Five satellites and not four

A single satellite sees a circle of the Earth, and the circle a station can see it from grows with altitude and then stops growing. Even at the geostationary altitude it holds 42 per cent of the surface. So a service that has to reach every point on the planet at every moment — a navigation system, a global telephone network, a warning system — needs several satellites, and the question is how many, in what orbits.

The question has an exact form, and the form is the useful part. At any instant each satellite sits above one point, and the whole Earth is covered if every point on the surface lies within one footprint radius of at least one of those points. Turn that round: the Earth is not covered if there is some circle on the surface, larger than a footprint, with no satellite’s sub-point inside it. So the quantity that decides everything is the radius of the largest empty circle the satellites leave — the covering radius of their directions — and continuous coverage means that radius never exceeds the footprint at any instant of the orbit.

The largest empty circle, computed exactly #

The covering radius of a set of points on a sphere sounds like something that has to be estimated on a grid, and it does not. The largest empty circle is always centred at a point equidistant from three of the satellites — a vertex of the pattern of regions each satellite is nearest to — so it is the largest circle through any three satellites that contains no fourth. With a handful of satellites that is a few hundred triples, and the answer is exact. The figures here compute it that way, and check the construction once against a brute-force search over twenty thousand surface points, which can approach the exact value from below and never pass it.

Five satellites at 11,605 kilometres were not chosen at random. That altitude is where a horizon footprint is 69.2 degrees, and 69.2 degrees is the worst covering radius the best five-satellite pattern ever reaches. At the instant drawn above the empty circle is only 67.6 degrees, but the pattern turns, and at other instants it opens further.

The worst instant decides it #

Lower the constellation and the footprints shrink, while the pattern’s geometry, being a matter of angles, stays exactly the same.

A tenth of a per cent of the planet is a hole about the size of a large country, and it moves. That is typical of a constellation just below its coverage threshold: the gaps are small, transient and wandering, which makes them easy to miss in a snapshot and impossible to ignore for a service that promises continuity. The test has to be taken at every instant, and the figure that does it plots the covering radius through a whole orbit.

The curves make the point that a snapshot cannot. A pattern’s coverage at its best instant is irrelevant; what matters is the peak of this curve. The five-satellite pattern spends most of each orbit a degree or so inside its footprint and rises to meet it only briefly, and it is those brief instants, repeated forever, that set the altitude. The six-satellite pattern’s peak, 66.5 degrees, is lower, so it could fly lower — or keep the altitude and tolerate a mask.

Four is not enough, at any altitude #

Running the same search for every constellation size gives the table the whole subject rests on: for each number of satellites, the best pattern over number of planes, phasing and inclination, and the altitude at which it first covers the Earth.

The first row is the most striking. Four points can cover a sphere: arranged as a regular tetrahedron, they leave no empty circle larger than 70.53 degrees, which a satellite at a high enough altitude could reach. But four satellites cannot hold a tetrahedron. Each has to travel round a great circle through the Earth’s centre, and however the four circles and phases are chosen, the search finds that at some instant of every orbit all four pass close to a single great circle, leaving the hemisphere on one side of it empty. A worst hole of 90 degrees cannot be closed from any finite altitude, because no footprint reaches past a hemisphere. Continuous global coverage with circular orbits needs at least five satellites, and John Walker’s analysis of these patterns in the 1970s found the same five-satellite answer, 69.2 degrees, that the search reproduces here.

The reason four cannot hold a tetrahedron is worth a paragraph, because it is not a failure of ingenuity. Two orbital planes through the Earth’s centre always intersect, along a line through both poles of their intersection, and every satellite in circular orbit crosses every other plane’s line of nodes twice an orbit. With four satellites there are six pairs of planes and the crossings are unavoidable. At the instants when the four sub-satellite points happen to lie near one great circle — and with four bodies turning at the same rate on tilted circles, such instants recur however the phases are set — the whole hemisphere on one side of that circle is empty. The search tries every plane count, phasing and inclination on its grid and finds no pattern whose worst instant stays below 90 degrees. A designer can make the bad instant brief. Continuous coverage does not care how brief it is.

The table falls steeply at first and then slowly. Five satellites need 69.2 degrees and twelve need 47.7 — seven more satellites to take twenty-one degrees off the worst hole — while the altitude needed falls from 11,605 kilometres to 3,107. Because a footprint saturates, the last steps of altitude buy almost nothing, and past a few Earth radii the only way to cover more is more satellites.

One entry in the table is out of order, and it is worth saying so rather than smoothing it. The best twelve-satellite pattern found has a worst hole of 47.7 degrees and the best eleven-satellite pattern 47.6. A true optimum cannot get worse when a satellite is added, so the twelve-satellite value is an upper bound that the search has not driven down to the optimum. Each number in the table is the best found on a grid of inclinations, not a proof that nothing better exists.

Why nobody flies the minimum #

Five satellites covering the Earth is a theorem, not a design. With no elevation mask they have to be at 11,605 kilometres, and a station watching a satellite on its horizon is looking through the whole atmosphere. With the ten-degree mask a working link needs, the same five have to climb to 27,225 kilometres, and at that range everything else about the system becomes expensive.

The signal is the first cost. Received power falls as the square of the distance, and a satellite 27,000 kilometres away delivers about seven hundred times less power to a given antenna than one at a thousand. The delay is the second: a signal relayed from the ground up to such a satellite and back down travels more than 54,000 kilometres and takes about 0.18 seconds, against under seven milliseconds through a satellite at a thousand kilometres. A telephone call through the first is audibly awkward; a network of computers talking through it is slow in a way no amount of bandwidth repairs.

So real constellations sit at the other end of the table and beyond it, with dozens or hundreds of satellites low down, trading the count for range. The table’s value is not that anyone flies its first rows. It is that the whole curve is set by one quantity — the largest empty circle — and a design is a choice of where on the curve to stand.

The navigation systems stand somewhere in between and ask for more than single coverage. A navigation-style pattern of twenty-four satellites in six planes at about 20,000 kilometres leaves a worst single-coverage hole of under forty degrees against a ten-degree footprint of sixty-six: far more margin than covering the Earth once needs, and that margin is exactly what lets four satellites be in view everywhere at once.

What it costs to keep moving #

The last sentence of that figure’s caption is the surprising connection. The problem of placing a given number of points on a sphere so that the largest empty circle is as small as possible is a classical one in geometry, and for some numbers its answers are the regular polyhedra. Four points want to be a tetrahedron, six an octahedron, twelve an icosahedron. Satellites are points on a sphere too, with one extra constraint: they have to keep moving round circles, in step.

The table puts a number on that constraint. Six static points leave a hole of 54.74 degrees; six satellites cannot do better than 66.5. Twelve static points leave 37.38; twelve satellites, 47.7. Five static points arranged as a triangular bipyramid leave 63.43; five satellites, 69.2. The price of motion is between six and twelve degrees of covering radius for these sizes, and for four points it is the difference between a solvable problem and an impossible one. Every constellation designer is solving the polyhedron problem with a handicap, and the handicap is orbital mechanics.

The same count, arranged two ways #

The patterns in the table are delta patterns: planes of equal inclination with their ascending nodes spaced evenly round the full circle of right ascension. That is a good arrangement for inclined orbits and a poor one for polar orbits, and the difference is large enough to see on a map.

Take sixty-six satellites in six planes of eleven, at 86.4 degrees and 780 kilometres, with an 8.2-degree mask — the numbers of the Iridium telephone constellation — and put them in a delta pattern.

A fifth of the planet is uncovered, and nine satellites are crowded over some places. With near-polar planes spread round the whole circle, every plane crosses every other twice — once near each pole, where every ground track of a near-polar orbit converges — and the ascending half of one plane runs alongside the descending half of another. How a set of planes is oriented is three rotations applied in a fixed order, and a delta pattern chooses the one of them that spreads the nodes, without regard to which way each half of each plane is moving. The satellites bunch where the planes meet and leave wide lanes between the planes elsewhere.

The polar arrangement that avoids this spreads the planes’ nodes over only half the circle. Each plane’s ascending half then covers one side of the Earth and its descending half the other, and neighbouring planes run in the same direction everywhere except along one line, where the last plane’s satellites meet the first plane’s coming the other way. That is a star pattern, and with the same sixty-six satellites, spread evenly, its worst hole falls by ten degrees.

The remaining gap is at the seam. Neighbouring planes that move in the same direction can have their satellites staggered, each one filling the space between two in the next plane, and a staggered pair of planes can sit further apart without leaving a gap. Across the seam the two planes move in opposite directions, their satellites pass each other, and no stagger can be held. Those two planes must therefore sit closer together than the rest. Moving the co-rotating planes to 31.6 degrees apart and closing the seam to 22 degrees — the spacing the Iridium constellation actually uses — changes nothing about the number of satellites.

A fifth of the Earth uncovered, then a hole a couple of degrees too wide, then a sliver less than a tenth of a degree too wide — with the same sixty-six satellites at the same altitude and inclination each time. The sliver that remains belongs to the geometry drawn here, which treats a footprint as a sharp circle at a fixed mask. A real system has a little link margin at the edge of its beams, and a tenth of a degree is inside it. The arrangement moved the answer further than adding satellites could have: closing a ten-degree hole by adding planes to a delta pattern would have taken many more spacecraft.

An inclined shell leaves the poles to someone else #

The inclined patterns in the table cover the whole planet only because the search allowed any inclination. A constellation that fixes its inclination to serve populated latitudes gives up the rest by construction.

No satellite in a 53-degree shell ever passes over a latitude higher than 53 — its tracks turn there, and crowd together just short of it — and a footprint of 24 degrees reaches only to 77, so the polar caps are empty at every instant — the worst hole is the cap itself, 37.8 degrees, and no number of satellites in that shell will shrink it. The population the shell serves lives almost entirely below sixty degrees, and the large broadband constellations of the present decade are built this way, with shells near 53 degrees carrying most of their satellites and separate high-inclination shells added for the rest. The same sixty satellites at 88 degrees would cover the poles and leave a worst hole of about 33 degrees elsewhere, so the choice is not between good and bad coverage but between two different maps of who is served.

Keeping a pattern a pattern #

A pattern computed on paper survives in orbit only if every satellite in it drifts in exactly the same way, and two perturbations threaten that.

The first is the Earth’s flattening, which turns every orbital plane about the polar axis at a rate that depends on altitude and inclination. If every plane has the same altitude and the same inclination, every node regresses at the same rate, the planes turn together, and the pattern is preserved; a plane a few kilometres higher or a tenth of a degree more inclined drifts out of place over months. So the altitudes and inclinations of a constellation are held identical far more tightly than the coverage requires, because the law that ties a period to a size also ties a phasing to an altitude.

The second is drag, which lowers each satellite and, perversely, speeds it up by an amount that depends on its own area and mass, and which an operator pays for every year in propellant to hold the pattern’s phasing. Hundreds of satellites sharing one shell also share one collision environment, and the rate at which such a shell produces debris rises with the square of the number in it.

What the geometry leaves out #

Single coverage is the weakest requirement. Every figure here asks whether at least one satellite is above the mask. A navigation receiver needs four at once to solve for its position and its clock, which is why the navigation constellations fly around two dozen satellites at 20,000 kilometres, far more than single coverage from that altitude would need.

The footprint is a sharp circle. A real service degrades gradually towards the edge of a beam, and a mask of eight or ten degrees is a convention rather than a cliff. The residual tenth of a degree in the seam-corrected pattern is real in this geometry and absorbed by that gradualness in practice.

The patterns are exactly circular and two-body. Eccentric orbits change the problem, and it is not only a nuisance: orbits that linger over one hemisphere can do things circular ones cannot.

And the search is a search. The table’s values are the best found on a grid of inclinations and phasings; the twelve-satellite entry shows that the grid does not always find the optimum, and none of the values is proved minimal.

Still open: four satellites in view, and four satellites that are not on circles #

Two questions sit directly beyond this. The first is multiple coverage — how many satellites, and how arranged, so that every point sees at least two, or three, or the four a navigation fix needs — which is the same covering problem asked about the second, third and fourth nearest satellite rather than the nearest. The second is the loophole in the result that four cannot do it: it holds for circular orbits, and John Draim showed in 1987 that four satellites in suitably chosen elliptical orbits can cover the Earth continuously after all, because an eccentric orbit can move a satellite quickly away from the configuration in which four points lie on a great circle.