A core that shut the door on the pebbles

A pebble in a gas disc is always in a headwind. The gas is held up partly by its own pressure, which falls outward, so it orbits a little slower than a solid body at the same distance would; a solid orbiting through it feels the slower gas as a wind against its motion, loses angular momentum to it, and spirals inward. For bodies of a centimetre or so at a few astronomical units — pebbles, in the language of planet formation — the drift is fast: they cross the disc in tens of thousands of years, far quicker than the disc lives. A young disc is a conveyor belt carrying solids from its outer parts to its inner ones.

The snow line is where the belt’s cargo changes: pebbles drifting in carry ice until they cross it, then lose their ice to vapour and carry only rock. A planet that forms near the line sits in the path of everything drifting past. If it is massive enough to catch pebbles as they go by, it grows quickly; and if it grows enough, it does something more drastic than eating the flux. It stops it.

How fast the belt runs #

The drift speed has a simple form, and it explains why pebbles in particular are the cargo. A solid’s coupling to the gas is measured by its Stokes number, the time drag takes to change its velocity divided by the orbital time. Dust grains, with tiny Stokes numbers, are dragged along with the gas and barely drift relative to it. Boulders, with large ones, plough through the gas almost unaffected and lose angular momentum only slowly. In between, at a Stokes number near one, the drift peaks at a speed of about the difference between the gas’s orbit and a pure Keplerian one — tens of metres a second, which at five astronomical units carries a body inward by an astronomical unit in a few hundred orbits. At the sizes and densities of a young disc, a Stokes number near one corresponds to centimetres to decimetres: pebbles.

The same size-dependent drag sorts the debris discs around older stars by size, with radiation in place of gas. In a young disc it means that the mass carried by pebbles is not stored in place but flows, and a planet anywhere downstream sees the whole of the disc’s solids pass it on their way to the star. That flux is what pebble accretion feeds on and what pebble isolation cuts off.

A pressure bump made by a planet #

A planet in a gas disc pushes the gas around it. Its gravity excites spiral waves that carry angular momentum away from its orbit, and the gas just inside and just outside is pushed away, opening a shallow dip along the planet’s track. A shallow dip is enough to matter. At the dip’s outer edge the gas’s density, and so its pressure, rises outward instead of falling, and where the pressure gradient changes sign the gas orbits at exactly the speed a solid would. There is no headwind there. A pebble drifting in from outside meets the edge, finds its drift reversed, and stays.

The mass at which this happens was measured in simulations by Lambrechts and collaborators in 2014: about twenty Earth masses for a disc whose thickness is a twentieth of its radius, scaling as the cube of that aspect ratio. The cube comes from the physics: a planet has to push the gas by a fraction of its own pressure to reverse the gradient, and the pressure support relative to gravity goes as the square of the aspect ratio, while the planet’s ability to push goes as its mass relative to a mass set by the disc’s thickness. In a disc that thickens outward the isolation mass grows outward, from a few Earth masses at the Earth’s distance to tens at Jupiter’s and over a hundred at the outer planets’.

Reaching that mass has two consequences at once. The core stops growing from pebbles, because no more reach it; and it is already heavy enough to start pulling in the surrounding gas, which it does increasingly fast, becoming a giant planet. The isolation mass is where a rocky-icy core turns into the start of a Jupiter — and where the conveyor belt behind it is cut.

A dam whose height depends on the gas #

The figure’s numbers are for one disc, and the most important thing they depend on is the least known.

Turbulence fills in the dip a planet tries to dig, so a turbulent disc needs a heavier planet to raise a bump; Bitsch and collaborators fitted that dependence, and the disc’s own pressure gradient, from simulations in 2018. The disc’s turbulence is the quantity whose value decides the direction of planet migration too, and observations of the dust in real discs — thin, settled layers and sharp rings that survive — have pushed the inferred values down towards a ten-thousandth. In such a quiet disc the dam closes at a smaller core, and so earlier. The uncertainty is a factor of three in the mass, which is a factor of a few in the time.

The dam is also not perfectly tight. Small grains, too small to drift fast, are carried along with the gas through the dip by diffusion, and some fraction of the finest dust leaks past even a strong bump. What is stopped is the pebbles that carry most of the mass; what leaks is the dust that carries little.

How quickly the door closes #

The question that connects the dam to the meteorites is when it closed.

A core embedded in a flux of pebbles catches a fraction of them as they drift past, a fraction that rises with its mass because a heavier core’s gravity reaches further across the stream and drag helps it capture what it deflects. Growth is therefore slow at first and then runs away, and it ends abruptly at the isolation mass. With pebble fluxes of the order inferred for young discs — about a hundred Earth masses per million years at Jupiter’s distance — a core of a hundredth of an Earth mass reaches thirty in under a million years. Pebble accretion was proposed largely because the older picture, in which cores grew by colliding with kilometre-sized planetesimals, could not build Jupiter’s core before the gas dispersed; pebbles do it with time to spare, and they put the door’s closing in the first million years of the solar system.

A core that moves while it grows #

The core was not standing still while it grew. A planet of a few Earth masses embedded in a gas disc exchanges angular momentum with the gas through two wakes whose torques nearly cancel, and the small residual drives it inward or outward at rates that can carry it across several astronomical units in a few hundred thousand years. A core migrating inward while it grows sweeps the pressure maximum it creates along with it, and the reservoir boundary is not where the core finished but where it was when the dam closed. If Jupiter’s core formed further out and migrated in, as several models of the early solar system propose, the barrier between the two meteorite families was further out too at first, and moved.

Once the core is massive enough to open a deep gap, its migration slows and becomes tied to the disc’s own accretion, and the boundary settles. The two families’ isotopes say only that a boundary existed; where exactly it sat is not recorded in them, and the asteroid belt’s present mixture of the two was arranged later by the giant planets’ scattering.

The planets that never shut a door #

The isolation mass grows outward, and that has a consequence for the outer solar system. Uranus and Neptune have about fifteen and seventeen Earth masses of rock and ice with modest envelopes of gas, while the isolation mass at their present distances is well over a hundred Earth masses — and still several tens at the distances where they may have formed, closer in. Neither reached it. Their cores grew from pebbles until the flux ran out, or until the gas dispersed, without ever becoming heavy enough to raise a pressure bump and start the runaway accretion of gas that made Jupiter and Saturn. They are, on this reading, the cores of giant planets that never finished, and their composition — mostly ice and rock — is what a core looks like when the door was never closed.

That makes the timing of the outer disc’s pebble flux decisive. A flux that lasted a little longer, or a disc a little thinner at twenty astronomical units, would have given the solar system four gas giants; one that ended a little sooner would have left Saturn an ice giant too. Where planets cannot form in place is one constraint on a system’s architecture; where they reach isolation in time is another.

Two reservoirs from one door #

Once the door is closed, the disc is divided.

Inside Jupiter’s orbit the disc keeps what it had, with no further supply from outside. Outside it, pebbles pile up at the pressure maximum, and the material from the outer disc that keeps arriving — material that was formed from a different mixture of the stellar debris the Sun’s cloud contained — accumulates there. Any difference in isotopic composition between the inner and outer disc’s solids is frozen in from that moment: the two sides evolve separately, each forming its own planetesimals from its own stock.

The pile-up at the bump is also a place where planetesimals form easily, because a high concentration of pebbles in still gas is exactly what the streaming instability needs to collapse clumps of pebbles into bodies kilometres across. The material beyond Jupiter’s orbit may have formed its planetesimals faster and more abundantly because the dam was there.

The meteorites remember #

The proposed record of all this is in the meteorites, and it was found before the explanation.

Every meteorite whose isotopes have been measured falls into one of two groups, distinct in the isotopes of chromium, titanium, molybdenum, nickel and several other elements, with nothing between them. The non-carbonaceous group includes the ordinary and enstatite chondrites and most iron meteorites, and comes from the inner solar system; the carbonaceous group includes the carbonaceous chondrites — the water-bearing rocks whose deuterium matches the oceans — and its own iron meteorites, and comes from beyond Jupiter. The difference cannot have been made after the bodies formed; it reflects two reservoirs of dust with slightly different mixtures of the nucleosynthetic debris from the stars that seeded the Sun’s cloud.

The ages come from clocks that run only for a few million years. Hafnium-182 decays to tungsten with a half-life of nine million years, and when a planetesimal melts and its iron sinks to form a core, the tungsten goes with the iron and the hafnium stays in the rock; the tungsten isotopes in an iron meteorite therefore record how much hafnium-182 was still alive when its core separated, which is a time. The same logic, with aluminium-26 and magnesium, dates the chondrites, and the method is an isochron, a scatter plot of the daughter isotope against the parent element whose slope is the age. Short-lived clocks are the only ones fine enough to resolve the first few million years, and they resolve them to a few hundred thousand.

Kruijer and collaborators showed in 2017 that the two groups’ iron meteorites — the cores of early, melted planetesimals — formed at overlapping times: non-carbonaceous irons from well under a million years to about one and a half, carbonaceous irons around two to three, with the chondrites of both families later. Both reservoirs therefore existed side by side for millions of years without mixing, starting within the first million. In a disc where pebbles drift freely across any radius in tens of thousands of years, that requires a barrier, and it requires the barrier early. A core at Jupiter’s orbit that reached pebble isolation within a million years is the barrier that requires nothing else.

What the barrier did not stop #

The dam stopped the pebbles; it did not stop everything. Once Jupiter grew to its full mass and the gas disc dispersed, the planetesimals that had formed on either side could be scattered across the barrier by gravity, and some carbonaceous bodies were thrown inward into the asteroid belt, which is why the outer belt is full of them today. A few per cent of the Earth’s mass, arriving late from the carbonaceous reservoir, is the leading explanation for its water and for the small carbonaceous signature in its mantle’s isotopes. The two families are mixed in the belt now because of events after the door closed — the barrier kept the dust apart, and the giant planets’ later motions mixed the bodies made from it.

There are other proposed barriers. A pressure maximum at the snow line itself, or at the inner edge of a region where the turbulence changes, could also have kept the reservoirs apart, and some models of the solar system’s isotopes favour a barrier that existed before Jupiter’s core did. What distinguishes them is time: a barrier made by the disc’s structure could have existed from the start, while one made by Jupiter appears only when the core reaches isolation, and the earliest non-carbonaceous irons already forming at a few hundred thousand years leave little room between the two. The isotopic data and the clocks constrain when the separation began to within a few hundred thousand years, which is about the precision with which the core’s growth time can be predicted.

Other systems, other doors #

The same mechanism should operate in every disc that forms a giant planet, and it predicts what the inner parts of such systems receive. A system whose giant forms early and far out receives little from beyond it, and its inner planets should be drier and smaller; a system whose giant forms late, or not at all, receives the full flux and may build larger inner planets. Small planets are more common around smaller stars, whose discs are less likely to have made a giant early, which is at least the direction the argument predicts. Systems of tightly packed super-Earths, the most common planetary systems known — though how many planets a star has is not a direct measurement — are rarer around stars that also host cold giant planets in some surveys and more common in others, and the disagreement may be about which surveys see which giants. The rings and gaps seen in the dust of young discs at tens of astronomical units are, in many cases, exactly the pressure maxima the theory describes — though whether they are made by planets or by the discs’ own structure is argued case by case.

What the figures assume #

The isolation mass is a fit to simulations of a planet in an idealised disc, accurate to tens of per cent; its dependence on the disc’s thickness and turbulence is the content of the first two figures, and both quantities are uncertain for the Sun’s disc. The growth figure uses an accretion efficiency of a simple form and constant pebble fluxes, where real fluxes decline as the outer disc is depleted. The flux figure’s split between inner and outer disc depends on when the door closed relative to how fast the flux decayed. The meteorite ages are grouped and rounded from published chronologies whose individual uncertainties are a few hundred thousand years. Each of these moves the numbers. None moves the conclusion that a core of a few tens of Earth masses at five astronomical units, reached within a million years, would have divided the disc in the way the meteorites show it was divided.

Still open: whether Jupiter’s core was the barrier, or the disc was #

The two families of meteorites prove that the inner and outer solar system were kept apart from very early on. They do not by themselves prove that Jupiter kept them apart. The test is in the precise timing: whether the separation began before any core could have reached isolation, which would favour a barrier built into the disc, or at about the time the core would have, which would favour Jupiter. Tightening the ages of the oldest irons of both families, and pinning down the disc’s turbulence from the dust rings of young discs elsewhere, are the two measurements that bear on it — one from meteorites in the laboratory and one from discs around other stars, about the same event.