The circle a station can see

A satellite and a station on the ground can exchange a signal only along a straight line that does not pass through the Earth. That is the whole of the constraint, and everything a ground network is built around follows from it: how far apart the stations have to be, how long each conversation lasts, and which parts of the planet are worth putting an antenna on.

The constraint is easiest to see from the satellite’s side. Look down from the spacecraft and the Earth is a disc whose edge is the horizon as seen from orbit. Every point inside that edge can see the satellite; every point outside it cannot. On the ground the edge is a circle centred on the point directly beneath the spacecraft, and the size of that circle — the footprint — is the single number that decides how much of the planet a satellite is in contact with at any moment.

In practice the edge is not the geometric horizon. A signal skimming the ground crosses a long path of atmosphere, reflects off terrain and buildings, and competes with everything else near the horizon, so a station works only above some minimum elevation angle — the elevation mask — commonly ten degrees for a tracking antenna and more for a small terminal. A mask shrinks the circle, and the essay’s first figure is how much.

Two regimes in one formula #

The footprint is a single line of trigonometry. The ray from a station to a satellite at the edge of visibility leaves the ground at the mask angle $\varepsilon$, and the triangle formed by the Earth’s centre, the station and the satellite gives the central angle between the station and the point beneath the satellite:

$$\lambda = \arccos\!\left(\frac{R\cos\varepsilon}{R+h}\right) - \varepsilon.$$

Two limits carry most of the meaning, and they are opposite in character.

When the satellite is low, $h$ is small beside $R$, the cosine is close to one, and for the horizon mask the angle is approximately $\sqrt{2h/R}$. The footprint grows as the square root of the altitude. Quadrupling the height of a low orbit only doubles the radius of the region it can see. This is the same geometry that sets how far a lighthouse can be seen at sea, and for the same reason: the horizon from a height is fixed by how quickly a sphere curves away from its tangent plane, and that departure is quadratic in distance.

When the satellite is high the cosine goes to zero and the angle approaches $90° - \varepsilon$. There is a hemisphere’s worth of Earth facing any distant object, and no altitude can see past it; a mask removes a band of width $\varepsilon$ from the edge of that hemisphere. The geostationary footprint, 81.3 degrees for the horizon mask, is already within nine degrees of the limit, and going further out buys almost nothing.

The transition between the regimes happens around an Earth radius of altitude. Below it, altitude is the cheapest way to see more ground. Above it, the only way to see more ground is more satellites.

The square-root form is good enough to reason with and not good enough to design with. At 420 kilometres it gives 20.8 degrees against the exact 20.2, three per cent high, and the error grows with altitude because the approximation keeps growing where the true footprint has begun to level off.

The saturated end has a consequence that is easy to state and was designed around from the start. Because the geostationary footprint is at most 81.3 degrees, no station further than 81.3 degrees from the point beneath a geostationary satellite can see it at all, and that point is always on the equator. So a station poleward of 81.3 degrees of latitude cannot see any geostationary satellite, whichever longitude it sits over. With the ten-degree mask a working antenna needs, the same formula gives 71.4 degrees, which excludes most of Svalbard, the whole of the high Canadian Arctic and the northern coast of Siberia. Arthur C. Clarke’s 1945 proposal of three relay stations spaced round the geostationary ring covered the populated world, and it was never going to cover the polar caps; nothing parked over the equator can.

A mask costs most where the orbit is lowest #

The second reading of the figure is about the mask, and it runs against the intuition that ten degrees is a small allowance.

The mask itself is not arbitrary. A ray leaving the ground at ten degrees of elevation crosses nearly six times as much atmosphere as a vertical one, and at five degrees about eleven times, so the absorption, the scintillation and the delay of a radio signal all rise steeply in the last few degrees above the horizon, and the ground clutter and reflections that confuse an antenna are concentrated there too. Ten degrees is where those costs stop being dominant for most links, which is why it recurs.

From low orbit most of the visible ground is near the visible horizon, because the disc of the Earth seen from four hundred kilometres is mostly a narrow ring foreshortened towards its edge. A ten-degree mask removes that ring, and with it three fifths of the area the satellite could otherwise reach. From a navigation satellite’s altitude the Earth is a much smaller disc in the sky, the ring near its edge is a smaller fraction of it, and the same mask costs a fifth.

That asymmetry is why low-orbit communication systems need far more satellites than the bare geometry suggests, and why they push their user terminals to work at low elevations despite the cost. Every degree of mask a low constellation can tolerate is worth more than a degree would be to a high one. It is also why the refraction that lifts the setting Sun matters to a radio link near the horizon: the ray that decides whether a satellite is just visible is the ray that spends the longest in the air, and it is bent, delayed and absorbed by the media between the antenna and the spacecraft more than any other.

How long a pass lasts, and the altitude where it never ends #

A footprint is an area. A pass is the time a station spends inside one, and it is a rate problem: the station is carried through the circle by the difference between the satellite’s motion and the Earth’s rotation.

The longest pass is the one straight overhead, where the station crosses the full diameter. For a satellite in the equatorial plane passing over a station on the equator the calculation is exact, because both motions are along the same line: the satellite’s angular rate is its mean motion $n$, the Earth’s is $\omega_\oplus$, and a prograde satellite crosses the footprint’s $2\lambda$ at the relative rate $n - \omega_\oplus$. A retrograde one crosses at $n + \omega_\oplus$. Every other pass, for every other geometry, is a shorter chord of the same circle.

The low end of the curve is the working life of every low-orbit ground station. A pass of the International Space Station’s altitude above ten degrees lasts under seven minutes even when it is directly overhead, and most passes are chords that last less. A station that wants to download a day’s data from such a satellite has a handful of these windows, each a few minutes, and every Doppler curve used to measure a spacecraft’s position from its frequency is confined to one of them.

The difference between the prograde and retrograde passes is the Earth’s rotation doing visible work. At 420 kilometres it is about thirteen per cent: the ground turns under a prograde satellite in the same direction as the satellite moves, so the station lingers in the footprint, and against a retrograde one, so it leaves sooner.

The high end is where the figure says something no formula for a period says directly. As the altitude rises the satellite’s angular rate falls, and at one altitude it falls to the Earth’s own rate. There the relative motion is zero, the station never leaves the footprint, and the pass lasts for ever. That altitude is 35,786 kilometres, and it is usually introduced as the altitude at which the harmonic law gives a period of one sidereal day. The two descriptions are the same statement. Reaching it from the pass duration makes plain what the geostationary orbit is for: it is the one orbit whose contact with a ground station does not have to be scheduled.

The two duration figures together show the mask’s price in time rather than in area. From 11.2 minutes to 6.9 is a loss of four minutes on every overhead pass, and a larger fraction on every chord that only clips the footprint’s edge — many of which disappear entirely once a mask is imposed, because they never rise above it.

The two ends of the duration curve are also the two halves of a solution. A low satellite’s passes are minutes long and a geostationary satellite’s pass never ends, so a low satellite that talks not to the ground but to a relay in geostationary orbit is in contact for most of its own revolution rather than for a few minutes a day. The only interruptions are the stretches when the Earth itself stands between the two spacecraft. That is the arrangement NASA’s Tracking and Data Relay Satellite system was built to provide, and it replaced much of a worldwide network of ground stations whose whole purpose had been to string short passes together. The divergence at 35,786 kilometres was turned, deliberately, into a way round the brevity at 400.

Where an inclined orbit is seen most often #

A footprint and a pass duration describe one pass. A ground network needs to know how many passes a station gets in a day, and that depends on where the tracks go, which is where the geometry of the line under a satellite comes back.

The count cannot be taken from a formula without approximations that hide the interesting part, so the figures below count it directly. The orbit is flown for six days with its node regressing under the Earth’s flattening, a station is placed at each latitude on four meridians, and a pass is recorded every time the satellite rises through the mask.

The first expectation is that the equator is best, because every track crosses it. The second, after the tracks’ crowding towards their turning latitude has been seen, is that the turning latitude is best, because the tracks bunch there. Neither is right.

A station on the equator is crossed by every track, but the tracks cross it steeply and far apart, and only the few that happen to pass within its circle count. A station exactly at the turning latitude sees the bunched northern ends of the tracks, but only from one side: its circle extends twelve degrees further north, where no track goes. The station that does best sits about one footprint radius equatorward of the turning latitude. Its circle then reaches exactly up to the latitude where the tracks bunch, so every bunched northern end falls inside it, while the steep legs of the same tracks still cross the southern half of the circle on their way up and down. At 40 degrees that station gets more than twice the passes of the equator and twice the contact time.

The mask moves the best latitude #

The explanation makes a prediction that can be tested by changing one number. If the best station sits one footprint radius short of the turning latitude, then a smaller footprint should move it closer.

It does, by almost exactly the amount the footprint shrank. The peak is at 46 degrees with a 5.7-degree footprint and at 40 with a 12.5-degree one. The cost of a steep mask is not only fewer passes everywhere — the equator loses half its passes and four fifths of its contact time — but a best latitude that has moved.

The same rule applied to a different inclination should put the peak somewhere else entirely, and it does.

Sixteen degrees is 28.5 less 12.5, which is the rule holding exactly. A low-inclination orbit also spends more of its time near the equator, so the equatorial station does better here than under the steeper orbit — 4.5 passes against 2.6 — and the contrast between the equator and the best latitude is less pronounced. What does not change is that the best-served ground is a band just inside the orbit’s reach, not the middle of it.

A polar orbit has its best station at the pole #

For an orbit whose turning latitude plus its footprint passes ninety degrees, the rule runs out of room. There is no latitude beyond the footprint’s edge for the circle to waste, and every station sees the satellite.

The orbit makes about fourteen and a half revolutions a day, and a station near 82 degrees sees essentially every one. A station at 78 degrees — the latitude of Svalbard, where the largest commercial ground stations for polar satellites are built — sees eleven and a half of them and has over an hour and a half of contact a day, against twenty-two minutes on the equator. That is the whole economic case for putting an antenna on an Arctic island, and it follows from a circle of seventeen degrees drawn round every point the satellite passes over.

Near the other pole the geometry is the same, which is why Antarctic stations serve the same purpose, and why a satellite that must return its data after every orbit needs a station at one end of the Earth or the other rather than anywhere between.

What the footprint leaves out #

Terrain and the mask are not uniform. A real station has mountains on some azimuths and open sea on others, so its mask is a curve round the horizon rather than a single angle, and its footprint is a lopsided shape rather than a circle. The figures use a single angle for every direction.

A pass is a link, not only a line of sight. Whether data can be exchanged depends on the antenna’s gain, the range, the atmosphere’s absorption and the transmitter’s power, and a satellite near the edge of the footprint is further away and seen through more air than one overhead. The footprint is the region where a link is geometrically possible, which is an upper bound on where it is useful.

The pass counts are averages. Four station longitudes and six days smooth out the day-to-day pattern, which for a repeating orbit is exactly periodic: a particular station may get six passes one day and three the next, in a cycle set by the repeat of the track. The averages are what a network is sized on; the daily pattern is what an operator schedules.

And the overhead-pass durations assume the equatorial case. For an inclined orbit the satellite’s path across the footprint is not parallel to the Earth’s rotation, so the relative rate is neither $n - \omega_\oplus$ nor $n + \omega_\oplus$ but something between, and the figure’s two curves bracket every real overhead pass at the same altitude.

What the circle is for #

The footprint turns a satellite from a point into an area, and the three figures of this essay are that area read three ways: how big it is, how long a station stays in it, and how often the tracks sweep it over a given place. Each reading contains a surprise that follows from nothing but the circle and the rotation. The area saturates, so altitude stops being the answer above an Earth radius. The time diverges at one altitude, which turns out to be the geostationary one. And the best place to build a station is one footprint short of where the orbit turns.

That last result has a practical form that anyone planning a rendezvous has to work around: the passes that matter are the ones a network can see, and a network built for one inclination is badly placed for another. A polar mission and a crewed-station mission launched from the same country do not share their ideal ground stations.

Still open: whether a geosynchronous satellite is ever really still #

The divergence at 35,786 kilometres was reached by setting the relative rate to zero, which is exact only for a circular orbit in the equatorial plane. A real geostationary satellite is never quite either, and its footprint therefore moves: a few degrees of inclination and a thousandth of eccentricity are enough to trace a small closed figure under it every day, one that a narrow ground antenna has to follow and that a satellite drifting towards one of two longitudes adds a slow walk to. Past that is the question no single footprint answers: how many circles, arranged how, are needed so that some satellite is always above every point on the Earth at once.