Astronomy & Celestial Mechanics
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Orbital debris — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A quadratic and a linear, crossing at 149 objects. The two rates that decide whether a shell at 900 km is stable, against how many objects are in it. Production goes as N² — every collision needs two objects, so the number of collisions is proportional to the square of the population, and each one is taken here to make 1600 trackable fragments. Removal goes as N, because drag acts on each object independently and takes 1,195 years to do it at this altitude. A quadratic and a linear cross exactly once, at 149 objects in this shell, and above that crossing the population grows with nothing launched. The shell presently holds about 2,280, which is 15 times the crossing. Every number on the production side is uncertain by a factor of a few — the fragment yield most of all, and the cross-section is calibrated against an observed collision rate rather than measured — so the position of the crossing carries that uncertainty with it. The shape does not, and the shape is the argument: a quadratic overtakes a linear once and never comes back, the crossing falls as the altitude rises because the lifetime is in the denominator, and what results is a threshold rather than a trend.

    A collision rate that needs no collision

    The flux through an orbital shell is a gas-kinetic calculation with no orbits in it. Production goes as the square of the population and removal goes as the first power, so a quadratic overtakes a linear once and never comes back — and which side of that a shell is on is decided by its altitude.

    part 1 · spaceflight
  2. 1,236 fragments anybody can see, and 63,397 that can kill. The cumulative fragment size distribution from the standard breakup model, for a catastrophic collision involving 1500 kilograms and for an explosion of the same object, both axes logarithmic. The exponents are −1.71 and −1.60, measured off the drawn curves; they are empirical, fitted to ground tests and to the observed clouds of real events, and they are steep. What follows is the reason a catalogue of tracked objects is not a catalogue of the hazard. Above ten centimetres — the size a ground radar can follow in low orbit — the collision makes about 1,236 pieces. Above one centimetre, which is the size that goes through a spacecraft at ten kilometres a second and cannot be shielded against, it makes 63,397. Above a millimetre, which shielding does stop but which erodes a surface, 3,251,385. The tracked population is under two per cent of the lethal one, and the difference is not a gap in the catalogue that better radars will close — objects of a centimetre at a thousand kilometres are beyond any sensor that has been proposed. An explosion makes fewer large pieces than a collision and a comparable number of small ones, because an explosion divides one object and a collision destroys two.

    The fragments nobody can see and cannot shield against

    A catastrophic collision in low orbit makes about a thousand pieces big enough to track and sixty thousand big enough to destroy a spacecraft. The catalogue is under two per cent of the hazard, the gap is not one better radars will close, and the fragments' lifetimes span a factor of twenty-five within a single event.

    part 2 · spaceflight
  3. How much future debris each removal prevents. The share of the derelict population's expected future fragment production removed, against the number of objects removed, out of 1858, for three orders of removal: by the ranking weight — collision rate times fragments times fragment lifetime — by mass alone, and at random. Removing the 200 top-ranked objects removes 70 per cent; the top fifty alone, 45 per cent; the top ten, 19. Removing by mass does nearly as well at first and falls behind, because mass is most of the weight and altitude the rest. Removing at random, fifty objects remove 3.5 per cent. The population is synthetic, built to the published inventory's broad shape; the concentration of risk in a few tens of heavy upper stages is the finding the real analyses also make.

    Fifty objects that hold most of the future

    Low Earth orbit holds thousands of dead satellites and spent rocket stages, and the debris of the future will come mostly from their collisions. But not in equal shares. A derelict's contribution is the chance it is hit, times the fragments it would make, times how long they would stay up — and all three favour the heaviest objects in the most crowded, highest orbits. Ranked that way, a few tens of rocket bodies near 850 kilometres carry nearly half the risk, and the arithmetic of cleaning space becomes the arithmetic of choosing them.

    part 3 · spaceflight
  4. Removals a year, set almost entirely by one percentage. The number of derelicts that must be removed each year to hold a shell at 50 per cent of its critical count, against the fraction of disposals that fail, for a 650-satellite constellation at 800, 1000, 1200 km. Because a dead satellite at these heights stays up for centuries, almost every failure has to be removed by hand, and the removal rate is very nearly the failure rate: at 1200 km, 13 a year if ninety per cent of disposals succeed and 1.3 if ninety-nine do. The removal budget is therefore set by an engineering reliability rather than by the orbital environment, and a percentage point of reliability is worth a fixed number of removal missions a year for as long as the constellation flies. Five removals a year is enough for this shell only above 96.2 per cent reliability.

    A removal rate set by one percentage

    How many dead satellites have to be taken out of a high orbit each year is decided less by the orbit than by how often a satellite fails to dispose of itself. Above eight hundred kilometres a derelict stays for centuries, so the removal rate needed to keep a shell stable is very nearly the failure rate, and moving disposal reliability from ninety per cent to ninety-nine cuts the work tenfold.

    part 4 · spaceflight
  5. Too slow to shatter. The kinetic energy a projectile brings per gram of a 2,000 kg satellite, against the impact speed on logarithmic axes, for projectiles of 1, 10, 100 kg. Above 40 joules per gram (dashed) the standard breakup model counts the collision as catastrophic — the whole target is shattered; below it, only a crater is made. The speeds needed are 12.6, 4.0, 1.3 km/s respectively. In the geostationary ring, where nothing crosses faster than about 0.8 km/s (shaded), only a projectile of hundreds of kilograms — another satellite — can shatter a satellite; in low orbit, at ten kilometres a second, a kilogram can.

    A ring where nothing leaves and little shatters

    Every argument about orbital debris in low orbit ends in the atmosphere, which pulls fragments down in years or centuries. The geostationary ring has no atmosphere, so its removal term is exactly zero and anything left there stays for millions of years. It is saved by a second fact of the same geometry. Everything in the ring moves at nearly the same speed in nearly the same direction, so collisions there are slow, and a slow collision usually leaves a crater where a fast one leaves a cloud.

    part 5 · spaceflight

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