A rendezvous paid for in air

The first lesson of orbital rendezvous is that it runs backwards. To catch a target ahead in the same orbit, a spacecraft must slow down: a lower orbit is a faster one, so braking drops it to a quicker path that gains on the target, and a later burn raises it back. The Gemini 4 crew learned this by doing the intuitive thing — thrusting towards the target — and watching it recede, at the cost of most of their fuel.

The same lesson has a version that needs no fuel at all. The thing that lowers a satellite’s orbit need not be a rocket. In low orbit the residual atmosphere does it continuously, slowing a satellite in a way that speeds it up, and the size of that drag depends on the area the satellite presents to the flow. A satellite that turns broadside to the airflow loses altitude faster than one that turns edge-on, and therefore moves ahead of it. Turn the pair round, and the drift reverses.

That is differential-drag phasing, and it is how the largest constellations of small satellites are spread round their orbits. It replaces a manoeuvre measured in metres per second with one measured in weeks, and a fuel budget with a dependence on the weather in the thermosphere.

A force switched on by an attitude #

The drag deceleration on a satellite is $\tfrac12\rho v^2\,C_DA/m$: the air density, the orbital speed squared, and the ballistic coefficient $C_DA/m$, which is the only part the satellite controls. A small satellite with deployed solar panels can present ten times the area flat-on to the flow that it presents edge-on, and the difference between the two attitudes is a force it can switch on and off by turning.

Those accelerations are tiny. A cold-gas thruster on a small satellite gives a few millimetres per second squared, ten thousand times more. But a thruster runs for seconds and a drag difference runs for weeks, and over weeks a micrometre per second squared adds up to metres per second and to thousands of kilometres of along-track displacement.

Why the gap grows as the square of the time #

The drag does not move a satellite along its orbit directly. It lowers the orbit: the semi-major axis decays at a rate $\dot a = -\sqrt{\mu a}\,\rho B$, with $B$ the ballistic coefficient, and two satellites with different $B$ decay at different rates. Their difference in semi-major axis grows linearly in time.

A difference in semi-major axis is a difference in orbital period. The lower satellite has the higher mean motion, $\Delta n = -\tfrac32 (n/a)\Delta a$, and so gains on the higher one at a rate that grows with $\Delta a$. Integrate a rate that grows linearly, and the separation grows as the square of the time:

$$s(t) = \tfrac34\, n\sqrt{\mu a}\,\rho\,\Delta B\; t^2 .$$

To stop, the pair swaps attitudes halfway: the satellite that was broadside turns edge-on and its partner broadside, the difference in semi-major axis shrinks back to zero at the same rate it grew, and the drift stops with the gap at twice what it was at the halfway point. It is a bang-bang manoeuvre — full control force one way, then full the other — with the attitude as the throttle. In the frame riding with the target, the relative orbit of the chaser is the same family of drifting ellipses that every rendezvous works in, with drag rather than a burn deciding which member of the family the chaser is on.

The quadratic growth is the counter-intuitive rendezvous geometry at its slowest. A tangential burn changes the semi-major axis at once and the gap then grows linearly; a continuous drag difference changes it gradually and the gap grows as its integral. Both are the same physics — along-track motion bought with a change in the orbit’s size — and differential drag is the limit of a burn so gentle it takes a month.

Both satellites are falling #

The figures draw the separation, which is what the manoeuvre controls, and leave out that both satellites are descending throughout. Differential drag acts on a difference, and the sum keeps working: a satellite flying edge-on still decays, more slowly than its broadside partner but not at all slowly at a few hundred kilometres. A phasing manoeuvre is therefore a manoeuvre between two orbits that are both shrinking, and the target slot is not a fixed point but a point riding down with the pair.

That matters in two ways. The along-track rate depends on the mean motion, which rises as the orbits shrink, so the same attitude difference moves the pair a little faster at the end of a long manoeuvre than at the start. And the density rises as the pair descends, roughly tenfold per hundred kilometres, so a manoeuvre that takes a pair down by a few kilometres strengthens its own control force by several per cent as it goes. Both effects are small over one phasing and are absorbed by the feedback loop described below; over a constellation’s lifetime they mean that the drag available grows steadily as the satellites age, and the last manoeuvres are the quickest.

The same descent is also a measurement. Every satellite whose decay is tracked is a probe of the density along its path, and orbits are one of the ways the thermosphere’s density is measured in the first place. A flock of satellites manoeuvring by drag is also a flock of density sensors, each reporting the air it flies through, with the attitude history needed to interpret it already on board.

How long half an orbit takes #

The useful measure is how long it takes to move one satellite a given fraction of the orbit relative to another — half an orbit being the largest phasing a constellation ever needs, since beyond that it is shorter to go the other way.

The trade with altitude is unforgiving in both directions. Low down the manoeuvre is fast, but the satellites are decaying fast too and will not stay up long. High up they last for decades, but there is too little air for drag to move them in a useful time. The method is practical in a band of a couple of hundred kilometres, which happens to be where most small Earth-observation satellites fly, because they are launched as secondary payloads into whatever low orbit the primary needs and are designed to decay within a few years to avoid adding to the debris population.

The largest use: spreading a flock #

A launch that carries dozens of small satellites releases them all at nearly the same point in the same orbit, a few seconds and a few centimetres per second apart. To image the whole Earth every day they need to be spread evenly round the orbit, and none of them carries enough propellant to do it — many carry none.

Differential drag does the spreading. The satellites separate slowly at first from the small differences in their release velocities, and each is then commanded to fly broadside or edge-on according to where it is and where it should be, so that the differences in their decay rates carry each to its slot. The process takes months. It has been used at that scale since the middle of the 2010s, and the proposal is older: drag-based station-keeping of a constellation was worked out in the late 1980s and used by some of the first commercial constellations.

The constellation case differs from a rendezvous in one important way. A rendezvous drives the gap between two satellites to one value; a constellation drives the gaps between dozens to many values at once, with every satellite both a target and a chaser. Each attitude command affects every gap involving that satellite, and the optimisation is over the whole set. But the dynamics of each pair are the ones drawn here.

Why not carry a thruster #

The obvious objection is that a thruster would do all of this in hours, and the answer is a set of constraints that have nothing to do with orbital mechanics and everything to do with how small satellites reach orbit.

A satellite launched as a secondary payload rides on someone else’s rocket, and the primary customer’s tolerance for risk sets the rules. Pressurised tanks, toxic or energetic propellants and anything that could ignite are restricted or forbidden on many rideshare launches, or require safety reviews whose cost is a large fraction of a small satellite’s budget. A three-unit cubesat has a volume of three litres, and every cubic centimetre given to a propulsion system is taken from the camera, the radio or the batteries. And a thruster is a component that can fail, on a satellite designed to be cheap enough that a few failures among dozens are acceptable.

Against that, differential drag needs nothing the satellite does not already have. It needs solar panels, which every satellite carries; an attitude-control system able to turn them, which an imaging satellite needs to point its camera; and tracking, which every operator does anyway. The control force is the by-product of hardware bought for other reasons.

So the comparison is not between a drag manoeuvre and a burn of the same size. It is between a manoeuvre that costs weeks and a few kilometres of altitude, and no manoeuvre at all. For a constellation whose purpose is coverage, a satellite in the wrong slot for two months is a small loss; a satellite that never reaches its slot is a permanent one. The engineering argument for drag is that it turns satellites that would otherwise be passive into satellites that can be placed.

Flying a plan in the wrong air #

The weakness of the method is the density, and the thermosphere’s density is hard to forecast. It depends on the Sun’s ultraviolet output, which varies with the eleven-year cycle and with the 27-day rotation of active regions across the solar disc, and on geomagnetic storms that can raise it by tens of per cent in a day. Models of it are wrong by tens of per cent routinely and by factors of two during storms.

A manoeuvre planned with one density and flown in another does not go as planned.

The open-loop failure is the expected one. With a denser atmosphere than planned, the gap grows faster in the first half than the plan assumed, and the swap comes too late; the second half’s braking is stronger too, but not by enough to undo the overshoot, and daily variations leave a residual drift that never stops.

The closed loop works because it does not need the density to be right. It needs the satellite’s position and velocity, which tracking provides, and a rule for when to swap attitudes: brake when the remaining gap is less than the distance it would take to stop at the planned rate. If the real braking is stronger than planned, the satellite stops short and the rule turns it round again; if weaker, it overshoots slightly and the rule reverses it. The density error becomes a slower or faster convergence rather than a miss. This is how differential drag is actually flown, and it is why the method needs navigation more than it needs a forecast.

What the free manoeuvre costs #

A manoeuvre with no propellant is not free. It is paid for in altitude, because drag that moves a satellite along its orbit also takes it down.

Four metres a second is a small amount of velocity change. A satellite with a thruster would spend a fraction of a kilogram of propellant on it. But many of the satellites using differential drag have no thruster at all, and for them four metres a second is not cheap or expensive — it is unavailable by any other means, and the altitude it costs is a few per cent of their orbital lifetime. The manoeuvre is a conversion of lifetime into position, at an exchange rate set by the solar cycle.

The figure also says when to do it. The same phasing at solar minimum takes three times as long and costs a third of the altitude. A constellation operator with time to spare spreads the flock in the quiet years and saves lifetime; one in a hurry does it at solar maximum and pays.

A second rendezvous geometry #

Differential drag can do more than phase. Turning the satellites so that their drag difference has a component across the orbit is not possible — drag acts along the velocity — but attitude changes can control a second quantity: the difference in their radial positions, by using the drag difference at the right moments in the relative orbit. In principle the full in-plane rendezvous — matching both along-track position and radial offset — can be done with drag alone, which is the subject of a body of control-theory work aimed at bringing a small satellite all the way to a target without thrusters.

What it cannot do is change the orbital plane. Two satellites in slightly different planes stay in different planes whatever their attitudes, because nothing in the drag points out of the plane. The orbit’s slow precession under the Earth’s flattening can be exploited — satellites at slightly different altitudes precess at slightly different rates, and differential drag changes their altitudes — but the resulting plane changes are tiny and take years. A flock released into one plane stays in it.

The same lever, pulled to leave #

The attitude that speeds a satellite along its orbit also shortens its life, and at the end of a mission that is the point. A satellite that flies broadside for its last months, or deploys a drag sail to multiply its area, reenters years sooner than one left edge-on, and the difference matters for how fast the low orbits clear themselves of dead objects. The physics of differential drag and of drag-augmented disposal are the same; what differs is whether the drag is applied to one satellite of a pair or to a satellite alone, and whether the goal is a position or an altitude.

A drag sail does for disposal what the attitude change does for phasing: it trades area for time with no propellant. And it has the same dependence on the Sun, so that a disposal planned for a few years at solar maximum can take twice as long through a deep minimum. The guidelines that limit how long a dead satellite may stay in orbit are written in years, and a satellite relying on drag to meet them is relying on a forecast of the solar cycle as much as on its own design.

Where the model stops #

Drag along the velocity, area switched instantly. The figures treat the drag difference as a step that the satellites switch at once, and ignore any lift or side force from inclined surfaces. Real satellites take minutes to turn, which is negligible against weeks, and their surfaces generate small forces across the flow that a careful controller can exploit or must correct.

A density model of three values. The atmosphere is drawn from representative solar-minimum and solar-maximum densities, log-interpolated in altitude, with no daily variation except in the control figure. The real density has a day–night bulge, latitude structure and storm response, all of which the satellites average differently according to their orbits.

Two satellites, one plane, circular orbits. The quadratic gap and the bang-bang manoeuvre are the linearised relative motion of two satellites in nearly the same circular orbit. Large separations, eccentric orbits and the Earth’s flattening add terms that matter for precise station-keeping and not for the picture.

The drag coefficient is a number. It is taken as 2.2. The drag coefficient belongs to the surface as much as to the shape, and changes with the gas composition at the satellite’s altitude; for differential drag what matters is the difference between two attitudes, which inherits the same uncertainty.

The surprising connection #

The control law that makes differential drag work is the one every driver uses to stop a car at a line: brake when the remaining distance is less than the distance needed to stop. It does not need to know the brakes’ strength precisely, only to keep measuring where the car is. The satellites are cars with brakes whose strength the Sun changes tenfold, steering by a feedback loop that converges whatever the strength is.

And the gap grows quadratically because, as in every rendezvous, the satellite that is being slowed is the one that pulls ahead. A satellite braked by air gains on a satellite that is not — exactly as the Gemini crew would have gained on their target had they braked instead of chasing. The physics that cost them their fuel is the physics that moves a flock of satellites with none.

Still open: drag-only rendezvous in a crowded orbit #

The orbits where differential drag works best are the ones filling fastest: hundreds of kilometres up, where tens of thousands of satellites are planned. Phasing by drag is slow and its timing depends on the Sun, which makes a satellite manoeuvring by it harder to predict for anyone tracking it and harder to move quickly out of the way of debris on a collision course. Whether a satellite that can only manoeuvre by changing its drag can meet the collision-avoidance practices now expected in low orbit — move within a day, by a predictable amount — or whether drag-only satellites will need a small thruster for that alone, is a question being answered by operators rather than by orbital mechanics.