ω\omega the spin rate. So a planet’s obliquity dynamics depend on its own shape, which depends on how fast it spins, which is itself changing.

The wobble inside the wobble. Left: the two components of nutation over 40 years, from the four largest terms of the standard series. The long wave is the regression of the Moon's node in 18.613 years, which is where nearly all of it comes from; the ripple on it is the semi-annual solar term at 1.3″ and the semi-monthly lunar one at 0.2″. Right: the loop the pole actually traces over one node cycle, in arcseconds on the sky, with the mean pole at the centre. The loop is 14.89″ by 19.90″ — taller than it is wide, because the longitude term is foreshortened by sin ε while the obliquity term is not, which is the one thing a schematic of this is always drawn getting wrong. Over the same 18.6 years precession itself carries the pole 936″ along its circle, 47 times the loop's own height, so nutation is a wobble on a path and not a path. It is nonetheless 99,480 times the 0.2 mas astrometry of a modern catalogue, which is why a position has to say whether it is referred to the mean pole or the true one.
Fig. 6 And the periodic terms riding on the slow circle. Nutation is the precession’s own wobble: the torque depends on the Moon’s orbital plane, which itself precesses in 18.6 years, so the pole traces a small ellipse superimposed on the great one. The largest term is nine arcseconds — a thousand times smaller than the precession it modulates and a thousand times larger than the astrometric precision that has to model it. Every one of these motions is the same torque on the same bulge, separated only by the period of whatever is doing the pulling.

The small variation that matters anyway

The Earth’s obliquity oscillates between about 22.1 and 24.5 degrees with a period of about 41,000 years. That is a libration, a bounded oscillation about an equilibrium, and by the standards of Mars it is nothing.

It is nevertheless one of the three astronomical cycles that pace the ice ages, alongside the ones that make a year an awkward number of days, and it is the one whose signature is cleanest in the record. Obliquity changes the contrast between summer and winter at high latitudes without changing the annual total much, which is exactly the quantity that decides whether a summer is warm enough to melt the previous winter’s snow. A 41,000-year periodicity is present in deep-sea sediment records over the last few million years, and it dominates the older part of the record.

The period the record has and the forcing does not

The 41,000-year signal is the clean part of the story, and the rest of the ice-age record contains a discrepancy that the astronomy alone does not explain.

For the last eight hundred thousand years the dominant periodicity in the ice volume record is not 41,000 years but about 100,000. That is close to the period at which the Earth’s orbital eccentricity varies, and the natural reading is that eccentricity is doing the forcing.

The difficulty is that eccentricity is by far the weakest of the three astronomical influences on insolation. Changing the eccentricity from 0 to 0.06 changes the annual mean insolation by about a tenth of a per cent, against the several per cent that obliquity moves the high-latitude summer by. So the record’s strongest period corresponds to the weakest forcing, which is a problem rather than a confirmation.

It is made sharper by the fact that the record did not always look like this. Before about a million years ago the dominant period was 41,000, matching obliquity, and the switch to 100,000 happened over a few hundred thousand years with no corresponding change in the orbital forcing. The astronomy did not change; the response did.

The favoured explanations are all about the climate system rather than the orbit: that the ice sheets grew large enough to have their own internal timescale, that a threshold in ice thickness or in bedrock response introduced a nonlinearity, or that the 100,000-year signal is not a response to eccentricity at all but a beat produced by the system skipping obliquity cycles.

A forcing measured to nine figures, a response that does not track it, and a transition with no forcing behind it is an uncomfortable position for an argument that is often presented as a triumph of celestial mechanics. The astronomy is not in doubt; what the climate does with it is.

What the model here cannot show

The two-frequency Colombo model in the figures is a caricature in one specific way, and it is the way that matters. It has one orbital precession frequency, and the solar system has many. Every real chaos in this subject comes from the overlap of neighbouring resonances, which a model with one resonance cannot contain. The figures show why a planet near the boundary is delicate; they cannot show what happens to it, and only a numerical integration of the whole system can.

That integration has its own limit, which is the expiry date on any prediction in a chaotic system. The Martian obliquity is predictable for a few tens of millions of years and not beyond; the Earth’s orbital elements are predictable for about sixty million. Past that, the statistics are the result and the trajectory is not.

Measuring an obliquity that is not the Earth’s

Everything above treats a planet’s obliquity as a known number, and it is worth setting out how it is known, because the methods differ enormously in what they require.

For a body with a visible surface the obliquity is geometric. Track a feature as the body rotates, over enough of an orbit that the viewing geometry changes, and the axis’s orientation in space falls out. That is how Mars’s obliquity was known long before anything went there, and it is good to a fraction of a degree.

For a body with no features the axis is found from its gravity field instead. A spacecraft’s orbit responds to the planet’s flattening, and the flattening is symmetric about the rotation axis — so tracking an orbiter determines the axis without any surface being seen. That is how the obliquities of the giant planets are known to the precision they are.

For an exoplanet neither is available, and the measurements that exist are indirect. A transiting planet on an eccentric orbit has an asymmetric thermal phase curve, and an obliquity changes that asymmetry in a way a circular orbit cannot mimic — so a full-orbit infrared light curve carries an obliquity signature at the level of tens of parts per million.

The other route is the one that will eventually be used routinely: the reflected-light curve of a directly imaged planet. A tilted planet presents a different fraction of its illuminated hemisphere through the year, and the resulting seasonal modulation is a direct measurement of the tilt. Neither has yet produced a secure detection.

So the dynamical arguments in this essay are, for every planet outside the solar system, arguments about a quantity nobody has measured — which is worth remembering when a habitability claim rests on an obliquity being stable.

There is one exception, and it is a partial one. A transiting planet’s orbital orientation relative to its star’s spin is measurable, and for a system with more than one transiting planet the mutual inclinations are measurable too. Neither is the planet’s own obliquity — the tilt of the planet’s spin axis relative to its own orbit — but the two are connected by the same secular machinery this essay describes, since a planet’s spin precession responds to the orbit’s precession exactly as the Earth’s does.

So a well-characterised multi-planet system constrains the frequencies a planet’s spin axis would have to avoid, without constraining where that axis actually is. That is enough to say which systems are dynamically dangerous and not enough to say what happened in any of them, which is a familiar position: the resonance structure is computable and the initial condition is not.

The same reasoning applies to the solar system’s own history and is the reason the argument is not merely hypothetical there. The frequencies were different in the past, because the planets have migrated, and a spin axis that is comfortably off resonance now may have been captured and released.

Where the tilt came from

Nothing above says what the tilt was originally, and that is a separate question with a different kind of answer.

A planet built out of a flat disc of material, accreting smoothly, would end up with its spin axis perpendicular to its orbit — the disc has one preferred direction and everything in it shares that direction. The obliquities in the solar system are not like that. The Earth is at 23.4 degrees, Mars at 25.2, Saturn at 26.7, Neptune at 28.3, Uranus at 98, and Venus at 177, which is to say upside down.

Two mechanisms are on offer and they leave different fingerprints. One is impacts: the last few bodies a planet swallows are large enough that their individual angular momenta are a substantial fraction of the planet’s, so the final tilt is a random walk with a few steps. That naturally produces a wide, roughly isotropic distribution, which is what the outer planets look like, and it is the standard account of Uranus.

The other is resonance capture of exactly the kind this essay has been about. A planet whose spin precession rate drifts into commensurability with an orbital mode can be captured into a Cassini state and carried to a large obliquity as the resonance moves — no impact required, and the tilt arrives smoothly. Saturn’s obliquity is the worked example: its spin precession is close to the node frequency of Neptune’s orbit, and a capture driven by the outward migration of the giant planets reproduces the observed 26.7 degrees.

The two accounts are distinguished by whether the planet is still in the resonance now, which is a question about a precession constant, which is a question about the planet’s internal mass distribution — so an obliquity’s origin is measured by weighing the interior with a wobble.

For Saturn that test has been made and the answer moved. The precession constant depends on the planet’s moment of inertia, and the moment of inertia was revised when a spacecraft’s final orbits measured the gravity field from inside the rings — which showed a deeper, more diffuse core than the older models assumed. The revision shifts the spin precession rate enough that Saturn now appears to sit just outside the resonance rather than inside it, and the favoured reading is that it was captured, carried to its present tilt, and then released when Titan’s outward migration moved the frequencies apart. That is a more elaborate history than either simple account, and it is the kind the machinery in this essay is for: a tilt is not a constant, and neither is the resonance that set it.

The equilibrium the whole argument turns on depends on the ratio of two precession rates, and both of the quantities that set it are worth moving.

Where a spin axis can sit still. Level curves of the Colombo Hamiltonian for a spin axis precessing at α cos ε about an orbit normal that is itself precessing at rate g about the invariable pole, with an orbit inclination of 1.5 degrees and α/g = 0.5. The horizontal axis is the longitude of the spin axis measured from the orbit's node and the vertical axis is the obliquity; the curves are traced by finding where the Hamiltonian crosses each level on a grid rather than drawn as ellipses that look right. The 2 marked points are the Cassini states, the obliquities at which the two precessions keep step so that the axis holds a fixed geometry — 3.0°, 179.0° — and each one is checked to satisfy α sin ε cos ε + g sin(ε − I) = 0 to a part in a billion. How many there are is not a matter of degree: four exist when α/g exceeds the three-halves power of the sum of sin I and cos I each raised to two thirds, which here is 1.135, and two when it does not, so a planet whose spin slowly changes can find two of its equilibria annihilate. Closed curves around a state are libration, and an axis on one of them oscillates in obliquity for ever without drifting; the curves that run the full width are circulation. What this cannot show is the real Solar System, which has not one orbital precession frequency but a dozen, and it is their overlap rather than any one of them that makes an obliquity chaotic.
Fig. 7 The Colombo Hamiltonian for a body whose spin precession is slower than its orbit’s rather than faster. The number of equilibria changes, which is the bifurcation the Cassini states pass through — and which side of it a body sits on is decided by a ratio it does not control.
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 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| = 1.305 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.
Fig. 8 And the obliquity map for an orbit inclined five degrees rather than one and a half. The resonance widens with the inclination, so a planet on a more inclined orbit has a wider band of ratios within which its obliquity is chaotic — which is why the Earth’s stability depends on the Moon and Mars’s absence of one does not save it.

Where the ladder goes

The next rung is the other free motion of the same axis: not its orientation in space but its position within the body. The Earth’s rotation pole wanders across its own crust by a few metres a year, on a period that is not the rigid-body one, and that wobble should have damped away long ago.

Further out, the same Cassini geometry describes something quite different. The Moon’s own spin axis sits in one of these states, which is why its equator, its orbit plane and the ecliptic share a common line of nodes — the observation Cassini actually made. The same states have been proposed for the obliquities of exoplanets that have been tidally processed, where capture into a high-obliquity state would leave a body permanently tilted and dissipating tidal heat for as long as it stayed there.

Where this goes next

The path of the celestial pole over 25,772 years. The circle the Earth's rotation axis traces among the stars, at a radius equal to the obliquity, with the bright stars that fall near it and the years at which each is closest. Polaris is the pole star for a few centuries either side of now, and nothing else on the circle is nearly as close. The pole star has a shelf life, and the sky has a slow hand The Earth's axis traces a circle among the stars once every 25,772 years. Polaris is at the pole now, was not four thousand years ago, and will not be in two thousand more. The Sun's altitude through the day at latitude 52°. Solar altitude against the hour of the day, at one latitude, for the solstices and the equinox. Where a curve crosses zero is sunrise or sunset, and the width between the crossings is the length of the day. The Sun's path, and the tilt that makes the seasons Summer is not when the Earth is closest to the Sun — that happens in January. It is when the Sun climbs higher and stays up longer, and both come from a 23.4° tilt. 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.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Cassini stateChaosDynamical ellipticityInvariable planeLibrationMilankovitch cyclesNodal precessionObliquityOrbital inclinationPrecession constantResonance overlapSeasonsSecular resonanceSpin axis precession