A swath is sized at the equator

An imaging satellite does not photograph the Earth. It photographs a strip. The instrument looks down and a little to either side, and as the spacecraft moves the field of view sweeps out a band of fixed width along the ground track — the swath — while everything outside it goes unseen on that pass. Whether the satellite eventually sees everything is a question about how those strips fit together, and it has an exact answer that is not the one the picture of a globe wrapped in tracks suggests.

The Landsat orbit is the clearest case to work with. It makes 233 revolutions in 16 days, it is inclined at 98.2 degrees so that its orbital plane turns at the rate the Sun appears to move, and its main imager sees a swath 185 kilometres wide. The repeat fixes the altitude, because a ground track closes only at isolated altitudes: a revolution count per day is a period, and a period is a size. This one lands at 699.6 kilometres in a first-order calculation with the Earth’s flattening included, within a few kilometres of the 705 the mission quotes — a difference that comes from how an altitude is referred to a flattened Earth, and that moves no spacing in this essay.

So the three numbers that describe the mission were chosen for three unrelated reasons. The inclination was chosen for the lighting, the altitude for the repeat, and the swath for the resolution. What follows is the question of whether those three choices, made separately, leave any of the planet unseen — and the answer depends almost entirely on one latitude.

The picture carries the whole argument in its shape. A day’s passes leave the tropics mostly unseen and pile up at the top and bottom of the map. The satellite spends the rest of its sixteen-day cycle filling the tropical strips, and it spends none of that effort on the polar caps, which were full after a few hours. Every question about coverage is therefore a question about the equator, and the rest of this essay is the reason why, the arithmetic of it, and a result from number theory that turns out to govern the order in which the gaps close.

The spacing that matters is measured across the track #

Two neighbouring ground tracks are the same curve displaced in longitude. Everything about them is identical except that one crosses the equator a little further east. The obvious measure of how far apart they are is that displacement read along the equator, and it is the wrong measure, because a swath is laid down across the track, not along a parallel of latitude.

The right one is the east–west offset projected onto the direction perpendicular to the track. At latitude $\varphi$ the offset is $R\cos\varphi\,\Delta\lambda$, and the fraction of it that lies across the track is set by the track’s heading. The heading has to be the heading on the ground, with the Earth’s rotation subtracted from the satellite’s own motion:

$$d(\varphi) = R\,\Delta\lambda\,\cos\varphi\;\frac{|\dot\varphi|}{\sqrt{(\cos\varphi\,\dot\lambda)^2 + \dot\varphi^{2}}}.$$

If the Earth did not turn, that expression would collapse to something tidy, $R\,\Delta\lambda\sqrt{\cos^2\varphi - \cos^2 i}$, and that tidy form is worth having for one reason: it shows at a glance that the spacing is largest where $\cos\varphi$ is largest, which is the equator, and falls to zero at the latitude the orbit turns at, where the tracks run side by side. The rotation changes the number by a per cent or so for a sun-synchronous orbit and by several per cent for a prograde one, and a per cent is the size of the margin a swath is designed with, so the calculation behind the figures keeps it. The formula is checked against the tracks themselves: a point on one track, and the nearest point of a copy shifted three degrees east, measured on the sphere, agree with it to within the tolerance the drawing can resolve.

Why the equator decides it #

Put the two spacings that matter for Landsat on one plot, against latitude: the spacing between successive revolutions, which is what a single day gives, and the spacing between the 233 tracks of the whole cycle, which is what sixteen days give.

Three things can be read straight off it.

The first is the design margin. At the equator the cycle’s tracks are 168 kilometres apart across the track, and the swath is 185. The ten per cent of overlap is not generosity; it is what absorbs the drift of the real track about its nominal position between the burns that hold a repeat orbit in place against drag, and what lets adjacent scenes be registered against each other. Shrink the swath by a tenth and the equatorial strips open.

The second is that nothing else is close. At thirty degrees of latitude the same swath is 27 per cent wider than the cycle’s spacing; at sixty, 2.3 times it; at eighty, eleven times. The polar regions are imaged on every pass that comes near them. A swath is not sized for the Earth; it is sized for a single line of latitude, and every other latitude is the beneficiary of a margin it did not ask for.

That over-coverage is the reason polar ice sheets were among the first things mapped completely from orbit and tropical forests among the last, with no difference in the instruments. It is also the reason the naive intuition about a polar orbit — that it sees the poles because it flies over them — is only half of the story. It sees the poles often because every one of its tracks converges there, and it sees the equator rarely because the tracks there are as far apart as the orbit ever puts them.

The third is how little one day does. Successive revolutions are twenty-seven hundred kilometres apart at the equator and do not come within a swath’s width of each other until 80.9 degrees — a thin cap around each pole. Everywhere else on the planet, a single day’s imaging is a set of disjoint strips. That is not a limitation of this instrument. It is a consequence of the altitude that the repeat requirement chose and the swath that the resolution requirement chose, and the two were chosen for different reasons.

A day of passes is not a sample of the cycle #

The latitude plot gives the spacing after one revolution and after a whole cycle. What happens in between is the interesting part, and it is not a smooth interpolation.

Each day’s descending passes cross the equator at longitudes that step west by one revolution’s walk. Over the cycle those crossings eventually land evenly, 1.545 degrees apart, but they do not arrive in order of longitude. They arrive in the order the orbit flies them, so the second day’s tracks fall between the first day’s at offsets set by how far a day’s revolutions overshoot a whole number. The useful question is not the average spacing after a given number of days but the widest gap left, because that is the swath a mission would need in order to have seen everything by then.

The first day’s worst gap, 4,205 kilometres, is larger than the 2,691-kilometre walk between successive revolutions, and the reason is simple bookkeeping. A nodal day — one turn of the Earth relative to the orbital plane, which for a sun-synchronous orbit is the ordinary solar day rather than the sidereal one four minutes shorter — holds 14.5625 revolutions of this orbit, so a whole day completes only fourteen descending crossings. Fourteen crossings span thirteen steps of 24.72 degrees, and the arc left over between the last and the first is 38.6 degrees. The widest strip after a day is the one the fifteenth pass would have split, a few hours into the next.

After that the curve moves in stairs. A day that happens to drop a pass into the widest gap shortens it; a day whose passes all land in narrower gaps leaves the widest one exactly as it was. The plateaus are as informative as the drops: from the ninth day to the fifteenth the swath needed does not change at all, because each day’s new tracks are filling narrower strips while the widest ones wait for the last day’s passes. A mission that needs complete coverage by day nine and a mission that needs it by day fifteen need exactly the same instrument on this orbit, and a mission that can wait one day longer than that needs one with a swath half as wide.

Three widths and no more #

The flat steps in that figure come from a result that has nothing to do with satellites.

Take a circle, choose a fixed angle, and mark the multiples of it: the angle, twice the angle, three times, and so on, each wrapped round. After any number of marks the circle is cut into arcs. The three-distance theorem says those arcs come in at most three different lengths, whatever the angle and however many marks — and when there are three, the largest is the sum of the other two. It was conjectured by Hugo Steinhaus and proved independently by Vera Sós, János Surányi and Stanisław Świerczkowski in 1957 and 1958.

The descending crossings of a ground track are exactly that construction. Each one is one revolution’s walk further round the equator than the last, so after any number of passes the equator is divided into gaps of at most three widths. Counted on every day of the cycle, the widths never number more than three. The theorem also explains the staircase. When a new mark lands, it always splits one of the largest arcs, and a largest arc disappears only when every arc of that length has been split — so the widest gap holds its value until the last of its kind is gone, then falls in one step to the next length down.

What decides when those steps happen is the continued fraction of the ratio of revolutions to days. For 233/16 the expansion is $14 + \cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{3+\cfrac{1}{2}}}}$, and its convergents — the best rational approximations with small denominators — are 14, 15, 29/2, 102/7 and 233/16. Each convergent is a near-repeat hidden inside the cycle. Twenty-nine revolutions take 1.991 days, so the orbit nearly repeats in two days, and that is the sixty-four per cent drop on the second day. A hundred and two revolutions take 7.004 days — a few minutes more than seven — so that near-repeat is completed just after the seventh day ends, and the drop it produces is recorded on the eighth.

This is the same mathematics that sets the calendar. A year that is not a whole number of days is reconciled with the day by the convergents of 365.2422 — four years for one extra day, then the finer corrections — and an eclipse that repeats a third of a world away is a near-commensurability of three lunar months whose remainder is a third of a day. In each case two periods that do not divide evenly are brought into step by the best small ratio between them, and the imperfection of that ratio is what moves the pattern. Here the pattern is a strip of unimaged ground in the tropics, and the imperfection decides which day it closes on.

The same small ratios have a darker use in dynamics. A moon whose period stands in a ratio of small whole numbers to another’s receives its kicks at the same place on every circuit, and a resonance of that kind clears gaps and locks moons. A repeat ground track is a resonance chosen on purpose — the orbit and the Earth’s rotation locked at 233 to 16 — and the near-resonances at 29 to 2 and 102 to 7 are the ones it passes through on the way.

A wide swath trades the cycle for days #

The Landsat instrument is narrow because it is sharp: 185 kilometres at thirty metres a pixel. A coarser instrument on the same orbit can see far more of the ground at once, and the Terra satellite, which flies the identical 233/16 track, carries one with a swath of 2,330 kilometres.

The sixteen-day cycle has become irrelevant to coverage. What matters is the second day, because the day-two near-repeat is what first brings the widest gap under 2,330 kilometres — and the staircase figure above already recorded that: the wide swath closes every gap on day 2. Poleward of thirty degrees it does not even need the second day.

The trade is exact enough to state as a product. For a given orbit the swath needed shrinks roughly as the reciprocal of the number of days allowed, because a cycle of $m$ days lays down about $m$ times as many tracks, spread round the same equator. A thirty-metre imager and a five-hundred-metre one on one spacecraft bus are therefore not two instruments with different resolutions. They are two points on one curve, revisit against resolution, with the orbit fixed and the swath the only free variable.

The same arithmetic for a different mission #

The Sentinel-2 satellites were designed later, with a wider imager at a comparable resolution, and their orbit makes 143 revolutions in 10 days at 98.62 degrees. The repeat places them higher than Landsat, and their swath is 290 kilometres.

Two things carry over and one does not. The equator is again the binding latitude, and the margin is again a few per cent — 290 against 274 — which is the same design habit applied to a different orbit. The staircase again has plateaus. What does not carry over is the shape of the stairs: the convergents of 143/10 are 14, 43/3 and 143/10, so the hidden near-repeat is at three days rather than two and seven, and the steps fall on different days. Two orbits a hundred kilometres apart in altitude close their gaps on unrelated schedules, set not by their altitudes but by the digits of a fraction.

The mission’s answer to the ten-day cycle is not a wider swath but a second satellite in the same orbit, half a cycle behind, which lays its tracks exactly between the first one’s and halves the revisit. That is the constellation version of the same trade, and it moves the problem from a single track to a pattern of them.

An inclined orbit leaves its tropics open #

Every example so far has been nearly polar, and a nearly polar orbit is the special case in which the tracks turn near the poles and cover the whole globe between them. An orbit inclined at 51.6 degrees — the inclination of the International Space Station — turns at 51.6 degrees, and the spacing curve ends there.

The shape is the same and it is compressed into a narrower band of latitude, which makes the crowding near the turning latitude far more abrupt. Successive revolutions are 2,109 kilometres apart across the track at the equator, 913 at 45 degrees, and 281 at 51 — a sevenfold squeeze in the last six degrees before the orbit turns — and a region lying just below that latitude is imaged far more often than a tropical one of the same area. Nothing north of 51.6 degrees is imaged at all. That is why an inclined orbit is chosen for a mission about the mid-latitudes, and a polar one for a mission about the planet.

The low orbit makes the point sharper in one other way. A two-day repeat at 344 kilometres is a short cycle, so its tracks never interleave finely: the whole cycle leaves the equator crossed at intervals of 1,054 kilometres across the track. A narrow imager on an orbit like this is not a mapping instrument at all. It is a sampling instrument, and what it samples is decided by where the tracks happen to fall.

What the calculation leaves out #

It counts one direction of passage. Every figure here uses descending crossings only. An optical imager on a sun-synchronous orbit genuinely works that way, because its ascending passes are over the night side; a radar or a thermal instrument can use both, which adds a second interleaved set of tracks and changes the staircase.

It assumes the instrument looks straight down. Many modern imagers can be pointed off-nadir, which lets a satellite image a strip well outside its nominal swath at the cost of resolution and a slanted view. A pointable instrument turns the swath from a fixed width into an accessible corridor much wider than any single image, and then coverage becomes a scheduling problem rather than a geometric one.

It treats the track as exactly repeating. The real track drifts about its nominal position between manoeuvres by up to a few kilometres, which is what the equatorial margin is for, and the orbit that has to be paid for every year is the price of keeping it within that band.

It says nothing about seeing the ground. A tropical scene imaged on schedule is frequently a picture of cloud, and the effective revisit time for a usable optical image of a rainforest is months, not days. The geometry sets an upper bound on how often the ground can be seen. The weather sets the number that matters.

And it uses a sphere. The swath edges are computed on a sphere of the Earth’s equatorial radius, and the altitude from first-order theory in the flattening. Both introduce errors at the kilometre level in position and none that would move a spacing by more than a small fraction of a per cent, which is below every margin quoted above.

The trade, stated once #

Collected in one place, the argument is short. The tracks of a near-polar orbit are separated across the track by an amount that is largest at the equator and falls to nothing at the turning latitude, so a swath that closes the gaps at the equator closes them everywhere. The number of days needed to close them is the number of days needed for the widest equatorial gap to fall below the swath, and that gap falls in steps whose timing is fixed by the continued fraction of the revolutions per day. The best a mission can do is choose a repeat whose near-repeats fall where it wants its coverage to improve, and a swath a few per cent wider than the spacing of the cycle it has.

The same reasoning — sampling a surface with a moving strip, and asking when the strips first meet — appears wherever an instrument builds up coverage from lines. An interferometer fills its plane of spatial frequencies with the tracks that its baselines draw as the Earth turns, and its gaps are placed by the same interplay between a rotation and a set of fixed spacings. The object being sampled is different and the arithmetic of strips is not.

Still open: how long a station waits for a satellite it can see #

Everything here has been about what the satellite sees. The reverse question — which stations on the ground can see the satellite, for how long, and how often — is the one every ground network is built around, and its answer is a circle drawn round each point beneath the spacecraft, whose radius depends on altitude in a way that saturates. Past that is the question no single track can answer at all: how to arrange several of them so that some satellite is always overhead everywhere, which is a problem about six numbers that fix an orbit chosen for many orbits at once.