Consider the transmission of signals from rate-adjusted
transmitters orbiting on GPS satellites. Let the gravitational
potential and velocity of the satellite be
, and
, respectively. Let the frequency of the satellite transmission,
before the rate adjustment is done, be
. After taking into account the rate adjustment discussed
previously, it is straightforward to show that for a receiver of
velocity
and gravitational potential
(in ECI coordinates), the received frequency is
where
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
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
while the eccentricity effect is of order
.
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Relativity in the Global Positioning System
Neil Ashby http://www.livingreviews.org/lrr-2003-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |