Two clocks in the dust of one collision

The terrestrial planets of the Solar System were assembled in a sequence of giant collisions. Bodies the size of the Moon and Mars, grown from the planetesimal disc in the first few million years, perturbed each other onto crossing orbits and merged, and the last of those mergers — a Mars-sized body striking the young Earth, between thirty and a hundred million years after the Sun formed — threw out the material that became the Moon. Simulations of terrestrial planet formation give each system of rocky planets tens of such impacts, spread over its first hundred million years.

Every one of them made dust. The essay on the ceiling a belt cannot exceed showed that a steady belt of planetesimals near a star grinds itself down so quickly that, at a few tens of millions of years, it cannot hold more than a millionth of the star’s light as dust at an astronomical unit — and that the few young stars with warm dust far above that ceiling must be showing something that is not steady. A giant impact is the obvious candidate. That essay ended on the question of how often the ceiling is broken. This one asks what an impact’s dust looks like as it fades, and what that says about how to count the impacts.

Debris that starts above the ceiling #

A giant impact between bodies the size of Mars and the Earth throws out a few hundredths of an Earth mass of material, and it does so in two forms. Part of the impactor and target is vaporised, and the vapour, expanding and cooling, condenses into spherules a millimetre or a centimetre across, like the glassy spherules found in lunar soils. The rest is solid fragments, from boulders to bodies tens or hundreds of kilometres across, spread round the star near the orbit of the collision.

Both forms are a collisional cascade the moment they exist. The fragments collide with each other at the speeds their orbits’ scatter gives them — kilometres a second at an astronomical unit — and grind themselves down into smaller fragments, and eventually into dust small enough for the star’s light to blow it out of the system. The cascade’s brightness is the area of its dust, and the dust’s area is set by how much mass is in the cascade and how quickly it is being ground: the same arithmetic as for a steady belt, with two differences. The cascade starts with all its mass in place at once, rather than replenished, and it starts close to the star, where everything happens fast.

Why the ceiling does not apply at first #

The ceiling of a steady belt comes from a cancellation: a more massive belt is brighter, but it grinds itself away faster, and after a time comparable with its age the two effects leave every belt at the same brightness, inversely proportional to its age. The cancellation needs time. A belt younger than its own collisional lifetime is still at its initial brightness, and that can be anything.

A giant impact resets the clock. Its debris is born at the moment of the collision, with an age of zero, so for a while it is far younger than its own collisional lifetime and can be as bright as its mass allows — orders of magnitude above the ceiling for a star of tens of millions of years, which is set by the age of the star rather than the age of the debris. The ceiling then catches up with it: as the debris ages, its own collisional lifetime passes, and it settles onto the line that a belt of its age would follow. What an observer sees is debris that is young in a system that is old, and the mismatch between the two ages is what puts it above the ceiling.

That is also why the arithmetic of the steady belts can be reused unchanged. Nothing about the cascade is different except its birthday.

Two clocks in one impact #

The same amount of debris fades at very different rates depending on the size of its largest pieces.

The condensate is brightest at first. Its spherules are the smallest pieces the impact makes and the most numerous by far, and smaller particles have more area per unit mass. A hundredth of an Earth mass of centimetre spherules has an area large enough to intercept a substantial fraction of the star’s light, and the system briefly glows in the infrared as brightly as a disc that is still forming planets. But centimetre bodies collide with each other within months, and the cascade they start runs down through the sizes in years. Its brightness falls as one over the time, and within three centuries it is below anything a telescope could detect against the star.

Fragments last longer, because a cascade’s lifetime is set by how long its largest bodies take to be destroyed, and that grows with their size. Kilometre fragments keep the dust detectable for about eight thousand years. Hundred-kilometre fragments, which are not only larger but much stronger — held together by their own gravity — keep it above the threshold for about nine million.

So a single impact carries two clocks. The condensate is a flash: a brightening of the star’s infrared excess on a timescale of months, fading over years to decades. The fragments are a glow, lasting millennia to millions of years, whose length depends on what the collision broke the bodies into. An observer catching a young star in the flash sees an impact that happened within the last few years. One catching it in the glow sees an impact that happened at any time in the last million.

Closer in, a shorter glow #

Where the impact happens matters as much as what it breaks.

At a third of an astronomical unit the orbital period is a sixth of the Earth’s and the collision speeds are higher, so every cascade runs through its sizes faster, and the time above the threshold falls as the radius to the seven-thirds power. The same impact visible for nine million years at an astronomical unit is visible for under a million at a third of one. The warm dust found round young stars is mostly at temperatures of a few hundred kelvin, which for a Sun-like star means a few tenths to one astronomical unit — the zone where the Earth and Venus formed — and the steep dependence on radius means that the impacts closest to the star are systematically underrepresented in any census.

Mass that buys brightness and not time #

The most useful property of these curves is the one they share with the steady belts.

A collision that throws out ten times as much debris makes dust ten times brighter, and grinds it down ten times faster, because a denser cascade collides more often. The two effects cancel once the cascade has had time to run: every curve joins the same line, brightness inversely proportional to time, and crosses the detection threshold at the same moment. That is the ceiling of the steady belts reappearing, in a different guise. A belt’s brightness at late times is independent of its initial mass; an impact’s visible lifetime is independent of the debris mass, so long as there was enough debris to start above the threshold at all.

It is a surprising simplification. The masses thrown out by giant impacts range over orders of magnitude depending on the impact angle and speed and the sizes of the bodies, and none of that is visible in how long the impact can be seen. What is visible is the debris’s largest pieces and its distance from the star.

Set by the largest pieces #

The visible lifetime depends on the largest fragment size through two factors, one geometric and one physical.

The geometric factor is mild: a cascade whose largest bodies are larger takes longer to grind down, roughly as the square root of their size. The physical factor is steep, and it is the strength of the bodies. Below a few hundred metres a rock is held together by its material strength, which weakens slightly with size because larger bodies contain larger flaws; above that, it is held together by its own gravity, which strengthens rapidly with size because the fragments of a shattered body have to be flung apart as well as broken. That is the size at which a body is easiest to break, and beyond it the energy needed to disrupt a body rises as roughly the 1.4 power of its size. The visible lifetime rises with it. A cascade fed by hundred-kilometre fragments lasts a thousand times longer than one fed by kilometre fragments.

There is a complication in the other direction. The fragments of a giant impact are not intact rocks; many of them reaccumulate from smaller pieces within hours of the collision, and are loose aggregates held together by gravity alone — the same structures as the asteroids whose spins hit a barrier with no size in it. An aggregate has no material strength to speak of, and below a few kilometres it is much weaker than an intact rock of the same size, which shortens the lower end of the curve. Above a few tens of kilometres, where gravity dominates either way, it makes little difference. The steepest part of the figure — the rise from kilometre to hundred-kilometre fragments — is the part least affected by what the fragments are made of.

So the most uncertain property of a giant impact’s debris — the size distribution of the fragments, which depends on the impact’s angle, speed and the bodies’ internal structure, and which simulations can predict only roughly — is exactly the property that controls how long the impact is visible.

Counting impacts from a census #

Infrared surveys of young Sun-like stars find warm dust round a few per cent of those between ten and a hundred million years old, with about one per cent showing extreme excesses, bright enough that no steady belt could make them. If those excesses are impact debris, the share of glowing stars is the number of impacts per star multiplied by the time each stays visible, divided by the length of the window.

The census therefore measures a product. Two per cent of stars glowing needs about two hundred impacts per star if their debris is kilometre fragments, seven if it is ten-kilometre fragments, and a fraction of one if it is hundred-kilometre fragments. The simulations of terrestrial planet formation that give tens of impacts per system sit in the middle of that range, which means the census is consistent with them — and also with very different histories, because every curve passes through the band at some impact rate. The census cannot separate a few long-lived impacts from many short-lived ones. That is its fundamental limit, and it is the same limit the mass and the light of a debris disc impose everywhere: the brightness is in the smallest grains and the information about the history is in the largest bodies, which are never seen.

Counting impacts from flashes #

The condensate clock breaks the degeneracy. Every impact makes a flash, whatever its fragments do, and the flash lasts months to years — short enough to be caught happening.

The rate of flashes in a monitored sample is the number of stars times the number of impacts per star divided by the window’s length, with no dependence on the fragments at all. Monitoring a hundred thousand young stars for a decade would see a third of a flash if each star suffers thirty impacts in its first hundred million years, and three if it suffers three hundred. The numbers are small, because a hundred million years is long compared with a decade, but they are counts of the thing itself rather than of a product.

The first such events have already been seen, in a different way. Monitoring of stars already known to have extreme warm dust — where the excess itself suggests the system is in the middle of its giant-impact phase — has caught brightenings of the infrared excess over months, followed by declines over a year or two, in stars of about thirty-five million years; the spectrum of one showed the silica features expected of condensed vapour. Another star’s large excess disappeared almost entirely within two years. These are the condensate clock, read directly. Because they were found among stars already glowing, they do not give an unbiased rate; but they show that the flash exists, lasts about as long as the cascade arithmetic predicts, and can be caught with infrared monitoring at a cadence of months.

What the Solar System says #

The Solar System is one sample of the process, and its record is partial but quantitative. The Moon-forming impact is the best known, and the Moon’s own tilt may still carry traces of how it settled afterwards; Mercury’s large iron core may record another impact, which stripped much of its mantle — the same process that leaves some exoplanets with more iron than their star could give them; Mars’s crustal dichotomy and the Earth’s own late accretion record more. Simulations that reproduce the terrestrial planets’ masses and orbits give the Earth alone something like ten giant impacts during its growth, and the system as a whole several tens.

On the figures above, a few tens of impacts with debris dominated by kilometre-to-ten-kilometre fragments would have kept the young Sun’s inner system glowing in warm dust for a few per cent of its first hundred million years, which is about what the census finds for other stars. The Solar System, seen from another star at the right moment, would have been one of the glowing ones — a few per cent of the time, during the epoch when the Earth was being assembled from collisions — and dark otherwise.

What the cascade leaves out #

The cascade is the steady-state formula applied to a transient. The fading curves assume the debris settles quickly into a collisional size distribution. In the first few orbits after an impact the debris is a clump on the collision’s orbit — and since every fragment comes back through the same point, the point of the collision, the debris stays pinched there for thousands of orbits, colliding most often where the stream is narrowest, while differences in orbital period shear the rest into a ring; its brightness varies with the clump’s geometry as seen from the observer, which is one of the features the monitored flashes show.

Not all debris stays. Small condensates are pushed outward by radiation pressure, and drag sorts the remaining grains by size, carrying the smallest inward; some fragments are re-accreted by the planets within a few million years, and some are scattered onto eccentric orbits. Both shorten the glow.

Fragment strengths are laboratory laws extrapolated. The breaking energies come from impact simulations calibrated on laboratory experiments, extended to bodies tens of kilometres across. Their uncertainty, a factor of a few, enters the visible lifetimes as its five-sixths power.

The threshold is illustrative. A fractional luminosity of $10^{-4}$ is roughly where warm dust becomes detectable against a Sun-like star’s photosphere at mid-infrared wavelengths; the true threshold depends on the instrument, the wavelength and the star’s distance.

Still open: how many collisions build a planet #

The share of young stars glowing in warm dust is consistent with the giant-impact phase that simulations predict for systems of rocky planets, and it cannot by itself say how many impacts that phase contains, because each impact’s visibility depends on the size of its largest fragments. The flashes can. A survey monitoring tens of thousands of young stars in the mid-infrared, at a cadence of months, over years, would count fresh impacts directly — and the count, divided by the number of stars and the window, would be the first measurement of how many collisions it takes to finish a system of terrestrial planets, made on systems where it is still happening.

Combined with the census, the count would also measure what the census alone cannot: the size of the fragments the collisions leave. Two numbers, the share glowing and the rate of flashes, would fix both the number of giant impacts per star and how violently each one broke the bodies involved — the most direct measurement possible of the epoch in which the Earth acquired its Moon.