Astronomy & Celestial Mechanics
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Spaceflight

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.

Assumes Atmospheric drag, Ground tracks and Station-keeping.

16 min read 8 figures Scales that defeat intuitionSmall perturbations, large consequences

There is no orbital mechanics in this essay. That is the point of it.

Two objects sharing a shell around the Earth are on determinate trajectories, each computable to metres, and asking whether a particular pair will collide is a question about those two trajectories. Asking how often any pair in the shell will collide is a different question, and it has an answer that treats the whole population as a gas: a number density, a mean relative speed, a cross-section, and a rate that is their product.

The gas-kinetic answer is the one that matters, because the interesting quantity is not whether a given satellite is hit but whether the population as a whole is stable.

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.
Fig. 1 The two rates that decide whether a shell at 900 kilometres is stable, against how many objects are in it. Production goes as N2N^2 — every collision needs two objects — and each is taken here to make 1,600 trackable fragments. Removal goes as NN, 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; the shell presently holds about 2,280, which is fifteen times the crossing. Every number on the production side is uncertain by a factor of a few, so the position of the crossing carries that uncertainty. The shape does not.

The rate on one object

The standard gas-kinetic result is that a particle of cross-section σ\sigma moving at relative speed vv through a medium of number density nn suffers collisions at rate

R=n σ v.R = n\,\sigma\,v.

Each factor needs a value, and each has a subtlety.

The number density is the count in a shell divided by the shell’s volume. A 25-kilometre shell at 900 kilometres altitude has a volume of 1.7×10101.7\times10^{10} cubic kilometres — the Earth is large and shells around it are larger — and holds about 2,280 tracked objects. That is 1.4×10−71.4\times10^{-7} per cubic kilometre, or one object per seven million cubic kilometres. Low Earth orbit is empty by any ordinary standard, and every alarming statement about it has to survive that fact.

The relative speed is not the orbital speed and is not zero. Two objects in the same shell have nearly the same speed, 7.4 km/s, and quite different directions: inclinations in low orbit run from zero to retrograde, so a head-on encounter closes at 15 km/s and a co-planar overtaking at nearly nothing. Averaged over the observed inclination distribution the mean relative speed is about 10 km/s. It is a property of the population’s inclinations rather than of any orbit.

The cross-section is the difficult one, and the honest treatment is to calibrate it rather than to assume it. Taking σ\sigma as the geometric area of two satellites gives a collision rate an order of magnitude above what is observed, and for a reason worth stating: most tracked objects are not satellites but fragments of tens of centimetres, and a collision between two of those is not catastrophic and makes no new trackable debris. So σ\sigma here is an effective catastrophic cross-section, fixed by requiring the whole tracked population to produce the roughly one catastrophic collision every eight years that is actually seen. It comes out at about 3,700 square centimetres — well under a square metre, which is the model admitting what it has averaged over.

The population peaks at 825 km, and so does the risk. Tracked objects per 25-kilometre shell against altitude, with the collision rate on a single object computed from each bin as nσv — a gas-kinetic rate, not an orbital calculation. The cross-section is not a satellite's area but an effective one, 3698 square centimetres, fixed by requiring the whole tracked population to produce the one catastrophic collision every 8 years that is observed — most tracked objects are fragments, and two fragments meeting make nothing new. The distribution is not smooth and its shape is history: the peak of 3,010 objects near 825 km is four decades of launches into sun-synchronous orbit plus the debris of two deliberate destructions, and the second rise past 1,300 km is the Soviet-era navigation constellation. At the peak one such object waits 46,409 years between strikes — which sounds safe until it is multiplied by the 3,010 objects sharing that shell, giving one collision every 31 years among them, and by the 23,680 in the whole of low orbit, giving one every 8. The rate on one object is reassuring and the rate on the population is not, and they are the same number.
Fig. 2 The population the rate is computed from, and its shape is history rather than physics. Tracked objects per 25-kilometre shell against altitude: the peak of 3,010 near 825 kilometres is four decades of launches into sun-synchronous orbit plus the debris of two deliberate destructions, and the second rise past 1,300 kilometres is an obsolete navigation constellation. At the peak one object waits 46,000 years between strikes — which sounds safe until it is multiplied by the 3,010 sharing that shell, giving one collision every 31 years among them, and by the 23,680 in the whole of low orbit, giving one every eight.

That last sentence is the argument’s whole rhetorical difficulty. The rate on one object is reassuring and the rate on the population is not, and they are the same number multiplied by different things.

Why the two rates have different powers

The structure that produces a threshold is arithmetic and does not depend on any of the numbers.

A collision requires two objects. The number of pairs in a population of NN is N(N−1)/2N(N-1)/2, so the number of collisions per unit time in a fixed volume is proportional to N2N^2. Each collision makes some number of new trackable fragments, so the production rate of debris goes as N2N^2.

Removal requires one object and the atmosphere. Each object decays independently on its own timescale, so the removal rate is N/τN/\tau.

A quadratic and a linear cross exactly once, at

Ncrit=2Vσv f τ,N_{\rm crit} = \frac{2V}{\sigma v\,f\,\tau},

with ff the fragment yield. Below the crossing, removal wins and any injection of debris decays away. Above it, production wins and the population grows with nothing launched at all.

Donald Kessler published that argument in 1978. It is one page of arithmetic and it has not been improved on, only quantified.

The altitude does everything

The critical population carries τ\tau in its denominator, and τ\tau — the orbital lifetime — is the quantity that varies most across the region in question.

A year at 400 km and 1195 years at 900. Orbital lifetime against starting altitude for a circular orbit, integrated from da/dt = −ρav/β with a ballistic coefficient of 100 kg/m² and a piecewise-exponential fit to the 1976 standard atmosphere. Only the density profile is tabulated; the decay is computed. The curve rises by a factor of 1397 between 400 and 900 kilometres — 10 months against 1,195 years — because the density falls by four decades across that span while nothing else in the expression changes much. That single ratio is why an altitude either cleans itself or does not. Below about 600 km a fragment is gone before it can find anything; above 800 it is there for centuries, and the twenty-five-year disposal rule is a statement about which side of this curve an operator is required to leave the vehicle on.
Fig. 3 Orbital lifetime against starting altitude, integrated from da/dt=−ρav/β\mathrm da/\mathrm dt = -\rho a v/\beta with a ballistic coefficient of 100 kg/m² and a piecewise-exponential fit to the standard atmosphere. Only the density profile is tabulated; the decay is computed. The curve rises by a factor of 1,400 between 400 and 900 kilometres — ten months against 1,195 years — because the density falls by four decades across that span while nothing else in the expression changes much. That single ratio is why an altitude either cleans itself or does not.
A quadratic and a linear, crossing at 6,246 objects. The two rates that decide whether a shell at 600 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 26 years to do it at this altitude. A quadratic and a linear cross exactly once, at 6,246 objects in this shell, and above that crossing the population grows with nothing launched. The shell presently holds about 1,160, which is 0 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.
Fig. 4 The same two rates at 600 kilometres rather than 900. Nothing about the arithmetic has changed and the crossing has moved from 149 objects to 6,246 — a factor of forty, from a factor of forty in the orbital lifetime. The critical population is not a property of low Earth orbit; it is a property of an altitude, and at 600 kilometres a shell can hold thousands of objects and still clean itself, which is why the large constellations were put there.

So the same shell arithmetic gives opposite answers at two altitudes 500 kilometres apart. At 400 kilometres a fragment is gone before it can find anything, and the critical population is enormous. At 900 kilometres it is there for a millennium, and the critical population is small.

That is why the International Space Station’s altitude is not a problem and the 800-to-1,000-kilometre band is. It is also why the twenty-five-year post-mission disposal rule exists, and why it was recently shortened to five: the rule is a statement about which side of that curve an operator is required to leave a vehicle on.

A quadratic and a linear, crossing at 23 objects. The two rates that decide whether a shell at 1200 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 8,398 years to do it at this altitude. A quadratic and a linear cross exactly once, at 23 objects in this shell, and above that crossing the population grows with nothing launched. The shell presently holds about 430, which is 19 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.
Fig. 5 And the other direction. At 1200 kilometres the drag lifetime is long enough that the crossing falls to twenty-three objects — fewer than are up there now by two orders of magnitude, and few enough that the threshold has no practical meaning: the shell is above it and will stay above it for as long as anything can be planned for. The three drawings of this figure are the same equation at three altitudes, and between them they cover a range in which the answer goes from “irrelevant” to “already exceeded”.

What the model gets wrong, deliberately

Three simplifications are made above and each is worth naming, because the argument’s strength is that it survives all three.

The population is not well mixed. Objects are concentrated at particular inclinations — 98° for sun-synchronous, 51.6° for the station, 82° for a large family of Russian payloads — and two objects at the same inclination and altitude rarely close on each other. Treating the shell as a uniform gas overstates the encounter rate for the clustered part of the population and understates it for the rest.

A quadratic and a linear, crossing at 597 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 400 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 597 objects in this shell, and above that crossing the population grows with nothing launched. The shell presently holds about 2,280, which is 4 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.
Fig. 6 What a factor of four in the fragment yield does to the same shell at 900 kilometres. The crossing moves from 149 objects to 597 — exactly the factor of four, because the yield multiplies the production term and the crossing is where production equals removal. The uncertainty in the yield is a linear uncertainty in the threshold and not in the conclusion, which is the whole reason this argument survives having a factor-of-two number in the middle of it: the shell holds 2,280 objects, and 597 is still well under that.

The fragment yield is not a constant. A catastrophic collision’s fragment count depends on the impact energy per unit target mass, and the standard NASA breakup model gives a distribution rather than a number. Taking 1,600 trackable fragments is a representative figure — Iridium 33 and Cosmos 2251 together produced about 2,300 catalogued — and it is uncertain by a factor of two either way.

Only tracked objects are counted. The catalogue holds objects larger than about ten centimetres in low orbit. The population between one and ten centimetres, which is large enough to destroy a satellite and too small to track, is estimated at a million or so and is not in the arithmetic above at all. It contributes to the destruction and not to the count.

The gap that leaves is the most consequential thing the arithmetic omits, and it is worth stating in the terms an engineer would use. Below about a centimetre, debris can be shielded against: a spaced double wall — a thin outer sheet that shatters the projectile into a spray, and a rear wall that absorbs the spray over a larger area — defeats a particle that a single thick plate would not. Above about ten centimetres, an object can be tracked and avoided. Between the two there is neither remedy: too large to stop, too small to see.

That band holds an estimated million objects in low orbit, and it is where most of the destructive encounters happen, because the population rises steeply as size falls while the energy required to destroy a satellite does not rise nearly as fast. A one-centimetre aluminium fragment at ten kilometres a second arrives with the kinetic energy of a small car at motorway speed, concentrated into a square centimetre.

So the catalogue-based arithmetic above understates the hazard to any individual spacecraft and correctly states the hazard to the population, and the two are different questions. The cascade is driven by the objects large enough to be counted, because only those carry enough mass to make a new generation of fragments; the risk to a working satellite is dominated by objects nobody can count at all.

Each of those moves the crossing. None of them changes the fact that there is one, because a quadratic overtakes a linear regardless of the constants.

The two events that made it a policy question

The arithmetic was published in 1978 and treated as a long-term concern for thirty years. Two events in fourteen months changed that.

In January 2007 China destroyed its own Fengyun-1C weather satellite with a kinetic interceptor at 865 kilometres altitude. The test produced more than 3,000 trackable fragments — at the time, the largest single debris-generating event in history — deposited squarely in the most populated shell, with lifetimes of centuries.

In February 2009 the operational communications satellite Iridium 33 and the derelict Cosmos 2251 collided at 789 kilometres, closing at 11.7 km/s. It was the first accidental collision between two intact satellites, it produced about 2,300 more trackable fragments, and it happened despite conjunction warnings having been issued: the predicted miss distance was several hundred metres, well inside the uncertainty of the prediction and well outside what anybody manoeuvres for.

Between them the two events increased the tracked population in low orbit by roughly a third. The arithmetic did not change and the initial condition did, and the debris from both is still there.

The population peaks at 825 km, and so does the risk. Tracked objects per 25-kilometre shell against altitude, with the collision rate on a single object computed from each bin as nσv — a gas-kinetic rate, not an orbital calculation. The cross-section is not a satellite's area but an effective one, 3698 square centimetres, fixed by requiring the whole tracked population to produce the one catastrophic collision every 8 years that is observed — most tracked objects are fragments, and two fragments meeting make nothing new. The distribution is not smooth and its shape is history: the peak of 3,010 objects near 825 km is four decades of launches into sun-synchronous orbit plus the debris of two deliberate destructions, and the second rise past 1,300 km is the Soviet-era navigation constellation. At the peak one such object waits 46,409 years between strikes — which sounds safe until it is multiplied by the 3,010 objects sharing that shell, giving one collision every 31 years among them, and by the 23,680 in the whole of low orbit, giving one every 8. The rate on one object is reassuring and the rate on the population is not, and they are the same number.
Fig. 7 The distribution as those events left it. The peak near 825 kilometres carries the fragments of both — Fengyun-1C’s at 865, Iridium–Cosmos’s at 789 — and the lifetime at those altitudes is measured in centuries, so the peak is not a transient. A single event moved a shell’s population by a substantial fraction of its own value, which is the practical meaning of a system near a threshold: the state depends on individual events rather than on trends.

A probability that falls when the knowledge gets worse

Conjunction assessment deserves a section of its own, because the quantity it produces behaves in a way that catches people out and that has led to real manoeuvres not being made.

The procedure is to propagate two objects’ state vectors with their covariances to the time of closest approach, and to integrate the probability that their combined hard-body volume is occupied. The output is a collision probability, and an operator manoeuvres when it exceeds a threshold — typically one in ten thousand.

The trap is that this probability is not monotonic in the danger. Hold the predicted miss distance fixed and let the position uncertainty grow: at first the probability rises, because the uncertainty ellipsoid begins to cover the other object. Past a point it falls again, because the same fixed probability mass is being spread over a larger and larger volume, and the fraction of it inside the hard-body radius shrinks.

So a badly tracked object can produce a lower computed probability than a well-tracked one at the same miss distance. That is probability dilution, and it is perverse in exactly the way it sounds: the less that is known about a conjunction, the safer it can be made to look, and the remedy — obtaining more tracking data — can raise the number that decides whether to act.

It is not an artefact to be corrected away, because the probability is a correct answer to the question asked. It is the question that is wrong: an operator wants to know the risk given what could be learned, not given what happens to be known. The practical response is to treat a large uncertainty as itself a trigger, and to require that a conjunction be re-tracked rather than merely re-computed before it is dismissed.

The Iridium–Cosmos event sits inside this. The conjunction was screened, the miss distance was predicted at several hundred metres, and the computed probability was not remarkable — because the covariance on a derelict tracked routinely is large, and a large covariance dilutes.

What can actually be done

Four things, and they attack different terms in the same expression.

Reduce NN by not adding to it. Passivating spent stages so they do not explode, deorbiting satellites at end of life, and choosing disposal orbits below the long-lifetime band. This lowers the production term quadratically and is by far the cheapest intervention.

Reduce τ\tau by adding drag. Deployable sails and tethers that raise a derelict’s area-to-mass ratio move it down the lifetime curve. A tenfold increase in ballistic coefficient at 800 kilometres cuts the lifetime by roughly the same factor, which moves the critical population up by the same factor.

Reduce NN by removal. Active debris removal — capturing and deorbiting existing large derelicts — attacks the population directly, and the modelling suggests removing five to ten of the largest objects a year would be enough to stabilise the worst shell. Nothing of the kind has yet been done at scale, and the difficulty is legal and financial rather than technical: the objects that most need removing belong to somebody.

Reduce σv\sigma v by manoeuvring. Conjunction assessment and avoidance manoeuvres reduce the rate for the small fraction of the population that can manoeuvre, and not at all for the rest. It protects individual satellites and does nothing for the shell.

The arithmetic run on a constellation

The population that the threshold argument was written for has changed in kind, and it is worth putting the new numbers through the same expression rather than arguing about them qualitatively.

A constellation of ten thousand satellites at 550 kilometres is a number density about forty times the tracked density in the worst shell today, and the N2N^2 term responds accordingly. What saves it is the other side of the ledger: the lifetime at 550 kilometres is a few years rather than a millennium, so the removal term is larger by a factor of several hundred. Multiply both through and the shell is comfortably below its own critical population — provided the satellites are manoeuvring, and provided that when they stop they come down.

Both provisos are doing an enormous amount of work. A constellation satellite is stable against the cascade because it is actively flown, so the relevant failure rate is not the collision rate but the rate at which vehicles are lost while still in orbit. At a failure rate of one per cent and a fleet of ten thousand, a hundred uncontrolled objects are present at any time, and each is a full-cross-section target with years of decay ahead of it.

The second consequence is one the gas-kinetic model does not contain at all. Avoidance manoeuvres scale with the number of pairs, so the operational burden grows as the square of the population even where the collision rate does not — and a constellation that must screen and act on tens of thousands of conjunctions a week is relying on automation whose failure modes are correlated across the fleet in a way that two independent satellites’ are not.

A shell can be dynamically stable and operationally fragile at the same time. The threshold argument answers a question about the population’s evolution over centuries; whether a specific altitude is usable next decade is a question about failure rates, disposal compliance and automation, none of which appears in NcritN_{\rm crit}.

A year at 400 km and 240 years at 900. Orbital lifetime against starting altitude for a circular orbit, integrated from da/dt = −ρav/β with a ballistic coefficient of 20 kg/m² and a piecewise-exponential fit to the 1976 standard atmosphere. Only the density profile is tabulated; the decay is computed. The curve rises by a factor of 1400 between 400 and 900 kilometres — 2 months against 240 years — because the density falls by four decades across that span while nothing else in the expression changes much. That single ratio is why an altitude either cleans itself or does not. Below about 600 km a fragment is gone before it can find anything; above 800 it is there for centuries, and the twenty-five-year disposal rule is a statement about which side of this curve an operator is required to leave the vehicle on.
Fig. 8 Orbital lifetime again, for a much lighter object per unit area — twenty kilograms per square metre rather than a hundred, which is roughly a fragment rather than a satellite. Every lifetime divides by five, so a fragment at 900 kilometres decays in 240 years against a satellite’s 1,195. The removal term is not one number for a shell, and the fragments the cascade actually makes are removed several times faster than the intact objects that make them — which softens the threshold in the right direction and is one of the reasons the simple model is a bound rather than a forecast.

Where the model stops

The largest limitation is not in any of the terms but in the framing.

The gas-kinetic model treats the population as stationary. It is not: launch rates have risen by more than an order of magnitude since 2018, driven by constellations of thousands of satellites, and the shells being filled are at 500 to 600 kilometres — below the worst band, with lifetimes of years rather than centuries. Whether that is a stabilising development or a destabilising one depends on the disposal reliability of the constellations, and that is a number about spacecraft engineering rather than about orbits.

The second limitation is that a threshold in a mean rate is not a prediction about a trajectory. Above the critical density the expected population grows, and the growth timescale is centuries; individual shells can sit above the threshold for a very long time without anything visibly happening, and then produce a cascade after a single event. That is the ordinary behaviour of a system whose driving term is quadratic and whose events are rare, and it is why the arithmetic is argued about rather than observed.

Where this ladder goes next

This rung establishes the gas-kinetic rate, the quadratic-against-linear structure, and the altitude dependence that decides which side of the threshold a shell is on.

Above it lies the evolution rather than the threshold: the coupled equations for the population in each shell with sources, sinks and transfer between shells, which is what the operational models solve and which produces a population history rather than a stability statement.

Beside it lies conjunction analysis, which is the opposite problem — a specific pair, a specific time, and a probability of collision computed from two covariance ellipsoids rather than from a density. The two disciplines share nothing but the objects.

And below it lies the measurement the whole subject rests on: the catalogue itself, maintained by radar and optical tracking, whose completeness limit is the ten centimetres that decides what is counted and what merely happens.

Where this goes next

Down by 280 km, and faster by 164 m/s. A circular orbit at 400 km with a ballistic coefficient of 100 kg/m², integrated down to 120 km through a tabulated atmosphere at solar minimum and solar maximum. At solar min it takes 1.2 years; at solar max it takes 147 days — a factor of 2.9 for the same satellite in the same orbit, decided by an eleven-year cycle nobody controls. The rising curves are the orbital speed on the right-hand scale, and they are the point: the drag force is opposite the motion and takes energy out, and the body goes faster, from 7673 to 7836 m/s. There is no contradiction in it. The specific energy is −μ/2a, so removing energy shrinks a, and the circular speed √(μ/a) rises when a falls; the kinetic energy gained is exactly half the potential energy lost, and the other half is what the air took. Every point on every curve was integrated from da/dt = −(ρ/β)√(μa), and the speed at each point is √(μ/a) at that point rather than a separate model. An orbit that speeds up as it is slowed down Drag takes energy out of a satellite and the satellite goes faster. There is no paradox in it, only a sign — and the same sign makes a re-entry date a space-weather forecast rather than an orbital computation, which is why Skylab was predicted for 1983 and came down in 1979. Where the fuel goes, and it is not where a satellite points. Left, the orbit pole of a geostationary satellite, in degrees from the Earth's. The Sun and the Moon between them carry it round a circle of radius 7.4° in 53 years, and a satellite launched into the equatorial plane starts on the rim of that circle rather than at its centre — so its inclination climbs from zero at 0.88° a year, reaches 14.8° after 27 years, and comes back. Right, what holding it costs. A plane change of 0.88° at 3.07 km/s is 47.1 m/s a year; holding the longitude against the equatorial bulge, computed from the same resonant term that makes the longitude a pendulum, is 1.8 m/s a year. North–south is 96% of the budget, and a satellite that gives up on it does not fail — it starts tracing a figure of eight on the sky 1.8° tall in the first year, which a fixed dish cannot follow and a steerable one can. Retiring at the end of the propellant is therefore a choice about which service ends first. The orbit that has to be paid for every year A geostationary satellite is not in equilibrium in any direction. The Sun and Moon tilt its plane by 0.85 degrees a year, the Earth's equatorial ellipticity makes two longitudes stable and two unstable, and the end of a satellite's life is the end of its propellant. three revolutions, on a turning Earth. The ground track of a circular orbit at 420 km and 51.64° inclination, over 3 revolutions, on an equirectangular graticule. The latitude is a sine wave bounded by ±51.64° exactly — sin φ = sin i sin u, so the inclination is the highest latitude the orbit ever passes over, and it is reached twice per revolution. Each successive pass is displaced west by (ω⊕ − Ω̇) × 92.90 min = 23.61°, of which 0.32° is the orbital plane's own regression and the rest is the planet turning underneath: the vehicle comes back to nearly the same place in inertial space and the place has moved. The period used is the nodal one, 92.899 min against the Keplerian 92.970: J₂ makes the two differ by 4.31 s, which is 0.018° of walk per revolution and 102° in a year — the difference between a repeat track and a track that used to repeat. The map is equirectangular and therefore wrong about area everywhere; what it is right about is longitude difference, which is the whole of what this figure measures. The line under a satellite A ground track is an orbit seen from a frame that is turning, so every pass lands west of the last one. The track closes only when two periods are commensurable — which turns "look at the same place every day" into a condition on the altitude.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

The 8 of 15 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

Ballistic coefficientCollision cross-sectionConjunction analysisCritical densityFragmentationKessler syndromeMean free pathNumber densityOrbital debrisOrbital lifetimePost-mission disposalRelative velocity