A geostationary communications satellite moves on a circular orbit of radius m…
PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Relativity & Gravitation · The equivalence principle and gravity as geometry
The Geometry of Gravity, Ch Redshift, local inertial frames and the failure of flat spacetime
Problem
A geostationary communications satellite moves on a circular orbit of radius r=4.216×107 m (measured from the Earth's centre), so that its orbital speed satisfies v2=GM⊕/r. Ground clocks sit at radius R⊕=6.371×106 m. Take GM⊕=3.986×1014m3s−2.
((a)) Estimate the fractional rate difference contributed by height alone, GM⊕(1/R⊕−1/r)/c2, and convert it to microseconds gained per day.
((b)) Estimate the fractional rate difference contributed by orbital speed alone, −GM⊕/(2rc2), and convert it to microseconds per day.
((c)) Give the net daily offset of the satellite clock relative to a ground clock, and state whether the satellite runs fast or slow.