A denominator that emptied the inner belt

Strip the short-period wobbles out of the planets’ motion and what remains is a linear system: eight coupled oscillators whose modes are shared by the whole Solar System, so that no planet has an eccentricity of its own — only a reading of how far each mode has turned. The same theory, applied to a body too small to disturb anything, describes an asteroid. The planets’ modes push on it; its own orbit precesses at a rate set by where it sits; and its eccentricity divides into a part the planets impose, which it shares with every neighbour at that distance, and a part that is its own.

The imposed part is a fraction, and fractions can have zero denominators. The proof of stability failed where two planetary frequencies nearly cancelled. The same kind of near-zero appears for asteroids, far more simply, and in one place it is not near-zero but exactly zero.

The eccentricity a body is given #

For a massless body at distance $a$, the linear theory gives two things. The first is its own apsidal precession rate — how fast the planets’ combined pull turns its orbit’s long axis —

$$A(a) = \frac{n}{4}\sum_j m_j\,\alpha_j\,\bar\alpha_j\,b^{(1)}_{3/2}(\alpha_j),$$

a sum over the planets of their masses times a Laplace coefficient that depends on the ratio of distances, with $n$ the body’s orbital frequency. The second is how strongly each planet drags on the body’s eccentricity. Because the planets’ own eccentricities are sums of modes, each oscillating at one of the eight frequencies $g_i$, the drag is a sum of eight terms at those frequencies with strengths $\nu_i$, and the solution that follows the forcing is

$$e_{\rm forced} = \left|\sum_i \frac{\nu_i}{A(a) - g_i}\,e^{\,i(g_i t + \beta_i)}\right|.$$

Everything an asteroid’s orbit does beyond this is free: a circle of radius equal to its proper eccentricity traversed at the rate $A$ around the forced value, which is how asteroid families are recognised long after the forced part of their members’ orbits has scattered them. Across the main belt the forced part is modest, as the figure shows: a few hundredths, rising slowly outward towards Jupiter. It is also nearly the same for every body at a given distance, which is what makes subtracting it meaningful.

A precession that matches Saturn’s #

Each term in the sum has a denominator, $A(a) - g_i$, and each is the difference between two frequencies: the body’s own precession and one of the planetary modes.

The body’s precession rate rises steeply towards Jupiter, which dominates it, and more gently towards the Sun, where the terrestrial planets add to it; between them it has a minimum, just outside the orbit of Mars. The planetary frequencies are horizontal lines, because they belong to the planets and not to the body. Beyond Mars the curve meets exactly one of them — the mode dominated by Saturn, $g_6$ — and it meets it twice, once on the way down into the minimum and once on the way out. Where the curve crosses the line the denominator vanishes: the body’s orbit precesses in step with the planetary mode, the forcing no longer averages out over a cycle, and its eccentricity is driven steadily in one direction. That is a secular resonance, named $\nu_6$ after the frequency it matches. The outer crossing sits at 2.04 AU, and that is where the main asteroid belt begins.

The prediction is not that the eccentricity becomes infinite. The linear theory breaks down long before, as the eccentricity grows beyond the small values its expansion assumes. What real orbits do near $\nu_6$ is grow in eccentricity over a million years or so until their perihelia reach the orbit of Mars, then of the Earth, and they are removed — by close encounters, by collisions with the terrestrial planets, or by falling into the Sun, which is where most of them end. The inner edge of the belt is a boundary drawn by a frequency.

A frequency the simple theory gets wrong #

The figures use the measured frequencies of the two giant-planet modes, 4.257 and 28.245 arcseconds a year, in place of the ones the linear theory computes, and the reason is instructive.

The linear theory — the theory of Laplace and Lagrange, first order in the planetary masses and second order in the eccentricities — gives the mode dominated by Saturn a frequency of about 22 arcseconds a year, a fifth slower than the truth. The same shortfall in Jupiter’s mode moves nothing important in the planets’ own history: the modes’ shapes and amplitudes are nearly right, and a frequency error of a fifth changes when things happen, not what happens. For a secular resonance it changes whether the resonance exists at all. The body’s precession has a minimum of about 23 arcseconds a year just outside Mars; a $g_6$ of 22 barely touches it, and the linear theory would put the belt’s inner edge nowhere in particular. With the correct frequency, the crossing lands at 2.04 AU, within a few hundredths of where the belt’s inner edge is observed for low-inclination orbits.

The missing speed comes from the near-commensurability of Jupiter and Saturn: their orbital periods are close to a ratio of five to two, and the terms of second order in the masses associated with that near-resonance — the great inequality that puzzled astronomers for a century before Laplace explained it — speed up the secular frequencies substantially. The lesson is general. A theory can be good enough everywhere except at a resonance, and at a resonance it has to be exactly right, because a small error in a frequency becomes a large error in a denominator.

The real $\nu_6$ is also not a single distance. Its location depends on the body’s own inclination and eccentricity, because those enter the body’s precession rate at higher order, and in the space of proper elements it is a curved surface: at 2.05 AU for orbits of low inclination, moving outward past 2.5 AU for inclinations near twenty degrees. The belt’s inner edge follows that surface, which is why there are many asteroids at 2.2 AU with low inclinations and almost none there with high ones.

A zero that is not a gap #

It is worth being clear about how different this boundary is from the belt’s other famous features. The Kirkwood gaps, at 2.50, 2.82 and 2.95 AU, are mean-motion resonances: places where an asteroid’s orbital period is a simple fraction of Jupiter’s, so that the two meet at the same points of their orbits over and over and the kicks accumulate. They depend on orbital periods of a few years, they are narrow, and they appear as holes with asteroids on both sides.

A secular resonance involves no orbital periods at all. It is a commensurability between two precession rates of tens of arcseconds a year — periods of tens of thousands of years — and it acts on the slow turning of orbits rather than on the bodies’ positions along them. Its width in distance is set by how steeply the body’s precession rate changes with distance, which near the minimum of the curve is not steeply at all; so $\nu_6$ does not cut a narrow gap but clears a broad region, and because the rate’s minimum lies just inside it, the region inside the resonance is not refilled from that side. The belt does not have a gap at 2.04 AU. It has an edge, and nothing inside it.

The two kinds of resonance also overlap in places, and where they do the motion is chaotic rather than merely resonant: an asteroid near the 3:1 mean-motion resonance feels secular resonances in the same neighbourhood, and the combination produces the rapid eccentricity growth that makes the 3:1 gap the second great delivery route to the Earth.

A delivery route to the Earth #

The same resonance that empties the inner edge of the belt fills the space near the Earth. An asteroid that drifts into $\nu_6$ has its eccentricity pumped within a million years to the point where its orbit crosses the Earth’s, and it becomes a near-Earth asteroid — a short-lived one, because such orbits last only a few million years before a collision or an ejection ends them.

The supply into the resonance is slow and steady. Sunlight absorbed on an asteroid’s day side and re-emitted in the evening pushes its orbit outward or inward by a few ten-thousandths of an AU per million years for a kilometre-sized body, faster for smaller ones, and over a few hundred million years that is enough to carry a body from the inner belt into $\nu_6$. The near-Earth population is therefore not a relic of the early Solar System but a flow, continuously replenished from the belt by thermal drift and continuously drained by the resonance’s delivery onto planet-crossing orbits. A large fraction of the near-Earth asteroids, and of the meteorites in museum collections, came to the Earth by way of this single zero in a denominator.

The same zero, tilted and further out #

The inclinations have a parallel theory, with their own precession rates and their own planetary frequencies, and it has its own zero. Where a body’s nodal precession matches the mode Saturn dominates in the inclination system, $s_6$, a second secular resonance, $\nu_{16}$, pumps inclinations as $\nu_6$ pumps eccentricities. It lies near 1.9 to 2.0 AU at low inclination, just inside $\nu_6$, and between the two of them the inner edge of the belt is a boundary in inclination as well as in distance. The small populations that survive inside it — the Hungaria asteroids near 1.9 AU and the Phocaea group near 2.3 AU — have high inclinations, above twenty degrees, which is precisely what places them on the far side of the curved resonance surfaces and shelters them.

The same arithmetic operates at the edge of the Solar System. In the Kuiper belt, beyond Neptune, a body’s precession is driven mostly by the four giant planets together, and between about 40 and 42 AU it matches the frequencies of the modes Uranus and Neptune dominate. The low-inclination Kuiper belt has a conspicuous deficit there: the same zero in a denominator, drawn at forty astronomical units by the ice giants instead of at two by Saturn.

Where the zero was when Saturn was elsewhere #

The resonance sits at 2.04 AU because Saturn sits at 9.54. Move Saturn and the zero moves.

The frequency $g_6$ is dominated by the mutual coupling of Jupiter and Saturn, which grows rapidly as the two approach. With Saturn at 8 AU instead of 9.5, the coupling is several times stronger, $g_6$ is 78 arcseconds a year instead of 28, and the body whose precession matches it lies much further out, where Jupiter’s pull on its orbit is stronger: at 3.0 AU, in the middle of the belt. As Saturn moves outward, the resonance slides inward across the belt.

Saturn has not always been where it is. The present architecture of the outer Solar System — the eccentric, inclined orbits of the Kuiper belt, the capture of Jupiter’s Trojan asteroids, the timing of the cratering record — is most naturally explained if the giant planets formed closer together and migrated apart, scattering the remnant disc of icy planetesimals that lay beyond them, and the migration moved Saturn outward by one or two astronomical units. If the asteroid belt was already in place when that happened, the $\nu_6$ resonance swept through it from the middle to its present inner edge.

Slowly, and the belt would be gone #

A resonance that sweeps past a body does not leave it where it was. The body is driven while the frequencies are close, and the longer they stay close, the more it is driven.

The calculation in the figure integrates a single body at 2.6 AU, starting on its forced orbit with no free eccentricity of its own, while Saturn is moved outward at a steady rate and the secular frequencies change with it. If Saturn takes a hundred thousand years to migrate, the resonance passes the body quickly and leaves it with a free eccentricity of 0.10 — within the range the belt has today. If Saturn takes a million years, the body is left at 0.31, close to crossing the orbit of Mars. If Saturn takes several million years or longer, the linear theory’s answer passes the point at which the body’s perihelion reaches Mars and then exceeds one, which means only that the body has left the theory’s range and, in reality, the belt.

The dependence has a simple form. A body passing through a resonance at a rate $\dot g$ is driven for a time proportional to $1/\sqrt{\dot g}$, so the eccentricity it acquires grows as the square root of the sweep time. The slower the migration, the more violently each part of the belt is disturbed as the resonance passes.

Across the whole inner belt the contrast is stark. A migration of three hundred thousand years leaves free eccentricities between about 0.12 and 0.2 — close to the proper eccentricities of the inner belt today. A migration of ten million years leaves nearly every body above the Mars-crossing line, which would have emptied the inner belt entirely, and the belt is not empty. The observed belt is the evidence: whatever moved Saturn outward did it in well under a million years. The inference runs the opposite way to the one drawn from a resonant chain of exoplanets that could not have been assembled in place, where only slow, smooth migration through the gas disc can explain the planets’ exact commensurabilities. Both are arguments from a configuration that the wrong rate of migration could not have left behind, and between them they say that planets migrate slowly while the gas is present and can be rearranged abruptly after it has gone.

That conclusion is one of the main reasons the favoured model of the giant planets’ rearrangement includes a violent episode rather than a smooth drift: a close encounter between Jupiter or Saturn and an ice giant, probably a third ice giant later ejected from the Solar System, which kicked the two gas giants apart in a few tens of thousands of years. The planetary encounter was proposed partly to spare the inner planets, whose orbits a slow sweep of the same resonances would also have excited beyond what is observed; the asteroid belt’s orbits say the same thing independently.

What the linear theory cannot show #

The calculation is a linear secular theory with two of its frequencies replaced by measured values, and it inherits every limitation of that theory. It is second order in the eccentricities, so it describes the approach to large eccentricity and not what happens there; the eccentricities above a few tenths in the figures are the theory’s extrapolation, useful as a statement that the body has been removed and not as a value. The migrations are drawn as steady outward drifts of Saturn alone, with Jupiter fixed; a real migration moved both planets, changed their eccentricities, and — in the jumping version — happened in discrete steps. The frequency correction is fixed on the present configuration and carried unchanged to Saturn’s other positions, an assumption a full calculation does not need to make. The body’s inclination is ignored, and with it the second family of secular resonances — the nodal ones, of which the $\nu_{16}$ at the inner belt pumps inclinations as $\nu_6$ pumps eccentricities. And the belt is represented by bodies that begin on their forced orbits with no free eccentricity, whereas the real belt had been stirred by embryos and by Jupiter’s growth before any of this happened.

What survives those limitations is the structure of the argument: the forced eccentricity has a denominator; the denominator vanishes where the body’s precession matches a planetary mode; the location of that match depends on the giant planets’ spacing; and a resonance that passes slowly excites far more than one that passes quickly.

Still open: when the belt met the resonance #

The argument that Saturn moved quickly assumes the asteroid belt was already there, cold, when it moved. If the giant planets rearranged themselves within the first few million years, while the gas disc still damped orbits, the sweep would have left no mark; if they did it hundreds of millions of years later, the sweep’s constraint applies in full and connects to the timing of the Moon’s basins. The asteroids’ own ages and the dates of their collisional families, the meteorites’ records of heating and shock, and the cratering chronology of the Moon all bear on when it happened, and they do not yet agree. The resonance at the inner edge of the belt is where it is because Saturn is where it is; the belt’s orbits say how fast Saturn got there, and the rocks have still to say when.