Astronomy & Celestial Mechanics
the sky, computed
Spaceflight

A stationary satellite that draws a figure of eight

A satellite with a period of exactly one sidereal day returns over the same ground every day, but only an orbit in the equator with no eccentricity returns over a single point. A few degrees of tilt draw a figure of eight, a little eccentricity a swing in longitude, and the two together draw the figure the Sun draws in the sky over a year.

Assumes Ground tracks and Equation of time.

17 min read 9 figures The sky as a clockSmall perturbations, large consequences

A geostationary satellite is described as hanging over one point on the equator, and the description is exact for an orbit that no real satellite flies. What the name promises needs three things at once: a period of one sidereal day, an orbit lying in the plane of the equator, and an orbit that is a perfect circle. The first can be held to a fraction of a second. The other two are never quite zero, and the moment either is not, the satellite stops being a point in the sky and starts tracing a closed curve over the ground once a day.

The period is what matters most, and it is the one that survives. Any orbit with a period of exactly one sidereal day is geosynchronous: it comes back to the same place over the rotating Earth every day, so its ground track closes after one revolution. Geostationary is the special case of that in which the closed track has shrunk to a single point. Everything else in the family is a small closed figure, and the figures are the subject here.

The figure of eight a geosynchronous orbit draws at 5°, 15°, 30° of inclination. The ground track over one sidereal day of a circular orbit whose period is exactly a sidereal day, at inclinations of 5, 15, 30°, centred on its own mean longitude. It is not a point. The latitude swings to ±i and back twice a day, and the longitude falls behind and then runs ahead of the Earth's rotation, because the rate at which an inclined orbit gains longitude is u̇ cos i / cos²φ: slowest at the nodes, where part of the motion is north–south, and fastest at the extremes of latitude, where all of it is eastward and a degree of longitude is shorter — so the track closes as a figure of eight. Its half-width in longitude is ±0.109° at 5°, ±0.993° at 15°, ±4.117° at 30° — exactly arcsin(tan²(i/2)) — against the small-inclination form i²/4 = ±0.109°, ±0.982°, ±3.927°: quadratic in the inclination, so the eight is tall and very thin. The longitude axis is stretched 8 times relative to the latitude axis, and without that stretch every one of these curves would be drawn as a vertical line. This is why a few degrees of inclination, which a geostationary operator spends most of its propellant preventing, moves the satellite a long way north and south and almost not at all east and west.
Fig. 1 One sidereal day of track for a circular orbit with a period of exactly a sidereal day, at inclinations of 5°, 15° and 30°, centred on its own mean longitude. The latitude swings to ±i twice a day and the longitude falls behind and runs ahead of the Earth’s rotation, so the track closes as a figure of eight. Its half-width in longitude is ±0.109° at 5°, ±0.993° at 15° and ±4.117° at 30° — exactly arcsin(tan²(i/2)) — against the small-inclination form i²/4 of ±0.109°, ±0.982° and ±3.927°. The longitude axis is stretched 8 times relative to the latitude axis; without the stretch every curve would be a vertical line.

Why a tilted circle falls behind and then runs ahead

A satellite in an inclined circular orbit with a one-day period covers its orbit at exactly the rate the Earth turns. If all of that motion were eastward it would stay over one meridian. It is not all eastward, and how much of it is changes round the orbit.

At a node, where the orbit crosses the equator, the satellite is climbing or descending at the inclination angle. Part of its motion is north–south, and only the fraction cos⁡i\cos i of it carries the satellite east. It gains longitude more slowly than the ground beneath it and falls behind. At the northernmost and southernmost points the motion is entirely eastward, and the satellite is at a latitude where a degree of longitude is a shorter distance. There it gains longitude faster than the ground and catches up.

The two effects combine into one expression. With uu the angle travelled round the orbit from the ascending node, the satellite’s longitude runs at u˙cos⁡i/cos⁡2φ\dot u\cos i/\cos^2\varphi against the Earth’s u˙\dot u, and integrating that gives the offset from the mean:

Δλ=arctan⁡(cos⁡i tan⁡u)−u.\Delta\lambda = \arctan(\cos i\,\tan u) - u.

The offset is westward through the first quarter of the orbit, crosses zero at the northern extreme, is eastward through the second quarter, and repeats with the opposite latitude in the southern half. A curve that goes west and back while climbing, and east and back while descending, is a loop; two such loops joined at the equator are a figure of eight.

The width has a closed form. The offset is largest where its derivative vanishes, and its value there is arcsin⁡(tan⁡2(i/2))\arcsin\big(\tan^2(i/2)\big), which for small inclinations is i2/4i^2/4 in radians. The width is quadratic in the tilt while the height is linear, so the eight is always far taller than it is wide: at thirty degrees it is sixty degrees tall and eight wide, and at five degrees ten tall and a fifth of a degree wide.

The real case is a thousandth of the drawing

Thirty degrees is not a geostationary satellite. The figure at the inclinations that actually occur is so thin that it has to be drawn with an absurd stretch.

The figure of eight a geosynchronous orbit draws at 1°, 3°, 5° of inclination. The ground track over one sidereal day of a circular orbit whose period is exactly a sidereal day, at inclinations of 1, 3, 5°, centred on its own mean longitude. It is not a point. The latitude swings to ±i and back twice a day, and the longitude falls behind and then runs ahead of the Earth's rotation, because the rate at which an inclined orbit gains longitude is u̇ cos i / cos²φ: slowest at the nodes, where part of the motion is north–south, and fastest at the extremes of latitude, where all of it is eastward and a degree of longitude is shorter — so the track closes as a figure of eight. Its half-width in longitude is ±0.004° at 1°, ±0.039° at 3°, ±0.109° at 5° — exactly arcsin(tan²(i/2)) — against the small-inclination form i²/4 = ±0.004°, ±0.039°, ±0.109°: quadratic in the inclination, so the eight is tall and very thin. The longitude axis is stretched 49 times relative to the latitude axis, and without that stretch every one of these curves would be drawn as a vertical line. This is why a few degrees of inclination, which a geostationary operator spends most of its propellant preventing, moves the satellite a long way north and south and almost not at all east and west.
Fig. 2 The same construction at 1°, 3° and 5°. The half-widths are ±0.004°, ±0.039° and ±0.109°, and the small-inclination form i²/4 agrees with every one of them to the precision printed. The longitude axis is stretched 49 times relative to the latitude axis. At one degree of inclination the satellite moves a degree north and south of the equator every day and less than a hundredth of a degree east and west.

The asymmetry between the two directions decides how a real satellite is flown. The Sun and the Moon pull the orbital plane away from the equator by about 0.85 degrees a year, and holding the inclination near zero costs roughly fifty metres per second annually — the largest single item in a geostationary satellite’s propellant budget. What that propellant buys is almost entirely the height of the figure of eight. Its width at a degree of tilt is too small to matter to anybody.

The drift is not unbounded, which is the other reason operators are willing to let it happen. The Sun and the Moon do not pull the orbital plane towards some fixed tilt; they make it precess about a plane lying between the equator and the ecliptic, tilted about seven and a half degrees to the equator. An orbit left alone at zero inclination swings away from the equator, reaches a maximum of about fifteen degrees, and comes back, over roughly half a century. A satellite abandoned at the end of its life does not wander off; it traces a slow cone about that plane, and it will be back near the equator decades later.

The height matters because a ground antenna is narrow. A large dish at the frequencies used for broadcasting has a beam a tenth to a fifth of a degree across, and a satellite moving a degree north and south every day leaves that beam twice a day unless the antenna is steered to follow it. So satellites near the end of their lives are often allowed to drift into inclined orbits deliberately, saving the north–south propellant to extend their operation for years, and the ground stations that still use them are fitted with tracking mounts. The east–west position, which is what separates one operator’s slot from the next, is held as before, by the much cheaper manoeuvres that counter the slow walk of a satellite towards one of two longitudes.

Eccentricity swings the longitude, and linearly

The second imperfection is shape. An orbit with a one-day period and a little eccentricity is not always at the same distance, and by the law of equal areas it moves faster than average near perigee and slower near apogee. The ground beneath it turns at a constant rate, so the satellite runs east of its mean position around perigee and falls back west around apogee, and a satellite with no inclination at all traces a line back and forth along the equator.

A day's swing in longitude from an eccentricity of 0.0005, 0.001, 0.005. The longitude of a satellite with a period of exactly one sidereal day and no inclination, measured from its mean, through one day starting at perigee. With no inclination it never leaves the equator, so the whole of its ground track is this back-and-forth: faster than the Earth turns near perigee, so it runs east, and slower near apogee, so it falls back west. The swing is ±0.057° at e = 0.0005, ±0.115° at e = 0.001, ±0.573° at e = 0.005, against the first-order value of 2e radians, ±0.057°, ±0.115°, ±0.573° — linear in the eccentricity, where the inclination's figure of eight is quadratic in its east–west width. So for the east–west half of a station-keeping box it is the eccentricity that binds: a thousandth of it swings the satellite ±0.115° of longitude, while it takes 5.1° of inclination to widen the figure of eight that much.
Fig. 3 The longitude of a satellite with a one-sidereal-day period and no inclination, measured from its mean, through one day starting at perigee, for eccentricities of 0.0005, 0.001 and 0.005. The swing is ±0.057°, ±0.115° and ±0.573°, against the first-order value of 2e radians, ±0.057°, ±0.115° and ±0.573°. A thousandth of eccentricity swings the satellite ±0.115° of longitude, and it would take 5.1° of inclination to widen the figure of eight that much.

The amplitude is 2e2e radians, and where it comes from is worth one line. The satellite’s angle round its orbit, the true anomaly ν\nu, differs from the uniformly advancing mean anomaly MM by ν−M≈2esin⁡M\nu - M \approx 2e\sin M to first order — the orbit’s own “equation of the centre” — and the ground turns in step with MM, so that difference is the longitude offset. It is linear in the eccentricity, where the inclination’s contribution to the width is quadratic, and that difference in order is why the two imperfections bind different halves of a station-keeping box. A slot a tenth of a degree wide on either side tolerates an eccentricity below about 0.0009 and a width-equivalent inclination of several degrees, so it is the eccentricity that sets the east–west limit and the inclination that sets the north–south one. Sunlight pressing on large solar panels pumps the eccentricity of a geostationary orbit round a small circle every year, which is why the eccentricity is actively managed even though nothing about it looks dramatic on a map.

The two small imperfections also have a use inside the slot, and it is the least obvious thing in this essay. Several satellites belonging to one operator often share a single orbital position — seven or eight of them in a box a tenth of a degree wide. They cannot be separated by longitude, because the box is too small, so they are separated by giving each a slightly different eccentricity and inclination, oriented differently round the orbit. Each then traces its own tiny figure, and the figures are arranged so that at every moment of the day the satellites are kilometres apart even though their mean positions coincide. The quantities fought as errors on a single satellite become the coordinates that keep a cluster from colliding.

A day's swing in longitude from an eccentricity of 0.0167, 0.05, 0.1. The longitude of a satellite with a period of exactly one sidereal day and no inclination, measured from its mean, through one day starting at perigee. With no inclination it never leaves the equator, so the whole of its ground track is this back-and-forth: faster than the Earth turns near perigee, so it runs east, and slower near apogee, so it falls back west. The swing is ±1.914° at e = 0.0167, ±5.731° at e = 0.05, ±11.472° at e = 0.1, against the first-order value of 2e radians, ±1.914°, ±5.730°, ±11.459° — linear in the eccentricity, where the inclination's figure of eight is quadratic in its east–west width. So for the east–west half of a station-keeping box it is the eccentricity that binds: a thousandth of it swings the satellite ±0.115° of longitude, while it takes 5.1° of inclination to widen the figure of eight that much.
Fig. 4 The same swing at much larger eccentricities, 0.0167, 0.05 and 0.1. The amplitudes are ±1.914°, ±5.731° and ±11.472°, against 2e radians of ±1.914°, ±5.730° and ±11.459°. The first-order form is still good to a tenth of a per cent at 0.1, which is far past anything a geostationary operator would tolerate and well inside the range the orbits of the next section use deliberately.

The first of those three eccentricities is not chosen at random. It is the eccentricity of the Earth’s own orbit, and its ±1.914 degrees will reappear below as a number that has been known for centuries under another name.

Putting apogee where the service is

A figure of eight with its loops symmetric about the equator serves both hemispheres equally and neither well at high latitude: from a station far from the equator, a satellite over it is low in the sky, and above about 71 degrees of latitude it is below a ten-degree mask altogether. Add eccentricity to inclination, place the apogee over one hemisphere, and the symmetry breaks in a useful direction. Near apogee the satellite is slow and high; near perigee it is fast and low. The loop on the apogee side shrinks and the satellite lingers in it, and the loop on the perigee side swells and is crossed quickly.

A geosynchronous orbit that loiters over one hemisphere. The ground track over one sidereal day of an orbit at i = 42°, e = 0.075, with its perigee at ω = 270°, centred on 135° E. Inclination alone draws a symmetric figure of eight and eccentricity alone a swing along the equator; together they draw an asymmetric loop — a small, slow lobe on the side of the equator where apogee is, where the satellite lingers, and a large, fast one on the other side that it crosses quickly. With apogee over the north, the orbit at ω = 270° spends 13.1 hours of each day north of the equator and ranges ±14.41° about its mean longitude. This is how a geosynchronous satellite is made to serve a high latitude that a geostationary one sees only near its horizon: the period keeps it over the same longitudes, and the argument of perigee decides which hemisphere gets the time.
Fig. 5 One sidereal day of track for an orbit at 42° inclination and 0.075 eccentricity with its perigee at ω = 270°, so that apogee falls at the northernmost point, centred on 135° E. The track is an asymmetric figure of eight: a small, slow northern loop and a large, fast southern one. The satellite spends 13.1 hours of each day north of the equator and ranges ±14.41° about its mean longitude.

That is very nearly the orbit of Japan’s Quasi-Zenith Satellite System, whose satellites spend the northern part of their day high over Japan and supplement navigation signals in cities where tall buildings hide the lower satellites of other systems. The period keeps each satellite over the same longitudes, the inclination carries it far enough north to be near the zenith, and the eccentricity, with the argument of perigee set at 270 degrees, turns an even split of the day into thirteen hours to eleven in favour of the hemisphere that wants it.

Push both numbers further and the effect becomes the design.

Two geosynchronous orbits, and the hemisphere each one loiters over. The ground tracks over one sidereal day of an orbit at i = 63.4°, e = 0.27, with its perigee at ω = 270° and an orbit at i = 63.4°, e = 0.27, with its perigee at ω = 90°, each centred on 100° W. Inclination alone draws a symmetric figure of eight and eccentricity alone a swing along the equator; together they draw an asymmetric loop — a small, slow lobe on the side of the equator where apogee is, where the satellite lingers, and a large, fast one on the other side that it crosses quickly. With apogee over the north, the orbit at ω = 270° spends 16.0 hours of each day north of the equator and ranges ±40.20° about its mean longitude; with apogee over the south, the orbit at ω = 90° spends 7.9 hours of each day north of the equator and ranges ±40.20° about its mean longitude. This is how a geosynchronous satellite is made to serve a high latitude that a geostationary one sees only near its horizon: the period keeps it over the same longitudes, and the argument of perigee decides which hemisphere gets the time.
Fig. 6 Two orbits at 63.4° inclination and 0.27 eccentricity centred on 100° W, identical except for the argument of perigee. With apogee over the north (ω = 270°) the satellite spends 16.0 hours of each day north of the equator; with apogee over the south (ω = 90°), 7.9 hours. Both range ±40.20° about their mean longitude. One number decides which hemisphere gets two thirds of the day.

These are the parameters of a Tundra orbit, and a satellite radio system for North America flew three satellites in orbits of this kind, spaced so that one was always high over the continent. The inclination is not arbitrary. At 63.4 degrees the Earth’s flattening stops turning the line of apsides — it is one of the two inclinations at which that motion stands still — so the apogee stays over the northern hemisphere without propellant being spent to keep it there. At any other inclination the apogee would creep round the orbit, and the hemisphere the satellite favours would slowly change.

The two tracks in that figure are the same loop reflected through the equator. That is exactly what changing ω\omega by 180 degrees should do, and it is a useful check that nothing in the construction prefers one hemisphere: the preference comes entirely from where perigee is put.

The Soviet Molniya orbit, which the Tundra orbit is often confused with, makes the contrast clear. It uses the same critical inclination and an even larger eccentricity, but its period is half a sidereal day rather than a whole one, so it is not geosynchronous at all. Its track does not close after one revolution; it closes after two, with two apogees a day over two longitudes half a world apart, each serving its own region for about eight hours. A Tundra orbit gives one region sixteen hours of a single satellite; a Molniya orbit gives two regions eight hours each. Which is better depends on how many regions there are to serve and how high a satellite is wanted over each.

Both exist because of one limitation already met. From the latitudes of Russia, Scandinavia and northern Canada, a geostationary satellite sits low on the southern horizon or below it, behind buildings and hills and through a long path of air. An orbit that parks its apogee over those latitudes puts the satellite near the zenith instead, and every argument in this section is a way of buying that elevation with eccentricity.

The figure the Sun draws

Now take the three numbers the Earth’s own orbit has — an inclination of 23.44 degrees between its axis and its orbit, an eccentricity of 0.0167, and a perigee about 283 degrees round from the ascending node — and hand them to the same construction.

Nothing about this is a metaphor. Seen from the Earth, the Sun goes once round the sky in a year along a circle tilted 23.44 degrees to the equator, at a rate that varies because the orbit is slightly eccentric. A mean Sun that went round the equator at a perfectly uniform rate would keep clock time. The difference between where the real Sun is and where the mean Sun is, read in right ascension, is the equation of time, and read together with the Sun’s declination it is the analemma — the figure of eight traced by the Sun’s position at the same clock time on every day of a year. The geometry is identical to a geosynchronous satellite’s: a body going round a tilted, slightly eccentric circle, compared with a reference that turns uniformly in the equator. Only the period is different, a year instead of a sidereal day.

The figure of eight a geosynchronous orbit draws at 23.44° of inclination. The ground track over one sidereal day of a circular orbit whose period is exactly a sidereal day, at inclinations of 23.44°, centred on its own mean longitude. It is not a point. The latitude swings to ±i and back twice a day, and the longitude falls behind and then runs ahead of the Earth's rotation, because the rate at which an inclined orbit gains longitude is u̇ cos i / cos²φ: slowest at the nodes, where part of the motion is north–south, and fastest at the extremes of latitude, where all of it is eastward and a degree of longitude is shorter — so the track closes as a figure of eight. Its half-width in longitude is ±2.467° at 23.44° — exactly arcsin(tan²(i/2)) — against the small-inclination form i²/4 = ±2.397°: quadratic in the inclination, so the eight is tall and very thin. The longitude axis is stretched 10 times relative to the latitude axis, and without that stretch every one of these curves would be drawn as a vertical line. This is why a few degrees of inclination, which a geostationary operator spends most of its propellant preventing, moves the satellite a long way north and south and almost not at all east and west.
Fig. 7 The inclination half of the analemma: a circular one-day orbit tilted at 23.44°. The half-width is ±2.467° — exactly arcsin(tan²(i/2)) — against the small-inclination form of ±2.397°. At four minutes of time to a degree, 2.467° is 9.87 minutes, and it is the amplitude of the obliquity term in the equation of time. The longitude axis is stretched 10 times.

The mapping between the two problems is exact, term by term. The satellite’s inclination is the obliquity. Its eccentricity is the Earth’s. Its argument of perigee is the angle from the point where the Sun crosses the equator going north — the March equinox — to the point where the Sun is nearest, which is the Earth’s perihelion seen from the other end: 283 degrees further along the ecliptic, in early January. And the sidereal day that the satellite’s period matches is replaced by the year that the Sun’s does, so a longitude offset of one degree becomes four minutes of time, because the mean Sun covers fifteen degrees an hour.

The obliquity term of the equation of time, which separates into a fictitious Sun moved to the equator, reaches just under ten minutes twice a year, and the figure of eight at 23.44 degrees is ±9.87 minutes wide. The eccentricity term reaches about seven and a half minutes once a year, and the eccentricity swing at 0.0167 was ±1.914 degrees, which is 7.66 minutes. Those are the two terms, computed here as ground tracks. Their sum should be the analemma.

A geosynchronous orbit that loiters over one hemisphere. The ground track over one sidereal day of an orbit at i = 23.44°, e = 0.0167, with its perigee at ω = 283°, centred on the prime meridian. Inclination alone draws a symmetric figure of eight and eccentricity alone a swing along the equator; together they draw an asymmetric loop — a small, slow lobe on the side of the equator where apogee is, where the satellite lingers, and a large, fast one on the other side that it crosses quickly. With apogee over the north, the orbit at ω = 283° spends 12.2 hours of each day north of the equator and ranges from 4.11° west to 3.56° east of its mean longitude — 16.4 and 14.2 minutes of time at four minutes a degree. This is how a geosynchronous satellite is made to serve a high latitude that a geostationary one sees only near its horizon: the period keeps it over the same longitudes, and the argument of perigee decides which hemisphere gets the time.
Fig. 8 The track of a one-day orbit with the Earth’s obliquity, eccentricity and perigee: i = 23.44°, e = 0.0167, ω = 283°. It is an asymmetric figure of eight with apogee over the north, spending 12.2 hours of the day north of the equator and ranging from 4.11° west to 3.56° east of its mean longitude — 16.4 and 14.2 minutes of time at four minutes a degree.
The equation of time, and its two causes. The difference between a sundial and a clock over the year, in minutes, computed from Kepler's equation and the tilt. The eccentricity term has one cycle a year and the obliquity term has two; their sum runs from −14.2 to 16.4 minutes.
Fig. 9 The equation of time over a year, computed directly from Kepler’s equation and the tilt, with its eccentricity and obliquity terms drawn separately. Their sum runs from −14.2 to 16.4 minutes — the same two extremes, to the tenth of a minute, as the geosynchronous track above, reached by a calculation that never mentions a satellite.

The two calculations share no step and no intermediate quantity. One integrates a satellite’s position against a rotating Earth over a day; the other computes the Sun’s right ascension against a mean Sun over a year. They agree on 16.4 and 14.2 minutes because they are the same problem. The 12.2 hours the track spends north of the equator agrees too: as a fraction of a year it is about 186 days, which is the length of northern spring and summer, longer than autumn and winter because the Earth is at aphelion in July. The loop on the northern side is the smaller one for the same reason the Quasi-Zenith loop is: the body is slow at the far end of its orbit, and it lingers there.

The analemma is the ground track of a geosynchronous satellite whose inclination is the obliquity and whose eccentricity is the Earth’s. The sundial’s error and the station-keeping budget of a broadcast satellite are one piece of spherical trigonometry applied to two clocks.

What the tracks leave out

They are two-body orbits. The figures use a Keplerian ellipse with a period of exactly one sidereal day and nothing else. The Earth’s flattening, its slightly elliptical equator, the Sun and the Moon, and the pressure of sunlight all perturb a real geosynchronous orbit, and over months those perturbations move the whole figure — the inclination growing, the eccentricity circling, the mean longitude drifting. The figures are a snapshot of the shape a set of elements produces on one day.

The latitude and longitude are geocentric. A ground station does not see a figure of eight in latitude and longitude; it sees the satellite move in elevation and azimuth by amounts that depend on where the station is, and for a station far from the sub-satellite point the apparent figure is distorted and foreshortened.

And the analemma comparison is exact only to the order drawn. The equation of time used for comparison includes the Earth’s eccentricity and obliquity and the two terms’ interaction, but not nutation, not the slow drift of perihelion, and not the difference between the tropical year and the anomalistic one. Those move the extremes by seconds, which is below the tenth of a minute to which the agreement is quoted.

What the imperfections are for

Every item in the first half of this essay is a nuisance to be removed, and every item in the second half is the same effect used on purpose. Inclination is fought on a geostationary satellite because it makes a narrow beam miss; it is chosen on a Tundra satellite because it carries the satellite to a high latitude. Eccentricity is pumped out of a broadcast orbit because it walks a satellite out of its slot; it is put into a navigation augmentation orbit because it buys a hemisphere two extra hours. The figure of eight is the same figure in both cases, and the difference between a defect and a design is where the apogee has been put.

The Sun has been drawing the same figure in every sundial’s shadow since there were sundials, and the correction of up to sixteen minutes that turns a sundial’s reading into clock time is the width of it.

Still open: how many circles cover the whole Earth at once

A single geosynchronous satellite, whatever its figure of eight, serves one region, and no single orbit of any kind keeps a satellite above every point on the Earth at the same time. The question of how to arrange several so that the planet is covered continuously — how many are needed, in how many planes, at what altitude — turns out to have an exact geometric answer and a minimum below which it cannot be done at all.

Where this goes next

How far across the ground a satellite can be seen from, at 0, 10, 30° masks. The footprint of a satellite — the largest angle at the Earth's centre between the point beneath it and a station that can still see it above an elevation mask — against altitude, on a logarithmic axis. λ = arccos(R cos ε / (R + h)) − ε. At low altitude it grows as the square root of the height, λ ≈ √(2h/R), drawn dashed for the horizon mask, so doubling a low orbit's altitude widens its footprint by only about forty per cent; far out it saturates, and no altitude sees past 90° − ε. At 420 km (a space station) the 0° footprint is 20.2°, a circle 2,254 km in radius holding 3.1 per cent of the Earth's surface. At 20,180 km (a navigation satellite) the 0° footprint is 76.1°, a circle 8,472 km in radius holding 38.0 per cent of the Earth's surface. At 35,786 km (geostationary) the 0° footprint is 81.3°, a circle 9,050 km in radius holding 42.4 per cent of the Earth's surface. Raising the mask costs most at low altitude: at 420 km a 10° mask takes 38 per cent off the footprint's radius and 62 per cent off its area, because most of what a low satellite could see is near the horizon, while at 35,786 km the same mask takes 20 per cent of the area. The circle a station can see A ground station can talk to a satellite only while the satellite is above its horizon, and the region it can do that from is a circle drawn round the point beneath the spacecraft. The circle grows as the square root of the altitude and then stops growing, the time spent inside it diverges at one altitude, and the stations that get the most contact are not where anyone would first put them. The equation of time, and its two causes. The difference between a sundial and a clock over the year, in minutes, computed from Kepler's equation and the tilt. The eccentricity term has one cycle a year and the obliquity term has two; their sum runs from −14.2 to 16.4 minutes. Two fictitious suns in sequence The equation of time is usually described as one quantity with two causes. It is better read as two separate reductions, each with its own imaginary body — one that moves uniformly along the ecliptic and one that moves uniformly along the equator — and the second is not the first. 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

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AnalemmaArgument of perigeeCritical inclinationEccentricityEquation of timeGeostationary orbitGeosynchronous orbitGround trackInclinationStation-keeping