A removal rate set by one percentage

A few tens of rocket bodies near 850 kilometres hold nearly half of the low orbits’ future debris, and the simulations that ranked them found that removing about five of the worst a year would stop the population growing. That conclusion carried an assumption that has since been overtaken. It supposed that nearly every new satellite would be disposed of properly at the end of its life — deorbited, or moved to an orbit that decays within twenty-five years — so that the derelicts to be removed were the legacy of the past rather than a continuing supply.

The large constellations have changed the supply. In five years they have put more satellites into low orbit than flew in the previous sixty, most of them low enough to fall within a few years if they die, some of them in shells near 1,200 kilometres where a dead satellite stays for thousands. For those shells the question of whether five removals a year is enough has a definite answer, and the answer turns almost entirely on one percentage: the fraction of satellites that fail to dispose of themselves.

Two populations and a threshold #

The model behind the figures is deliberately simple, with every parameter stated, and its claim is a scaling rather than a forecast. A constellation of a fixed size retires a fixed number of satellites each year. A fraction of them — the disposal reliability — leave as intended; the rest stay where they died, as derelicts, and leave only when drag brings them down. Derelicts are struck by one another and by the fragments of earlier strikes, each strike between a fragment and a dead satellite breaking the satellite into about a thousand pieces large enough to track, and the fragments decay by drag too, faster than intact satellites because they have more area for their mass.

Working satellites are left out of the collisions, which is the optimistic assumption: an operating satellite manoeuvres away from conjunctions it is warned about, and the constellations perform thousands of such manoeuvres a year. A dead satellite cannot. The population that matters for the long-term environment is the one that cannot dodge.

The structure of the fragment equation produces a threshold. Each derelict is a target, and each fragment striking a target makes a thousand more fragments; the fragments leave at a rate set by their drag lifetime. If there are enough targets, a fragment on average makes more than one new fragment before it decays, and the fragment population grows on its own whatever anyone launches. Below that number of targets, fragments die out faster than they breed. The threshold is a count of derelicts — the critical count — and it is the same comparison of a quadratic production with a linear loss that sets the altitude above which a shell’s debris is self-sustaining, written per shell rather than per altitude.

Low shells clean themselves #

The critical count depends overwhelmingly on how long fragments last, and that depends overwhelmingly on altitude.

The thermosphere’s density falls roughly exponentially with height, with a scale height that grows from tens of kilometres near 400 to hundreds near 1,000, and the time an object takes to decay goes inversely with the density it meets. An object at 550 kilometres, where most of the largest constellation flies, is down in a few years whatever happens to it. An object at 800 is up for a century or two, and one at 1,200 for thousands of years. The density also swings by a factor of ten over the solar cycle, so a shell’s cleaning rate is faster at solar maximum and slower at minimum, but the ranking of altitudes never changes.

The consequence is that low shells have room for an enormous number of dead satellites and high shells for almost none. At 500 kilometres the fragments of any collision fall out within a few years, faster than they could find a target, and the critical count is in the tens of thousands — more dead satellites than will ever be put there. At 1,200 kilometres a fragment lasts for millennia, and a few tens of derelicts sharing a shell are enough for it to breed. The numbers depend on the yield and on the effective cross-section assumed, which here is set so that the present population of low orbits gives about the one catastrophic collision a decade that has been observed. The ratio between low and high shells does not depend on either; it is the lifetime ratio, and that is measured.

What a constellation leaves behind #

A constellation’s derelicts are its failure rate multiplied by how long each stays.

At 550 kilometres the dead satellites come down within seven years, so the shell reaches a balance between new failures and natural decay at a few hundred derelicts even if a third of the constellation fails, far below a critical count in the tens of thousands. The low constellations’ design choice is the one that makes them safe: they are flown where the atmosphere is a removal service that works for free.

At 1,200 kilometres there is no balance on any timescale that matters. The derelicts simply accumulate, at the failure rate, for as long as the constellation is replenished. A 650-satellite constellation retiring 130 satellites a year adds 1.3 derelicts a year at ninety-nine per cent reliability and 13 at ninety, and a critical count of 34 is passed in a few years at the second rate and in a quarter-century at the first. Reliability of ninety-nine per cent, which would be an excellent figure for any spacecraft system, is not enough on its own for a shell that high to stay below its threshold.

Removal is the failure rate #

Removal changes the arithmetic in one simple way, and the first figure is its consequence. To hold a high shell at a fixed number of derelicts, every derelict that arrives has to be taken away, because drag will not do it in any useful time. The removal rate is therefore very nearly the failure rate: the constellation’s retirements multiplied by the fraction that fail. At 800 kilometres drag contributes a little and the line bends at high reliability; at 1,000 and 1,200 the lines are straight, a factor of ten in removals for every factor of ten in failure fraction.

That turns a policy question into an engineering one with an exchange rate. Every percentage point of disposal reliability in a high shell is worth a fixed number of removal missions a year, for as long as the constellation is flown and replenished — 1.3 removals a year per point for the case drawn. The cost of a removal mission, which involves rendezvous with and capture of an uncooperative tumbling object, is far larger than the marginal cost of making a satellite’s deorbit system more reliable: redundant propulsion, a passive drag sail that deploys if the satellite stops answering, or simply carrying the propellant for disposal from the start and using it before the satellite’s other systems age. The figure says what the reliability is worth.

It also says why five removals a year is a statement about one shell and one constellation, not about the environment. The simulations that produced that number assumed ninety per cent reliability across all operators and a population whose growth came from old derelicts; for a single high constellation of a few hundred satellites five a year is enough above about ninety-six per cent, and for a constellation of a few thousand at the same height it would take ninety-nine and a half.

Why a satellite fails to leave #

A disposal failure is not usually a failure of the disposal. It is a failure of the satellite before disposal begins: the power system degrades, a reaction wheel seizes, the radio stops answering, and a satellite that was due to fire its thruster in a year’s time is now unable to receive the command or execute it. The satellites most likely to fail are therefore the ones nearest the end of their lives, which is exactly when disposal is due. An operator who waits for the planned end of life before disposing accepts the full ageing risk; one who disposes at the first sign of trouble, while the satellite can still act, buys reliability with a shorter service life.

The other remedy is a disposal that needs nothing from the satellite. A drag sail stowed at launch and deployed on a timer, or by a circuit that triggers when the satellite stops hearing from the ground, multiplies the area by tens or hundreds. Drag depends on area over mass, so a sail moves the satellite down the lifetime curve by the same factor: a centuries-long lifetime at 800 kilometres becomes a few years. At 1,200 kilometres even a large sail leaves a lifetime of centuries, because the air there is too thin for any area to matter, and the satellite has to carry its own propellant down. The altitude decides not only how long a derelict stays but which kinds of disposal can work there at all.

What one removal costs #

The other side of the exchange rate is the price of a removal, and that is set by orbital mechanics before any question of capture. A servicing spacecraft has to reach each derelict’s orbit, and derelicts in a shell share an altitude but not a plane: their orbital planes are spread in the angle of the node, and those planes drift apart at rates set by the Earth’s oblateness, differently for different inclinations. Changing plane in low orbit is the most expensive manoeuvre there is — a degree costs about 130 metres a second at 1,200 kilometres — so a servicer visiting several derelicts either waits for the oblateness drift to bring their planes to it, which takes months to years, or pays for the turn in propellant. Once there, it must match the target’s motion closely enough to grab it, and a derelict that has been tumbling uncontrolled for years is rotating at degrees per second about an axis nobody chose.

So each removal is a mission, or a substantial fraction of one, and a removal rate of thirteen a year is a fleet. A removal rate of 1.3 a year is one servicer. The tenfold difference in the first figure is the difference between a programme and a single spacecraft.

A century of three reliabilities #

The threshold is not a cliff edge in time, and the last figure shows how the populations actually diverge.

Nothing happens at first. The derelicts accumulate quietly for a decade or more, too few to strike one another often, and the fragment count stays near zero. Then the first collisions come, the fragments begin to strike other derelicts, and the fragment population turns upward — after about ten years at ninety per cent reliability, twenty at ninety-seven, forty-five at ninety-nine. A century later the least reliable case holds a hundred and fifty times as many fragments as the most reliable, and its fragments are still growing faster than in proportion to time, because each strike adds more targets’ worth of debris than the drag removes.

The delay is the dangerous part. A constellation operator, a regulator or an insurer looking at the shell in its first ten years sees no collisions under any of the three reliabilities and has no evidence from the environment that one of them is building a problem. The difference is decided by an engineering figure that can be measured from the start — how many satellites fail to deorbit — and expressed in the environment only decades later, when the derelicts are already there and the only cure is the expensive one.

The same arithmetic, one level down #

The constellation figures repeat the argument the earlier derelict rankings made with a different variable. There, the removals that mattered most were the heaviest objects in the most crowded, highest orbits, because each contributed its collision rate times its fragments times their lifetime. Here the constellation contributes its failure rate times its derelicts’ lifetime, and for a high shell the lifetime is so long that the failure rate is all that remains. In both cases the drag lifetime is the multiplier that turns an ordinary number into a large one, and in both the cheapest point of intervention is the one before the object is abandoned: a rocket body that is deorbited at the end of its launch, or a satellite that is disposed of while it can still act, never enters the ranking at all. Removal is the remedy for a decision that was not made in time.

What has actually been measured #

The constellations publish, or have had measured for them, the numbers the model needs. The largest at 550 kilometres has reported failure rates of satellites that could not be controlled to disposal at around one to a few per cent, and its failed satellites have come down within a few years as the drag curve says. The constellation near 1,200 kilometres has a few hundred satellites, and the derelicts from its early years are tracked; how many of its satellites will fail to dispose of themselves over its life is not yet known, because most have not yet reached their end. The twenty-five-year disposal guideline that most operators follow has in practice been met by about half to two-thirds of objects reaching end of life in low orbit over the past two decades, by the space agencies’ own annual assessments — a reliability far below the figures the constellations need at their heights.

The model’s other parameters are less directly measured. The breakup yield comes from the standard breakup model, fitted to ground tests and to the clouds from real events; the cross-section is set to reproduce the observed collision rate; and the drag lifetimes come from a standard atmosphere that is itself wrong by a factor of two at solar extremes. None of those uncertainties moves the conclusion that matters, which is the proportionality between removal and failure fraction: it holds whatever the yield and the cross-section, because in a shell where drag does nothing every failure has to be cleared by hand.

What the model leaves out #

It treats one shell in isolation, and real shells exchange debris: fragments from a collision spread over hundreds of kilometres of altitude, and the fragments from a high shell decay through every shell below it on their way down. A collision at 1,200 kilometres is therefore a supply of fragments to the shells near 800 for centuries. It treats all derelicts as the same size and all fragments as equally dangerous, where real objects span the range from the large rocket bodies that dominate the mass to the fragments below tracking size that dominate the hazard. And it lets working satellites dodge perfectly, which they do not: a conjunction warning depends on knowing where both objects are, and fragments too small to track give no warning at all.

Each of these makes the high shells worse rather than better, and none changes the shape of the first figure.

Still open: what reliability a high constellation should be required to reach #

The arithmetic is clear and the decision is not. A licensing condition that requires ninety-nine per cent disposal reliability for satellites above 1,000 kilometres, or that requires an operator to pay for removing its own failures, would set the removal burden directly; neither exists in a form that binds across the operators and countries flying in the same shells. The reliability a constellation actually achieves will be known only as its satellites reach end of life over the next decade, which is the decade in which its derelicts begin to accumulate. Whether the shells near 1,200 kilometres stay below their thresholds depends on a number that is being measured in real time, by satellites that either answer their last command or do not.