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This paper is motivated by the current development of several space missions(e.g. ACES on International Space Station) that will fly on Earth orbit lasercooled atomic clocks, providing a time-keeping accuracy of the order of5~10^{-17} in fractional frequency. We show that to such accuracy, the theoryof frequency transfer between Earth and Space must be extended from thecurrently known relativistic order 1/c^2 (which has been needed in previousspace experiments such as GP-A) to the next relativistic correction of order1/c^3. We find that the frequency transfer includes the first and second-orderDoppler contributions, the Einstein gravitational red-shift and, at the order1/c^3, a mixture of these effects. As for the time transfer, it contains thestandard Shapiro time delay, and we present an expression also including thefirst and second-order Sagnac corrections. Higher-order relativisticcorrections, at least O(1/c^4), are numerically negligible for time andfrequency transfers in these experiments, being for instance of order 10^{-20}in fractional frequency. Particular attention is paid to the problem of thefrequency transfer in the two-way experimental configuration. In this case wefind a simple theoretical expression which extends the previous formula (Vessotet al. 1980) to the next order 1/c^3. In the Appendix we present the detailedproofs of all the formulas which will be needed in such experiments.Comment: 11 pages, 2 figures, to appear in Astronomy & Astrophysic

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