Astronomy & Celestial Mechanics
the sky, computed
Concept

Kessler syndrome — where it appears

The condition in which collisions between orbital debris produce fragments faster than the atmosphere removes them, so the population grows without further launches. It follows from production going as the square of the population and removal as the first power.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

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.

spaceflight · Orbital debris 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.

spaceflight · Orbital debris 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.

spaceflight · Orbital debris 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.

spaceflight · Orbital debris

Named alongside it

The objects these essays reach for when they reach for this one.

Orbital debrisOrbital lifetimePost-mission disposalActive debris removalBallistic coefficientBreakup modelArea-to-mass ratioAtmospheric dragCollision cross-sectionCollision probabilityCollision rateConjunction analysis

All concepts