Relativistic theory for time and frequency transfer to order $\vec {c}^{-3}$

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Abstract
This paper is motivated by the current development of several space missions (e.g. ACES on International Space Station) that will use Earth-orbit laser cooled atomic clocks, providing a time-keeping accuracy of the order of 5 10-17 in fractional frequency. We show that to such accuracy, the theory of frequency transfer between Earth and Space must be extended from the currently known relativistic order (which has been needed in previous space experiments such as GP-A) to the next relativistic correction of order . We find that the frequency transfer includes the first and second-order Doppler contributions, the Einstein gravitational red-shift and, at the order , a mixture of these effects. As for the time transfer, it contains the standard Shapiro time delay, and we present an expression also including the first and second-order Sagnac corrections. Higher-order relativistic corrections, at least , are numerically negligible for time and frequency transfers in these experiments, being for instance of order 10-20 in fractional frequency. Particular attention is paid to the problem of the frequency transfer in the two-way experimental configuration. In this case we find a simple theoretical expression which extends the previous formula (Vessot et al. [CITE]) to the next order . In the Appendix we present the detailed proofs of all the formulas which will be needed in such experiments.