Relativity

11Jun07

According to the theory of relativity, due to their constant movement and height relative to the Earth-centered inertial reference frame, the clocks on the satellites are affected by their speed (special relativity) as well as their gravitational potential (general relativity). For the GPS satellites, general relativity predicts that the atomic clocks at GPS orbital altitudes will tick more rapidly, by about 45,900 nanoseconds (ns) per day, because they are in a weaker gravitational field than atomic clocks on Earth’s surface. Special relativity predicts that atomic clocks moving at GPS orbital speeds will tick more slowly than stationary ground clocks by about 7,200 ns per day. When combined, the discrepancy is 38 microseconds per day; a difference of 4.465 parts in 1010. To account for this, the frequency standard onboard each satellite is given a rate offset prior to launch, making it run slightly slower than the desired frequency on Earth; specifically, at 10.22999999543 MHz instead of 10.23 MHz.

GPS observation processing must also compensate for another relativistic effect, the Sagnac effect. The GPS time scale is defined in an inertial system but observations are processed in an Earth-centered, Earth-fixed (co-rotating) system, a system in which simultaneity is not uniquely defined. The Lorentz transformation between the two systems modifies the signal run time, a correction having opposite algebraic signs for satellites in the Eastern and Western celestial hemispheres. Ignoring this effect will produce an east-west error on the order of hundreds of nanoseconds, or tens of meters in position.

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