7 Doppler Effect

Since orbiting clocks have had their rate adjusted so that they beat coordinate time, and since responsibility for correcting for the periodic relativistic effect due to eccentricity has been delegated to receivers, one must take extreme care in discussing the Doppler effect for signals transmitted from satellites. Even though second-order Doppler effects have been accounted for, for earth-fixed users there will still be a first-order (longitudinal) Doppler shift, which has to be dealt with by receivers. As is well known, in a static gravitational field coordinate frequency is conserved during propagation of an electromagnetic signal along a null geodesic. If one takes into account only the monopole and quadrupole contributions to earth’s gravitational field, then the field is static and one can exploit this fact to discuss the Doppler effect.

Consider the transmission of signals from rate-adjusted transmitters orbiting on GPS satellites. Let the gravitational potential and velocity of the satellite be V (r ) ≡ V j j, and v j, respectively. Let the frequency of the satellite transmission, before the rate adjustment is done, be f0 = 10.23 MHz. After taking into account the rate adjustment discussed previously, it is straightforward to show that for a receiver of velocity vR and gravitational potential VR (in ECI coordinates), the received frequency is

[ 2 ] f = f 1 + −-VR-+-vR∕2-+-Φ0-+--2GME--∕a-+-2Vj- (1-−-N-⋅-vR∕c), (46 ) R 0 c2 (1 − N ⋅ vj∕c)
where N is a unit vector in the propagation direction in the local inertial frame. For a receiver fixed on the earth’s rotating geoid, this reduces to
[ ( )] fR = f0 1 + 2GME--- 1-− 1- (1 −-N-⋅ vR-∕c). (47 ) c2 a r (1 − N ⋅ vj∕c)
The correction term in square brackets gives rise to the eccentricity effect. The longitudinal Doppler shift factors are not affected by these adjustments; they will be of order 10–5 while the eccentricity effect is of order e × 10–10.

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