Then, let a signal be transmitted from satellite nr.
i, at position
and having velocity
in ECI coordinates, at satellite clock time
, to satellite nr.
j, at position
and having velocity
. The coordinate time at which this occurs, apart from a constant
offset, from Eq. (38
), will be
The coordinate time elapsed during propagation of the signal
to the receiver in satellite nr.
j
is in first approximation
l
/
c, where
l
is the distance between transmitter at the instant of
transmission, and receiver at the instant of reception:
The Shapiro time delay corrections to this will be discussed in
the next section. Finally, the coordinate time of arrival of the
signal is related to the time on the receiving satellite's
adjusted clock by the inverse of Eq. (48
):
Collecting these results, we get
In Eq. (50) the distance
l
is the actual propagation distance, in ECI coordinates, of the
signal. If this is expressed instead in terms of the distance
between the two satellites at the instant of transmission,
then
The extra term accounts for motion of the receiver through the
inertial frame during signal propagation. Then Eq. (50) becomes
This result contains all the relativistic corrections that
need to be considered for direct time transfer by transmission of
a time-tagged pulse from one satellite to another. The last term
in Eq. (52) should
not
be confused with the correction of Eq. (40
).
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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 |