Runaway planetesimal growth

Why do a few bodies become planets while trillions of their neighbors never grow at all?

The early Solar System was a disc of roughly similar rocks. Out of that population came eight planets and a leftover swarm — an asteroid belt whose entire mass is a fraction of the Moon's, and a comet reservoir of trillions of bodies that never became anything.

If the starting rocks were much alike, that outcome is strange. A process operating uniformly on similar objects should produce a spread of intermediate sizes, not a handful of enormous winners and an ocean of untouched debris. Something in the growth rule must convert a tiny head start into an unbridgeable one — and then, apparently, stop.

Start with the simplest growth rule you could write down. A body sweeps up whatever it runs into, so its growth rate is the rate at which material crosses its cross-sectional area: growth proportional to radius squared, and therefore to mass to the two-thirds power. Now check what that implies for relative growth. Divide by mass and you get a per-unit-mass growth rate that falls as mass to the minus one-third. Bigger bodies grow more slowly in proportion. Under this rule the population converges: small bodies catch up. That is called orderly growth, and it produces exactly the boring spread of sizes we do not observe.

So the real rule must be different, and the missing ingredient has to be something that rewards being large. Gravity is the obvious candidate. Ask what a large body's gravity does to a small one passing nearby — not a direct hit, but a near miss. It bends the trajectory inward. So the target the body effectively presents is wider than its physical outline, and the effective cross-section is the geometric one multiplied by a factor of roughly one plus the square of the escape velocity divided by the square of the relative velocity. That factor is gravitational focusing.

Now the crucial step, and it is worth doing slowly. Escape velocity grows with the body's own mass. So the focusing factor is not a constant bonus; it increases as the body grows. When random velocities in the disc are small compared with a body's escape velocity, the focusing term dominates, and the standard result is that growth goes roughly as mass to the four-thirds power rather than two-thirds.

Test that against the same relative-growth check. Divide by mass and the per-unit-mass rate now goes as mass to the plus one-third — it rises with size. Two bodies that started one percent apart do not stay one percent apart; the gap widens without limit, and in finite time. This is not ordinary exponential growth, where every body doubles on the same clock and ratios are preserved. It is stronger than exponential: the leader's doubling time keeps shortening while the follower's does not. That is why the outcome is winner-take-all rather than merely uneven, and it is why the tiny initial differences do not need explaining — any difference at all is enough.

But if the feedback is that violent, why is the Solar System not one enormous rock? Something has to brake it, and the brake is hidden in the same formula. Look at the denominator: focusing is strong only while relative velocities stay small. As a body grows, its gravity does not only pull neighbours in — it also stirs the ones it misses, raising their random velocities. That stirring is fed by the growing body's own mass, so the winner is steadily degrading the very condition that made it win. Once relative velocities approach the escape velocity of the growing embryo, the focusing factor collapses toward one, and runaway ends.

What comes after is the phase usually called oligarchic growth, worked out in the 1990s: a set of comparable embryos, each dominating and clearing a feeding zone spaced some tens of its own Hill radii from the next, growing at similar rates because each has stirred its own neighbourhood into the slow-growth regime. And so the answer to the original question falls out. The trillions that never grew are not failures of the process; they are the material left in a disc that had been dynamically heated past the point where accretion is efficient, in regions where the embryos ran out of reach — the asteroid belt being additionally stirred by Jupiter.

One honest caveat about the state of the field. This account is the classical planetesimal-accretion picture, and it is well established, but it is not the whole story. It struggles to build the cores of the giant planets before the gas disc disperses, and a body of work since around 2010 argues that much of the growth came instead from pebble accretion — centimetre-scale grains aerodynamically drawn in, which is far faster. How the original planetesimals formed is a separate open problem, with the streaming instability the leading candidate. Runaway growth is real and probably necessary, not complete.

Three points worth pinning down.

First, runaway does not mean fast in absolute terms. Its distinguishing feature is that the ratio between the largest body and a typical one diverges, not that the clock is short.

Second, gravitational focusing is not "extra gravity". Nothing about the force changes; what changes is the fraction of nearby trajectories that terminate on the body — a geometric consequence of a fixed force acting for longer on a slower passer-by, which is why velocity sits in the denominator.

Third, resist reading the leftovers as debris that failed. They are the population the mechanism predicts: once stirring has raised velocities, collisions between small bodies stop building and start shattering.

• How pebble accretion changes the timing, and what evidence favours it.
• Why the asteroid belt is so depleted even for a stirred region.
• What sets the spacing of oligarchs, and how that spacing shows up in observed exoplanet systems.

Every term the collection defines is gathered in the glossary.