Before we hunt traps, this page defines every symbol it uses, on this page, so nothing is borrowed blind. Read the toolkit below once; the figures make each idea concrete.
The sidereal day, in one line: the orbit plane sits still against the stars, so we measure Earth's spin against the stars too — that is the sidereal day, 86164 s, about 4 minutes shorter than the 86400 s solar day (which extra bit comes from Earth orbiting the Sun). This is why the "days" D in the repeat ratio N/D are sidereal, not solar. See Earth Rotation & Sidereal Time.
Look at the figure: east is positive λ, west is negative λ. Earth spins eastward, so between two crossings the ground has slid east under the fixed orbit — meaning the next crossing lands to the west, a negative change Δλ. That is exactly why Δλ=−ω⊕T carries a minus, and its size is the nodal spacing S=∣Δλ∣=ω⊕T.
The orbit is a great circle tilted by i. As you march along in longitude λ, the latitude ϕ rises and falls smoothly between +i and −i. The exact relation from spherical geometry (projecting the tilted great circle onto latitude) is
ϕ=arcsin(sinisinλ),
and near the equator (small λ, or small i) this reduces to the approximate sine wave
ϕ(λ)≈isinλ,
which is literally a sine wave on a flat lat–lon map. The red curve is the exact form with i=50°; the amplitude (its height) isϕmax=i (put λ=90°: arcsin(sini)=i) — that is why max latitude depends only on inclination, never on altitude.
Three points make a triangle: Earth's center O, the satellite Sat (at distance RE+h), and the ground edge G the camera just reaches. The look-angle at the satellite is η; the central angle at O is λs; the third angle at G is 180°−(η+λs). The sine rule (side/sine-of-opposite-angle is equal for all three) gives
REsinη=RE+hsin(η+λs)⇒λs=arcsin(RE(RE+h)sinη)−η,
and since arc length = radius × angle-in-radians, the full swath is W=2REλs (with λs in radians). See Remote Sensing Sensor Geometry for sensor details. Why sine rule and not cosine rule: the sine rule links each side to the sine of the angle opposite it, and here we know one side (RE, opposite the known angle η) and the side we want (RE+h, opposite the unknown η+λs) — a known side/opposite-angle pair matched to an unknown side/opposite-angle pair is exactly the sine-rule pattern. The cosine rule instead needs two sides and the included angle between them, which we don't have.
Left: after the coprime cycle N/D (with D counted in sidereal days), the many passes interleave into N equally spaced strips of width δ=360°/N; the swath W must satisfy W≥δ for gap-free coverage. Right: J2 makes Earth's equatorial bulge slowly rotate ("regress") the whole orbit plane — over weeks this shifts the entire groundtrack pattern in longitude, and only when the regression is tuned (as in sun-synchronous design) does the repeat stay locked.
The satellite's orbit plane physically moves west a little each orbit, causing the westward drift.
False. The orbit plane is (to first order) fixed in inertial space; it is the Earth that rotates eastward underneath it, so the next crossing appears farther west. The plane itself doesn't chase the ground.
Increasing altitude h widens the swath but does not change the maximum latitude.
True. Swath grows with h (the sensor's fixed view-angle η subtends more ground from higher up), but ϕmax=i is purely geometric — set by inclination alone, altitude-independent.
A polar orbit (i=90°) can image every point on Earth given enough time.
True. With ϕmax=90° the track reaches both poles, and westward drift (S) plus Earth's spin eventually walks the tracks past every longitude, so full coverage is achievable.
If two orbits have the same period they must have the same groundtrack.
False. Same T gives the same nodal spacing S, but different inclination gives a different latitude envelope (±i), and a different starting node shifts the whole pattern in longitude. Same period ≠ same track.
Using the 86400 s solar day instead of the 86164 s sidereal day gives a slightly-off but harmless answer.
False. The ~4 min/day error is small per orbit but accumulates over the repeat cycle and corrupts the integer N/D closure — the track never truly overlays. Small bias, ruined repeat math.
For a repeat orbit with N/D orbits per sidereal day, N and D must share no common factor.
True. If they shared a factor, the pattern would close in fewer orbits than N; the reduced (coprime) fraction is the true fundamental cycle.
A wider swath always shortens the worst-case (full-cycle) revisit time.
False. Once W≥δ (the track spacing) coverage is already gap-free, so widening further only piles on overlap — it does not shorten the full repeat cycle, which is fixed by N/D. Below that threshold it does help; hence "always" is what makes the statement false.
The groundtrack of a retrograde orbit (i>90°) reaches higher latitude than a prograde one with the same numerical i.
False. For i>90°, ϕmax=180°−i, which is less than 90°. A retrograde orbit at i=100° tops out at 80° latitude, not 100° (latitude can't exceed 90° anyway).
"Revisit time equals the orbital period, because the satellite comes back around every orbit."
The satellite returns to the same orbital position, but Earth has rotated ~S underneath, so it's over a different place. Revisit is governed by the repeat cycle N/D and swath, not T alone.
"Swath width and track spacing are the same thing — both describe how the passes cover the ground."
No. SwathW is what the sensor sees across-track; spacingδ=360°/N is the geometric gap between neighboring passes. Gap-free coverage needs W≥δ — they are compared, not equal.
"Since the orbit plane is fixed in space, the equator crossing longitude is the same every orbit."
The plane is fixed, but the rotating Earth presents a new longitude at each crossing; the crossing longitude shifts west by S=ω⊕T per orbit. (Also, J2 nodal regression slowly drifts the plane too.)
"Max latitude =i always, so a 70°-inclination orbit reaches latitude 70° and a 110° one reaches 110°."
The first is right; the second is wrong. For i>90° use ϕmax=180°−i=70°. Latitude physically can't exceed 90°, so the ±i rule must switch to ±(180°−i).
"To fill the globe faster, just raise the altitude — you get more orbits per day."
Backwards. Higher altitude means longer period (Kepler's third law), hence fewer orbits per day and wider track spacing S — generally slower fill, not faster.
"A sun-synchronous orbit repeats its groundtrack every day by definition."
Not necessarily. Sun-synchrony fixes the local solar time of crossings (via matched nodal regression), which is a separate condition from the integer repeat N/D. A sun-sync orbit can have a 16-day repeat, like Landsat.
"The swath formula uses the sine rule, so we could equally have used the cosine rule and gotten the same numbers."
The sine rule is chosen because we have a known side with its opposite angle (RE opposite η) matched to the unknown side with its opposite angle (RE+h opposite η+λs) — the exact sine-rule pattern. The cosine rule needs two sides and the included angle between them, which we don't have, so it would not directly isolate λs.
"Since λs is a small angle in degrees, I can just multiply 2RE by it in degrees to get the swath in km."
No — W=2REλs is arc length and demands radians. Multiplying by degrees inflates the answer by the factor 180/π≈57. Convert λs→λs×π/180 first.
Why is the sidereal day, not the solar day, the correct spin period for groundtrack math?
The orbit plane lives in an inertial (star-fixed) frame; the groundtrack compares Earth's spin to that same frame. Earth's spin relative to the stars is the sidereal day (86164 s); the extra 236 s of the solar day comes from Earth's orbital motion around the Sun, which is irrelevant here.
Why does the groundtrack look like a sine wave on a flat (rectangular) map?
The tilted great circle projects to latitude as ϕ=arcsin(sinisinλ), which oscillates smoothly between +i and −i as longitude λ advances; near the equator it flattens to ϕ≈isinλ, and on a rectangular lat–lon grid that traces a sinusoid of amplitude i.
Why does the nodal spacing S carry a minus sign in Δλ=−ω⊕T?
With east taken as positive longitude, Earth rotates eastward, so relative to the ground the next crossing lands to the west — a negative longitude change Δλ. The minus encodes the westward convention (see the sign-convention figure); S=∣Δλ∣ is its magnitude.
Why must ϕmax be independent of altitude?
Because it's set by how far the orbit plane tilts from the equator — a fixed geometric angle i (put λ=90° in ϕ=arcsin(sinisinλ) to get ϕmax=i). Raising or lowering the satellite within that plane never changes the plane's tilt.
Why is a target near the equator generally harder to revisit than one at high latitude?
Near the poles, converging meridians pack tracks closer together (denser coverage), so high-latitude targets are seen more often; equatorial targets sit where tracks are most spread out — they rely on swath overlap to avoid gaps.
Why does gap-free global coverage require W≥δ rather than W≥S?
δ=360°/N is the spacing between all interleaved tracks after the full repeat cycle, which is finer than the single-orbit spacing S. Coverage is judged against the final packed pattern δ, not the coarse per-orbit shift S.
Why does the swath's central half-angle λs grow faster than the sensor half-angle η near the horizon?
In sin(η+λs)=RERE+hsinη, as η grows the right side approaches 1 and the arcsine steepens — near the limb the line of sight grazes Earth's curvature, so a small extra look-angle sweeps a large ground arc.
What is the groundtrack for a perfectly equatorial orbit, i=0°?
ϕmax=0°, so ϕ=arcsin(sin0°sinλ)=0 for all λ: the track is a straight line along the equator, drifting west by S each orbit but never leaving latitude 0 — no sine wave.
What happens to the "sine wave" as i→90°?
The latitude swing grows to ±90°, so the track runs almost straight north–south, reaching both poles; the exact curve ϕ=arcsin(sinisinλ) becomes maximally "tall," nearly vertical near the poles on a rectangular map.
If the swath exactly equals the track spacing, W=δ, what coverage do you get?
Exactly seamless, zero-overlap coverage — every point is imaged once with no gaps and no redundancy. It's the critical threshold; any narrower and stripes of ground are missed.
For a geostationary orbit (T=1 sidereal day, i=0), what does the groundtrack degenerate to?
S=360° per orbit means the crossing returns to the same longitude — the groundtrack collapses to a single stationary point on the equator. (Nonzero i would make it a small figure-eight.)
What is the revisit if the swath is so wide it overlaps every neighboring track on the very next orbit?
Best-case revisit approaches one orbital period — consecutive swaths already cover the target's neighborhood, so you don't wait for the full N/D cycle.
What happens to the repeat cycle if the period T is irrational relative to the sidereal day?
No integers N,D satisfy NT=DTEarth exactly, so the track never exactly repeats — it densely fills a band and only approximately revisits. True repeat orbits require a rational (ideally small-integer) ratio.
What is ϕmax for the degenerate case i=180° (fully retrograde equatorial)?
ϕmax=180°−180°=0° — an equatorial track again, but the satellite travels westward relative to a prograde one; the groundtrack still hugs the equator.
How does J2 nodal regression change a long-term groundtrack, and why can it be an asset?
The equatorial bulge slowly rotates the orbit plane, drifting the whole pattern in longitude over weeks; if left uncorrected it breaks the exact repeat, but if tuned it powers sun-synchrony, locking crossing times to the Sun.
Recall Self-check before you leave
Which single quantity is set purely by inclination and nothing else?
Maximum latitude ϕmax ::: ϕmax=i for i≤90°, else 180°−i — altitude, period, and swath are all irrelevant to it.
