🕒 Not Geostationary Time on Geostationary Orbit 🚀
In GPS satellites, which orbit the Earth every 12 hours at the orbit radius 4 times the Earth radius, time runs at about half a nanosecond per second faster than on the Earth. That creates major problems for geolocation, if time passage difference is ignored. Another very popular orbit is geostationary orbit, where a satellite hangs over the same meridian, because such a satellite has the same 24-hour rotation period as the Earth. The period determines the radius of geostationary orbit (Kepler’s third law), which is about 7 times the Earth radius. What about the “time speed” there? We can do the same estimate we did for GPS orbit, by replacing 12-hour period with 24-hour period and 4-times-the-Earth radius with 7-times- the-Earth radius. Combined gravitational and relativistic time differences give us:
6/7 GM/(Rc²) — 96 π²R²/(24×3600×c)².
With G = 6.67/10 ¹¹, M = 5.97×10 ²⁴, R = 6.37×10 ⁶, c = 3×10 ⁸, π = 3.14, it is:
5.95/10 ¹⁰–0.57/10 ¹⁰ = 0.538/10⁹ sec.
Which means: for 1 second passed on the Earth, 1.000000000538 seconds pass in a geostationary satellite. Time difference is still about half a nanosecond per second (not far from the estimate of the time difference for GPS: 0.445/10⁹ sec).
