A satellite that drifts to one of two longitudes

A geostationary satellite is in a resonance: its orbital period equals the Earth’s rotation period, so it stays above one longitude. That is what makes the orbit useful and it is also what makes it sensitive to a part of the Earth’s gravity field that no other orbit notices.

For any other satellite, the longitude-dependent structure of the Earth’s field averages away — the satellite passes over every longitude in turn and the bumps cancel. A geostationary satellite never moves relative to the Earth, so whatever bump it is sitting over acts on it in the same direction forever.

An orbit that has to be paid for every year is the general statement; this essay is about the term that makes the east–west half of that bill unavoidable, and about what it does to the satellites that stop paying.

Where the two minima come from #

The Earth’s gravity field is expanded in spherical harmonics, and the terms that depend on longitude are the tesseral ones. The largest is $J_{22}$, whose amplitude corresponds to an equatorial ellipticity of about one part in $10^5$ — some seventy metres of difference between the equatorial radii along two perpendicular directions.

A term that varies as $\cos 2\lambda$ has two maxima and two minima around the equator, and the minima of the potential are the stable points. They sit near 75° east and 255° east — over the Indian Ocean and over the eastern Pacific — and the unstable maxima are ninety degrees away from them.

The physics is a pendulum. A satellite displaced from a stable longitude feels a restoring along-track acceleration, which changes its semi-major axis, which changes its period, which makes it drift back. The full cycle takes about two years for a small displacement and longer for a large one, exactly as a pendulum’s period lengthens with amplitude.

It is worth noticing which quantity the harmonic acts on. The along-track acceleration changes the satellite’s energy and therefore its semi-major axis, which by Kepler’s third law changes its period — and a period a fraction of a second different from a sidereal day is a longitude drift of a fraction of a degree a day. So the chain from a seventy-metre bump on the equator to a satellite leaving its slot runs through the harmonic law, and nothing about it is specific to spaceflight. Which component of a small force moves which element is the general machinery, and here it is the along-track component acting on the semi-major axis, which is the most efficient of the six couplings there are.

There is a detail of the geometry worth having, because it explains why the stable longitudes are where they are. The minima of the potential sit over the long axis of the equatorial ellipse — the directions in which the Earth bulges most — because a satellite there is closest to the excess mass and therefore deepest in the well. The long axis runs roughly through the Indian Ocean and the eastern Pacific, and it is not aligned with anything geologically obvious at the surface: the equatorial ellipticity is a bulk property of the mantle’s density distribution, dominated by the same deep structure that produces the largest features of the global geoid. So a communications satellite’s fuel bill is set by the arrangement of density anomalies a couple of thousand kilometres below the Earth’s surface.

What it costs #

The peak along-track acceleration is about $1.7\times10^{-8}$ metres per second squared, which sounds negligible and accumulates. Over a year it is half a metre a second of velocity change; the operational east–west budget for a geostationary satellite is a couple of metres a second a year once the correction cycle and the margin are included.

That is not the largest term in a geostationary budget. North–south station-keeping — fighting the lunisolar torque that tips the orbital plane — costs about fifty metres a second a year, two orders of magnitude more. But the two are different in kind: the north–south budget can be abandoned, at the cost of the satellite’s ground track becoming a figure-of-eight rather than a point, and many satellites do exactly that at the end of their lives. The east–west budget cannot, because abandoning it means drifting out of the slot and into somebody else’s.

The economics follow directly. A satellite’s operational life is set by its propellant, and a geostationary satellite typically carries a decade and a half of station-keeping. The value of the slot, the mass of the propellant and the cost of the launch are all connected by the numbers in this essay.

There is a second, subtler cost that does not appear in a propellant budget. A slot in the geostationary ring is an internationally allocated resource, and the allocation specifies a longitude and a tolerance — typically a tenth of a degree in each direction, because the ring is crowded and adjacent satellites are separated by fractions of a degree. Holding a tenth of a degree against an acceleration that would move a satellite tens of degrees in a year requires a correction every week or two, and a satellite that misses a few corrections is not merely off-station but is a hazard to its neighbours. The tolerance rather than the acceleration is what sets the operational cadence.

The comparison with the other perturbations a geostationary satellite feels is worth making explicit, because it decides what a mission plans for. Solar radiation pressure produces an acceleration of order $10^{-7}$ metres per second squared for a modern satellite with large panels — several times the tesseral term — but it is periodic over a year and largely averages away in longitude, contributing instead a forced eccentricity of a few times $10^{-4}$ that has to be controlled separately. The lunisolar torque is larger still and acts on the inclination. So the three main perturbations act on three different elements, and the station-keeping strategy is three loosely coupled control problems rather than one.

Where the uncontrolled satellites are #

The stable longitudes have a consequence that nobody planned and everybody has to live with.

A satellite that runs out of fuel, or fails, stops holding its slot. It drifts towards the nearer stable longitude, arrives, and librates about it — sweeping through a range of longitudes hundreds of degrees wide over a period of years, at a rate of up to a degree a day.

So the geostationary ring’s uncontrolled population is not uniformly distributed. It is concentrated near the two stable longitudes, and it moves. An operational satellite at any longitude will be passed by drifting objects, and the encounter rate is highest near the minima.

That is why the accepted practice at end of life is to raise the orbit a few hundred kilometres into a graveyard region above the ring, where the object is out of the way and — because it is no longer resonant — no longer accumulates a longitude drift. The manoeuvre costs about eleven metres a second, and it has to be budgeted before the propellant runs out, which requires knowing how much is left in a tank that has no gauge.

There is a further consequence for anybody trying to catalogue the ring. A librating object’s longitude is a slow function of time with a turning point at each end of its swing, so it spends most of its time near the extremes of its libration and passes quickly through the middle. The observed distribution of uncontrolled objects therefore has peaks at the turning points rather than at the stable longitudes themselves — a purely kinematic effect, the same one that puts a planet near the ends of its retrograde loop for longer than in the middle. Reading that distribution as evidence about where satellites were abandoned would be a mistake.

What was actually measured #

The equatorial ellipticity was known before there were satellites to be perturbed by it, and the satellites measured it far better.

From the drift itself. The most direct measurement is to place a satellite at a known longitude, stop controlling it, and watch. The along-track acceleration is read off the second derivative of the longitude, and the whole potential can be mapped by doing it at several longitudes. Early geostationary satellites did this involuntarily and the results refined the coefficient.

From the global field. Modern values come from dedicated gravity missions, which measure the field to hundreds of harmonic degrees from low orbit. At geostationary altitude nearly all of that is invisible, so the mission’s value for one low-degree coefficient is what the operators use.

And the libration is observed. Uncontrolled objects in the ring are tracked, their longitude histories are known over decades, and they librate at the predicted periods with the predicted amplitudes. That is a check of the potential’s shape rather than only its amplitude.

Be explicit about what a station-keeping manoeuvre actually does, because the mechanism is not obvious. The satellite is not pushed sideways to hold it in place. It is given a small along-track burn that changes its semi-major axis by a few tens of metres, which changes its period by a fraction of a second, which makes it drift slowly in the opposite direction to the tesseral acceleration. The satellite is then allowed to drift back across its slot, and the cycle repeats every week or two. What is being controlled is a drift rate rather than a position, and the box the satellite is held inside is a region in longitude and longitude-rate rather than a region in space — which is why the operational quantity is a “drift box” rather than a station.

Where the picture stops #

Three of them, and the second is the operational one.

The pendulum picture assumes a small perturbation. For a satellite librating over a wide range of longitude the potential is not quadratic and the period depends on the amplitude. Objects with libration amplitudes of a hundred degrees or more take five to seven years for a cycle, and their motion has to be integrated rather than described.

Solar radiation pressure competes. A modern satellite has a large area-to-mass ratio because of its solar panels, and radiation pressure produces an acceleration comparable to the tesseral one. For high-area objects — a defunct satellite with deployed panels, a piece of insulation — radiation pressure dominates and the longitude behaviour is quite different, including a coupling to the eccentricity that can grow it substantially.

And the resonance is not perfect. A satellite whose period is slightly off geostationary drifts steadily rather than librating, and the boundary between drifting and librating is a separatrix of the same kind that appears in every resonance in this collection. Objects near it behave unpredictably over decades, which is exactly the regime an abandoned satellite finds itself in.

A fourth belongs with them because it is the one that is changing. The geostationary ring now contains several hundred operational satellites and rather more uncontrolled objects, and the conjunction rate between them is high enough that operators receive warnings regularly and manoeuvre in response. Those manoeuvres cost propellant that was budgeted for station-keeping, so the population density in the ring has become a term in every satellite’s operational lifetime — a feedback of exactly the kind a collision rate that needs no collision to measure describes for low orbit, arriving in a regime where there is no drag to clear the debris and the objects stay forever.

Why a resonance turns a small term into a large one #

The general point is one this collection meets in several forms, and it is worth stating in its cleanest.

A perturbation matters in proportion to how long it acts in one direction. A term that averages to zero over an orbit contributes almost nothing however large it is; a term that does not average contributes in proportion to the time, however small it is. The tesseral harmonics of the Earth are a hundred thousand times smaller than its monopole and a hundred times smaller than its oblateness, and for one particular orbit they are the dominant perturbation, because that orbit does not average them.

That is the same statement as the difference between secular and short-period terms in a planetary theory, and the same as the reason a resonance clears a gap in one place and locks a moon in another. In each case a resonance converts an oscillating perturbation into a constant one, and a constant perturbation integrates.

The practical corollary for spaceflight is that the important perturbations on an orbit cannot be listed by size. They have to be listed by whether the orbit’s geometry lets them accumulate — which is why the equations that say which component moves which element are the first thing a mission designer reaches for and the magnitude of the field’s harmonics is the second.

A parting observation about how this behaves compared with the low-orbit case. A satellite in low orbit that stops manoeuvring re-enters within years to decades, because the atmosphere removes it; the problem there is transient. A satellite in geostationary orbit that stops manoeuvring stays for millions of years, because there is nothing to remove it. The two regimes have completely different disposal problems and completely different time constants, and the whole of the difference is whether there is any atmosphere left at that altitude. The line under a satellite traces the same geometry in both cases; what differs is what eventually happens to the satellite drawing it.

End on a comparison that puts the term in perspective. The Earth’s monopole holds a geostationary satellite in orbit; its oblateness is a thousandth of that and produces no effect on a satellite in the equatorial plane at all, because there is nothing to regress; and its equatorial ellipticity is a hundred-thousandth of the monopole and is the dominant perturbation. Three terms, five orders of magnitude apart, and the smallest one is the one the operators think about daily. That inversion is entirely a consequence of the orbit’s geometry, and it is the single most useful thing to know about perturbations: what matters is not how large a term is but what it does to the elements the mission cares about, and for how long it does it in the same direction.

There is one more thing the two minima do that is worth recording, because it is a slow-motion consequence nobody designed. Since uncontrolled satellites accumulate at the stable longitudes and librate about them with periods of a couple of years, the graveyard population is not spread uniformly around the ring — it is concentrated at two longitudes and passes through them twice per libration. Every operational satellite parked near a stable point therefore sees a higher flux of derelict objects drifting past than one parked elsewhere, and the drift is slow, which means the encounter velocities are low and the relative geometry is nearly repeatable. Low relative velocity makes a collision less energetic and makes it far easier to predict and avoid; the concentration makes there be more of them to predict. Whether the net effect is favourable is genuinely unclear, and the arithmetic depends on the population’s growth rate rather than on anything in the dynamics. The critical altitude below which drag clears debris has no analogue here: nothing at all removes an object from geostationary orbit.

One more number puts the whole effect in perspective. The equatorial ellipticity responsible for all of this is a departure from circularity of about seventy metres in a radius of six and a third million — one part in ninety thousand. Everything in this essay follows from that, and it follows only because the orbit’s period matches the rotation exactly enough for the small term to accumulate instead of averaging away.

Where the ladder goes next #

The rung directly above is the north–south budget: the lunisolar torque that tips a geostationary plane, why it costs a hundred times more, and what an inclined-orbit satellite gives up. The one above that is the end-of-life problem — how much propellant is left in a tank with no gauge, and what happens to the ring when the disposal manoeuvre is not affordable.