The orbit that has to be paid for every year

A geostationary satellite is described as parked. It sits over a fixed longitude, a dish on the ground points at it and never moves again, and the arrangement has the look of something that has been solved.

Nothing about it is at rest. The orbit is a compromise between four perturbations that pull in different directions, none of which is negligible, and holding the satellite where the dish expects it costs propellant continuously. When the propellant runs out the spacecraft is still perfectly functional and its mission is over, which is a striking way for an engineering system to end.

Why the plane will not stay put #

The Earth’s equator is inclined 23.4° to the ecliptic, so the Sun’s pull on a satellite in the equatorial plane has a component out of that plane for most of the year. The Moon’s orbit is inclined about 5° to the ecliptic, so it does the same and with a different period.

Averaged over a year, the effect of both is a torque that carries the orbit’s angular momentum vector — its pole — around a fixed direction, in the same way that the lunisolar torque on the Earth’s own equatorial bulge carries the rotation pole around. The direction it circles is not the Earth’s pole; it is a compromise between the Earth’s pole and the ecliptic pole, lying about 7.4° from the first, and the circuit takes about 53 years.

The consequence is geometric and unavoidable. A satellite launched into the equatorial plane has its pole at the Earth’s pole, which is a point 7.4° from the centre of that circle — that is, on its rim. Carried round the rim, its inclination is the distance from the Earth’s pole, which starts at zero, rises to twice 7.4°, and returns.

An inclination of zero is not a stable state; it is the worst possible starting point on a circle it is being carried round.

The longitude is a pendulum #

The east–west problem is quite different in character, and it is the more interesting one.

The Earth’s equator is not a circle. Its ellipticity is small — the equatorial radii differ by about 70 metres — but a geostationary satellite orbits at exactly the rotation rate, so the perturbation from that asymmetry is resonant: the same small force acts in the same direction on every revolution instead of averaging out. The result is a longitude acceleration

$$\ddot\lambda = 18\,n^2 J_{22}\left(\frac{R_\oplus}{a}\right)^2\sin 2(\lambda - \lambda_{22}),$$

peaking at $1.7\times10^{-3}$ degrees per day squared.

The distinction between the two stable points and the two unstable ones is the whole of east–west station-keeping. A satellite stationed near a stable longitude is being pulled back towards it and needs very little correction; one stationed near an unstable longitude is being pushed away and needs the most. Most of the commercially desirable longitudes — over the Atlantic, over Europe, over the Americas — are neither, and the cost is somewhere between.

The box, and what it is a statement about #

An operator holds a satellite within a station-keeping box, quoted as $\pm 0.05°$ or $\pm 0.1°$ in latitude and longitude. That box is not a tolerance in the engineering sense; it is a statement about how much of each perturbation is being tolerated before a correction is made.

The latitude half-width is the inclination allowed to accumulate before a north–south manoeuvre. At 0.88° a year, holding $\pm 0.05°$ means a manoeuvre roughly every three weeks; holding $\pm 0.1°$ halves the manoeuvre count and doubles the excursion. The total Δv is the same either way — it is set by the drift rate, not by the box — so a wider box saves operational effort and not fuel.

The longitude half-width is the amplitude of the libration allowed before an east–west burn, and here the box does affect the cost, weakly: a satellite allowed to drift further spends more time away from its nominal longitude but is corrected less often, and since the acceleration is quadratic in the excursion the total cost falls slightly with a wider box.

Only a few tens of metres of pointing on the ground correspond to a tenth of a degree at geostationary distance, so the boxes are set by the beamwidth of the dishes on the ground rather than by anything about the spacecraft.

What was actually measured #

The numbers in this essay are all operational rather than theoretical, and they are among the best-checked in astrodynamics because thousands of spacecraft have paid them.

North–south: 45 to 51 m/s per year. The variation is real and it is the Moon’s node. The lunar orbital plane precesses with an 18.6-year period, so the combined lunisolar torque varies over that cycle and the annual inclination drift runs between about 0.75 and 0.95 degrees. An operator budgeting for a fifteen-year mission has to take the high end.

East–west: 0.5 to 2 m/s per year, depending on longitude, plus a comparable amount for eccentricity control. Solar radiation pressure drives the eccentricity round an annual cycle whose amplitude depends on the area-to-mass ratio; a satellite with large solar arrays and low mass — which is every modern communications satellite — sees an eccentricity excursion of a few times $10^{-4}$, corresponding to a daily longitude oscillation of a few hundredths of a degree.

Total, over a fifteen-year life: about 750 m/s. For a spacecraft with a mass of five tonnes and an apogee engine exhaust velocity of 3 km/s, that is 1.1 tonnes of propellant carried for station-keeping alone — a fifth of the launch mass, entirely to stay still.

The end, which is a manoeuvre #

A satellite that simply stops is a hazard, because it librates into one of the two stable longitudes and joins the population already there. The IADC guideline, and now the licence condition in most jurisdictions, is that a geostationary spacecraft must be raised at least 235 km + $1000\,C_R A/m$ above the geostationary ring before it is passivated — a graveyard orbit high enough that solar radiation pressure will not bring it back down through the ring.

That manoeuvre costs about 11 m/s, or roughly three months of station-keeping propellant, and it has to be performed while enough propellant remains to perform it. Estimating the remaining propellant in a fifteen-year-old spacecraft, from tank pressures and thruster-firing histories, is one of the genuinely uncertain quantities in the business, and the usual practice is to retire with a margin — which is to say, to end a working mission early because the fuel gauge cannot be trusted.

Four perturbations, one table #

Setting them beside each other is the fastest way to see why the budget is shaped as it is.

The last row is the one worth explaining. $J_2$ is by far the largest perturbation on a low orbit and it costs a geostationary satellite nothing at all, because its effect at synchronous radius is a slow rotation of the node within a plane that is already being defined by the operator. The nominal radius of 42,164.17 km already includes the $J_2$ correction to the period, so the perturbation has been absorbed into what “geostationary” means rather than being fought.

A perturbation is only a cost if it moves the spacecraft relative to what the mission needs, and three of these four do.

The same reading explains why the inclination is fought and the longitude libration is merely managed. A drifting inclination breaks the geometry the ground segment was built around — a fixed dish sees the satellite move — and there is no cheap way to accommodate it. A librating longitude, by contrast, is a slow oscillation about a point, and a satellite can simply be stationed at a stable longitude and left to swing within its box, which is why the two graveyard longitudes are also the two cheapest places to operate.

The ring is a scarce resource, and the scarcity is not gravitational #

Everything so far treats a longitude as a place a satellite can be put. Whether it may be is a separate question, settled by treaty rather than by dynamics, and the reason is interference.

Two satellites at nearby longitudes are seen from the ground at nearby angles, and a dish has a finite beamwidth. A three-metre antenna at C band has a main lobe around two degrees across, so a satellite two degrees away is inside the beam and its transmissions arrive alongside the wanted signal. The spacing of the ring is therefore set by the size of the receiving antennas rather than by anything about orbits, and the historic figure of two degrees comes from exactly that calculation.

Two degrees of spacing gives 180 positions in a full circle, and the useful arc is much smaller than a circle for any given service area — a satellite serving Europe has to be visible from Europe at a workable elevation angle, which confines it to perhaps 70 degrees of longitude. The result is that the commercially valuable arc holds a few dozen slots, and they are allocated by the International Telecommunication Union through a filing and coordination process that has produced its own economy of speculative registrations.

The station-keeping box connects to this directly. A satellite held to $\pm0.05°$ occupies a tenth of a degree; one that has stopped correcting its longitude wanders through several degrees and through its neighbours’ slots. So the box is not only about keeping a dish pointed — it is the condition on which the slot was granted, and a satellite that cannot hold it has to leave.

Steering the eccentricity rather than fighting it #

Solar radiation pressure entered the budget above as a one-metre-a-second annual nuisance, and the way it is handled is a good example of working with a perturbation instead of against it.

The pressure acts in the anti-solar direction, so as the Sun goes round the sky over a year, the force direction rotates with it. Its effect is to push the orbit’s eccentricity vector — the vector pointing towards perigee with length $e$ — around a circle, once a year, with a radius set by the spacecraft’s area-to-mass ratio. For a modern platform with large arrays that radius corresponds to $e \approx 3\times10^{-4}$, which is a daily east–west oscillation of a few hundredths of a degree.

Fighting that means burning to keep $e$ near zero, continuously, all year. The alternative — the sun-pointing perigee strategy — is to place the eccentricity vector deliberately so that the annual circle traced by the radiation pressure is centred on the origin rather than starting there. The eccentricity then goes round a closed loop of fixed radius, never growing, and the longitude oscillation it produces is a constant known amplitude that the box is sized to accommodate.

Nothing is corrected and nothing accumulates. The manoeuvres that remain are the east–west drift corrections, which are then timed so that each one also places the eccentricity vector where the coming year’s solar cycle wants it. One burn, two purposes, and a perturbation converted from a cost into a constraint on when the burn is scheduled.

Several satellites in one box #

The scarcity above has an obvious response — put more than one spacecraft at the same longitude — and it creates a collision problem inside a volume a few tens of kilometres across.

Two satellites in the same box cannot simply be flown to the same nominal position, because their errors are independent and their separation would be uncontrolled. What is done instead is eccentricity and inclination vector separation: the two spacecraft are given eccentricity vectors pointing in opposite directions, or inclination vectors similarly opposed, so that whenever one is at the east side of the box the other is at the west, or one is north when the other is south.

The geometry is arranged so that the two separations are never simultaneously zero. A satellite whose radial offset vanishes has, at that moment, a maximal cross-track offset, and the minimum distance between the two over a day is a few kilometres rather than a few metres — held by the shape of the two orbits rather than by any active avoidance.

Four or five satellites have been colocated this way at a single longitude. The scheme’s cost is that it consumes part of the box: the separations use up the tolerance that would otherwise absorb station-keeping error, which means tighter control on each spacecraft and more frequent manoeuvres. It is the ring’s scarcity being paid for in propellant.

There is a limit to how far the trick scales. Each additional spacecraft needs its own pair of separations, and the available room in the box is fixed, so the practical ceiling is around four or five — beyond which the required control accuracy on each satellite costs more propellant than the shared slot is worth. The arithmetic that makes colocation attractive is the same arithmetic that stops it.

None of this applies to the constellations in low orbit, where the satellites are numerous, individually cheap and short-lived, and the separation is maintained by continuous propulsion and active avoidance rather than by orbit design. The geostationary ring is the last place in spaceflight where a single spacecraft’s position is a scarce and individually negotiated asset, and the practices in this essay are what that scarcity produced.

It is also the one regime where a satellite’s operators can say precisely what its remaining life is, because the limiting consumable is measured in metres per second and the drain is known to a few per cent a year. Everywhere else in spaceflight a mission ends when something breaks; here it ends on a schedule that was written before launch.

Two practical matters follow from all of this and are worth stating, because they are where the arithmetic meets an operator’s decisions.

The first is that the two station-keeping budgets are not comparable in size. East–west keeping, against the triaxiality that drives the longitude pendulum, costs of order two metres a second a year. North–south keeping, against the lunisolar torque on the plane, costs about fifty — twenty-five times as much, and it is the whole of the difference between a satellite’s fuelled lifetime and the fifty years its hardware might last. So a satellite near the end of its propellant is often taken out of north–south keeping and allowed to drift in inclination, reaching a degree or two over a few years. It is then still useful: a ground station with a steerable antenna can follow the resulting daily figure-of-eight, and the satellite is leased for services that can tolerate it. Inclined-orbit operation is the industry’s ordinary way of trading a ground-segment cost against an orbital one, and it exists because one of the two budgets is far larger than the other.

The second is that the disposal orbit is not a fixed altitude. The standard requires a satellite to be raised far enough above the ring that solar radiation pressure cannot bring it back, and since that pressure acts on area while gravity acts on mass, the required height depends on the vehicle’s area-to-mass ratio: a few hundred kilometres for a compact satellite, more for one with large arrays. The formula is written into the guidelines rather than a single number, and the manoeuvre costs about eleven metres a second — a few months of north–south keeping, spent at the end on the one manoeuvre that returns nothing.

Where the model stops #

The account above treats the four perturbations as independent, and at the level of a budget they are. They are not independent in detail.

The north–south manoeuvre changes the plane, which changes the resonant longitude acceleration slightly; the east–west manoeuvre changes the semi-major axis, which changes the drift rate; and both are performed by thrusters mounted on a spacecraft whose attitude is itself being controlled, so every burn has a component in a direction nobody wanted. Modern operations combine the two into a single manoeuvre with the thrust vector chosen to deliver both components, which saves propellant and complicates the analysis.

The 7.4° figure for the forced plane is also not a constant. It depends on the satellite’s own semi-major axis, and for orbits away from geostationary the balance between the Sun’s and the Moon’s contributions shifts — at low altitudes the Earth’s own oblateness dominates and the whole picture is different, which is what makes a sun-synchronous orbit possible at 800 km and impossible at 36,000.

One asymmetry runs under all four perturbations and is worth naming. Three of them — oblateness, triaxiality, lunisolar — are conservative: they move an element back and forth, and a satellite that could wait long enough would find its orbit returned. Solar radiation pressure is not, because the photons are absorbed and re-emitted rather than reflected elastically, and because the satellite’s own attitude changes what area it presents. So the eccentricity it drives is genuinely pumped rather than merely oscillated, which is why the strategy for it is to choose an orientation that makes the drift close on itself rather than to cancel it. The distinction between a perturbation that oscillates and one that accumulates decides whether a budget is a rate or a total, and it is the first question to ask of any term in the table above.

Where this ladder goes next #

This rung is the geostationary case, which is the one with the largest fleet and the clearest economics. The rungs above it generalise in two directions.

One is other stations. A spacecraft at Sun–Earth $L_2$ is at an unstable equilibrium and drifts off exponentially with an $e$-folding time of about three weeks, yet holds its position for a few metres per second a year — a thousandth of the geostationary cost, because the instability can be cancelled while it is still small rather than fought continuously. The tubes that make that point a gateway are the same structures that make it cheap to sit at, and the contrast between the two budgets is a good illustration of what an equilibrium is worth.

The other is what happens when the budget is not paid. The graveyard population, the inclined-orbit secondary market for ageing satellites, and the growing question of whether refuelling or robotic servicing changes the arithmetic — the first such docking, of MEV-1 to Intelsat 901 in 2020, added five years to a satellite that had run out of nothing but propellant.