A moonlet that weighs a pile of rubble

Kepler’s third law is a weighing machine. Newton’s form of it says that the cube of an orbit’s size divided by the square of its period is the gravitational constant times the total mass of the two bodies, and every mass in astronomy that is known well — the Sun’s, the planets’, a binary star’s — is known because something was seen to orbit it with a measurable period. The planets were weighed by their moons a century before any spacecraft reached them.

Asteroids were not, for most of the twentieth century, because none was known to have a moon. Their masses came from the tiny tugs they give each other or Mars, measurable only for the largest, and their densities — the quantity that would say what they are made of and how — were guesses. The guesses assumed solid rock. Then, in 1993, the Galileo spacecraft flew past the asteroid Ida on its way to Jupiter, and in the images returned months later there was a second object: a moonlet a mile and a half across, about ninety kilometres from Ida’s centre. It was named Dactyl. It was the first asteroid satellite ever seen, and it weighed Ida.

A law in which the size cancels #

Put the third law round an asteroid of radius $R$ and bulk density $\rho$, with a satellite small enough that its own mass can be neglected. The asteroid’s mass is $\tfrac{4}{3}\pi R^3 \rho$, and the period of a satellite at orbital radius $a$ is

$$P = \sqrt{\frac{3\pi}{G\rho}} \left(\frac{a}{R}\right)^{3/2}.$$

The asteroid’s radius has disappeared, except as the unit in which the orbit is measured.

A moonlet five radii out from an asteroid of twice the density of water goes round in twenty-six hours, whether the asteroid is a kilometre across or a hundred. That is the law that makes binaries such good instruments: what has to be measured is a ratio of lengths and a period, and both can often be measured from the binary’s own light — the period from the regular dips when one body eclipses the other, the size ratio from the depths of the dips — without knowing the distance to the system at all. Only the conversion of the ratio to an absolute density needs anything else, and it needs nothing but $G$.

At one radius the formula has a second meaning. A satellite skimming the surface has the same period as a point on the equator of an asteroid spinning so fast that its equatorial material is weightless. That is the spin period at which a strengthless pile of loose material starts to shed from its equator, about 2.3 hours for a density of two — and the observed fact that almost no asteroid larger than a few hundred metres spins faster than about 2.2 hours, a barrier with a corner in it where cohesion takes over for the small ones, is the same third law read at its lower limit. The spin barrier and the moonlet’s period are one curve.

A density from a single glimpse #

Dactyl was not seen to go round Ida. Galileo saw it over about five hours of a flyby, from one side, and its orbital period turned out to be about a day. There was no period to put in the formula. What there was, from the sequence of images, was an approximate position and velocity: Dactyl about ninety kilometres from Ida’s centre, moving at about six metres a second. A single snapshot of a satellite does not give a mass. It does give limits, because the moonlet is still there.

If Ida’s density were too low, its gravity at ninety kilometres would be too weak to hold a moonlet moving at six metres a second, and Dactyl would be on its way out of the system — which would make it a coincidence, an unbound passer-by caught in the one place the spacecraft happened to look, and the chance of that is negligible. If the density were too high, the same position and velocity would put Dactyl on an orbit so eccentric that its closest approach would bring it inside Ida’s surface, and a moonlet on such an orbit would have struck Ida within a day. Between those limits lies a band of densities for which Dactyl is on a bound orbit that clears the asteroid.

The band is wide, from about one and a half to nearly five in the figure, and it depends on the details of the snapshot. The published analysis narrowed it by asking more of the orbit: that it should survive not merely one revolution but many, against the irregular gravity of an elongated asteroid and the tide of the Sun, which together throw out moonlets on most of the orbits a single snapshot allows. What remained was a density of $2.6 \pm 0.5$ grams per cubic centimetre.

A metre a second more in the snapshot moves the lower limit by forty per cent, because the binding condition depends on the square of the velocity. That is the trouble with a single glimpse, and the reason densities from binaries got much better once binaries were found that could be followed for a full orbit and more.

What the number meant #

Ida is an S-type asteroid, the commonest kind in the inner asteroid belt, and its spectrum resembles the ordinary chondrite meteorites that make up most of the stones that fall to Earth. Those meteorites have grain densities — the density of the rock itself, without any empty space — of about 3.3 to 3.6 grams per cubic centimetre. If Ida were a solid block of that rock, its density would be about 3.4. It was 2.6.

The difference is empty space: about a quarter of Ida’s volume is not rock. Meteorites themselves have a few per cent of microscopic cracks, so most of that quarter must be larger voids — gaps between fractured blocks that have been shattered by impacts and held together by their own gravity rather than by the strength of the rock. Every density measured from a binary since has said the same thing more strongly. The S-type near-Earth binaries come out at two to two and a half, thirty to forty per cent empty. The dark C-type asteroids, whose grain density is lower to begin with, come out at about one, which can only mean half of them is void.

That is the evidence that converted the idea of the rubble pile from a hypothesis about how asteroids might be built into a description of how most of them are. An object a few kilometres across that has been broken by impacts and reassembled by gravity should be porous, should have no strength to speak of, and should spin no faster than its loose surface can stay put — and the densities, the spin barrier and the shapes of the bodies visited by spacecraft since all agree. When the Hayabusa and OSIRIS-REx missions arrived at Itokawa, Ryugu and Bennu, they found exactly what the binaries had said: heaps of boulders and gravel, with bulk densities between one and two.

A density that sets a size limit #

The low densities explain a second fact about asteroids that would otherwise be unconnected. Gravity in a body a kilometre across is tiny — a few ten-thousandths of a metre per second squared at the surface for a density of two — and a rubble pile has no tensile strength to add to it. Material at the equator stays only if the spin is slower than the orbital period of a satellite skimming the surface, which is the third law at one radius, and that period depends on the density alone. A body half as dense must spin about forty per cent more slowly to hold together. The observed barrier near 2.2 hours, which applies to asteroids from a few hundred metres to ten kilometres regardless of size, therefore implies a density of about two for the typical body near the barrier — the same number the binaries give independently. Two quite different observations, one of spins and one of orbits, return one density through one formula.

Below a few hundred metres the barrier fails: some small asteroids spin in minutes. They must have some cohesion, from fine grains held together by surface forces that matter at that scale and not at larger ones. The binaries, whose primaries are almost all larger than that, sit in the regime where gravity alone holds the pile, which is why their densities can be read so directly from the third law.

Where the error comes from #

Once binaries could be followed for whole orbits, mainly by radar for near-Earth pairs and by adaptive optics on large telescopes for main-belt ones, the density’s error came to be dominated by geometry rather than dynamics.

The period is the easy part. Mutual eclipses repeat on the binary’s period and can be timed over months to a fraction of a per cent, so even though it enters the density squared it contributes almost nothing. The orbit’s size enters cubed, so a three per cent error in the separation becomes nine per cent in the density. The volume of the primary — from a radar shape model, from a light curve inverted into a shape, or from a spacecraft’s images — enters once, and for a lumpy, irregular body a volume good to ten per cent is good. A density from a binary is therefore typically good to ten or twenty per cent, which is enough to tell rubble from rock and not enough to tell one kind of rubble from another. The law is exact to the mass of the planet; its error budget is in the ruler, not the clock.

A moon that is part of the mass #

The formula above neglected the satellite’s own mass, and the third law does not: it weighs the pair. For a small moonlet the difference is trivial, for a large one it is not.

The overstatement is the cube of the size ratio. Dactyl is a tenth of Ida’s size and a thousandth of its mass, and can be ignored. Dimorphos, a fifth of Didymos’s size, adds one per cent. But a fair fraction of binary asteroids in the main belt are pairs of nearly equal bodies — Antiope, two eighty-kilometre objects orbiting each other every sixteen hours, is the best known — and for those the third law gives the combined mass, which has to be divided between the two by their volumes, assuming they share a density. That assumption is natural for two halves of what was probably one body, split by spin, and it is the one place in the method where something has to be assumed rather than measured.

The binaries that came from spinning too fast #

Why there are so many binary asteroids is itself an answer the third law points to. About one in six near-Earth asteroids larger than a couple of hundred metres has a satellite, a far larger fraction than among the planets. The binaries’ primaries tend to spin close to the barrier the first figure draws at one radius, their satellites orbit within a few primary radii, and many primaries have a ridge round the equator like a spinning top. The picture these point to is that sunlight’s slow torque on an irregular body — the rotational cousin of the force that moves an orbit by heat — spun it up to the barrier, the loose material at the equator lifted off, and some of it gathered into a moonlet in orbit just outside, after the surface slid before the body shed — a split that decides whether the piece can leave, with a satellite as one possible outcome. The binaries are a by-product of the same weakness they were used to measure.

That makes the density from a binary a self-consistent story rather than a lucky measurement. A body that is loose enough to shed a moon at its spin barrier is a body that is porous, and a body that is porous has a low bulk density, which the moon then measures.

A test that was aimed at a moon #

The most deliberate use of the third law on an asteroid binary was an experiment designed round it. In 2022 the DART spacecraft struck Dimorphos, the moonlet of Didymos, at six kilometres a second, to test whether an impact could change an asteroid’s orbit. The test was measured by the third law in reverse. Dimorphos’s orbital period was known to a few seconds from mutual eclipses before the impact, and afterwards it was shorter by about thirty-three minutes — a change in orbit size, read through the period, of a few per cent. The momentum delivered was several times the spacecraft’s own — a collision moves the centre of a cloud of debris where an explosion cannot — because the impact threw out a plume of debris that carried momentum away in the opposite direction, and the size of that enhancement depends on how porous and weak the target is. A rubble-pile moon, weighed by its orbit before and moved by an impact after, turned out to be easier to push than a solid one would have been.

Masses without moons, and why they are worse #

The alternatives to a moon show how much the binaries changed. The largest asteroids — Ceres, Vesta, Pallas — tug on each other and on smaller asteroids during close passages, and the deflections, read over decades of positions, give their masses to a few per cent. A few dozen more are weighed by their pull on Mars, whose distance has been measured by radar and spacecraft ranging to metres; the asteroids are then nuisance parameters in the planetary ephemeris, fitted in bulk, and the fit and the table it produces carry their masses only as well as the ephemeris can separate them. Both methods reach only bodies hundreds of kilometres across.

For the small asteroids that make up almost all of the population, before the binaries, density was inferred from indirect arguments — the rate at which sunlight’s recoil drifts an orbit, which depends on what the surface is made of and on the mass it has to move — or simply assumed. The drift rate does give a density, given a model of the surface’s thermal properties, and for a few near-Earth asteroids it has been measured to agree with the binary method. But it carries the thermal model’s uncertainty with it, while the third law carries none. Of all the ways of weighing a small body at a distance, only a moon reduces the measurement to a ruler and a clock.

That is why the discovery of binaries was more important than its first number. Dactyl gave one density, uncertain by twenty per cent. The binaries that followed — about four hundred are now known, among near-Earth asteroids, main-belt asteroids, Jupiter’s Trojans and the Kuiper belt — give densities for every class of small body, each with nothing between the measurement and Newton’s law but the volume of the primary. The Kuiper-belt binaries, weighed the same way at forty astronomical units, come out at about one gram per cubic centimetre or less for the small ones: ice and void, the outer counterpart of the inner belt’s rubble.

Spheres, Keplerian ellipses and borrowed grain densities #

The figures treat the primary as a sphere of uniform density and the satellite’s orbit as a Keplerian ellipse round it. Real primaries are elongated, and their irregular gravity makes close satellite orbits precess and wobble; the third law then holds for the orbit’s average size and needs a correction for the primary’s flattening, which is how the shapes of some primaries have been measured in turn. The snapshot figure uses a single angle for Dactyl’s velocity where the real geometry was three-dimensional and uncertain. And porosity is computed by assuming a grain density from a meteorite analogue, which for many asteroid types is itself uncertain by ten per cent or more; the porosities are good to that level and no better.

Still open: whether the voids are large or small #

The densities say that a quarter to a half of a small asteroid is empty. They do not say how the emptiness is divided — whether it is a few large caverns between house-sized boulders, or a fine-grained porosity spread through compacted gravel, or a mixture graded from one to the other with depth. The difference matters for how an asteroid responds to an impact, for how it holds together as it spins, and for how its moonlet formed. Radar sounding from orbit, which can see the internal structure of a body as it sees ice layers in the polar caps of Mars, is planned for the Hera mission’s arrival at Didymos and Dimorphos. If the voids are large, the rubble piles are heaps of fragments that never compacted; if they are fine and uniform, the asteroids are closer to sandstone than to scree, and the third law will have weighed two quite different things under one number.