3.2.37 · D5Orbital Mechanics & Astrodynamics
Question bank — Orbit types — LEO, MEO, GEO, HEO, SSO, Molniya
Everything here rests on the same backbone you already met: gravity supplies the centripetal force, giving and (this is Kepler's 3rd law), plus the bulge that makes orbit planes and perigees drift (J2 Perturbation and Nodal Precession).
True or false — justify
A "true/false" is only worth points if you can name the physics. Read each, decide, then reveal.
T/F: A higher orbit always moves faster because it is farther from Earth.
False. Speed is , which shrinks as grows — GEO crawls at ~3.07 km/s while LEO races at ~7.7 km/s. The "farther = faster" feeling is car intuition, not orbit physics.
T/F: All six orbit types obey different laws of motion.
False. They are all the same two-body ellipse under gravity; only altitude, eccentricity and inclination differ. See Two-Body Problem and the Vis-Viva Equation.
T/F: A geosynchronous orbit is automatically geostationary.
False. Geosynchronous only requires period = one sidereal day; if it is tilted () or stretched () it traces a figure-8 in the sky. Geostationary needs and on top.
T/F: GEO's period is exactly 24 hours.
False. It is one sidereal day, 86 164 s (~23 h 56 min). Using 86 400 s overshoots the altitude by ~50 km because Earth also orbits the Sun, adding extra rotation per solar day.
T/F: A Sun-synchronous orbit spends most of its time over the poles.
False. SSO is near-polar, but its purpose is a constant Sun angle at each latitude via nodal precession, not pole coverage. The ~98° inclination is chosen to match the Sun's ~0.9856°/day, not to hover over a pole.
T/F: Molniya's inclination is a convenient round-ish number the designers picked by taste.
False. It is the exact root of — the critical inclination where stops the perigee from drifting. Physics-forced, not arbitrary.
T/F: A circular LEO and a GEO satellite have the same orbital energy per unit mass.
False. Specific energy is ; GEO has a much larger , so its is less negative (higher). That energy gap is why reaching GEO costs more (see Hohmann Transfer and Delta-v Budgets).
T/F: Two satellites in the same-size circular orbit but different inclinations have the same period.
True. Period depends only on via ; tilt doesn't change the orbit's size, so the period is identical.
T/F: A geostationary satellite over the equator can provide good coverage to a city near the North Pole.
False. From high latitude a GEO sat sits barely on the horizon; the signal skims the atmosphere and buildings block it. That very failure is why Molniya's high northern apogee exists.
Spot the error
Each line contains a plausible-sounding statement with one broken piece of reasoning. Find it, then reveal the repair.
"Molniya loiters over the north because its engine slows it there."
No engine. It slows at apogee because of Kepler's 2nd law — equal areas in equal times means it crawls when far out. The stretched ellipse puts that slow apogee over the northern hemisphere.
"SSO needs a prograde inclination just below 90° to track the Sun."
Wrong sign. The precession formula must give a positive eastward drift, so must be negative ⇒ (retrograde, ~98°).
"GPS sits in GEO so it can stay fixed over each user."
GPS is MEO (~20 200 km, ~12-h period), not GEO. GPS deliberately moves across the sky; you need several visible satellites at once to fix a position, which a single fixed dot could never give.
"We should use the solar day (86 400 s) to compute GEO because satellites serve people, and people live by the Sun."
The satellite must match Earth's rotation relative to the stars (its inertial frame), which is the sidereal day 86 164 s. Solar day includes Earth's orbital motion around the Sun and gives the wrong altitude.
"A satellite at apogee is at its fastest because gravity has been pulling it the whole way up."
Backwards. Climbing to apogee converts kinetic energy into potential energy, so it is slowest at apogee and fastest at perigee.
"To make an orbit Sun-synchronous just raise the altitude; higher orbits precess faster."
The opposite: , so higher orbits precess slower. SSO is tuned by choosing inclination (and altitude) together, not by simply going higher.
"Molniya and GEO can't share a period because their shapes differ."
Period depends only on , not on shape. A Molniya orbit is a 12-h orbit with the same as a circular 12-h MEO; it just spends that time in a stretched ellipse instead of a circle.
"Atmospheric drag is negligible in LEO because space is a vacuum."
At 400–600 km the residual atmosphere is thin but not zero, and the satellite ploughs through it at ~7.7 km/s, so drag steadily lowers the orbit. See Atmospheric Drag and Orbital Decay.
Why questions
Answer in one sentence of mechanism, not a definition.
Why does a lower orbit have a shorter period even though it is a smaller loop and faster?
Both effects push the same way: the path is shorter and is larger, so collapses doubly — this is .
Why can the bug of perigee drift become a feature at ?
At that critical inclination , so the drift term vanishes and the perigee/apogee stay frozen — Molniya parks its apogee permanently over the north.
Why does a Molniya ground track appear to "hang" over Russia for hours?
Near apogee the satellite moves slowest (Kepler's 2nd law) and apogee is placed over the north, so it lingers over that region for ~8 of its 12 hours — visible on its Satellite Ground Tracks.
Why does reaching GEO cost more energy than reaching LEO despite GEO moving slower?
You must climb far out of the gravity well; specific orbital energy rises with , and that potential-energy gain dominates the smaller kinetic energy at GEO.
Why does an SSO keep the same Sun angle (constant shadows) at each latitude?
Its orbit plane precesses eastward at ~0.9856°/day, exactly matching Earth's motion around the Sun, so the Sun-to-plane geometry stays fixed all year.
Why is imaging usually done from LEO rather than GEO?
LEO's small distance gives fine ground resolution and a tight footprint; GEO's huge range blurs detail even though it sees a whole hemisphere at once.
Edge cases
Boundaries and degenerate inputs — the scenarios that break naive rules.
What happens to the period as eccentricity at fixed semi-major axis ?
Nothing — period depends only on , so a circle and a thin ellipse of the same share the identical period; only the speed variation around the loop changes.
What is the limiting behaviour of orbital speed as altitude ?
; an infinitely distant orbit needs essentially no speed because gravity there is vanishingly weak.
Degenerate case: an orbit with inclination and at GEO altitude — what do you get?
A true geostationary orbit: it hangs motionless over one equatorial longitude, the special (non-degenerate-purpose) point of the whole GEO family.
Edge case: at exactly (pure polar), what does the nodal precession do?
becomes zero, so the plane does not precess at all — which is why a pure polar orbit can't be Sun-synchronous and SSO must tilt slightly past 90°.
Boundary: a satellite so low its perigee dips into denser atmosphere each pass — what is the fate?
Drag bleeds energy at perigee, lowering apogee, circularising and shrinking the orbit until re-entry; the orbit decays rather than stabilising (Atmospheric Drag and Orbital Decay).
Limiting case: if a "geosynchronous" orbit has but grows from , what does the ground track become?
It opens from a fixed dot into a north–south figure-8 (analemma); the period still equals a sidereal day but it no longer stays stationary.
Recall One-line self-test
Say out loud why "higher = faster" is wrong. ::: Because decreases with — farther out, gravity is weaker, so a slower orbit balances it.