Ground Tracks and Orbit Regimes
Ground-track construction, westward shift, repeat tracks and LEO/MEO/GEO/Molniya
The westward shift
Plot the point directly beneath a satellite, minute by minute, on a flat map of the world. What comes out looks almost like a sine wave: north to a maximum latitude, back down through the equator, south to the mirror-image minimum, and up again.
Almost. Follow it round for one full revolution and the curve does not return to where it started. It lands some distance to the west.
That gap is not an error, and it is not a perturbation. The orbit is exactly where it was — that is what an inertial frame means. The planet moved.
And that gap is the entire story of this lesson. It determines which countries a satellite can talk to, how many spacecraft a constellation needs, whether an imaging satellite sees the same terrain in the same lighting every week, and why the whole world's television is parked in a single ring 35,786 km up.
Watch first · presented by Shoaib Aziz
Eppur si muove. (And yet it moves.)
A ground track is the locus of the sub-satellite point: the spot on the surface directly below, where a line from Earth's centre to the satellite pierces the crust.
For a circular orbit the construction takes two steps and no calculus. Measure the argument of latitude — the angle travelled around the orbit from the ascending node. Then latitude and longitude follow from spherical trigonometry:
Latitude and longitude of the sub-satellite point. The final term is Earth's rotation — remove it and you get the track on a hypothetical non-rotating Earth, which is a closed curve.
💡Maximum latitude equals inclination
Set in the latitude formula and . That is not an approximation.
So a satellite at — the ISS — never passes over anywhere north of 51.6° N. Not over Stockholm, not over Anchorage, not over most of Scandinavia. The crew has never seen the Arctic from directly above.
This single fact drives a surprising amount of mission design. Want to image the poles? You need . Want to serve high-latitude Russia? A geostationary satellite at sits on the horizon from Murmansk, which is why the Soviets built the Molniya orbit at instead. Want global coverage from a low orbit? The inclination is not a free parameter — it is set by the highest latitude you must reach.
For retrograde orbits () the maximum latitude is : a satellite at reaches 82°, high enough for polar imaging but not quite over the pole itself.
Figure 3.6
Now the westward shift. During one orbital period , Earth turns by . The orbital plane does not move, so the track restarts that much further west:
Westward shift per revolution. Note the denominator is the sidereal day of 86,164 s, not 86,400 — using the wrong one drifts your track by a degree a day.
Figure 3.7
📐Worked Example: why the ISS pattern repeats every three days
ISS: km, so s = 92.8 minutes.
Revolutions per sidereal day: .
Westward shift per revolution: .
Now look at that 15.474. Only the fractional part decides whether the track ever repeats. A whole number would mean the pattern closes after exactly one day. Instead we have 0.474 of a revolution left over daily, so each day's track is offset from the last by about — most of the way round the planet.
A repeat needs to be a ratio of small whole numbers, and it very nearly is: , which would close after 31 revolutions in 2 days. Nearly. The 0.026 of a revolution left over walks the track about 9° per two-day cycle, so the pattern drifts rather than locking.
And the ISS could not hold an exact repeat even if it wanted to: atmospheric drag lowers continuously and reboosts raise it back in steps, so the period is never constant for long. Repeat tracks belong to satellites that are flown to keep them.
Compare a designed repeat orbit. Landsat 8 is specified as 233 revolutions in exactly 16 days — revolutions per day, an exact rational. The altitude was solved backwards from that requirement rather than chosen first, so every 16 days the satellite passes over the same field from the same geometry, and images can be differenced pixel for pixel across decades.
| Regime | Altitude | Period | Typical i | What it is for | The cost |
|---|---|---|---|---|---|
| LEO | 200–2000 km | 90–127 min | 0–98° | Imaging, ISS, Starlink — close means high resolution and low latency | Fast tracks, brief passes, drag |
| MEO | 2000–35,786 km | 2–24 h | 55° | GNSS: four satellites visible from anywhere | Radiation belts, higher launch cost |
| GEO | 35,786 km | 23 h 56 m | 0° | Broadcast, weather — one fixed dish, no tracking | No polar coverage, 240 ms round-trip delay |
| Molniya | 500 × 39,850 km | 12 h | 63.4° | High-latitude comms — hours of apogee dwell | Extreme eccentricity, crosses the radiation belts twice per orbit |
| SSO | 600–800 km | ~100 min | ~98° | Imaging with constant lighting every pass | Retrograde, so launch costs more |
Figure 3.8
💡Why GEO is a dot, and why Molniya is a figure-eight
GEO. Period exactly one sidereal day, , . The satellite goes round at precisely the rate the ground turns beneath it, so the westward shift is exactly zero and the ground track collapses to a single stationary point. This is why your satellite dish is bolted to a wall and never moves — and it is the reason a strip of orbit 35,786 km up is regulated real estate, with slots allocated by international treaty.
Geosynchronous but inclined. Same period, but tilt it. The satellite still returns to the same longitude every day, but now it swings north and south by during the day. The track becomes a figure-eight (an analemma), crossing itself at the equator. Several early comsats flew these, and inclined GEO satellites at end of life drift into exactly this pattern as station-keeping stops.
Molniya. Period is half a sidereal day, and . By Kepler's second law the satellite crawls through apogee, spending roughly eight of its twelve hours above 40,000 km over the northern hemisphere and then plunging through perigee in under an hour. Three satellites, phased 8 hours apart, give continuous coverage of the Arctic — from a country where geostationary satellites are useless above about 70° latitude.
🕰Wireless World, October 1945
Arthur C. Clarke — then a 27-year-old radar officer, not yet a novelist — published a four-page article called Extra-Terrestrial Relays pointing out that a satellite at 35,786 km would have a 24-hour period, and that three of them, 120° apart, could relay radio to essentially the whole planet.
This was twelve years before Sputnik. Nothing had ever been in orbit. Clarke assumed the relays would have to be crewed, because he could not imagine vacuum tubes lasting unattended for years.
He did not patent it. He later remarked that he had considered the idea too fanciful to be worth the fee.
The geostationary belt is now formally called the Clarke Orbit, and carries several hundred satellites worth tens of billions of dollars.
One more construction, because it is the one that turns the westward shift from a nuisance into a design tool.
A repeat ground track requires that after days the satellite has completed an exact whole number of revolutions relative to the rotating Earth. That is a rational-number condition on the period, and since period depends only on , it is really a condition on altitude:
Choose the repeat cycle first, then solve backwards for the altitude that delivers it. This is how every Earth-observation orbit is actually specified.
↓But there is a problem
Everything in this unit has quietly leaned on two shortcuts.
The first is in the very first equation we used: ECI to ECEF is "a single rotation about through , ignoring polar motion and precession." Ignoring them is not free. Earth's axis is drifting across the sky at 50 arcseconds a year, so an "ECI" position from a 2026 TLE and a J2000 position differ by tens of kilometres. The ground itself wanders on the axis by ten metres and slides on its plates by centimetres a year.
The second shortcut is the word "ECI" itself. Lesson 1 admitted it is a family — J2000, GCRF, TEME, mean-of-date, true-of-date — and then used it as if it were one thing.
There is also a deeper debt, about time: every ground track needed the satellite's position as a function of the clock, and for anything but a circle there is no formula for that. We will pay it. But first, the frames. Before you can ask when a satellite is somewhere, everyone involved has to agree on what where means — and for two thousand years, the reference everyone used was quietly moving.