Astronomy & Celestial Mechanics
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Orbits

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.

Assumes Secular theory, Angular momentum and The ellipse.

15 min read 6 figures Small perturbations, large consequencesMeasuring the unreachable

No planet has an eccentricity of its own: the secular theory of the Solar System turns each planet’s eccentricity into a sum over modes of the whole system, and what is conserved is not any planet’s eccentricity but a combination of all of them. That combination is the angular momentum deficit — how much angular momentum the planets’ orbits lack, compared with what the same planets would have on circular, coplanar orbits of the same sizes. Eccentricity and inclination both cost angular momentum, so the deficit is a budget out of which every planet’s eccentricity and inclination is paid.

Laplace’s proof that the eccentricities stay bounded holds only in the linear theory, and it fails because one combination of frequencies is nearly zero. The deficit does not depend on the frequencies. It is conserved by secular evolution to every order — by any interaction averaged over the orbits, however strong, provided no mean-motion resonance is involved — and so it gives a bound that survives exactly where the linear theory breaks: no planet can ever have more eccentricity than the whole deficit would buy it. This essay turns that bound into a stability criterion and applies it to the Solar System, where it gives a surprisingly sharp answer about where the danger comes from.

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.
Fig. 1 For each adjacent pair of planets, the deficit divided by the critical deficit at which their orbits could touch, on a logarithmic scale. Blue: the deficit of the pair’s own subsystem. Orange: the whole Solar System’s. On their own budget the inner pairs sit at 0.21, 0.12 and 0.34 — safe; on the whole budget at 112, 65 and 187. The giant pairs are safe either way.

The budget, and what it buys

The deficit of a planet of mass mm on an orbit of semi-major axis aa, eccentricity ee and inclination ii is

C=Λ(1−1−e2cos⁡i),Λ=mGM⊙a,C = \Lambda\left(1 - \sqrt{1 - e^2}\cos i\right), \qquad \Lambda = m\sqrt{GM_\odot a},

and the system’s deficit is the sum over planets. Every term is positive, and every term vanishes for a circular orbit in the reference plane, so a planet with a given share of the deficit has a definite largest eccentricity — the one it would have with no inclination and no share for anyone else.

Secular evolution moves the deficit between planets and between eccentricity and inclination. It does not change the total, because the averaged interactions conserve the semi-major axes, and with them each Λ\Lambda, as well as the total angular momentum. The deficit is the difference between the sum of the Λ\Lambda’s and the total angular momentum, and both are fixed. So the budget is fixed, and only its distribution changes.

A budget is useful only if it can be compared with a price, and the price that matters is a collision. Two orbits can touch when the inner planet’s farthest point reaches the outer planet’s nearest: a(1+e)=a′(1−e′)a(1+e) = a'(1-e'). The deficit that configuration costs depends on how the eccentricity is shared between the two planets, and there is a cheapest way of sharing it.

The price of a collision

The deficit a pair needs before its orbits can touch. The critical angular momentum deficit of a pair of planets, in units of the outer planet's circular angular momentum, against the ratio of their semi-major axes, for inner-to-outer mass ratios of 0.1, 1, 10. It is the least deficit, shared between the two in the cheapest way, with which the inner planet's farthest point can reach the outer planet's nearest. Widely spaced pairs need a large fraction of their angular momentum; a pair at 0.9 needs 2.7 × 10⁻³ for equal masses. A light inner planet makes the pair cheaper to bring together, because its eccentricity costs less — the minimum is reached by giving the light planet almost all of it. Mercury and Venus, at a ratio of 0.54 and a mass ratio of 0.068, need 1.8 × 10⁻² of Venus's, with Mercury carrying an eccentricity of 0.70 at the minimum.
Fig. 2 The critical deficit of a pair, in units of the outer planet’s angular momentum, against the ratio of their semi-major axes, for inner-to-outer mass ratios of 0.1, 1 and 10. Closer pairs need far less; a light inner planet makes the pair cheaper. Mercury and Venus need 1.8 × 10⁻² of Venus’s, with Mercury carrying an eccentricity of 0.70 at the minimum.

The critical deficit falls steeply as the two orbits are brought closer — a pair at nine-tenths of each other’s distance needs a few thousandths of its angular momentum to touch, where a pair at half needs a few hundredths — and it falls when the inner planet is light. The second dependence is the interesting one. Eccentricity costs a planet in proportion to its mass, so the cheapest way to bring two orbits together is to give almost all of the eccentricity to the lighter planet. For Mercury and Venus, whose mass ratio is about one to fifteen, the cheapest collision has Mercury at an eccentricity of 0.70 and Venus at 0.09.

This is the comparison Laskar and Petit made the basis of a classification: a system is AMD-stable if its deficit is less than the critical deficit of every adjacent pair, because then no redistribution the secular dynamics can produce brings any two orbits into contact. The criterion needs only the present orbits. It needs no integration, no frequencies and no assumption about which resonances operate, and it holds for as long as the dynamics is secular.

Mercury and Venus in the plane of their eccentricities. Mercury's eccentricity against Venus's. Above the dashed line Mercury's farthest point reaches beyond Venus's nearest, and the orbits can touch. Each solid curve holds the pair's deficit fixed: the combinations of eccentricities the pair could share if it had that much and kept it to itself. The critical curve just touches the dashed line, at Mercury 0.70 and Venus 0.09 — the cheapest way for the orbits to meet is almost entirely Mercury's. The deficit the four terrestrial planets actually hold (blue) is 4.8 times too small to reach it, and lets Mercury reach at most 0.38 even if Venus gives up all of hers. The dot is the present pair.
Fig. 3 Mercury’s eccentricity against Venus’s. Above the dashed line the orbits can touch. Each solid curve holds the pair’s deficit fixed; the critical curve just touches the line, at Mercury 0.70 and Venus 0.09. The terrestrial planets’ deficit (blue) is 4.8 times too small to reach it and lets Mercury reach at most 0.38. The dot is the present pair.

The picture makes the criterion concrete. Each curve of constant deficit bounds a region of the eccentricity plane the pair can reach; the collision line cuts across the plane; the critical deficit is the curve that just touches it. The deficit the four terrestrial planets hold between them — even if all of it were given to Mercury and Venus — encloses a region far inside the collision line. Mercury could reach an eccentricity of 0.38 at the very most, and Venus 0.09. The terrestrial planets, on their own budget, cannot collide with each other under secular evolution, ever.

Safe on their own budget, not on the system’s

The hero figure applies the same comparison to every adjacent pair, twice: once with the deficit of the pair’s own subsystem, and once with the deficit of the whole Solar System. The difference between the two is the whole story.

The terrestrial planets hold a deficit of about 8×10−98 \times 10^{-9} in units of a solar mass times the square root of an astronomical unit. The four giants hold about 4×10−64 \times 10^{-6}, five hundred times more, most of it Jupiter’s and Saturn’s — because the giants are three hundred times more massive and their eccentricities, a few per cent, are comparable to the terrestrial planets’. Measured against their own budget, the inner pairs sit at a fifth, an eighth and a third of their critical deficits: safe. Measured against the whole system’s, they sit at a hundred, sixty-five and nearly two hundred times critical: if the giants’ deficit could flow freely into the inner system, every inner pair could collide.

The giants themselves are safe either way. Even the whole system’s deficit, given entirely to any pair of giants, is less than half of what it would take to bring their orbits together, because they are widely spaced and massive and the price of their collisions is high. The outer Solar System is AMD-stable outright; its stability is proved by the present orbits alone.

So the criterion says something sharper than “stable” or “unstable”. The terrestrial planets are stable because the deficit does not flow between the two subsystems — because the inner planets’ secular modes and the giants’ are nearly separate, the giants’ frequencies far from the inner ones’, and the exchange between them averaging to nothing. It is the isolation of the inner Solar System, not a shortage of deficit, that keeps it intact.

A loan from Jupiter

How much of the giants’ deficit would it take?

How much of Jupiter's deficit Mercury would need. The largest eccentricity Mercury could reach if it held all of the terrestrial planets' angular momentum deficit plus a fraction of Jupiter's, against that fraction, on a logarithmic scale. With the terrestrial planets' own deficit alone the bound is 0.38. Mercury's orbit begins to destabilise the inner planets near an eccentricity of 0.6 — which needs 0.5 per cent of Jupiter's deficit — and reaches Venus's orbit at 0.86, which needs 1.7 per cent. Jupiter's deficit is 329 times the terrestrial planets', so a leak of a per cent of it is a flood for Mercury. The leak needs a channel, and the one the integrations find is a resonance between Mercury's perihelion precession and Jupiter's.
Fig. 4 Mercury’s largest possible eccentricity, if it held all of the terrestrial planets’ deficit plus a fraction of Jupiter’s. Alone: 0.38. An eccentricity of 0.6, where the inner planets begin to destabilise, needs 0.5% of Jupiter’s deficit; reaching Venus’s orbit at 0.86 needs 1.7%.

Jupiter’s deficit is three hundred and thirty times that of all four terrestrial planets together. Mercury, the lightest of them, needs little to reach a large eccentricity — its angular momentum is a twentieth of Venus’s — so a transfer of half a per cent of Jupiter’s deficit would let it reach 0.6, where integrations find the inner planets starting to destabilise, and under two per cent would let it reach Venus’s orbit. For the inner Solar System a leak of a per cent from the giants is a flood.

The leak needs a channel, and the long integrations have identified it. Mercury’s perihelion precesses at a rate close to the rate at which Jupiter’s perihelion does; the combination of the two frequencies is one of the small denominators that the linear theory computes badly. When the two rates come into resonance — which the chaotic diffusion of Mercury’s own frequency allows it to do, occasionally — Jupiter’s deficit flows into Mercury’s eccentricity, and the inner planets’ isolation ends. That is why about one per cent of the integrated futures see Mercury’s orbit go unstable, and why including general relativity, which adds 43 arcseconds a century to Mercury’s precession and moves it away from the resonance, reduces that probability sixty-fold. The criterion did not predict the channel. It predicted that one was needed, and measured how wide.

The asymmetry deserves a number from Jupiter’s side. Half a per cent of Jupiter’s deficit, the amount that takes Mercury to an eccentricity of 0.6, would lower Jupiter’s own eccentricity from 0.0484 to about 0.0483 — a change smaller than the oscillation Saturn drives in it every few tens of thousands of years, and invisible in any record. The transfer that would end the inner Solar System as it is would leave Jupiter’s orbit indistinguishable from what it is now. Stability, in a system where one member holds nearly all of a conserved quantity, is a question about how well the small members are shielded, not about whether the large one can afford to give.

Nor does the deficit care whether it is spent on eccentricity or on inclination. Mercury’s orbit is tilted by six degrees to the invariable plane, more than any other planet’s, and that tilt is part of its share of the budget; a secular exchange that took Mercury’s inclination down to zero would free deficit for its eccentricity, and one that tilted it further would lock deficit away. The inclination modes and the eccentricity modes are separate in the linear theory — two independent sets of frequencies — and coupled in the full one. The resonance that destabilises Mercury in the integrations involves both: its eccentricity rises as the deficit arrives from Jupiter through the perihelion resonance, and its inclination exchanges with the eccentricity through a second resonance in the node. A criterion that counted eccentricity alone would understate how much of the budget Mercury already holds.

A small deficit as a record of a quiet past

The same budget can be read backwards, and it says something about the Solar System’s early history. The terrestrial planets’ deficit today is small — their eccentricities and inclinations are a few per cent — and it is about all they have, because almost none of the giants’ deficit reaches them. That smallness is a constraint on anything that could have pumped it in the past.

The giant planets are thought to have rearranged themselves early on, Jupiter moving slightly inward and Saturn, Uranus and Neptune outward, in an episode that also swept a secular resonance across the asteroid belt. As the giants’ orbits changed, their secular frequencies changed, and the frequencies of the terrestrial planets’ modes lay in the path. Each time a giant frequency swept through a terrestrial one, the two subsystems were briefly coupled — exactly the leak the hero figure shows the inner planets cannot afford — and deficit flowed inward. A slow migration, taking tens of millions of years, would have swept slowly enough for the coupling to pump the terrestrial planets’ eccentricities to several times their present values, and the present small deficit would be impossible to reach from there, since nothing in secular dynamics removes deficit once it has arrived.

So the terrestrial planets’ quiet orbits are evidence that the giants’ rearrangement was fast — fast enough that each resonance crossing was brief and transferred little — or that it happened before the terrestrial planets had finished forming, while gas or a disc of debris could still damp what was pumped in. The same argument that the asteroid belt makes from the location of its inner edge, the terrestrial planets make from the size of their budget. A deficit is a quantity that can be added to and not easily taken away, and a small one is a record of a past in which little was added.

Three ceilings

Three ceilings on each terrestrial planet's eccentricity. For each terrestrial planet: the largest eccentricity Laplace–Lagrange theory gives it, the sum of its secular modes' amplitudes (dark); the largest the terrestrial planets' angular momentum deficit allows if every bit of it went to that planet (light); and the eccentricity at which its orbit would reach its inner neighbour's, or for Mercury its outer neighbour's (dashed). The linear theory gives Mercury 0.23, Venus 0.07, Earth 0.06, Mars 0.14; the deficit allows 0.38, 0.09, 0.07, 0.20. The deficit's bound needs no integration and no frequencies, and holds to every order of secular theory; it lies above the linear theory's values — only a little for Venus and the Earth, which already hold most of the deficit between them, and by more for the two light planets — and for Mercury it is still well below the eccentricity that brings it to Venus. It proves the terrestrial planets cannot collide among themselves — and nothing about what the giants might give them.
Fig. 5 For each terrestrial planet: the largest eccentricity in Laplace–Lagrange theory (dark), the largest its subsystem’s deficit allows (light), and the eccentricity at which its orbit reaches a neighbour’s (dashed). Linear theory: Mercury 0.23, Venus 0.07, Earth 0.06, Mars 0.14. The deficit: 0.38, 0.09, 0.07, 0.20.

The three ceilings on each terrestrial planet’s eccentricity measure three different things. The linear theory’s value is what the modes add up to if their phases line up: a prediction, valid as long as the linear theory is. The deficit’s bound is what the budget allows if every bit of the terrestrial planets’ deficit went to one planet: a guarantee, valid as long as the evolution is secular and the inner subsystem keeps its own budget. The collision value is the geometry of the neighbouring orbit.

For Venus and the Earth, which between them hold most of the inner deficit, the guarantee is barely above the prediction — there is little deficit elsewhere to borrow from. For Mercury and Mars, light planets with neighbours that could give them much, the guarantee is well above the prediction, and Mercury’s guarantee of 0.38 is far below the 0.86 at which it would reach Venus. The guarantee is the stronger statement and the weaker number: it is what the planets cannot exceed under any secular evolution at all, which is much more than the linear theory claims to know, and it is correspondingly generous.

What the criterion says about other systems

The value of a criterion that needs only the present orbits is that it can be applied to every planetary system ever found, including ones whose masses and eccentricities are known too poorly for an integration to mean much.

How close two eccentric planets can sit and still be safe by the budget. The ratio of deficit to critical deficit for a pair of 5-Earth-mass planets around a Sun-like star, each with eccentricity 0.02, 0.05, 0.1, against the ratio of their orbital periods. The pair is safe by the budget below one. At eccentricities of 0.02 it is safe beyond a period ratio of 1.07; at 0.05, beyond 1.17; at 0.1, beyond 1.36. The compact systems found by transit surveys mostly have period ratios between 1.3 and 2 and eccentricities of a few hundredths, which puts most of their pairs on the safe side by this criterion alone — a statement about billions of years, made from the present orbits, with no integration.
Fig. 6 Deficit over critical deficit for a pair of 5-Earth-mass planets around a Sun-like star, each with eccentricity 0.02, 0.05 or 0.1, against their period ratio. Safe below one: beyond period ratios of 1.07, 1.17 and 1.36 respectively. Most compact systems from transit surveys lie between 1.3 and 2.

The compact systems found by transit surveys — several planets of a few Earth masses within the orbit of Mercury, with period ratios of 1.3 to 2 — have eccentricities of a few hundredths, measured from transit durations and timing variations. For a pair of five-Earth-mass planets with eccentricities of two hundredths, the budget is safe for any period ratio above 1.07; at five hundredths, above 1.17; at a tenth, above 1.36. Most of the observed pairs are on the safe side, which says that their stability over the age of their stars needs no explanation beyond the present orbits — and the few that are not are the ones worth integrating.

The comparison also says something about how such systems were assembled. Planets that formed with more eccentricity than the critical deficit allows would have reached orbit crossing, collided or scattered, and either merged or spread out until their deficit was below the critical value of their new spacing. Observed systems mostly sitting below their critical deficits, some of them not far below, with spacings set by the feeding zones that formed them, is what a population sculpted by such instabilities would look like.

The budget is also a different kind of criterion from the one most often quoted for pairs of planets. The Hill criterion, from the zero-velocity surfaces of the three-body problem, says that two planets on initially circular orbits can never have a close encounter if their separation exceeds about three and a half mutual Hill radii, because their energy and angular momentum do not permit it. It is exact for two planets and a star and makes no averaging assumption, but it applies only to a pair, says nothing about long-term diffusion in a system of several, and depends on the initial eccentricities in a way that becomes useless when they are more than a few hundredths. The deficit’s criterion trades the exactness for generality: it assumes averaging, and in return it applies to any number of planets and any eccentricities. Between them they cover the short-period and the secular sides of stability, and a pair that passes both is safe in a sense that requires no integration.

What the bound assumes

The deficit is conserved by secular evolution, which is evolution averaged over the orbits. It is not conserved through mean-motion resonances, where two planets’ orbital periods are commensurate and the averaging fails; there, energy and angular momentum are exchanged on the orbital timescale and the semi-major axes change, so the Λ\Lambda’s are no longer fixed. A pair in or near a mean-motion resonance can be unstable even when it is AMD-stable, and close pairs are the ones where such resonances overlap and chaos sets in. The criterion is therefore a statement about the secular part of the dynamics, and needs a separate check — the Hill criterion, or the spacing at which mean-motion resonances overlap — for the short-period part.

The collision condition is geometric: the orbits touch, which is necessary for a collision but not sufficient. The planets also have to be at the touching point at the same time, and a close encounter short of a collision usually scatters rather than merges. AMD-instability is permission for disaster, not a forecast of it — which is why the Solar System, AMD-unstable for its inner planets on the whole budget, has a one per cent chance of trouble in five billion years rather than a certainty.

And the planets are treated as points with fixed masses on Keplerian orbits about a fixed Sun. General relativity, tides, and the Sun’s slow loss of mass all act outside the secular averaging that conserves the deficit, though each is small enough here to change the budget by less than the margins the figures show.

Still open: how isolated the inner planets are

The criterion reduces the stability of the terrestrial planets to one question: how well isolated is their secular subsystem from the giants’? The answer is a number between nothing and everything — the fraction of the giants’ deficit that can reach the inner planets over a given time — and the long integrations measure it only through its consequence, the occasional destabilisation of Mercury.

What would measure it directly is the distribution of the deficit’s exchange between the two subsystems across an ensemble: how much flows inward over each billion years, and how often it flows in the large doses that matter. That distribution is the real content of the one per cent. Its shape — whether the leak is a slow trickle that sometimes accumulates, or rare large transfers through the resonance with Jupiter — decides whether the inner Solar System’s stability is a matter of time or of luck, and it is set by the same small denominators that carved the inner edge of the asteroid belt.

Where this goes next

Earth's eccentricity is a sum of 8 sinusoids. The eccentricity of Earth over 800 thousand years, from the Laplace–Lagrange solution for all eight planets — the secular matrix built from the JPL masses and semi-major axes, symmetrised, and diagonalised by Jacobi rotations. It runs between 0.0035 and 0.0436, and it has no period, because it is a sum of 8 incommensurable frequencies. The two largest contributions to this planet are the modes at 3.73 and 7.33 arcseconds per year, drawn as the flat lines: those are constants, and everything moving in the figure is their beat. The check is the quadratic form ½ΣΛe², which a symmetric secular matrix conserves exactly and which drifts by 6.7e-16 across the whole interval — computed from the same curves the figure draws and from nothing the eigenvalues were fitted to. The true angular momentum deficit, Σ Λ(1 − √(1−e²)), drifts by 7.1e-4, and that difference is not an error either: the two agree only to fourth order in e, and Mercury at 0.206 supplies almost all of the gap. No planet has an eccentricity of its own Strip the short-period terms out of the planetary equations and what is left is a linear system. Its eigenvectors are modes of the whole solar system, and the number a catalogue quotes for a planet's eccentricity turns out to be a reading of a clock rather than a property of the planet. The divisor is 0.129″/yr and the theory's own error is 0.24. Six frequencies of the secular solar system on one logarithmic axis, in arcseconds per year. The top two are the pair whose near-equality is the whole story: the perihelia of Mercury and Jupiter separate at 1.333″ a year, the nodes of Mercury and Venus at 1.462, and the difference of those two differences is 0.129 — a resonant argument that turns once every 10.0 million years. A term with that argument in the disturbing function acts in one direction for five million years at a stretch, which is what pumps Mercury's eccentricity, and it is the reason the inner solar system's Lyapunov time is what it is. The bottom three bars are why this figure exists. The divisor is smaller than the corrections the theory that computes it leaves out. Relativity contributes 0.4298″ a year to g₁ alone — the same 43 arcseconds a century that broke Newtonian gravity — which is 3.3 times the divisor; the fourth-order terms in the eccentricity that Laplace–Lagrange truncates come to about 0.24″, which is 1.8 times it; and the second-order solution computed on this page gets 0.35″, missing the published value by more than the value itself. A theory cannot bound what it cannot resolve. Laplace's proof that the eccentricities stay bounded is a proof about a system whose frequencies are constants, and the frequency that decides the question is not one. The bound that holds only in the linear theory Laplace proved the planetary eccentricities bounded, and the proof is a proof about a linearised system with constant frequencies. One combination of those frequencies is nearly zero — and it is smaller than the terms the linearisation threw away, which is why the stability of the solar system is a probability rather than a theorem. 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.

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Essays that link to this one from their own argument.

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Angular momentum deficitCompact planetary systemsEccentricityLaplace–Lagrange theorySecular resonanceSecular theorySolar system stability