| 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 $1/c^2$ (which has been needed in previous space experiments such as GP-A) to the next relativistic correction of order $1/c^3$. We find that the frequency transfer includes the first and second-order Doppler contributions, the Einstein gravitational red-shift and, at the order $1/c^3$, 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 ${\cal O}(1/c^4)$, 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 $1/c^3$. In the Appendix we present the detailed proofs of all the formulas which will be needed in such experiments.
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