A ring where nothing leaves and little shatters

The arithmetic of debris in low orbit has two terms. Production grows as the square of the population and removal as its first power, and the removal is the atmosphere: drag lowers every orbit, faster the lower it is, until the object burns up. Whether a shell is stable depends on which term wins, and the answer is set by the altitude, because the atmosphere’s density falls by orders of magnitude between four hundred kilometres and a thousand.

The geostationary ring, 35,786 kilometres above the equator, has no atmosphere at all. Its removal term is not small; it is zero. Every satellite that dies there and every fragment that is made there stays — not for centuries but for the life of the solar system, slowly perturbed by the Moon, the Sun and sunlight, never falling. If the low-orbit argument applied unchanged, the ring would be the most fragile orbit there is, and its first collision cascade would be permanent.

It does not apply unchanged, because the geometry that removes the sink also removes most of the energy from collisions.

A drag that is not there #

The low-orbit argument’s removal term is worth seeing once to measure how complete its absence is.

The curve rises a thousandfold over five hundred kilometres. Extended to geostationary altitude, it has no meaning: the residual gas density there is so low that drag would take far longer than the age of the universe to change an orbit noticeably. Other forces do act. The Moon and the Sun pull the orbit’s plane out of the equator, and the Earth’s slightly elliptical equator makes uncontrolled objects drift toward one of two stable longitudes and oscillate about it. Sunlight pushes on large, light objects and slowly changes their eccentricity. None of these removes anything. They rearrange the population; they do not deplete it.

So the ring’s population of uncontrolled objects can only grow. Each satellite that is not moved away at the end of its life adds one object for good. Each collision adds fragments for good. The low-orbit language of “equilibrium” — the population at which production and removal balance — has no geostationary counterpart.

How fast things meet #

A collision’s consequences depend on its speed, and in the ring the speed is set by geometry.

Every object at geostationary altitude orbits at about 3.07 kilometres a second, in nearly the same direction. Active satellites are held near the equator, at inclinations of a fraction of a degree. Uncontrolled ones are not: the Moon and the Sun swing their orbital planes through a cycle of 53 years, taking the inclination of an object that starts in the equator up to about fifteen degrees and back. Such an object crosses the equatorial ring twice a day, once going north and once going south, and at each crossing its velocity differs from an active satellite’s only by the tilt.

Twice the orbital speed times the sine of half the inclination is the length of the chord between two velocity vectors of equal length tilted by the inclination. At fifteen degrees it is about eight hundred metres a second; at the median, under six hundred. In low orbit, where objects cross at all angles — prograde and retrograde, polar and equatorial — the typical collision speed is ten kilometres a second. The ring’s collisions are slower by a factor of fifteen to twenty, and their energies by a factor of two or three hundred.

Too slow to shatter #

The consequence follows from how satellites break. The standard breakup model, calibrated on ground tests in which projectiles were fired into satellite structures, distinguishes two regimes by the energy delivered per gram of the target. Above about forty joules per gram the collision is catastrophic: the whole target is broken into fragments whose number depends on the combined mass. Below it the collision is a cratering event: the projectile and a patch of the target are broken up, the rest survives, and the number of fragments depends on the projectile’s mass and the square of the speed.

At ten kilometres a second, forty joules per gram of a two-tonne satellite is delivered by a projectile of under a kilogram. At eight hundred metres a second, it takes about two hundred and fifty kilograms — not a fragment but another satellite. Almost every plausible collision in the ring is therefore a cratering event, and almost every plausible collision in low orbit with anything larger than a bolt is catastrophic.

That factor of a hundred is the ring’s protection. A collision there adds about a dozen large fragments, not a thousand, and the fragments it does add move away from the collision at speeds of metres to tens of metres a second, spreading slowly round the ring rather than filling a shell. The cascade argument — each collision making enough fragments to cause more than one further collision — needs a fragment yield the ring’s speeds do not supply.

This is the essay’s surprising connection. The same fact that makes the ring permanent makes it gentle. There is no drag because the ring is far from the Earth, and everything there moves at the same speed in the same direction because the ring is a single orbit — the one whose period matches the Earth’s rotation. Low orbit has both a sink and violent collisions because it is not one orbit but a thick shell of many, crossing at every angle. The geometry gives the ring nothing to remove its debris and nothing to break it violently.

A single orbit, and how fragments spread round it #

The ring is, to a good approximation, one orbit. Its defining property is its period — one sidereal day — and the period fixes the radius through the harmonic law, so every object that keeps the Earth’s pace is at the same distance from the Earth. Objects a little above or below that radius have slightly longer or shorter periods and drift slowly in longitude relative to the turning Earth: an object fifty kilometres above the ring falls behind by about six-tenths of a degree a day, circling the whole ring in under two years.

That drift governs what happens after a collision. Fragments leave the collision with speeds of metres to tens of metres a second relative to their parents, which changes their orbital radii by kilometres to tens of kilometres. Within weeks they have spread along the ring ahead of and behind the collision point; within a year or two they are distributed all the way round it, in a band whose width is set by their spread in radius. In low orbit the same spreading turns a cloud into a shell around the whole Earth in months, but the shell is crossed by everything at every angle. In the ring the band is crossed only by objects at the ring’s own speed, slowly.

The narrowness of the ring cuts both ways. Fragments in the band are confined near the active satellites’ altitude, which raises their density there; but they move with the active satellites rather than across them, so they meet them at the low speeds drawn above. A fragment in the ring is like a car on a motorway that every other car is also on — dangerous only through the small differences in speed between lanes.

A crowd at two longitudes #

Uncontrolled objects do not drift uniformly round the ring forever. The Earth’s equator is slightly elliptical, and the resulting longitude-dependent pull has two stable points, near 75 degrees east and 105 degrees west, about which uncontrolled objects with small drift rates oscillate rather than circulate. A dead satellite drifts to one of two longitudes and swings back and forth around it for as long as it exists.

The population of uncontrolled objects is therefore concentrated near those two longitudes, and the active satellites stationed there — over the Indian Ocean and over the Americas — see a collision rate several times higher than the ring’s average. The kinetic estimate in the last figure treats the ring as uniform; an operator at one of the stable points must multiply it by that local concentration. The ring’s most crowded neighbourhoods are set not by where satellites are wanted but by where gravity collects the dead ones.

Slow encounters give long warnings #

Low collision speeds have a practical benefit beyond the fragment count. An encounter at five hundred metres a second lasts longer and is predicted further ahead than one at ten kilometres a second. Two objects closing at that speed take minutes to cross the few-kilometre region where their position uncertainties overlap, rather than seconds, and their relative motion near the ring is regular enough to be predicted days in advance from orbits measured by optical tracking.

So the geostationary operators’ collision avoidance is in some ways easier than its low-orbit equivalent: fewer objects, slower encounters, longer warnings, and the option of a tiny station-keeping adjustment to move a satellite a few kilometres along the ring. It is harder in one way. The objects that threaten a geostationary satellite are tracked less precisely than those in low orbit, because they are thirty-six thousand kilometres away and seen only by reflected sunlight, and the smallest dangerous fragments — tens of centimetres across and smaller — are too faint to track at all. For those, the avoidance is statistical: a satellite cannot dodge what it cannot see, and its risk is the product of the kinetic rate and the unseen population.

A rate that ignores the inclinations #

How often collisions happen is the other half, and it has a property worth stating because it would not be guessed.

The rate of collisions with a satellite is the density of objects near it, times its cross-section, times the speed at which they pass. An object with a small inclination stays close to the equatorial plane for a large fraction of each day, so it contributes a high density near the ring, but it passes slowly. An object with a large inclination spends only a small fraction of each day near the ring, so it contributes a low density, but it passes fast. In the small-angle limit the density goes as one over the sine of the inclination and the speed as the inclination, and the product is independent of it. The collision rate depends only on how many uncontrolled objects there are and how spread out they are in radius.

The number is small. A thousand objects larger than a few tens of centimetres, in a hundred-kilometre band, give an active satellite a chance of about one in a hundred thousand per year of being hit — about one in eight thousand over a fifteen-year life. For the few hundred active satellites in the ring, that is a collision somewhere every few centuries at present numbers.

The geometry also explains why that number grows linearly, not quadratically, as long as uncontrolled objects are not left in the ring itself. Collisions between two uncontrolled objects matter only if they put fragments where the active satellites are, and the uncontrolled population is spread over tens of kilometres of radius and fifteen degrees of inclination; the active satellites occupy a thin strip within it. The quadratic term of the low-orbit argument exists in the ring, but its coefficient is tiny, because collisions are rare, slow and spread out.

What the protection rests on #

The ring’s safety is therefore not natural in the way low orbit’s would be with enough drag. It rests on two practices, and both are matters of discipline rather than physics.

Satellites are moved away at the end of their lives. A geostationary satellite with fuel left at the end of its mission is raised by about three hundred kilometres into a graveyard orbit, where the active ring’s satellites can never meet it. The height of the graveyard is set by a formula with a term for sunlight, because sunlight changes the shape of a light object’s orbit and can bring it back down toward the ring. A satellite that is moved does not join the uncontrolled population at all.

Fuel is kept for it. The move costs about eleven metres a second, a few months of station-keeping propellant, and it must be reserved before the end of a mission whose length is uncertain; a satellite’s fuel gauge is least accurate when it matters most. Satellites that run out early, or fail suddenly, stay in the ring — and as in low orbit, a few large derelicts hold most of the future: a two-tonne satellite left in the ring is a permanent target and a permanent projectile.

The fraction of satellites successfully moved has risen over the decades to a large majority, and the uncontrolled population in the ring has grown more slowly than the number of launches. It still grows, because nothing else removes anything, and each satellite that fails to move is a permanent addition.

The fragments that were already made #

Several explosions have already happened in or near the ring — old upper stages and satellites whose batteries or leftover propellant failed — and they have added fragments that the collision arithmetic above does not count. An explosion delivers its energy from inside, at whatever speed the stored energy permits, and it can shatter an object completely whatever the orbital speeds around it. The low collision speeds of the ring protect it from shattering collisions; they do nothing about objects that shatter themselves.

Those fragments are too small to track from the ground at geostationary distance — the smallest objects routinely tracked there are tens of centimetres to a metre across, against ten centimetres in low orbit — and their number is estimated from optical surveys that sample the ring’s population statistically. They spread round the ring over months and their inclinations cycle with everything else’s. They are the main reason the uncontrolled population is uncertain by a factor of several, and the main reason the ring’s long-term outlook is argued about.

What the figures assume #

The inclination cycle is idealised. Every uncontrolled object is given the same maximum inclination and a uniform spread of ages over one cycle; real objects started from different planes and have been perturbed differently. The speed distribution’s shape depends on that, its upper limit does not.

The breakup model is a fit. The forty-joules-per-gram threshold and the fragment counts come from tests of particular structures at particular speeds, and they are applied far from the speeds at which some of them were measured. Modern satellites, built with more composite material and less aluminium, may break differently.

The collision rate is a kinetic-gas estimate. It treats the uncontrolled objects as a gas spread over a band of radius and inclination, with no concentrations; real objects bunch near the two stable longitudes, and an active satellite stationed near one of those longitudes sees a higher rate than one elsewhere.

Only objects large enough to break a satellite are counted. The far more numerous small fragments, which can disable a satellite by striking a solar panel or a cable without breaking it, are not in these figures.

Still open: whether discipline is enough for a thousand years #

The ring is valuable in a way no other orbit is: it is the only place where a satellite stays fixed over a point on the equator, and the number of useful slots in it is limited by how close together satellites can be placed without interfering. A permanent, slowly growing population of uncontrolled objects in the ring is a slowly growing tax on that resource, paid in collision avoidance and in the occasional loss of a satellite.

Whether the tax stays small depends on the fraction of satellites moved to the graveyard at the end of their lives — the same percentage that decides the removal rate a high low-orbit shell needs — and on whether the graveyard itself stays clear of the ring for as long as its objects exist. At a few hundred kilometres above the ring and with no removal, the graveyard is another permanent population, separated from the active one by orbital mechanics rather than by any physical barrier. Whether that separation holds for a thousand years, as sunlight and the Moon slowly reshape the graveyard orbits, is the question the graveyard’s designers are still answering.