A dodge made of air

A satellite in low orbit is flying through the top of the atmosphere, and the air, though a trillion times thinner than at sea level, drags on it. Drag lowers the orbit, and a lower orbit is a faster one, so a satellite that presents more area to the air falls behind its neighbours at first and then overtakes them — an orbit that speeds up as it is slowed down. Two satellites that turn different faces to the flow drift apart along their common orbit with no propellant at all. The essay on rendezvous paid for in air followed how that drift is used to space out a flock of small satellites, slowly and cheaply, and ended on a question the method’s users now face: whether a satellite that can only change its drag can also get out of the way.

The question is not hypothetical. The shells between four and six hundred kilometres hold most of the satellites launched in the last few years, and tens of thousands more are planned. Each of them receives warnings of close approaches with other satellites and with debris, and the practice that has grown up among operators is to move when the predicted probability of collision exceeds about one in ten thousand, and to move within a day or so of the warning. A satellite with a thruster can do that. A satellite with only its attitude to work with has to do it with the air.

This essay works out whether it can. The answer comes in two parts that point in opposite directions, and the second one is the one that matters.

A threshold for moving #

A collision warning is a statement of probability. Two objects’ orbits are each known with some uncertainty, and the predicted miss distance at their closest approach carries the combined uncertainty of both. The probability of collision is the chance that the true miss distance is smaller than the sum of their sizes.

For a well-tracked pair with a combined uncertainty of a hundred metres, a miss of under three hundred metres is enough to bring the risk below one in ten thousand. With three hundred metres of uncertainty — typical of an operational satellite against a tracked piece of debris a few days ahead — the dodge has to put the two about half a kilometre apart.

The third curve shows the trap in the method. With a kilometre of uncertainty the risk is spread over so large a region that it never exceeds the threshold, even for a predicted direct hit, and no manoeuvre is called for. Poorly tracked objects look safe because they are poorly tracked. This effect — called dilution — means that the threshold rule protects a satellite from the objects it knows best and leaves it exposed to the fragments nobody can see well. It also means that the size of a dodge is set by the quality of the tracking as much as by the predicted miss.

The shift a change of attitude buys #

A drag manoeuvre is simple: on receiving a warning, the satellite turns to its high-drag attitude and holds it. Its semi-major axis falls slightly faster than it would have, its mean motion rises, and it gains on its predicted position at an accelerating rate. The along-track shift grows as the square of the time since the turn.

For a small satellite of five kilograms that can swing its frontal area from a hundredth to a tenth of a square metre — a cubesat turning from end-on to face-on — the shift in a day is tens of kilometres at four hundred kilometres’ altitude and several kilometres at five hundred, even when the Sun is quiet and the upper atmosphere is at its thinnest. The half-kilometre a dodge needs is reached within hours.

That is the first part of the answer, and it is more generous than intuition suggests. Drag is weak, but along-track displacement is the one quantity in orbital mechanics that grows without limit from a small persistent force, and it grows as the square of the time. A force about a ten-millionth of the satellite’s weight, applied for a day, moves it kilometres. The density does the rest: at four hundred kilometres the air is dense enough that a cubesat’s natural lifetime is a year or two, and a change of attitude changes that lifetime appreciably.

Why the shift grows as the square of the time #

The quadratic growth is worth seeing from the inside, because it is what makes a weak force useful. Drag removes orbital energy at a steady rate, so the semi-major axis falls steadily: a decrease proportional to the time. The orbital period depends on the semi-major axis, so the period shortens steadily too, and the satellite’s angular speed rises in proportion to the time. Its position along the orbit is the accumulated angle, the integral of the angular speed — and the integral of something growing linearly is something growing quadratically.

The same chain is what makes catching up by slowing down work, and it is why a manoeuvre that would be useless over an hour is decisive over a day. A dodge one hour long achieves a five-hundred-and-seventy-sixth of what a dodge one day long achieves. The useful lever is the warning time, and the warning time is set not by the satellite but by the tracking network and by how early the encounter’s risk becomes clear enough to act on.

The chain also explains why the shift is along track and not radial. The orbit is lowered by only metres to tens of metres over a day, a negligible change in the radial separation at the encounter; the angular speed’s change, multiplied by a day’s worth of orbits, is what moves the satellite kilometres along its path.

An altitude limit that moves with the Sun #

The density falls by about a factor of ten for every hundred kilometres of height, and the dodge falls with it.

With a day’s warning the same satellite can make the needed shift up to about six hundred kilometres when the Sun is quiet and seven hundred when it is active. The difference is the solar cycle: ultraviolet light from the active Sun heats the thermosphere and expands it, and at five hundred kilometres the density at solar maximum is ten times its value at minimum. A drag-only satellite at six hundred and fifty kilometres can dodge during the active years of the cycle and loses the ability in the quiet ones.

The altitude limit also depends on the warning. The shift goes as the square of the time, so a warning of half a day buys a quarter of the shift, and the altitude limit falls by about sixty kilometres. Conjunction warnings are typically issued days ahead, but the predictions are refined as the encounter approaches and the decision to move is often made within the last day, when the covariance has shrunk enough to say whether the risk is real.

How much sail a dodge needs #

The governing quantity is the change in drag area per unit mass, multiplied by the drag coefficient.

A cubesat turning face-on gains about four hundredths of a square metre per kilogram. A drag sail — a thin membrane of a few square metres unfurled from a small satellite, designed to bring it down at the end of its life — gains about a square metre per kilogram, twenty-five times more, and extends the altitude to which a day’s dodge is possible by about two hundred kilometres. The requirement grows tenfold for every hundred kilometres of altitude, so each factor of ten in sail buys a hundred kilometres, and a heavier satellite with the same sail loses the same hundred.

The same physics that makes the dodge possible makes it a debris-mitigation tool. A sail deployed at the end of a satellite’s life raises its drag permanently and shortens its descent from decades to years — the removal rate that decides how crowded a shell becomes. A satellite that carries a sail for disposal carries, for free, the ability to dodge.

A manoeuvre that cannot be announced exactly #

The difficulty is the air itself. The density at four to six hundred kilometres is not known in advance to better than about twenty or thirty per cent a day ahead, because it depends on the Sun’s ultraviolet output and on geomagnetic activity, both forecast imperfectly, and because the density models themselves can be wrong by a factor of two during storms.

A drag manoeuvre therefore has an uncertain size. A turn begun a day before an encounter at four hundred and fifty kilometres moves the satellite about twenty-eight kilometres, give or take eight. The satellite certainly misses the debris; but it arrives at a position known only to several kilometres, which is larger than the position uncertainty that prompted the dodge in the first place. Its new trajectory has to be re-predicted from tracking after the fact, and in the meantime its own covariance is larger than before.

That matters because collision avoidance is not done alone. The satellite’s new position has to be screened against every other object, and in a crowded shell the dodge can create a new conjunction with something else. When both objects in a conjunction are active satellites, their operators coordinate: one moves, the other holds still, and each tells the other what it will do. A thruster burn can be announced to the centimetre per second. A drag manoeuvre can be announced only as an attitude and a time, with its effect known to thirty per cent. The other operator is left with a moving target whose motion nobody can state precisely, and the practice of screening every planned manoeuvre against the catalogue before executing it becomes a statement of probability on both sides.

The usual remedy is to make the dodge smaller: start later, so that the shift is just enough, and accept a larger relative error on a smaller number. But the absolute error then becomes comparable with the shift itself, and the dodge’s purpose — to move by a known margin — is lost.

The thruster’s millimetres #

Compare the same job done with a thruster.

A tangential velocity change of Δv moves a satellite along track by three times Δv times the elapsed time, the same factor of three that makes a burn move the wrong way at first and then gain. A half-kilometre shift a day later needs two millimetres per second. For a five-kilogram satellite that is a hundredth of a newton-second of impulse, a fraction of a gram of cold gas, delivered in a second — and its size is known to a per cent because the thruster’s impulse is calibrated and the burn is short.

The air supplies the same impulse readily below six hundred kilometres. What it cannot supply is a manoeuvre whose result can be stated in advance. That is the real cost of propellant-free manoeuvring: not distance, which the air provides in surplus, but certainty, which it cannot provide at all.

A flock that dodges together #

Drag-only satellites usually fly in groups: the constellations of small imaging satellites that pioneered differential-drag phasing hold dozens of identical spacecraft in one orbital plane, spaced by changes of attitude. A dodge by one member changes the spacing of the whole group, because the satellite that moves ahead has to be brought back into place by its neighbours drifting forward to meet it or by itself holding back while they catch up. Each dodge costs the group weeks of re-phasing.

There is a second, subtler cost. The satellites of a flock share an altitude and a density, so a density forecast error moves all of them by the same fraction. Their relative positions are robust against the error; their absolute positions are not. For collision avoidance against an outside object, what matters is the absolute position, and a flock whose members all drift by the same uncertain amount offers the outside world a set of moving targets with correlated errors — easier to model as a group than individually, but not more certain.

Who moves when both can #

When two active satellites approach each other, the conventional resolution is for one operator to move and the other to hold course. For a pair with thrusters, the choice is made by whoever has propellant to spare or whose mission tolerates the disruption. For a pair in which one can only change its drag, the choice is constrained: the drag-only satellite can move only in one direction along track — ahead of its unperturbed position, since more drag lowers and quickens its orbit — and can stop gaining but not undo the shift it has made, except by holding its low-drag attitude while its neighbours hold their high-drag ones.

So a drag-only satellite that dodges loses its place. It ends up ahead of its planned slot in the constellation, lower than its neighbours, and recovers it only by the slow phasing the method was designed for, over weeks. A constellation of drag-only satellites dodging debris frequently would spend much of its life out of formation. That is acceptable for some missions, where the satellites’ exact spacing matters little, and fatal for others, where coverage depends on it.

What the model leaves out #

The density is drawn as two values. The solar-minimum and maximum densities bracket a variation that is continuous through the cycle, with day-to-day changes of tens of per cent from geomagnetic activity superimposed. A dodge during a storm faces a density that can double within hours.

The drag coefficient is assumed constant. It depends on the satellite’s surface and how the gas molecules scatter from it, and on the attitude, so the change in drag between two attitudes is itself uncertain by ten per cent or more.

Only along-track motion is drawn. A drag manoeuvre also lowers the orbit slightly, which changes the radial separation at the encounter — usually the more sensitive direction for collision risk, since the uncertainty is smallest radially. The along-track shift dominates the geometry for encounters a day or more away.

Warnings arrive one at a time. The figures treat a single conjunction. A satellite in a crowded shell receives several warnings a week, and a dodge for one encounter changes the geometry of every later one. A thruster-equipped satellite resolves them one by one, each burn small and known; a drag-only satellite’s shifts accumulate with their uncertainties, and after several dodges its predicted position days ahead is uncertain by more than the tracking alone would make it — a collision rate that needs no collision to grow, because the uncertainty does the work.

The covariance is isotropic. Real position uncertainties are elongated along track, where errors grow fastest, so the shift needed depends on the encounter’s geometry. A dodge along the direction of largest uncertainty is the least effective one, and along-track is that direction.

Still open: whether drag-only satellites can meet the practices of a crowded orbit #

The arithmetic says a drag-only satellite below six hundred kilometres can get out of the way of almost anything, given a day’s warning, and that it cannot say exactly where it will end up. The second property is the one the emerging practices of collision avoidance are built against: they rely on manoeuvres being planned, screened against the catalogue and announced before they are executed, and a drag manoeuvre can be planned and announced only as an intention.

Whether that is acceptable will be decided as much by agreement as by physics. Better density forecasts — assembled from the orbits of satellites themselves, now numerous enough to measure the thermosphere continuously — would shrink the thirty per cent; operators could agree to treat drag-only satellites as non-manoeuvrable for coordination, and to move their own satellites instead; or small satellites could be required to carry a few grams of propellant for the purpose. Each route trades cost against certainty differently. The air will always be there to push against. What it does not offer, at any altitude, is a promise about where the push will end.