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Relativity And The Sagnac Effect


This article describes how relativistic clock and sagnac effects are accounted for in the simulated GNSS signals.


Gravitational Frequency Shifts and Time Dilation
General relativity dictates that a clock runs slower in a gravitational field, the stronger the field the slower the clock. A GPS satellite's clock runs faster when observed from a receiver on the earth as the satellite is in a weaker gravitational field. Conversely a consequence of special relativity is that a clock moving with respect to the observer appears to run slow. As a GPS satellite is moving in the reference frame of an observer on the earth, a time dilation effect occurs and the satellite's clock appears to run slower. So the general and special relativistic effects work in opposition to one another, with the gravitational effect being the more dominant for a GPS satellite. Thus an uncompensated satellite master clock would appear to run fast to the earth-bound receiver.
In the real GPS satellites the relativistic effect is nominally compensated by reducing the master 10.23 MHz clock down by 0.00457 Hz before launch. For an ideal satellite in a circular orbit, this would remove the effect. With an elliptical orbit the satellite clock will still not be correct to an earth observer as it speeds up and comes closer to the earth on one half of it's orbit (clock slows down) and slows down and goes further from the earth in the other half of it's orbit (clock speeds up). Thus the receiver must make compensation for this according to the eccentricity of the orbit and the satellite's position within the orbit at a given time. This correction is given in ICD-GPS-200D,
Thus to simulate relativistic effects the simulator only has to apply the modification to signal timing which is equal and opposite to the compensation given in the ICD.

Sagnac Effect
The Sagnac effect is not a relativity effect but is caused by the rotation of the ECEF reference frame during the signal propagation. The speed of light delay used in pseudorange calculation depends upon the Cartesian distance between the satellite at time of transmission and the receiver at time of reception. Naturally the satellite and receiver positions must be resolved in the same reference frame in order to calculate the correct distance. In solving the orbital equations of satellite motion with the ephemeris data transmitted, the result is the satellite xyz at the time of transmission. So the receiver must add an additional term due to the rotation of the ECEF frame during the propagation delay. Thus the simulator must make an adjustment to the signal timing so the receiver gets the correct answer when compensating for the Sagnac effect in this way.
Although it is not possible to turn off/on the Sagnac effect you might wish to experiment with SimGEN scenario conditions to observe the effect on range calculation. A possible approach is to set up a satellite’s inclination angle to be 90o so that it passes through the Earth’s Z-axis (north / south pole). Since the Sagnac effect corrects the satellite position for ECEF rotation during signal propagation time, when the satellite passes close to the Earth's axis of rotation the effect would be smallest. This can be observed using SimGEN’s quick-look data.

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