Q4Satellite Communication
Question
Q.4. A quasi-GEO satellite is in a circular equatorial orbit close to geosynchronous altitude. The quasi-GEO satellite, however, does not have a period of one sidereal day; its orbital period is exactly 24 hour-one solar day. Calculate: (i) The radius of the orbit (ii) Is the satellite moving towards the east or towards the west?
Answer
A quasi-GEO satellite with an orbital period of exactly one 24-hour solar day (rather than the 23h56m sidereal day of a true GEO satellite) requires a slightly larger orbital radius of approximately 42,240 km (compared to true GEO's 42,164 km), and since its period exceeds Earth's rotation period, the satellite gradually drifts westward relative to the ground.
Given: the quasi-GEO satellite's orbital period is exactly 24 hours (one solar day = 86,400 s), rather than the 23 hours 56 minutes 4 seconds (86,164 s) sidereal day period of a true geostationary satellite (a true GEO period matches Earth's actual rotation period relative to the fixed stars, not the sun).
(i) Radius of the orbit: using Kepler's third law relating orbital period T to orbital radius r for a circular orbit around Earth (mass M, gravitational constant G, standard gravitational parameter μ=GM≈3.986×10¹⁴ m³/s²):
Substituting T = 86,400 s and μ = 3.986×10¹⁴ m³/s²:
This is slightly larger than the true GEO orbital radius (computed the same way using the sidereal period T=86,164 s), which works out to approximately 42,164 km — since a longer orbital period corresponds to a larger orbital radius (slower orbital speed, as required by Kepler's third law) for a given central body.
(ii) Direction of drift: since the quasi-GEO satellite's orbital period (24 h, the solar day) is slightly longer than the time it takes the Earth to complete one full rotation relative to the fixed stars (23h 56m 4s, the sidereal day), the satellite completes each orbit slightly more slowly than the Earth rotates beneath it. This means that, relative to a fixed point on the Earth's surface, the sub-satellite point gradually falls behind the Earth's rotation — since the Earth rotates eastward faster than the satellite orbits, the satellite appears, from the ground, to slowly drift westward over time (the ground track slowly regresses westward), completing one full drift cycle around the Earth over an extended period determined by the small period mismatch between the solar and sidereal day (approximately 365.25 days, since this period mismatch accumulates to exactly one full extra sidereal rotation per year, which is precisely why the solar day and sidereal day differ in the first place).