Global Positioning System Clocks
Cesium atomic clocks operate by counting hyperfine transitions of cesium atoms that occur roughly 10 billion times per second at a very stable frequency provided by nature. The precise number of such transitions was originally calibrated by astronomers and is now adopted by internationa lagreement as the definition of one atomic second. To achieve high location precision, the ticks of the atomic clock must be known to an accuracy of 20-30 nanoseconds. Because the satellites are moving relative to and above ground observers, Relativity must be taken into account.The Global Positioning System is based on the principle of the constancy of
c in a local inertial frame: the Earth-Centered Inertial (ECI) frame.Time dilation of moving clocks is significant forclocks in the satellites as well as clocks at rest on the ground.Special Relativity predicts that the on-board atomic clocks should fall behind ground clocks by about 7 microseconds per day because of the slower ticking rate due to the time dilation effect. General Relativity predicts that satellite clocks will seem to tick faster than the surface clocks by 45microseconds per day. The total relativistic effect is about 38 microseconds per day. This is a huge difference compared to the required accuracy, that is, 38,000 ns as compared to 25 ns, the former being 1,500 times larger.
To compensate for the General Relativistic effect, GPS engineers slow down the satellite clock frequency at pre-launch so that when the satellites are orbiting the clocks will have the same rate as the reference atomic clocks at the Global Positioning System ground stations. A clock whose natural ticking frequency has been pre-corrected on theground for relativity changes in orbit is a “GPS clock.” A Global Positioning System clock can be used to determine local time in the surface frame atany point along the orbit. The satellite clocks are reset in rate before launch to compensate for relativistic effects by changing the international definition of the number of atomic transitions that constitute a one-second interval. With this re-definition, the clocks on board the satellites run at nearly the same rates as ground clocks.Global Positioning System receivers have a built in computer chip that does the necessary relativistic calculations to find the user’s location. Since the ground receivers rotate in ECEF, satellite positionschange with each measurement. So the receiver must perform a different rotation for each measurement made into some common inertial frame. After solving the propagation delay equations, a final rotation mustbe performed into the ECEF to determine the receiver’s position. This complexity – where ground and satellites are both moving – is simpler to describe in an inertial reference frame, ECI, centered at the earth’s center of mass, which center is moving at constant velocity. For the solar system, an International Celestial Reference Frame (ICRF) is similarly defined, centered at the solar system barycenter.It can be shown by sample configurations that path-dependent discrepancies in the rotating ECEF frame are inescapable by any practical means, while synchronization in the underlying ECI frame is self consistent.For the Global Positioning System this means that synchronization of the entire system of ground-based and orbiting atomic clocks is performed in the local inertial frame, or ECI coordinate system.