8 Crosslink Ranging

Consider next the process of transferring coordinate time from one satellite clock to another by direct exchange of signals. This will be important when “Autonav” is implemented. The standard atomic clock in the transmitter satellite suffers a rate adjustment, and an eccentricity correction to get the coordinate time. Then a signal is sent to the second satellite which requires calculating a coordinate time of propagation possibly incorporating a relativistic time delay. There is then a further transformation of rate and another “e sinE” correction to get the atomic time on the receiving satellite’s clock. So that the rate adjustment does not introduce confusion into this analysis, I shall assume the rate adjustments are already accounted for and use the subscript ‘S’ to denote coordinate time measurements using rate-adjusted satellite clocks.

Then, let a signal be transmitted from satellite nr. i, at position ri and having velocity vi in ECI coordinates, at satellite clock time T (i) S, to satellite nr. j, at position rj and having velocity vj. The coordinate time at which this occurs, apart from a constant offset, from Eq. (38View Equation), will be

√ ------ (i) (i) 2--GM--ai- T = TS + c2 eisinEi. (48 )
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: ΔT = T (j) − T (i) = l∕c. 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. (48View Equation):
∘ ------- (j) (j) 2 GM aj TS = T − ----2----ej sin Ej. (49 ) c
Collecting these results, we get
∘ ------- √ ------ (j) (i) l 2--GM--aj- 2--GM--ai- TS = TS + c − c2 ej sin Ej + c2 eisin Ei. (50 )
In Eq. (50View Equation) the distance l is the actual propagation distance, in ECI coordinates, of the signal. If this is expressed instead in terms of the distance |Δr | = |rj(ti) − ri(ti)| between the two satellites at the instant of transmission, then
l = |Δr | + Δr-⋅ vj-. (51 ) c
The extra term accounts for motion of the receiver through the inertial frame during signal propagation. Then Eq. (50View Equation) becomes
(j) (i) |Δr | 2√GM---a2- 2√GM---ai Δr ⋅ vj TS = T S + -----− -----2----ej sin Ej +-----2----eisin Ei + ---2---. (52 ) c c c c

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. (52View Equation) should not be confused with the correction of Eq. (40View Equation).

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