Astronomy & Celestial Mechanics
the sky, computed
Concept

Orbital debris — where it appears

The population of spent hardware and collision fragments in Earth orbit. Its number density and relative speed give a collision rate directly, and above the altitude where atmospheric drag removes objects the population can grow from collisions alone.

Named by 13 essays across 3 fields — 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 Mass at the top, area at the bottom, 10 decades apart. Two moments of a collisional cascade's size distribution, per logarithmic interval of diameter, over 10 decades from a ten-micron grain to a hundred-kilometre parent body. Both axes are logarithmic and the vertical scale is arbitrary; only the slopes carry the argument. A population in which every collision makes fragments that go on to collide reaches a steady state where the same mass flows through every size per unit time, and that fixes the differential number distribution at an index of 3.5. The two consequences pull opposite ways. Mass per decade goes as the diameter to the power 0.5, so it climbs and almost all the mass is in the largest few bodies. Cross-sectional area per decade goes as the diameter to the power -0.5, so it falls, and almost all the area — which is what scatters light, what is detected, and what anything passing through gets hit by — is in the smallest. Over the range drawn the small end carries 10⁵ times the area of the large end and 10⁻⁵ times its mass. A disc's brightness therefore measures a population whose mass it says nothing whatever about, and the two numbers are connected only through the index of this line.

Where the mass is and where the light is

A population that grinds itself up settles into a size distribution with a fixed slope, and that slope puts almost all the mass in the largest bodies and almost all the cross-section in the smallest. So a debris disc's brightness measures a population whose mass it says nothing about, and the two are connected only by the exponent.

orbits · Collisional cascade Two longitudes a satellite falls towards, and 0.5 m/s a year to stay elsewhere. The along-track potential a geostationary satellite feels, against longitude. The Earth's equator is slightly elliptical — about seventy metres between its long and short axes — and that one harmonic of the gravity field gives the geostationary ring two minima and two maxima. A satellite left unattended drifts towards the nearer minimum at 75° or 255° east, overshoots, and librates about it with a period of a couple of years. The peak acceleration accumulates 0.5 metres a second of velocity change a year, and an operational east–west budget is a small multiple of that once the correction cycle is accounted for — a fixed cost of every commercial slot in the ring, paid forever. The two minima are the graveyard of the geostationary population: uncontrolled satellites accumulate there, which is why they are the most crowded longitudes in the belt and why an uncontrolled object is most likely to be found near one.

A satellite that drifts to one of two longitudes

The Earth's equator is elliptical by about seventy metres. That one harmonic of the gravity field gives geostationary orbit a potential with two minima, so an unattended satellite slides towards the nearer one and stays — and every operational slot in the ring is paid for with fuel, every year, forever.

spaceflight · Station-keeping 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 A cloud of fragments that pinches where it was born. 240 fragments thrown from a satellite on a circular orbit 800 kilometres up, with kicks of about 100 metres a second in random directions in the orbital plane, each then following its own Kepler ellipse, drawn at 0.5, 1, 4.5, 20 orbital periods of the parent, with the Earth's centre marked and the break-up point at the top. Radial offsets from the parent's circle are exaggerated 2.5 times. Half a period after the break-up the fragments are spread 28 degrees along the orbit and 1374 kilometres in altitude, the kicks having carried some to higher apogees and others to lower perigees. After one period the along-track spread has grown to 53 degrees but the whole cloud is only 112 kilometres thick, and the 83 fragments within eight degrees of the break-up point span 35: every fragment's orbit passes through the break-up point, and the cloud pinches there once a period. By 4.5 periods it is 231 degrees long and 863 kilometres thick overall; by 20 it is 352 degrees long, still pinched to 11 kilometres where it began.

Every fragment comes back through the same point

When a satellite breaks up, its pieces fly off in every direction and every one of them returns, an orbit later, to the place where it broke. The cloud pinches there once a period, stretches along the orbit at a rate set by one component of each kick, and becomes a ring in hours.

orbits · Vis-viva The three terms of the minimum disposal height. The minimum height above the geostationary ring to which a satellite should be raised at the end of its life, by the international guideline ΔH = 235 km + 1000 · Cᵣ · A/m, broken into its three terms, for reflectivity times area-to-mass ratios of 0.005, 0.01, 0.02, 0.03, 0.05 square metres per kilogram. The first 200 km is the protected zone itself; the next 35 km is the most that the Moon, the Sun and the Earth's uneven gravity lower an orbit's perigee over time; the rest is the eccentricity that sunlight raises every year, which a derivation from first principles puts at 942 km per unit of Cᵣ·A/m for an orbit started circular — the guideline's round thousand. Totals: 240 km, 245 km, 255 km, 265 km, 285 km. The term that depends on the satellite is the one sunlight sets, and for a light satellite with large solar arrays it is the largest.

A graveyard height with a term for sunlight

A geostationary satellite at the end of its life is raised about three hundred kilometres and switched off. The height is written as a formula — 235 kilometres plus a thousand times its reflectivity and area-to-mass ratio — and each term is a separate piece of orbital mechanics, the last one sunlight bending the orbit a little every year.

spaceflight · Station-keeping 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 The kick speeds read back out of the periods: median 100 m/s against the 100 m/s put in. The distribution of kick speeds recovered from the fragments' periods alone, by p(v) = −2v g′(v) applied to the along-track kicks, for 300 and 3000 fragments (points), against the log-normal distribution actually used, median 100 m/s and width 0.3 dex (line), on a logarithmic speed axis with the density per logarithmic interval. With 3000 fragments the recovery follows the shape of the truth and puts the median at 100 m/s, though differentiating a measured distribution twice smooths its peak and roughens its tails; with 300 it is noisy but already in the right place. The inversion uses one assumption, that the kicks were isotropic, and one measurement per fragment, its period — the quantity tracking networks determine first and best.

A kick speed read off the slope of a histogram

Throw fragments at one speed in random directions and the component of each kick along the orbit is spread evenly between plus and minus that speed — Archimedes' theorem about the area of a sphere, applied to debris. Each fragment's period records that component, so the histogram of periods is a stack of flat bands, one per speed, and its slope gives the speeds back: a break-up weighed by its own violence from a catalogue of orbits, if the event was isotropic, and fragment by fragment if it was not.

orbits · Vis-viva Two clouds with the same kicks: the explosion's centre at −2.3 m/s, the collision's at 10.9. The along-track kicks read from the periods of 1500 fragments, for an explosion (blue) and for a collision (orange) whose fragments received the same random kicks plus a shared velocity change of 12 m/s along the track — the centre-of-mass motion a projectile's momentum leaves behind. The two histograms have the same width and shape; the collision's is displaced. The number-weighted mean of the explosion's kicks is −2.3 ± 2.6 m/s, consistent with zero, and the collision's 10.9 ± 2.5 m/s. The width of a cloud says how violent the event was; its centre says whether anything arrived from outside.

A collision moves the cloud's centre, and an explosion cannot

However violently a satellite blows itself apart, its fragments' momenta add up to the parent's: the centre of the cloud keeps moving exactly as the satellite did. A collision is different, because the projectile brings momentum in, and a collision energetic enough to shatter its target needs a projectile heavy enough to move the centre by at least 80,000 joules per kilogram divided by the impact speed — eight metres a second in low orbit. The width of a debris cloud measures how violent the event was. Its centre measures whether anything arrived from outside.

orbits · Vis-viva 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.

spaceflight · Orbital debris A glow that switches off when the shadow climbs past the shell. The brightness that 100,000 objects in orbit add to the zenith sky, as a share of a natural moonless sky of 22 magnitudes per square arcsecond, against how far the Sun is below the horizon, for shells at 550 and 1200 km. Each object reflects sunlight with an effective area of 0.5 square metres; together they spread a glow whose brightness does not depend on the shell's distance, because more objects are in each patch of sky the further away the shell is. At the end of astronomical twilight the two shells add 3.0% and 2.5%. The lower shell's zenith goes dark when the Earth's shadow passes it, at 23 degrees of depression; the higher one at 33.

A glow counted in square metres of sunlit metal

Satellites too faint to see one at a time still reflect sunlight, and spread across the sky they add a glow that no mask removes. Its brightness does not depend on how far away the satellites are, only on how much sunlit area is in orbit and whether the Earth's shadow has reached it. A hundred thousand satellites as bright as today's add a few per cent to the darkest sky — and the estimate that the sky is already a tenth brighter rests on an area of debris nobody has measured.

sky · Twilight How far a change of attitude moves a satellite before the encounter. The along-track shift a 5-kg satellite achieves by switching from 0.01 to 0.1 m² of frontal area when warned of a conjunction, against the warning time, at 400, 500, 600 km, between solar minimum (lower edge) and maximum (upper edge), logarithmic. The shift grows as the square of the time. In a day it is 26.1 to 156 km at 400 km; 3.9 to 39 km at 500 km; 0.6 to 10 km at 600 km. The dashed line is the 556 m needed to bring a 300-m position uncertainty's collision risk below one in ten thousand.

A dodge made of air

A small satellite with no thruster can still move along its orbit by turning broadside to the thin air, and the question is whether that is enough to get out of the way of debris. Below six hundred kilometres the answer, measured in distance, is easily yes: a day's warning buys kilometres when a few hundred metres would do. Measured in predictability it is no. The density of the air is forecast to about thirty per cent, so the dodge goes somewhere nobody, including the satellite's own operator, can say in advance to the precision the dodge was meant to achieve.

spaceflight · Rendezvous

Named alongside it

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

Kessler syndromeOrbital lifetimeBallistic coefficientBreakup modelGabbard diagramGeostationary orbitGraveyard orbitOrbital periodPost-mission disposalVis-viva equationActive debris removalArea-to-mass ratio

All concepts