A ceiling that forgets how much there was

About a fifth of Sun-like stars have debris discs — belts of dust detectable by the infrared light it re-emits after absorbing starlight. The dust cannot be primordial. Radiation drag and collisions remove small grains within a few hundred thousand years at most, so a disc seen around a star a billion years old is being continuously replenished, and the only reservoir large enough is a belt of unseen planetesimals whose collisions grind them down through every size to dust. The mass is in the bodies and the light is in the dust, and the two are connected by a cascade.

Those discs are found by their infrared excess — light at far-infrared wavelengths well above what the star’s photosphere emits — and resolved, for the nearest, as rings tens to hundreds of astronomical units across. The natural question to ask of a disc is how much mass is in its belt, and the extrapolation from dust to planetesimals is uncertain by orders of magnitude because the size distribution’s upper end is never seen. There is a different question, whose answer is surprisingly firm: how bright can a belt be, at a given distance from its star and a given age, if it has been grinding steadily since the star formed? The answer does not depend on the belt’s mass at all.

Why a heavier belt fades faster #

The cascade’s supply is set by its largest bodies. Small bodies are destroyed quickly, but they are also replenished by the breakup of larger ones, so in a steady cascade the number at each size is fixed by the rate at which the size above feeds it, all the way up to the largest bodies, which are fed by nothing. The belt’s total mass, and with it the amount of dust at every smaller size, falls as fast as its largest bodies are destroyed.

The rate at which a large body is destroyed is the rate at which it meets a projectile large enough to break it, and that is proportional to how many such projectiles there are — to the belt’s mass. Double the mass and the destruction rate doubles. So the belt loses mass at a rate proportional to the square of its mass, and a quantity that decays at a rate proportional to its own square decays as one over the time once the time exceeds the initial collisional lifetime. The solution is $M(t) = M_0/(1 + t/t_c)$, with $t_c$ inversely proportional to $M_0$.

At late times $M(t) \approx M_0 t_c / t$, and since $t_c \propto 1/M_0$, the product $M_0 t_c$ is independent of the initial mass. Every belt, once it has had time to grind, carries the same mass at the same age: the mass that could have survived that long. A belt born heavier simply reached that mass sooner. The dust brightness is proportional to the mass, and it converges in the same way.

More mass, and no more dust #

The convergence can be seen from the other side, by fixing the age and varying the initial mass.

That turns a debris disc’s brightness into a statement with an unusual logical shape. If a disc is below the ceiling for its age and radius, its brightness measures its mass, within the usual uncertainties of the dust-to-planetesimal extrapolation. If it is on the ceiling, its brightness measures only its age: the belt could have started with twenty Earth masses or two thousand, and the dust today would be the same. An old disc’s brightness sets a lower limit on how much mass its belt began with, and no upper limit at all.

The ceiling was worked out in this form by Mark Wyatt and colleagues in 2007, as a formula combining the belt’s radius, its width, the size and strength of its largest bodies, their orbital eccentricities and the star’s mass and luminosity. The dependence on the largest bodies’ size and strength is real, but weak — the square root of the size and the five-sixths power of the strength — and the dependence on the belt’s initial mass is absent. That is what makes the ceiling useful: it is set by quantities that vary by factors of a few across systems, not by the one that varies by orders of magnitude.

Close to the star the ceiling is low #

The collisional lifetime depends steeply on distance from the star. Closer in, orbital periods are shorter, so bodies meet more often; relative velocities are higher, so collisions are more destructive; and a belt of given mass is packed into a smaller volume. The lifetime of the largest bodies falls roughly as the thirteen-thirds power of the radius inward.

The ceiling on brightness follows. It rises as the seven-thirds power of the radius, so at a given age a belt at a hundred astronomical units can be tens of thousands of times brighter than one at one.

That is why the Solar System’s own belts look the way they do. The asteroid belt, at two to three astronomical units and four and a half billion years old, is far below the ceiling for its radius only in the sense that it has already been ground to a tiny remnant: its present mass, about five ten-thousandths of an Earth mass, is roughly what a belt at that distance could keep for that long. The Kuiper belt, at forty astronomical units, retains ten times more and grinds slowly. Seen from outside, both would be faint by the standards of other stars’ discs, the Kuiper belt’s dust a fractional luminosity of perhaps one in ten million.

The ceiling moves with age as well as with radius, and for young stars it is far higher close in.

The two panels together describe how the inner parts of planetary systems clear. At ten million years, while terrestrial planets are still forming from embryos, the region they occupy can hold substantial dust in a steady cascade, and warm discs around young stars are expected. By a billion years the same region can sustain almost nothing, and any warm dust that remains near the habitable zone must be continually resupplied from outside — by comets, as the Solar System’s own zodiacal cloud is today. That dust, whatever its origin, is what a future telescope has to look through to find an Earth, and the ceiling says that around old stars it cannot come from a local belt.

The asteroid belt is a test of the whole argument, because its history is partly known. Models of the early Solar System suggest the region between Mars and Jupiter began with at least an Earth mass of solids, perhaps much more, and ended with a thousandth of that. Most of the loss is attributed not to grinding but to dynamics: Jupiter’s resonances and the scattering by embryos that formed and were ejected emptied the belt of bodies wholesale, and only then did collisions take over. The collisional part of the history is recorded in the asteroid families, groups of fragments from single catastrophic collisions whose ages can be read from their spreading orbits, and in the size distribution, which shows the knee that a cascade with a strength-dominated small end produces. What remains is close to the mass a belt at that radius can keep for four and a half billion years, which is what the ceiling says any belt left alone would converge to — whatever it started with, and whatever else happened to it first.

Discs that should not exist #

The ceiling becomes a diagnostic when a disc lies above it. Infrared surveys have found a handful of stars, a few per cent of young Sun-like stars and far fewer old ones, with warm dust close in — at a few astronomical units or less, glowing at hundreds of kelvin — at fractional luminosities of one in a hundred or more.

A disc that far above the ceiling cannot have been grinding steadily since its star formed; a belt massive enough to be that bright at the start would have ground itself to nothing long ago. The dust must be recent. Two sources are possible. It could be the debris of a single giant collision — two planetary embryos, or a planet and a large body, smashing together and filling the region with fragments that grind down over thousands to a million years. Or it could be delivered from further out, by comets scattered inward from a colder belt, perhaps by a planetary instability that suddenly destabilised the outer system.

The first explanation has direct evidence in a few systems. The spectra of some warm discs show silica and other minerals formed at high temperature, as in impact melt, and at least one warm disc, around a young star in a nearby group, was seen to brighten suddenly in the infrared and then fade over a few years — as if watching the aftermath of a collision. The Moon is thought to have formed in exactly such an event, when a Mars-sized body struck the young Earth, and an observer elsewhere would have seen the Sun’s warm dust jump far above its ceiling for a million years afterwards.

What the Solar System’s history looks like on this diagram #

The same logic applies to the record written on the Moon and the planets. A surface dated by counting its craters records the impact rate over time, and that rate fell steeply through the first billion years, roughly as one over the time with bursts on top — the signature of a population of impactors grinding and scattering itself away, as the ceiling describes. Whether the early record contains a distinct spike, a late heavy bombardment some four billion years ago, or only the tail of a smooth decline, is the Solar System’s version of asking whether a disc is on its ceiling or above it.

If the spike was real, it is most naturally explained by a planetary instability that scattered the outer belt, sending comets inward and brightening the Solar System’s dust far above its ceiling for a few hundred million years. Other stars’ discs offer a population in which such events can be counted: the fraction of stars of a given age whose discs lie above their ceilings is the fraction currently in the aftermath of a catastrophe, and that fraction, multiplied by the duration of each event, gives the rate at which such catastrophes happen.

What the formula assumes #

The ceiling assumes that a belt’s largest bodies are of one size, have one strength and are on orbits of one eccentricity, and that the cascade below them is in a steady state. Each assumption is an idealisation. The largest bodies have a size distribution of their own; their strength depends on their size in a way that is measured only for rocks in the laboratory and inferred for asteroids; and the eccentricities, which set the collision speeds, depend on what stirred the belt — planets, or the largest bodies themselves — which is exactly what is not seen.

It also assumes the belt was stirred from the start. A belt whose bodies have nearly circular orbits collides gently and does not grind at all; it begins producing dust only when something raises the eccentricities, and its clock starts then. A belt stirred late, by planets that formed or migrated after it, is younger in collisional terms than its star, and its ceiling is correspondingly higher. That is one route by which old stars can host discs brighter than a naive ceiling allows without any catastrophe.

And the formula gives an order of magnitude, not a number. Its prefactor carries the uncertainties of the parameters above, which together amount to a factor of ten or so. What does not carry that uncertainty is the scaling — one over the age, and the seven-thirds power of the radius — and the independence from initial mass. A disc a factor of three above the ceiling is not remarkable; a disc thirty thousand times above it is not a steady cascade under any choice of parameters.

The dust itself is a further step removed. What a telescope measures is the fraction of starlight the dust re-emits, which depends on the grains’ cross-section and temperature, and the conversion from that to a belt’s mass passes through the size distribution, through the smallest grains’ removal by radiation pressure, and through the grains’ ability to absorb and emit. The ceiling compares like with like — a predicted fractional luminosity with a measured one — so those uncertainties enter only through the prefactor, but they enter, and they are one reason the published ceilings are quoted as upper envelopes of the observed population rather than as sharp limits.

The mass problem, seen from the ceiling #

The ceiling also sharpens a puzzle that the dust-to-planetesimal extrapolation raises. The brightest cold discs around young stars, at fractional luminosities of a part in a thousand at a hundred astronomical units, sit near their ceilings. Being on the ceiling requires that the belt started heavy enough to have begun grinding by now, and for a belt of sixty-kilometre bodies that means hundreds of Earth masses of solids. A protoplanetary disc of the usual kind holds perhaps a few tens to a hundred Earth masses of solids in total, most of it inside the orbits where planets form, and far less at a hundred astronomical units.

So the brightest discs seem to demand more solid mass than their parent discs could have had, which is known as the debris disc mass problem. The ceiling does not create the problem — it comes from the extrapolation to large bodies — but it narrows the ways out. A belt whose largest bodies are smaller than sixty kilometres grinds faster and reaches the ceiling with less mass, so the problem eases if the largest bodies are a few kilometres across. A belt stirred by planets rather than by its own largest bodies, with higher collision speeds, does the same. A belt that is not in steady state at all — brightened by a recent event — escapes the problem entirely. Each of those is a statement about planet formation, and each predicts a different distribution of disc brightness with age.

A mass that erases itself #

The general point is a property of any population that destroys itself at a rate proportional to its own abundance. Such a population forgets its initial size: after a time, the survivors number about what the time allows, whatever there were at the start. The same is true of a population of reacting particles in a chemistry problem, of the stars in a dense cluster evaporating in some regimes, and of the planetesimals in a debris belt. A large initial population only means the forgetting happens sooner. The observation that distinguishes a population on its ceiling from one below it is the observation of how it depends on age, and for debris discs the one-over-age decline has been seen in the statistics of discs around stars of different ages — with a scatter of more than an order of magnitude at each age, which is itself the evidence that not all belts are on their ceilings yet.

Still open: how often the ceiling is broken #

The warm discs above their ceilings are rare, and their rarity is the measurement. If giant collisions like the Moon-forming impact happen in most young planetary systems, and each fills the inner system with dust for a million years, the fraction of young stars showing such dust should be a few per cent — which is about what infrared surveys find for stars of ten to a hundred million years. The number is small and the sample smaller, and it is not yet known whether the warm discs around older stars, rarer still, are late giant impacts, comets delivered by instabilities, or something else. The space telescopes now monitoring young stars in the infrared, watching for discs that brighten and fade, are counting the catastrophes directly; each one seen is a planet being built or broken, observed as dust above a ceiling that no steady process can reach.