Astronomy & Celestial Mechanics
the sky, computed
Spaceflight

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.

Assumes Station-keeping and Perturbations.

16 min read 5 figures Small perturbations, large consequencesEverything is a conic

The geostationary ring is a single circle, 35,786 kilometres above the equator, where an orbit’s period matches the Earth’s rotation and a satellite hangs over one longitude. It is the most valuable orbit there is and it cannot be cleaned: at that height there is no air to drag a dead satellite down, and anything left there stays for millions of years, drifting round the ring and settling into one of the two longitudes where the Earth’s uneven gravity traps it. The only thing that can be done with a geostationary satellite at the end of its life is to move it out of the way, a few hundred kilometres higher, and switch it off.

How much higher is written into international guidelines as a formula. The Inter-Agency Space Debris Coordination Committee recommends a minimum perigee increase of

ΔH=235 km+1000 CR Am,\Delta H = 235\ \mathrm{km} + 1000\,C_R\,\frac{A}{m},

where CRC_R is the satellite’s reflectivity coefficient, between one and two, and A/mA/m its area-to-mass ratio in square metres per kilogram. For a typical satellite the answer is a little under three hundred kilometres. The disposal reserve that height requires is held back from years of station-keeping and measured by a fuel gauge that is least accurate when it is needed. This essay takes the formula apart, term by term, because each term is a different piece of orbital mechanics, and the last one is the most surprising.

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.
Fig. 1 The minimum disposal height above the geostationary ring, broken into its three terms, for reflectivity times area-to-mass ratios from 0.005 to 0.05 square metres per kilogram. The first 200 kilometres is the protected zone; the next 35 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 sunlight raises every year. A first-principles derivation puts that last term at 942 kilometres per unit of the ratio for an orbit started circular — the guideline’s round thousand. Totals run from 240 to 285 kilometres.

Two hundred kilometres of protected zone

The first term is a definition. The protected region round the geostationary ring is a band 200 kilometres above and below it, and within fifteen degrees of latitude either side of the equator — a torus that holds every operational geostationary satellite and the room they need to manoeuvre. A satellite’s end-of-life orbit must never enter it.

That is a convention, not a law of physics, and the number is generous. Operational satellites are held within about 75 kilometres of the ring’s nominal radius by their station-keeping, and the width of the zone is meant to cover the manoeuvring of satellites moving between longitudes, which do so by raising or lowering their orbits slightly to drift east or west. Two hundred kilometres is enough that a satellite parked at 250 kilometres above the ring drifts round it, relative to the operational satellites, at a rate of a few degrees a day — slow, but never stationary relative to anything it could hit.

The geostationary rule is the high-altitude half of a pair. In low orbit the opposite is done: a satellite at the end of its life is lowered, not raised, so that the atmosphere will bring it down within a set time — twenty-five years under the long-standing guideline, five years under the rule the United States adopted for its licensed satellites in 2022. Low orbit cleans itself through drag, and disposal there means speeding the cleaning up; the geostationary ring has no such process, and disposal there means moving the problem somewhere it will do no harm. One region is disposed of by time, the other by distance, and the arithmetic of each is set by the force that is present — air in the one case, sunlight in the other.

Thirty-five kilometres of gravity

The second term is the largest amount by which the long-term gravitational perturbations can lower the perigee of a disposal orbit over its whole future, with nobody there to correct it.

Three effects contribute. The Moon and the Sun pull the orbit’s plane out of the equator, driving its inclination up to about fifteen degrees and back over a cycle of about fifty-three years. They also perturb its eccentricity, by small amounts that depend on where the Moon and Sun are relative to the orbit’s perigee. And the Earth’s equatorial ellipticity, the same slightly non-circular equator that makes geostationary satellites drift towards two stable longitudes, perturbs the orbit’s shape a little. Integrations of disposal orbits over centuries, with all three included, show the perigee wandering by up to about thirty-five kilometres from where it was left. The second term is an envelope, computed once, that does not depend on the satellite, because gravity pulls on every satellite in the same way whatever its shape or mass.

A term that depends on the satellite: sunlight

The third term is proportional to the satellite’s area-to-mass ratio, which means it depends on what the satellite is made of — and the force that depends on area rather than mass is radiation pressure. Sunlight carries momentum, and a surface absorbing or reflecting it is pushed, with a pressure at the Earth’s distance of 4.56×10−64.56\times10^{-6} newtons per square metre. On a satellite with 50 square metres of solar panels and a mass of three tonnes that is an acceleration of a hundred billionths of a metre per second squared — tiny, and applied continuously, from one direction, for years.

Its effect on an orbit is not what intuition expects. A constant push from one direction does not move a circular orbit bodily away from the Sun. Averaged round each orbit, it does no net work and leaves the semi-major axis — the orbit’s size and energy — unchanged. What it changes is the eccentricity. On the half of the orbit where the satellite moves towards the Sun the push slows it; on the other half it speeds it up; and the net effect over an orbit is to stretch the orbit, raising the side where the satellite moves away from the Sun and lowering the other, a change in the eccentricity vector at right angles to the direction of the Sun.

The Sun does not stay in one direction. It moves round the sky once a year, and the direction of the push turns with it. The eccentricity vector is therefore carried round a circle, once a year, following the Sun at right angles, with a radius set by the balance between the push and the Sun’s angular speed:

es=3f2v ω⊙≈0.011 CR Am,e_s = \frac{3f}{2v\,\omega_\odot} \approx 0.011\,C_R\,\frac{A}{m},

with ff the radiation acceleration, vv the orbital speed and ω⊙\omega_\odot the Sun’s annual angular rate.

The eccentricity vector of an orbit above geostationary height, carried round by sunlight in a year. The eccentricity vector — pointing to perigee, with length equal to the eccentricity — of a near-geostationary orbit over one year of solar radiation pressure, for area-to-mass ratios times reflectivity of 0.01, 0.02, 0.04 square metres per kilogram, started circular, at the origin. The vector moves on a circle of radius 0.011 × Cᵣ·A/m, following the Sun round the sky once a year, and the orbit's eccentricity is the vector's length. Started at the origin it reaches twice the radius half a year later: maximum eccentricities of 2.23·10⁻⁴, 4.47·10⁻⁴, 8.94·10⁻⁴, which at 42,164 kilometres dip the perigee by 9, 19, 38 km. That dip is why a disposal orbit must be raised by more than the protected zone alone requires.
Fig. 2 The eccentricity vector — pointing to perigee, with length equal to the eccentricity — of a near-geostationary orbit over one year of sunlight, for reflectivity times area-to-mass ratios of 0.01, 0.02 and 0.04 square metres per kilogram, started circular at the origin. The vector moves on a circle of radius 0.011 CRA/m0.011\,C_R A/m, following the Sun round the sky, and started at the origin it reaches twice the radius half a year later: eccentricities of 2.2×10−42.2\times10^{-4}, 4.5×10−44.5\times10^{-4} and 8.9×10−48.9\times10^{-4}, which at 42,164 kilometres dip the perigee by 9, 19 and 38 kilometres.

The coefficient of a thousand follows directly. A disposal orbit is left circular, so its eccentricity vector starts at the origin, which lies on the circle it will follow; half a year later it is at the opposite side, at twice the radius. The perigee is then lower than the circular radius by the semi-major axis times the eccentricity: 2a es=2×42,164×0.011 CRA/m≈940 CRA/m2 a\, e_s = 2 \times 42{,}164 \times 0.011\,C_R A/m \approx 940\,C_R A/m kilometres. The guideline’s thousand is that number, rounded up, and it is what sunlight will take away from the perigee every year of the satellite’s afterlife.

Why the push changes the shape and not the size

The claim that a steady push leaves the orbit’s size alone, and changes only its shape, is worth checking against the general rules for how a small force changes an orbit, because it is the reason the whole problem reduces to an eccentricity.

Which direction of force moves which element is set by Gauss’s equations. A push along the direction of motion changes the orbital energy, and so the semi-major axis, at a rate proportional to the push times the speed. A push from a fixed direction in space is along the motion on one side of the orbit and against it on the other, and on a nearly circular orbit the two halves cancel: the energy gained on one side is lost on the other, and the semi-major axis oscillates slightly within each orbit without drifting. The same push changes the eccentricity vector at a rate that does not cancel, because the eccentricity responds to the push with a sign that depends on where in the orbit it is applied, and the fixed direction and the orbital position combine so that every part of the orbit contributes in the same sense.

So a satellite in sunlight keeps its orbital elements not constant in a very particular way: its size fixed, its shape breathing on a yearly cycle, its orientation turning with the Sun. That is why the geostationary disposal problem is entirely a problem of the perigee and never of the mean height. The satellite’s average distance from the ring is whatever the disposal burns made it; only the lowest point of its orbit moves, and it moves by exactly the amount the eccentricity circle allows.

Watching a disposal orbit dip

The same arithmetic, followed through a year, shows the perigee sinking towards the ring and recovering.

The perigee of a disposal orbit dipping back towards geostationary height. The perigee altitude above the geostationary ring, in kilometres, of a satellite raised 300 km into a circular disposal orbit, over the following year, for reflectivity times area-to-mass ratios of 0.02, 0.12, 0.4 square metres per kilogram. Sunlight builds up an eccentricity that peaks half a year later, and the perigee dips to 281 km, 186 km, −80 km relative to the ring. The dashed line is the top of the protected zone, 200 km above it. At 0.02 the orbit stays clear of the protected zone; at 0.12 the orbit dips into the protected zone it was moved out of; at 0.4 the orbit crosses back through the geostationary ring itself. Whole satellites have ratios of a few hundredths; the fragments of a broken one — flakes of insulating blanket, pieces of solar panel — have ratios ten or a hundred times larger, and sunlight carries their perigees back down through the ring once a year, every year, for as long as they are up there.
Fig. 3 The perigee altitude above the geostationary ring of a satellite raised 300 kilometres into a circular disposal orbit, over the following year, for ratios of 0.02, 0.12 and 0.4 square metres per kilogram. The perigee dips to 281, 186 and −80 kilometres relative to the ring. At 0.02 it stays clear; at 0.12 it dips into the protected zone it was moved out of; at 0.4 it crosses back through the geostationary ring itself.

A whole satellite has a ratio of a few hundredths: its mass is concentrated in propellant tanks, structure and electronics, and its area in solar panels. The fragments of a broken one do not. A flake of multi-layer insulation blanket, the gold-coloured foil wrapped round spacecraft to control their temperature, has a ratio of several square metres per kilogram, a hundred times a whole satellite’s; a piece of solar panel, a few tenths. For those, sunlight’s yearly eccentricity is not a few hundredths of a per cent but several per cent, and their perigees swing through hundreds or thousands of kilometres.

Optical surveys of the geostationary region by the European Space Agency, beginning in the early 2000s, found a population of faint objects with exactly that behaviour — orbits whose eccentricities and inclinations changed from month to month far faster than anything massive could — and inferred ratios of up to tens of square metres per kilogram. Light debris is not held above the ring by any disposal height; sunlight carries it back down through the ring every year, and the only defence is not to create it, by disposing of satellites intact and preventing the explosions and break-ups that shed it.

Pointing the perigee at the Sun

The factor of two in the coefficient comes from starting the disposal orbit circular, at the edge of the circle the eccentricity vector will follow. It can be started elsewhere.

The eccentricity vector of an orbit above geostationary height, carried round by sunlight in a year. The eccentricity vector — pointing to perigee, with length equal to the eccentricity — of a near-geostationary orbit over one year of solar radiation pressure, for area-to-mass ratios times reflectivity of 0.02, 0.04 square metres per kilogram, started circular, at the origin, with the same orbits started instead at the centre of their circles — the eccentricity sunlight imposes, perigee pointed at the Sun — drawn dashed. The vector moves on a circle of radius 0.011 × Cᵣ·A/m, following the Sun round the sky once a year, and the orbit's eccentricity is the vector's length. Started at the origin it reaches twice the radius half a year later; started at the centre it stays at the radius all year, halving the largest eccentricity: maximum eccentricities of 4.47·10⁻⁴, 8.94·10⁻⁴, which at 42,164 kilometres dip the perigee by 19, 38 km. That dip is why a disposal orbit must be raised by more than the protected zone alone requires.
Fig. 4 The eccentricity vectors of two disposal orbits, ratios 0.02 and 0.04, started circular (solid) and started instead at the centre of their circles — given, at the disposal burns, the eccentricity sunlight would impose, with perigee pointed at the Sun (dashed). Started at the centre, the vector stays at the circle’s radius all year, and the largest eccentricity, and the perigee dip, are halved.

That is the Sun-pointing perigee strategy, and it is used in practice. The disposal burns are arranged to leave the orbit with a small eccentricity of just the right size and orientation — perigee in the direction of the Sun on the day of disposal — so that sunlight rotates the eccentricity vector round a circle centred on the origin rather than through it. The perigee then stays at a constant height above the ring, lower than the average by aesa e_s instead of dipping by twice that. The same trick is used for operational geostationary satellites, whose eccentricity is controlled all their lives by small manoeuvres that keep the vector on a circle centred on the origin.

What the height costs

Raising an orbit by a small amount ΔH\Delta H takes two burns, one to begin the transfer and one to circularise, and together they cost about v ΔH/2av\,\Delta H/2a — 3.65 metres a second for every hundred kilometres at geostationary height.

The velocity change and propellant a disposal manoeuvre costs. The total velocity change of the two small burns that raise a geostationary satellite to its minimum disposal height, in metres a second, against the reflectivity times area-to-mass ratio that sets that height. Raising by ΔH costs about v·ΔH/(2a), 3.65 metres a second per hundred kilometres. For Cᵣ·A/m = 0.02 the guideline height is 255 km and the manoeuvre 9.3 m/s; for 0.05, 285 km and 10.4 m/s. The dashed curve is the same manoeuvre when the disposal orbit is given, from the start, the eccentricity sunlight would impose, with perigee pointed at the Sun, which halves the sunlight term: 9.5 m/s at 0.05. With engines of 300 seconds of specific impulse the reserve is 0.35 per cent of the satellite's mass — small, but it has to be held back from years of station-keeping, which is why the uncertainty in how much propellant is left matters so much.
Fig. 5 The velocity change for the two burns that raise a geostationary satellite to its minimum disposal height, against the reflectivity times area-to-mass ratio that sets the height. For a ratio of 0.02 the height is 255 kilometres and the manoeuvre 9.3 metres a second; for 0.05, 285 and 10.4. Dashed: the same manoeuvre with the perigee pointed at the Sun, halving the sunlight term, 9.5 metres a second at 0.05. With engines of 300 seconds’ specific impulse the reserve is about a third of a per cent of the satellite’s mass, small in propellant and large in the months of operation it displaces.

A third of a per cent of a satellite’s mass does not sound like much, and in propellant it is a few kilograms. The cost lies elsewhere. The disposal reserve is held back from station-keeping, which on a geostationary satellite is dominated by the fifty metres a second a year needed to counter the Moon and Sun’s pull on its plane; ten metres a second of reserve is two or three months of station-keeping, which at the revenue of a large communications satellite is a substantial sum. And because the reserve has to be large enough with confidence, and the gauge of remaining propellant is least precise near empty, operators hold back more than the formula requires. The formula sets the floor; the uncertainty of the fuel gauge sets what is actually held back.

The same push elsewhere

Radiation pressure on a spacecraft is one instance of a force that shapes orbits wherever surfaces meet sunlight, and the other instances make the geostationary case look less like a peculiarity of engineering.

Small asteroids are pushed by sunlight too, but the force that matters for them is not the pressure of the sunlight they reflect: it is the recoil from the heat they re-emit. An asteroid absorbs sunlight on its day side and radiates it in the infrared after a delay, so the emission is strongest from its afternoon side, and the recoil from that heat moves its orbit — the Yarkovsky effect, which changes the semi-major axis, because the delayed emission has a component along the direction of motion that does not cancel round the orbit. Its size, like the sunlight term here, is set by the ratio of area to mass, and the rate of drift says what the asteroid’s surface is made of.

A solar sail is the same physics turned into propulsion: a spacecraft built to maximise its area-to-mass ratio, and angled so that the push has a component along the motion. Tilted one way it spirals outward, the other way inward, and it needs no propellant at all — a drive with no rocket equation, whose performance is measured by exactly the ratio that sets a dead satellite’s perigee dip. A sail with a ratio of a few tens of square metres per kilogram is, by this essay’s arithmetic, a piece of debris sunlight can move by thousands of kilometres a year — which is the point of it.

How often it is done

The guideline has no force of law in most countries, and compliance is measured by watching what satellites actually do at the end of their lives. The European Space Agency’s annual reports on the space environment track every geostationary satellite that reaches end of life. In recent years roughly nine in ten have attempted a disposal manoeuvre, and a smaller majority — around seven or eight in ten — have reached a disposal orbit that fully meets the guideline. The failures are partly satellites that ran out of propellant unexpectedly or failed before they could be moved, and partly satellites moved to a lower height than the formula asks, often because the operator underestimated how much the sunlight term matters for a satellite with large arrays.

The satellites that are not moved stay in the ring for good. There are several hundred dead objects in the geostationary region already, some uncontrolled satellites drifting through the ring twice a day as their orbits’ inclinations grow under the lunisolar pull, others trapped oscillating round the stable longitudes. Each is a hazard to the satellites operating at the longitudes it passes through, and the rate of close approaches, though far lower than in low orbit, is rising as the ring fills.

What the calculation leaves out

The eccentricity circle is computed for a constant radiation acceleration. A real satellite’s effective area changes as it rotates, its solar panels stop tracking the Sun when it is switched off, and its reflectivity changes as its surfaces age; the circle is then not a circle, and its radius changes. The guideline’s coefficient is an approximation that covers the typical case.

Earth’s shadow is ignored. For about forty days around each equinox a geostationary orbit passes through the Earth’s shadow once a day, for up to seventy minutes, and the radiation push switches off there; the effect on the yearly circle is a small distortion.

The gravity term is taken as a fixed 35 kilometres. The true long-term perigee variation depends on the disposal orbit’s inclination and on the date of disposal, and integrations for individual cases give values from a few kilometres to somewhat more than 35.

And the perigee figures start from a disposal at the moment of zero eccentricity, so that the dip happens in the first half-year. A satellite disposed of at a different time of year sees the same dip at a different date.

Still open: whether three hundred kilometres is enough for a thousand years

The disposal orbit is meant to keep a dead satellite out of the protected zone indefinitely, and the formula was derived for perturbations whose effects were computed over a century or two. Over longer times, resonances between the lunisolar perturbations and the Earth’s gravity field can build up eccentricity slowly in some disposal orbits, and long integrations have shown that a small fraction of orbits left at the guideline height drift back towards the protected zone over several centuries, depending on their initial eccentricity vector and inclination. Choosing the initial orbit with those long-term dynamics in mind — the Sun-pointing perigee is one such choice, and a favourable inclination another — reduces the risk further. Whether the geostationary graveyard is stable for the millions of years its occupants will stay there, or whether it is slowly leaking dead satellites back towards the ring on timescales longer than anyone has integrated, is a question about the long-term dynamics of a region of space that humanity has been filling for only sixty years.

Where this goes next

A gauge good to 7 per cent with a tenth of the load left, and to 23 per cent with three hundredths. The uncertainty in the propellant remaining in a spacecraft tank, as a percentage of what remains, against the fraction of the 450-kilogram load still in the tank, on a logarithmic uncertainty axis with the tank emptying to the right. Bookkeeping — summing every thruster firing through a flow-rate model — carries an error common to all burns of 2 per cent of the mass used, plus an independent 5 per cent per burn that averages down over 2000 firings; its absolute error grows with the mass used. Gauging by pressure and temperature infers the empty volume of the tank from the gas law applied to a known mass of pressurant, with a combined 0.66 per cent uncertainty in n R T / P and a 0.2 per cent uncertainty in the tank's volume; its absolute error grows as the gas fills the tank. The two methods are independent and are combined by inverse variance. With a tenth of the load left the combined estimate is uncertain by 2.9 kg, 7 per cent of what remains; with three per cent left, by 3.1 kg, 23 per cent. Near empty the absolute error barely changes, so halving what is left doubles the relative error — the gauge is at its worst exactly when the last manoeuvre has to be planned from it. A fuel gauge that is worst when it is needed A spacecraft's tank has no float and no dial. The propellant left is estimated by adding up every burn or by reading the pressure and temperature of the gas above the liquid, and both methods' errors grow with the propellant used. Relative to what remains, the error doubles every time what remains halves — so a geostationary satellite has to hold back months of station-keeping as a margin against a gauge that cannot see the last few kilograms. 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. 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.

About the same objects

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

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Area-to-mass ratioDisposal manoeuvreEccentricity vectorGeostationary orbitGraveyard orbitOrbital debrisProtected regionSolar radiation pressure