A tow with no rope

Every engine considered in this collection pushes on something it carries. A chemical rocket throws its combustion products out of a nozzle; an ion engine throws xenon out at tens of kilometres a second; even a sail, which has no rocket equation, pushes on sunlight. The gravity tractor is the strangest engine of all, because it pushes on nothing it carries and touches nothing it moves. It is a spacecraft that flies to an asteroid, stops beside it, and stays there. Its own gravity pulls the asteroid towards it. Its engines, firing gently away from the asteroid, keep it from falling in. The two together form a tow with no rope, and the asteroid is dragged along at a rate set by nothing but the spacecraft’s mass and how close it dares to hover.

The idea was proposed in 2005 as a way to deflect an asteroid on course for the Earth without touching it — without needing to know whether it is a solid rock or a pile of rubble that a push would scatter, without risking breaking it into pieces, and with a deflection that can be measured and adjusted as it happens. The price is that the tow is extraordinarily weak, and the arithmetic of how weak decides when it could ever be used.

A pull that ignores the asteroid’s mass #

The spacecraft, of mass $m$, hovers at distance $d$ from the asteroid’s centre. Its gravity gives the asteroid an acceleration $Gm/d^2$, and the asteroid’s own mass does not appear: a heavier asteroid is pulled harder in proportion to its mass, and accelerated by exactly the same amount. That is the equivalence principle at work on a towing problem, and it means the tractor’s effect on the asteroid’s velocity depends only on how heavy the tractor is and how close it can sit.

The numbers are very small. A twenty-tonne spacecraft a hundred and fifty metres from the centre of a two-hundred-metre asteroid pulls it at six hundredths of a billionth of a metre per second squared. Over a year that accumulates to a velocity change of about two millimetres a second — the speed of a garden snail. Halve the distance and the pull quadruples; double the tractor’s mass and it doubles. Nothing else helps.

The asteroid’s mass appears in the other half of the problem. To hover, the spacecraft must thrust away from the asteroid with exactly the force the asteroid’s gravity exerts on it, which is proportional to the asteroid’s mass. A two-hundred-metre asteroid of ordinary rubble weighs about eight billion kilograms and pulls a twenty-tonne spacecraft at a hundred and fifty metres with about half a newton — a modest force for an ion engine, but one that has to be sustained continuously for years.

The exhaust that must miss #

The engines cannot simply be aimed along the line to the asteroid. To hold the spacecraft up, the thrust must point away from the asteroid, which means the exhaust streams towards it. Any exhaust that strikes the surface pushes the asteroid in the wrong direction, away from the tractor, cancelling the gravitational tow. So the engines are canted outward, each angled away from the line to the asteroid by enough that its cone of exhaust passes beside the asteroid rather than onto it.

The cant needed is the angle the asteroid subtends as seen from the spacecraft plus the half-angle of the exhaust cone. Close in, the asteroid fills half the sky, the engines must point nearly sideways, and most of their thrust is wasted pushing the spacecraft sideways in two opposed directions; only the cosine of the cant angle holds it up. At one and a half radii, with twenty-degree plumes, the useful share is under half. Further out the waste falls, but the pull falls faster, as the square of the distance.

The figure also shows something less obvious. The propellant needed to give the asteroid a unit of velocity does not depend on the tractor’s mass at all: a heavier tractor pulls harder, and needs more thrust to hover, in exactly the same proportion. What it depends on is the asteroid’s mass, the engines’ exhaust speed and the cant. A two-hundred-metre asteroid needs about half a million kilograms of ion-engine propellant per metre per second of deflection — or about a tonne per two millimetres a second, which is the order of a year’s tow. The tractor is a device for spending propellant on an asteroid without touching it, at an efficiency of a few tens of per cent.

From millimetres a second to a miss #

A velocity change of two millimetres a second is useless unless it is given long before the encounter. The asteroid and the Earth meet at a particular point of the asteroid’s orbit, and what matters is how far the asteroid’s arrival at that point is displaced. A small change in velocity along the direction of motion changes the orbit’s period; the asteroid then arrives at the meeting point a little early or late, and the shift grows in proportion to the time elapsed. For a velocity change along the track, the along-track displacement after time $t$ is about three times the velocity change times $t$ — the same along-track drift that makes a satellite with slightly more energy fall behind.

The displacement accumulates twice: the velocity change grows during the tow, and the drift grows during and after it. A five-year tow followed by five years of coasting moves a two-hundred-metre asteroid’s arrival by an Earth radius; eight years of continuous towing does the same. Moving the arrival by an Earth radius is the minimum for a miss — the Earth’s gravity bends a near-miss inward, so the real requirement is somewhat larger — and the tractor needs most of a decade to achieve it.

A larger asteroid is much harder.

Doubling the asteroid’s size doubles the distance at which the tractor can hover, which quarters its pull. A four-hundred-metre asteroid needs decades of warning for a deflection by a tractor of practical size — towards the limit of how far ahead an impact can be predicted with confidence, since orbit predictions decades ahead depend on forces as small as the recoil of the asteroid’s own heat. For the largest asteroids that are a real threat, a kilometre or more across, a gravity tractor alone would need centuries.

The asteroids a tractor cannot move #

The scaling becomes stark for the asteroids that matter most.

A kilometre-wide asteroid is five times the size of the two-hundred-metre one, so the tractor must hover five times further from its centre and pulls it twenty-five times more weakly: under a tenth of a millimetre a second per year for twenty tonnes. Moving such a body by an Earth radius would take a tow of a century. The kilometre-sized asteroids are the ones whose impact would have global consequences, and they are also the ones that surveys have found most completely — more than nine in ten of those that come near the Earth are now catalogued, and none is on a course to hit it in the next century. The tractor is therefore matched, by its arithmetic, to exactly the population that is still being discovered: bodies of a hundred to a few hundred metres, large enough to devastate a region, numerous, mostly not yet found, and small enough to be towed if one were found with a decade or two of warning.

That matching is the strongest case for the method and also its main limitation. The tractor works best on bodies whose existence, orbits and threat would need to be established a decade or more ahead, which is precisely what the surveys now being built — wide-field telescopes in space operating in the infrared, where small dark asteroids are brightest — are designed to supply. Without that warning, the only methods left are fast ones.

Threading a keyhole instead #

There is a way round the arithmetic that makes the tractor much more useful than these numbers suggest. Many close approaches are preceded, years earlier, by another close approach, and the asteroid’s path during that earlier pass decides whether it will return to hit. The region of the earlier pass that leads to an impact — the keyhole — is small, typically a few hundred metres to a few kilometres across, rather than the Earth’s twelve thousand. An asteroid predicted to pass through a keyhole need only be moved by a kilometre or so at the earlier pass to miss the later impact entirely, and a kilometre of along-track shift is within reach of a tractor in months.

The keyhole is also the reason the tractor’s precision matters more than its strength. A deflection that misses the keyhole by a large margin may send the asteroid into another keyhole nearby; a kinetic impact, whose effect is uncertain until it happens, could do exactly that. A tractor, whose pull is known to the precision of its own position and which can be watched and adjusted as it works, can steer an asteroid between keyholes. It was proposed, in the first instance, as a precision trim to follow a coarser deflection, as much as a deflection in its own right.

One collision against a year of patience #

The obvious alternative is to hit the asteroid. In 2022 the DART spacecraft struck Dimorphos, a hundred-and-sixty-metre moonlet, and changed its orbit round Didymos measurably.

A five-hundred-and-eighty-kilogram impactor at six kilometres a second carries enough momentum to change a five-billion-kilogram asteroid’s velocity by seven tenths of a millimetre a second. The real change was three and a half times larger, because the impact threw out a plume of debris that carried away momentum in the opposite direction — a collision moves the centre of a debris cloud where an explosion cannot — and that enhancement factor, β, depends on the asteroid’s strength and porosity, which are not known in advance. A twenty-tonne tractor would match the impact in a year.

That comparison reverses the usual picture of the two methods. The tractor is not hopelessly weak against small asteroids; a year of hovering equals a well-aimed impact for a body the size of Dimorphos. What it gives up is speed, and what it gains is knowledge. The impactor’s effect has an uncertainty of a factor of several until it is measured afterwards; the tractor’s is known as it happens. The two are complements: an impact to deliver the bulk of a deflection, a tractor to measure and trim it.

Why the tractor flies close, and what that costs #

Hovering over a small asteroid is itself a demanding problem in control. The asteroid’s gravity is irregular, because the body is lumpy; it rotates, so the gravity field beneath the hovering spacecraft changes with the rotation period; and the equilibrium is unstable — a spacecraft that drifts slightly closer is pulled harder and falls further unless its thrust increases. Hovering is the opposite of an orbit: a spacecraft in orbit is in free fall and needs no thrust, but an orbit round the asteroid gives no net tow, because the pull averages to zero over each revolution. The tractor has to stay on one side of the asteroid, in the direction it wants the asteroid to go, and spend thrust every second it does so — an unstable point that costs less to hold than a stable orbit is the same trade in another setting, and here the instability is the price of a pull that does not average away.

A variant avoids hovering altogether: a tractor in a halo orbit, circling in a plane offset to one side of the asteroid, so that it spends all its time on that side and its pull, though weaker than at the closest hover, never reverses. The halo tractor needs far less thrust to maintain and is more tolerant of the asteroid’s irregular gravity, at the cost of a pull reduced by the larger mean distance.

The engine that a long mission needs #

The tractor’s propulsion has to be an engine that can thrust for years at a fraction of a newton, which in practice means electric propulsion: ion or Hall thrusters with exhaust speeds of twenty to fifty kilometres a second. Those are the same engines whose long, gentle burns cost more than impulsive ones for the same orbit change, and whose use is chosen by the calendar — by how long a mission can afford to take. For a tractor the calendar is set by the threat, and the engine’s virtue, its efficiency in propellant, is the one that matters: a chemical engine would need tens of times more propellant to hover for the same years.

There is a second virtue that is easy to overlook. Most deflection concepts move the asteroid once and then rely on predictions of where it will go. A tractor that has been hovering for months has tracked the asteroid’s position with radio precision all that time, and so has measured its orbit, including the slow drift from the recoil of its own heat, far better than any telescope on Earth. Part of what a tractor delivers is the orbit it is changing.

A deflection measured while it is made #

The asteroid’s response to a tow can be watched in a way that its response to an impact cannot. A spacecraft hovering beside an asteroid is tracked from the Earth by the Doppler shift and the delay of its radio signal, to fractions of a millimetre a second and metres of range. Because it is holding station on the asteroid, its own motion is the asteroid’s motion plus a small, known offset, and the tow’s effect — a change in the asteroid’s heliocentric velocity growing by a few hundredths of a millimetre a second each week — appears directly in the tracking as it accumulates. The deflection is not predicted from a model of the asteroid’s strength and then checked months later; it is read off week by week, and the tow can be lengthened, shortened or redirected in response.

That is a different kind of operation from any other method, and it suits the problem the tractor is best at: not stopping a certain impact at the last moment, but steering an asteroid with a small impact probability away from a keyhole, years ahead, by a measured amount. The probability of impact in those cases is often a few per cent or less, and the deflection needed is small; what is needed above all is confidence that the action taken has not made things worse. A tractor gives that confidence by construction, because its pull is gravity and gravity has no unknown parameters.

A round asteroid and a tractor on its axis #

The figures treat the asteroid as a uniform sphere and the tractor as hovering on the line through its centre in the direction of the asteroid’s motion. Real asteroids are elongated and rotate, so the hovering distance and the direction of the pull vary through each rotation; the along-track direction changes round the orbit, so a tow must be steered to keep its pull useful; and the drift rule — three times the velocity change times the elapsed time — is a near-circular approximation that changes for eccentric orbits and close planetary encounters. The plume is drawn as a sharp-edged cone, while real exhaust spreads, and a small fraction always strikes the surface. These change the numbers by tens of per cent. They do not change the scaling — the pull falling as the square of the asteroid’s size, the propellant set by the asteroid’s mass — which is what decides when a tractor can be used.

Still open: whether a tractor will ever be flown #

No gravity tractor has flown. Every mission that has hovered near an asteroid — Hayabusa, OSIRIS-REx, Hera when it arrives — has been far lighter than a tractor needs to be, and has hovered for hours rather than years. A demonstration would need to measure a pull of tens of picometres per second squared on a body whose own orbit is perturbed by sunlight’s recoil at a comparable level, which is possible with radio tracking of the asteroid through the spacecraft over a year or two, and has been proposed. If it is done, the tractor becomes a tool with a measured performance, and planetary defence gains a method whose result is known while it is being applied. If not, the tractor remains what it has been since 2005: the most precise deflection method on paper, and the only one that never touches the thing it moves.