[AntonioLao]

All successful data downloads of GPS signals must be processed through computer algorithms that took into account both special relativity for satellite’s velocity and general relativity for the differences in the gravitational potential.

These relativistic corrections can be divided into two distinct parts, see Jorgensen, P.S. (1986), “Relativistic correction in GPS user equipment,” Proc. Position location and navigation system 1986 (PLANS), 177-183, Las Vegas, Nevada, 4-7 November 1986, IEEE, New York. The less problematic part is proportional to the eccentricity. But for perfect circular orbits, this is zero. For eccentricity of 0.02, the time differences ∆t can be as large as 45 nanoseconds, corresponding to a measured distance of 15 meters or about 50 feet. However, there is an exact mathematical expression for solving it as a function of the eccentricity, the semi-major axis, the eccentric anomaly, with the gravitational constant of earthbound systems and of course the vacuum speed constant of light. The more problematic relativistic correction cannot be avoided by any satellites. But again fortunately, it is independent of orbital eccentricity. The corresponding relative frequency offset is ∆f/f=-4.4647x10^(-10). This is an increase of 38000 nanoseconds per day. Its measured distance error would be as much as 12700 meters or 41560 feet and indicating that clocks in earth’s orbit run faster. The apparent change in frequency is ∆f=0.0045674 cycle per second at the fundamental frequency of 10.23 megacycles per second. Fortunately, this is easily corrected by adjusting the frequency of the satellite clocks before launching to 10.22999999543 megacycles per second.

A big technical surprise is that while for absolute geocentric point positioning relativistic considerations are necessary to avoid errors in measured distances, for all practical purposes, the relativistic effects cancel out for all relative positioning.