Satellite Orbits Quiz

Core fact. GEO altitude h = 35,786 km corresponds to a semi-major axis a = 42,164 km measured from Earth's centre: h = a − R⊙ = 42,164 − 6,378.

Why the others are wrong:

• A (20,200 km) is the GPS/MEO altitude.
• B (26,578 km) is the semi-major axis of GPS and Molniya orbits — an orbital radius, not a GEO quantity.
• D (42,164 km) is the classic trap: it is the orbital radius from Earth's centre, not the altitude above the surface. Memory anchor: “42,164 from the centre, 35,786 from the ground.”

Core fact. The conventional LEO band is ~160–2,000 km: below ~160 km atmospheric drag makes orbits decay within one revolution (e.g., the ISS re-boosts at ~420 km); above 2,000 km the Van Allen inner belt begins and the regime transitions to MEO.

Why the others are wrong:

• A describes the sub-orbital / very-low-perigee region where no stable satellite can persist.
• C is the MEO band (2,000–35,786 km).
• D is GEO and beyond (supersynchronous orbits).

Anchor examples: ISS ~420 km • Starlink ~550 km • Iridium ~780 km • OneWeb ~1,200 km — all LEO.

Core fact. GPS flies at 20,200 km altitude (a = 26,578 km), inclination 55°, in six orbital planes — the defining MEO navigation architecture. The half-sidereal-day period (11 h 58 min) makes the constellation geometry repeat daily.

Why the others are wrong:

• B (780 km) is Iridium's LEO altitude — communications, not navigation.
• C (GEO) gives poor geometry at high latitudes and zero polar coverage — unsuitable for a global navigation system.
• D describes a Molniya-type HEO — Russian communications heritage, not GNSS.

Comparative anchor: all four GNSS chose MEO — GPS 20,200 • GLONASS 19,100 • Galileo 23,222 • BeiDou 21,528 km.

Core fact. The Molniya orbit's period is half the sidereal day: 43,080 s = 11 h 58 min, so the satellite completes exactly two orbits per (sidereal) day. Verify with Kepler III: T = 2π√(26,578³/3.986×10⁵) = 43,080 s. ✓

Why the others are wrong:

• A (90 min) is a LEO period — incompatible with a = 26,578 km.
• B (24 h) is a geosynchronous period — would require a = 42,164 km.
• C is close in clock time but the correct statement is half a sidereal day (86,164/2 = 43,082 s), not half a solar day (43,200 s) — the 2-minute difference matters for ground-track repeat.

Design payoff. With T = half a sidereal day and three satellites phased 8 h apart in the same orbital plane, one satellite is always near apogee over the Northern Hemisphere — continuous high-latitude coverage.

Core fact. Kepler's third (harmonic) law: T² = (4π²/μ)a³. Period depends only on the semi-major axis — not on satellite mass, and for elliptical orbits not on eccentricity either.

Why the others are wrong:

• A states the first law (ellipse law).
• B states the second law (equal-areas law).
• D is physically false: v = √(μ/r) for circular orbits — LEO ~7.8 km/s, GEO 3.07 km/s.

Exam tip. “Third law” ⇔ “the T²–a³ formula” — it is the only Kepler law containing a usable equation, and the one behind every period/altitude calculation.

Concept. Three conditions must hold simultaneously: (1) T equals the sidereal day (86,164 s); (2) the orbit lies in the equatorial plane (i = 0°); (3) motion is west-to-east, matching Earth's rotation. Then the satellite's angular rate exactly cancels Earth's, and its sub-satellite point stays fixed at one equatorial longitude. Option B is the complete statement.

Why the others fail:

• A — apparent motion depends on relative angular rate, not distance; the Moon (384,400 km) visibly drifts because its 27.3-day period differs from Earth's rotation.
• C — false: the satellite moves at 3.07 km/s in inertial space; only its velocity relative to the rotating ground is zero.
• D — no hover is used; GEO satellites thrust only for millimetres-per-second-class station-keeping corrections.

Connected idea. Same period but i ≠ 0° or e ≠ 0 → merely geosynchronous (figure-8 sky track), requiring a tracking antenna.

Concept — two compounding facts:

1. Small footprint: from 800 km the horizon-limited coverage radius is ~2,750 km — each satellite sees only a few percent of Earth's surface at useful elevation angles.
2. Short pass: with a ~100-minute period, a satellite rises, transits and sets over a given station in ~10–20 min. Another satellite must rise as one sets — a handover every few minutes.

Continuous service stitches these brief, small footprints together — hence Iridium's 66 satellites, OneWeb's ~648, Starlink's thousands.

Why the others fail: A — link margin is not the constraint, geometry is; C — ITU rules govern spectrum/filings, not constellation size; D — backwards: LEO satellites move fastest (~7.6 km/s), which is exactly why passes are short.

Concept — Kepler's second law as angular-momentum conservation. Gravity is a central force (no transverse component) ⇒ zero torque ⇒ h = r²ω conserved ⇒ the satellite sweeps equal areas in equal times, moving slowest where r is largest (apogee). For Molniya (e = 0.72) this stretches ~8 of the 11 h 58 min into the apogee half of the orbit.

Design payoff. The dwell is the mission: apogee (~39,700 km) sits over the Northern Hemisphere for ~8 h per orbit, so three satellites give continuous high-elevation coverage at latitudes where GEO fails.

Why the others fail: A — no apogee-hold thrust is used; B — at 39,700 km the atmosphere is hard vacuum, drag is zero; D — solar gravity perturbs the orbit secularly but is not the dwell mechanism (the dwell exists even in a pure two-body orbit).

Concept. Earth's equatorial bulge (J2) makes inclined orbit planes regress (nodes drift westward for prograde, eastward for retrograde). At i ≈ 98° the regression rate equals +0.9856°/day — exactly the Sun's eastward drift along the ecliptic. The orbit plane then maintains a fixed angle to the Sun–Earth line, so the satellite crosses each latitude at the same local solar time (e.g., Landsat's 10:30 a.m. descending node).

Why it matters: constant sun angle and shadowing make images comparable season after season — indispensable for crop monitoring, forestry and change detection in Kenya.

Why the others fail: B — SSO altitudes (~600–900 km) sit inside the inner Van Allen zone; radiation is managed by shielding, not inclination; C — describes GEO (a rotation-period match, a completely different mechanism); D — retrograde launches sacrifice the ~0.46 km/s eastward rotational boost.

Concept — orbital geometry of the shadow. The GEO ring lies exactly in Earth's equatorial plane. Near the equinoxes the Sun's apparent position also lies in that plane, so once per day the satellite, Earth and Sun align and the satellite enters the umbral shadow for up to ~72 minutes (the exact maximum occurs with the Sun near the ecliptic and a minimum at the satellite's longitude). Near the solstices the Sun is offset ±23.4° out of the equatorial plane, and the ring passes above/below the shadow cone entirely.

Engineering consequence. Eclipse seasons run ~45 days around each equinox; the 72-minute worst case sizes the battery (Q20), and the sudden thermal swing (solar heating to shadow) drives TCS design.

Why the others fail: A — eclipse geometry is controlled by orbital planes, not Earth–Sun distance (which varies only ±1.7%); B — lunar shadows do affect GEO (“lunar eclipses” of satellites) but are rare and brief, not seasonal; C — solar storms are a radiation/environment issue, not a shadowing mechanism.

Step-by-step:

Sanity check. The answer must sit inside the LEO band 90–127 min — it does. (OneWeb's real shells at ~1,200 km have exactly this period, ~13.1 orbits/day.)

Error watch. Forgetting to add R⊙ (using a = 1,200 km) yields a nonsense ~20-min period — always convert altitude to radius first. Option A (95 min) corresponds to ~450 km; C (118 min) to ~2,000 km; D (127 min) to ~3,500 km (just above the LEO ceiling).

Step-by-step:

Interpretation. Circular speed falls as 1/√r: LEO needs ~7.8 km/s, GEO only ~3.07 km/s. This inverse relationship surprises many students but follows directly from gravity supplying exactly the centripetal force.

Why the others fail: A (1.5 km/s) is near the Molniya apogee speed; B (2.5 km/s) corresponds to r ≈ 63,800 km (a supersynchronous disposal orbit); D (7.8 km/s) is the LEO speed, obtained by substituting r ≈ 6,600–6,900 km.

Concept — coverage geometry. From GEO (a = 42,164 km), each satellite sees ~42.5% of Earth's surface. Three satellites spaced 120° apart in longitude cover all longitudes between ~81°N and 81°S: at the edge of one satellite's coverage (central angle ~81° from its sub-satellite point) the next satellite at 120° away is only 39° from the station, comfortably above the horizon. Practical constellations fly 3–4 (plus spares) to allow for failures and overlapping footprints.

Why the others fail: B (6) and C (12) provide extra redundancy but are not the minimum; D (24) is the GPS MEO constellation size, a different orbit class with a different (revisit/geometry-based) requirement.

Precision note. “Near-global” excludes the polar caps above ~81°, which no GEO satellite can see — the reason Q18 needs HEO or polar LEO.

Step-by-step:

Cross-check with first-law geometry: r_p = a(1−e) = 7,442 km → h_p = 1,064 km — the classic Molniya values. Note r_a (45,714 km) exceeds the GEO radius (42,164 km): a Molniya apogee is higher than GEO.

Why the others fail: A (35,786 km) is the GEO altitude — the most tempting distractor; B (20,200 km) is the GPS altitude; C (26,578 km) is the semi-major axis itself — forgetting both the (1+e) factor and the R⊙ subtraction.

Step-by-step:

Interpretation. 7.59 km/s is 27,300 km/h — a Starlink satellite crosses Kenya's width (~1,100 km) in about 2.5 minutes, which is why LEO gateways need tracking antennas and constellations need frequent handovers.

Why the others fail: A (3.07 km/s) is the GEO speed; C (9.81 km/s) confuses g's numerical value with a velocity; D (11.2 km/s) is surface escape speed, requiring the vis-viva formula with the √2 factor at r = R⊙.

Requirement decomposition:

Only LEO SSO satisfies both imaging constraints simultaneously — this is why every agricultural/Earth-observation programme (Landsat, Sentinel-2, SPOT) flies sun-synchronous LEO. GEO's continuous stare cannot deliver metre-class resolution from 35,786 km (diffraction-limited aperture would be enormous).

Why the others fail: A — GEO resolution is kilometres per pixel, not 10 m; C — MEO geometry gives neither resolution nor illumination stability; D — HEO dwell is over high latitudes, and perigee imaging passes are minutes long with varying range.

Symmetric trade analysis:

The 28 dB loss difference (20log of the ~24–36× range ratio) is the entire economics of the terminal market; the ~250 ms latency difference is the entire interactivity story. There is no universal winner — interactive mass-market broadband favours LEO despite complexity; broadcast/trunk favours GEO. Option C is the only one that analyses both sides symmetrically.

Why the others fail: A — inverts the latency comparison (GEO is worst for gaming); B — violates t = d/c (same c, very different d); D — GEO cannot see the poles at all; polar-LEO shells can.

Quantitative geometry. The maximum elevation to the GEO arc from latitude φ is ε_max = 90° − φ − sin^-1(R⊙/a_GEO) = 90° − φ − 8.7°. At 78°N: ε_max ≈ 3.3° — the satellite sits on the horizon, where the path traverses ~19× the zenith atmospheric column, is blocked by terrain/vegetation, and suffers severe rain/snow attenuation. Above 81.3° the GEO arc is below the horizon entirely.

Power cannot fix geometry: raising EIRP does not lift the satellite above the horizon.

Solution analysis. Molniya HEO (i = 63.4°): apogee sub-satellite latitude reaches 63°N at ~39,700 km altitude with ~8 h dwell — three satellites give continuous, high-elevation service at 78°N. The polar-LEO alternative (e.g., Iridium's 86.4° shells) also works but needs tens of satellites and offers shorter passes.

Why the others fail: B — power doesn't change visibility; C — any single non-GEO satellite passes and sets, and MEO geometry at 78°N is no better than GEO; D — GEO is geostationary: it is either always visible or never (seasonal libration aside).

Analysis by orbit class:

• LEO: small footprints mean a ground user sees few satellites at once; guaranteeing 4-in-view globally needs many hundreds of satellites across many planes — costly and geometry changes rapidly.
• GEO: only 3–4 satellites cover the globe, but at high latitudes the visible GEO satellites crowd near the horizon with terrible GDOP, and above 81° none are visible — navigation-quality geometry is impossible at the poles.
• HEO: geometry varies enormously over the orbit; satellite availability and DOP swing between apogee dwell and perigee rush — unsuitable for a uniform global service.
• MEO (correct): altitude ~20,200 km gives each satellite a footprint covering ~40% of the globe; ~24 satellites in 6 evenly spaced planes guarantee ≥ 4 visible (usually 6–12) everywhere with good GDOP — the GPS/Galileo/GLONASS/BeiDou architecture.

Why the others fail: A — small footprints are precisely LEO's weakness for availability (many sats needed); B — GEO fails the polar/GDOP requirement despite perfect coverage geometry at the equator; C — HEO's time-varying geometry violates the simultaneous visibility requirement.

Analysis — worst-case sizing logic. Spacecraft subsystems are sized by the most demanding condition they must survive, not the average. For a GEO power system:

• At solstice: the Sun is offset ±23.4° out of the equatorial plane; the GEO ring passes entirely above (June) or below (December) the umbral cone — no eclipse.
• At equinox: the Sun lies in the equatorial plane; each satellite enters Earth's shadow once per day, for up to 72 minutes (the geosynchronous eclipse maximum).

The battery must therefore carry the full spacecraft load for 72 min with margin — that single event sizes the battery capacity (C = P × 1.2 h / (DOD · V · η)) and, through the recharge energy, the array. Designing to the solstice would leave the satellite powerless six weeks per year.

Why the others fail: A — confuses array illumination with eclipse occurrence; the array is sized to the load plus recharge, not the season; B — battery ageing is continuous and driven by DOD/temperature, not season; D — contradicts the equinox/solstice geometry entirely.