The line under a satellite

An orbit is a conic in a plane that does not turn. A ground track is that same orbit written down in a frame that turns once a day, and the whole subject is what the subtraction does.

The subtraction is not a small correction. A satellite in low orbit goes round in about ninety minutes, during which the Earth turns through twenty-three degrees, so successive passes are displaced by more than the width of the Atlantic. Nothing about the orbit changes between one pass and the next; the planet underneath has moved.

The two lines of trigonometry #

Everything follows from one rotation. With $u$ the argument of latitude — the angle from the ascending node, measured in the orbital plane — and $i$ the inclination,

$$\sin\varphi = \sin i\,\sin u,\qquad \lambda = \lambda_0 + \arctan\!\left(\cos i\,\tan u\right) - \omega_\oplus t.$$

The first says the latitude oscillates between $\pm i$ and is a pure sine in $u$ only for small inclinations. The second is the orbit’s own longitude minus the Earth’s rotation, and the minus sign is the entire content of the subject.

Three consequences are immediate and worth stating as such. A satellite passes over no latitude higher than its inclination, so nothing at $28.5°$ ever flies over London. A retrograde orbit at $i > 90°$ reaches latitude $180° - i$, so a $98°$ orbit reaches $82°$ and not $98°$. And the track crosses the equator twice per revolution — once northbound, once southbound — which is why an imaging satellite has two distinct sets of observing conditions.

The walk, and why it is what it is #

The westward displacement per revolution is $\omega_\oplus T$, with $T$ the period, and it deserves a moment because the choice of which $\omega$ and which $T$ decides whether a repeat orbit works.

$\omega_\oplus$ is the sidereal rotation rate: $360°$ in 23 h 56 m 04 s, not in 24 hours. The day that is four minutes short is short because a solar day includes one extra turn per year to keep facing the Sun, and a satellite’s inertial orbit knows nothing about the Sun. Using the solar day here is a $0.3$ per cent error, which is $0.07°$ per revolution and $400°$ per year — an entire circuit of the Earth’s worth of drift, from one wrong constant.

$T$ is the nodal period — the time between successive ascending-node crossings — and not the Keplerian one.

Repeating #

An imaging satellite that is to revisit the same ground, and a radar altimeter that is to difference one pass against another, both need the track to close. It closes only when the two rates are commensurable: $k$ revolutions in exactly $m$ sidereal days, with $k$ and $m$ whole numbers.

That is a condition on the period, and therefore on the semi-major axis, and therefore on the altitude. It is not a preference that can be traded against something else.

Longer cycles fill the gaps. A 16-day repeat has $k/m$ with $m = 16$, and there are sixteen times as many such altitudes; Landsat uses 233 revolutions in 16 days, and the gaps between adjacent one-day repeat altitudes are populated by these. The trade is coverage against revisit: a short cycle revisits often and leaves gaps between swaths, and a long one covers everything and comes back rarely.

The best-known choice is TOPEX/Poseidon and the Jason series, at 1336 km: 127 revolutions in 10 days. The altitude was not chosen for the altimeter’s convenience — it was chosen because 127/10 is a repeat that also avoids aliasing the principal tidal constituents, and a satellite altimeter that samples the ocean at a period commensurate with a tide measures the tide as a constant offset.

The other commensurability #

There is a second rate that can be made to match, and matching it produces the single most useful orbit in Earth observation.

Sun-synchrony and ground-track repetition are two separate conditions on two separate quantities, and a satellite that needs both has its inclination fixed by the first and its altitude by the second — which leaves nothing free. That is the situation of essentially every operational imaging satellite, and it is why so many of them are at 700 to 800 km and $98°$: those are not preferences, they are solutions.

Reading a track backwards #

The construction runs the other way too, and historically that was the important direction: given a ground track, what was the orbit?

Before radar tracking, an artificial satellite’s orbit was determined from optical sightings by observers scattered around the world, each reporting a time and a direction. The inclination follows immediately from the highest latitude at which the satellite is ever seen. The period follows from the interval between successive passes over one station, or more precisely from the westward walk, since the walk is the period times a known rotation rate. And the altitude follows from the period by Kepler’s third law.

That is a complete orbit — inclination, period, altitude — from nothing but the positions and times of a handful of naked-eye sightings, and it is how Sputnik’s orbit was established within days by amateurs. The ground track is not merely a consequence of the orbit; it is a sufficient statistic for most of it. What the track does not give is the eccentricity or the argument of perigee, both of which need the satellite’s distance rather than its direction. A circular-orbit assumption is being made every time an altitude is quoted from a period alone, and for most low satellites it is a good one.

Geostationary, which is the degenerate case #

Set $k/m = 1$ and $i = 0$ and the ground track collapses to a point. The period is one sidereal day, the altitude is 35,786 km, and the satellite hangs over one longitude.

It does not hang there by itself. The Earth’s equatorial cross-section is slightly elliptical — a $J_{22}$ term, with the long axis pointing at about $15°$ W and $165°$ E — so a geostationary satellite feels a longitudinal acceleration towards one of two stable points at $75°$ E and $105°$ W, and drifts unless it is held. East–west station-keeping costs a couple of metres per second a year.

North–south is far more expensive. The Sun and Moon torque the orbital plane away from the equator at about $0.85°$ a year, and holding the inclination near zero costs about 50 m/s annually — which dominates the propellant budget and therefore the operational lifetime of the spacecraft. Both of the last two sections concern a repeat orbit, and neither concerns the orbit as a geometric object: one is about the propellant it takes to keep a condition satisfied, and the other about what the satisfied condition is worth.

Neither is decided by the orbit designer alone, which is the other reason the two sit together: the propellant is a mission’s budget and the repeat interval is its science requirement, and the orbit is where the two are reconciled.

What a track is measured against #

A ground track is a computed object, and it is worth asking what fixes it in reality, because the answer is not the orbit.

The satellite’s position in inertial space comes from orbit determination — directions and ranges fitted to a dynamical model, or nowadays from GPS receivers on board, which give position directly to a few centimetres. Converting that inertial position into a latitude and longitude requires the Earth-orientation parameters: the rotation angle, the polar motion, and the precession–nutation of the axis. Those are measured by radio interferometry and published, and without them a position in space is not a position on the ground.

The error budget divides cleanly. For a low satellite the dominant term is along-track, because a timing error of one second is 7.5 km of track; cross-track errors are much smaller. For a geostationary satellite the reverse holds. And the frame conversion contributes at the metre level, which matters for altimetry and for nothing else.

That division explains a practical fact: a ground track’s predicted position is far more uncertain in when than in where. A pass predicted for a ground station is quoted to the second, and the uncertainty in the time of closest approach is what limits the pointing of a narrow-beam antenna, not the uncertainty in the direction.

What the track is for #

The reason to compute a ground track at all is that everything about an orbit’s usefulness is expressed on the ground rather than in space.

Coverage. An instrument sees a swath of some width either side of the track, and whether consecutive swaths overlap depends on the walk per revolution against the swath width at the relevant latitude. At the equator a $23°$ walk is 2,600 km, so a 200 km swath leaves enormous gaps; near the poles the tracks converge and overlap heavily — a distortion of coverage that no map projection creates and every map projection displays. Every polar-orbiting satellite images the poles many times a day and the tropics once every few days, which is the opposite of what most applications want.

Contact. A ground station sees a satellite only when it is above the local horizon — in practice above a mask of about ten degrees, which for a 420 km orbit makes the region it can be seen from a circle about 2,800 km across and the longest pass under seven minutes. The number of usable passes per day follows from where the track goes relative to the station, and it is typically four to six for a mid-latitude station and a low orbit — which is why rendezvous and phasing are planned around communication windows as much as around orbital mechanics.

Lighting. For an imaging mission the local solar time at the crossing is the whole design, as above.

Both of the last two sections are about the same thing from opposite sides: a repeat track is expensive to maintain, and it is worth maintaining because the pattern it produces is itself a measurement.

The frame, and what is being assumed #

Two assumptions are in every figure here and both are worth naming.

The first is that the orbit is circular. An eccentric orbit’s ground track is not symmetric: the satellite moves faster near perigee, so the track is stretched in longitude near apogee and compressed near perigee, and the whole pattern precesses as the argument of perigee drifts. A Molniya orbit — $63.4°$, twelve-hour period, eccentricity 0.74 — spends eight of its twelve hours over a small region near apogee, which is the entire point of it, and its ground track is a long thin loop rather than a sine wave.

The second is that the Earth’s rotation is uniform, which it is not: length of day varies by milliseconds, and the pole itself wanders. Both are negligible for a ground track and are not negligible for the geodesy that a ground track is used to do — a radar altimeter measuring sea level to a centimetre needs the Earth-fixed frame to a centimetre, and the frame is itself a product of measurement.

The drift that has to be flown against #

A repeat orbit is a solution to an equation, and the equation’s inputs do not stay put. Keeping a track repeating is an operational activity for the whole of a mission’s life.

The culprit is drag. Even at seven hundred kilometres the residual atmosphere removes energy, the semi-major axis falls, and a smaller orbit has a shorter period — so the satellite completes its revolutions slightly early and the ground track walks eastward relative to where it should be.

The arithmetic is unforgiving because the walk accumulates. A period short by a hundredth of a second puts the equator crossing about seventy metres east on the next revolution, four hundred metres a day, and a hundred and fifty kilometres a year. A mission that requires its passes to fall within a kilometre of a nominal track therefore has to intervene several times a year.

The intervention is a small prograde burn that raises the orbit slightly, which lengthens the period and starts the track walking west again. The manoeuvre is sized so that the track drifts across the allowed band and back, so that a single burn buys as long as possible before the next one.

The schedule is set by the Sun rather than by the spacecraft. Drag depends on the density of the upper atmosphere, which depends on the extreme-ultraviolet heating, which varies by a factor of several over the solar cycle and by more during a geomagnetic storm. So a satellite at solar minimum may go months between maintenance burns and at solar maximum need one every few weeks, with the same hardware in the same orbit.

There is a further subtlety that appears in the long run. Each maintenance burn is imperfect, and the errors accumulate in the orbital elements that the repeat condition does not constrain — chiefly the eccentricity, which drifts under the combination of drag and the oblateness terms. A mission with a strict track requirement therefore also has an eccentricity requirement, and the manoeuvre is planned in a plane rather than along a line.

A repeat orbit is not a place a satellite is put; it is a condition a satellite is flown to satisfy, and the propellant that maintains it is a substantial fraction of what a low-Earth mission carries.

Where the tracks cross #

There is a use for a ground track that turns its geometry into an instrument, and it exploits the one feature of the pattern that has been treated so far as incidental: the tracks intersect.

An ascending pass and a descending pass cross at a point, at two different times separated by anything from hours to days. At that crossover the satellite is looking at the same place twice, and if the instrument measures something that ought not to have changed, the difference between the two measurements is an error rather than a signal.

For a radar altimeter measuring sea surface height, that is the whole of the calibration. The sea surface at a crossover has changed by whatever the tides and the ocean have done between the two passes, which is modelled; what is left is the difference between two measurements of the same quantity, and it is dominated by the error in the satellite’s own radial position.

The number of crossovers is large. A repeat cycle covering the globe produces tens of thousands of them, distributed over the ocean, and fitting a correction to the whole set determines the radial orbit error as a function of position around the orbit.

That analysis is how altimeter orbits were validated before satellite navigation receivers flew, and it is still the check applied afterwards. The radial orbit error of a modern altimetry mission is quoted at a centimetre or two, and the crossover residuals are the evidence for it.

The technique’s limitation is geometric and follows from the ground track’s own shape. Crossovers are dense near the highest latitude the orbit reaches, because that is where ascending and descending tracks converge, and they are sparse near the equator, where the two families are nearly parallel. So the orbit error is best constrained where the ocean is least interesting.

A pattern computed from two rotations turns out to be a network of self-consistency checks, and the density of that network across the globe is fixed by the inclination and by nothing else.

The track’s shape is set by the inclination and its repetition by a commensurability, and each is worth drawing at a second value.

Where the ladder goes next #

The rung above is the swath and its geometry: how an instrument’s field of view maps onto the ground as a function of latitude, what the overlap condition is, and why a repeat cycle and a swath width jointly determine whether global coverage is achieved at all. Past that is the inverse problem — given a required revisit time and coverage, solve for the orbit — which is the constellation-design question, and whose answer for global continuous coverage is the Walker pattern rather than any single track.