The bound that holds only in the linear theory

The rung below made a planet’s eccentricity a reading of a clock rather than a property of the planet: strip the short-period terms out of the planetary equations and what is left is a linear system whose eigenvectors are modes of the whole solar system, oscillating at frequencies that belong to nobody in particular.

That construction came with a famous consequence attached. A linear system with real frequencies has bounded solutions, in the sense that the elements wander and do not run away. The eccentricities and inclinations wander, sometimes by a great deal, and they never grow without limit — which is what Laplace and Lagrange showed and what was taken, for most of two centuries, as a proof that the solar system is stable.

It is not a proof that the solar system is stable. It is a proof that a particular linearisation of the solar system is stable, and the difference matters at exactly one place.

What the theorem actually says #

The secular problem, written to second order in the eccentricities and inclinations, splits into two independent linear systems. One is in the variables $(h, k) = e(\sin\varpi, \cos\varpi)$ and its eigenvalues are the frequencies conventionally called $g_1$ through $g_8$. The other is in $(p, q) = i(\sin\Omega, \cos\Omega)$ and its eigenvalues are $s_1$ through $s_8$.

Both matrices come from averaging the disturbing function over the fast angles, in the way an orbital average discards what happens within one revolution; both are symmetrisable, so both have real eigenvalues, and a linear system with real eigenvalues has solutions that are sums of sinusoids. The eccentricity of any planet is then the vector sum of eight rotating terms with fixed amplitudes, and it is bounded by the sum of those amplitudes for ever.

The inclination system carries a check worth stating, because it is a conservation law wearing a linear-algebra disguise. One of its eigenvalues is exactly zero. Tilting the entire system rigidly changes no mutual inclination and therefore costs nothing, so a uniform tilt is a zero-frequency mode — which is the statement that the direction of the total angular momentum is fixed. That direction is the invariable plane — angular momentum is the conserved quantity behind equal areas — and a construction of the matrix that lost the zero would have lost its conservation without saying so.

What the theorem drops #

The linearisation keeps terms of second order in eccentricity and inclination and discards everything above. That is not a small approximation for the inner solar system, where Mercury’s eccentricity is 0.206 and its square is four per cent.

Four per cent of a frequency near five arcseconds a year is about a fifth of an arcsecond a year. Hold on to that number.

The theory also drops general relativity, which contributes a genuine additional perihelion advance to every planet and by far the largest to Mercury: forty-three arcseconds a century, which is 0.43 arcseconds a year added to $g_1$.

Now the frequencies themselves. The combination that matters is

$$\theta = (\varpi_1 - \varpi_5) - (\Omega_1 - \Omega_2),$$

the difference between Mercury’s perihelion relative to Jupiter’s and Mercury’s node relative to Venus’s. Its rate is $(g_1 - g_5) - (s_1 - s_2)$. The two halves are 1.333 and 1.462 arcseconds a year, and their difference is 0.129 — a resonant argument that goes round once every ten million years.

Set that beside the two numbers held on to above. Relativity’s contribution to $g_1$ is more than three times the divisor. The terms the second-order theory truncates come to about twice it. And the second-order calculation done from scratch, with the same planetary masses and the same Laplace coefficients, returns a divisor of the wrong sign and three times the magnitude.

A theory cannot bound what it cannot resolve. The quantity that decides whether the linear picture survives is smaller than the theory’s own error in computing it.

Why a small divisor is dangerous #

The reason a nearly-zero frequency matters is not that the linear theory breaks — the linear theory has no such term in it at all. It matters because of what the next order does with it.

At fourth order in the eccentricities the equations acquire terms proportional to $\cos\theta$ and $\sin\theta$, with $\theta$ the resonant argument above. Ordinarily such a term averages away: the argument circulates, the term changes sign, and over many circulations the net effect is a small oscillation. The size of that oscillation is the term’s coefficient divided by its frequency.

Divide by a frequency near zero and the oscillation is no longer small. The arithmetic of that division is worth doing once. A fourth-order term in the secular equations has a coefficient of order $e^2$ times a second-order frequency, so a few hundredths of an arcsecond a year for Mercury. Divided by an ordinary secular frequency of five arcseconds a year, it produces an eccentricity oscillation of order a per cent of the amplitude already there — negligible. Divided by 0.129, it produces something forty times larger, and forty times a per cent is not negligible at all.

The mechanism is not exotic and this collection already has it. Two resonances whose libration widths overlap produce chaos, and that is where the inner solar system’s chaos comes from. The secular resonances are close enough together, and wide enough, that the region they occupy is not a set of clean islands. What makes this particular pair the ones that matter is that a libration width scales as the square root of the term’s coefficient divided by nothing at all, while the spacing between the two resonances is the divisor — so a small divisor is exactly the condition for two resonances to overlap.

The other divisors, and why only this one is dangerous #

Eight $g$ frequencies and eight $s$ frequencies give a great many combinations, and a combination close to zero is not rare. What makes this one different is a matter of degree that becomes a matter of kind.

Most of the small combinations are small because two frequencies happen to be near each other, and they come out at a few tenths of an arcsecond a year at worst — periods of a few million years, which is long but not long compared with anything. The combination $(g_1-g_5) - (s_1-s_2)$ is small because two differences are near each other, and the cancellation is second-order: two quantities near 1.4 arcseconds a year agreeing to within a tenth.

There is a second reason it is the dangerous one, and it is about which planets are involved. A resonant term pumps the eccentricity of the planet whose amplitude in the relevant mode is largest, and both $g_1$ and $s_1$ belong to Mercury — the planet with the smallest angular momentum in the system, the largest eccentricity to begin with, and by far the least to lose. The same resonance acting on Jupiter would be a rounding error, because the angular momentum deficit available could not move Jupiter’s eccentricity by a per cent.

The metronome that is not chaotic #

The same theory has a striking success in the opposite direction, and it is worth the detour because it shows what the linear picture is good for.

The combination $g_2 - g_5$ — Venus against Jupiter — comes to about 3.2 arcseconds a year, which is a period of 405,000 years. It is the largest single term in the Earth’s eccentricity, and unlike almost everything else in the secular solar system it is stable: the two frequencies involved are dominated by the giant planets and the inner planets’ chaos barely touches them.

That stability has been used. Cyclic variations at 405,000 years appear in sedimentary records going back more than two hundred million years, matching a prediction made from planetary dynamics with no free parameters. It is now the longest-baseline clock in geology, and the reason it works is precisely the reason the Mercury argument does not: the term whose frequency is well determined is the one that can be used, and the term whose frequency is not is the one that decides the outcome.

The other orbital cycles in the same record — the shorter eccentricity terms near 100,000 years, the obliquity near 41,000, the precession near 21,000 — are all present and all less reliable further back, for the same reason. Their frequencies drift as the system’s chaos works on them, so a rock older than about fifty million years cannot be dated by them without assuming the answer.

What the numerical experiments say #

The question is therefore not answerable analytically, and it has been answered numerically instead: integrate the full equations, not a truncation, for five billion years, many times, with the initial conditions varied within their measurement uncertainty.

The result is a distribution rather than an answer. In about one per cent of such integrations Mercury’s eccentricity climbs past 0.7 within five billion years. At that eccentricity its orbit crosses Venus’s — and an orbit can look like a circle and still not be one, which is exactly what makes the present configuration look safer than it is. What follows a crossing is a collision, an ejection, or a close encounter that destabilises the rest of the inner system.

The other ninety-nine per cent are unremarkable. The eccentricities wander within the ranges the linear theory sketches and nothing happens. The one per cent is not a statement about how well the planets have been measured. Improving the measurements by a factor of a thousand would move the horizon out by the logarithm of a thousand times the Lyapunov time, which is about thirty-five million years, and would leave the five-billion-year question exactly where it was.

What is conserved even so #

Something does survive, and it is worth having because it bounds the damage rather than the wandering.

The quantity is the angular momentum deficit: the difference between the angular momentum the system would have if every orbit were circular and coplanar, and the angular momentum it actually has. To second order it is $\tfrac12\sum\Lambda_j e_j^2$, with $\Lambda$ the circular angular momentum of each planet, and the secular matrix’s symmetry makes it exactly conserved in the linear theory.

Past the linear theory it is still conserved, because it is a consequence of the exact conservation laws rather than of the truncation. And it is small. The total deficit of the solar system is a tiny fraction of what would be needed for the giant planets to do anything interesting, which is why the instability, when it happens in an integration, is always confined to the inner planets: there is not enough deficit in the system to disturb Jupiter, and the whole of it is enough to destroy Mercury. There is a lesson in that sequence which is not about celestial mechanics. A result quoted for a century as settled was known within its own field to depend on a truncation, and the reason it survived is that the alternative was silence rather than a competing claim. A field will hold on to a wrong statement in preference to no statement, and the correction arrives when something can be said instead — which here meant a computer rather than an argument.

What would settle it #

Three things would, and none of them is analysis.

A better ephemeris does not help, for the reason above. The horizon moves logarithmically and the question is five billion years out.

A better integration might. The one-per-cent figure comes from a few thousand integrations of an averaged model plus a few hundred of the full one, and the tail of a distribution estimated from a few hundred samples is exactly where a number is least reliable. Whether the answer is one per cent or three is not currently decidable.

And the initial conditions are not the only uncertainty. The mass of the asteroid belt — a set of nuisance parameters in every ephemeris — the tidal dissipation in Mercury, and the treatment of relativity all move $g_1$ by amounts comparable with the divisor. A calculation that changes the divisor’s sign changes whether the argument librates, and therefore changes the answer qualitatively rather than by a percentage.

The result’s history bears that out, and it is worth setting down before the modern position.

The proof, and how long it was believed #

The history of the stability result is worth following, because the failure was demonstrated analytically a century before the numerical work and was not absorbed.

Laplace’s theorem of 1773 concerns the semi-major axes rather than the eccentricities: it shows that, to first order in the planetary masses, the semi-major axes have no secular terms — they oscillate and return, so no planet’s orbital energy drifts. Lagrange then established the corresponding result for the eccentricities and inclinations in the linearised secular system, which is the bounded quasi-periodic solution this essay began with.

Poisson extended the semi-major-axis result to second order in the masses in 1809, finding terms that grow with time but only as the product of time and a periodic function — bounded in amplitude, so still no drift.

At third order the result fails. Spiru Haretu showed in 1885 that secular terms appear in the semi-major axes at that order which are not bounded by any periodic factor. The proof of stability, in the sense it had been quoted for a century, was known to be a proof about a truncation and known to fail when the truncation was extended.

Poincaré’s work five years later was more general and more damaging. He showed that the series of classical perturbation theory are in general divergent — that there is no convergent expansion of the kind the whole enterprise had assumed — and that the three-body problem admits solutions of a complexity no series could represent. The prize essay in which he showed it contained an error, discovered after printing, whose correction is where the modern understanding of chaos begins.

What is striking is how little that changed the received account. The stability of the solar system continued to be described as established, on the strength of a first-order theorem, for most of the twentieth century — and the reason is not that anybody disbelieved Poincaré. It is that no alternative statement was available: nobody could say what did happen, only that the proof did not prove it.

The question became answerable when it became computable, and the answer that arrived was not a theorem in either direction but a probability.

None of the three is close to being done, and the question is likely to stay a probability for some time yet.

A last remark on what the linear theory is still good for, since this essay has been mostly about where it fails. Every long numerical integration of the solar system is checked against it: the eight frequencies it predicts appear in the numerical output, to several digits, over the intervals where the numerical solution is trustworthy. So the theory is not superseded but bounded — it is exactly right about the frequencies and wrong about their permanence, and knowing which half is which is what lets a numerical result be believed. A theory that is precise and incomplete is more useful than one that is approximate and general, provided its boundary is known.

One more history covers the planet the bound fails for first.

Where this ladder goes next #

This rung has taken a proof of stability and located precisely where it fails: not in its algebra, which is correct, but in the size of a quantity it computes badly and treats as though computing it badly did not matter.

The rung above is the tool the numerical work actually runs on. Proper elements are the amplitudes rather than the instantaneous values — the constants of the linear theory, extracted by removing the forced terms — and they are what makes an asteroid family recognisable ten million years after it was made, because the members share proper elements and share nothing at all in their osculating ones.

Beside it lies the same arithmetic applied to a single body rather than a system: the secular resonances that sweep through the asteroid belt as the giant planets migrate, and what a resonance passing through a population does to it.

And below it, the habit: a bound is only as good as the constants it is a bound on. A theorem about a linearised system says nothing about the original one unless somebody checks that the neglected terms are small compared with the quantities the theorem turns on — and here they are not, by a factor of three, in a calculation nobody thought to make for a hundred and eighty years.