Numerical study of relativistic frequency shift for the cold-atom clock experiment in space

This paper is motivated by the development of several space missions using an Earth-orbit laser-cooled atomic clock, providing a time-keeping accuracy of the order of 10-16-10-18 in fractional frequency. We focus here on a particular part of the future data processing, namely the relativistic effects on frequency shift. These effects appear to be numerous and intricate, and it is important to precisely quantify their order of magnitude. Obviously, at this level of accuracy, a detailed analysis of all natural or artificial potential sources of error is required, and such a study is still missing at this time. We present here a numerical study of one-way relativistic frequency shifts of orders 1/c2, 1/c3 and 1/c4. These shifts are computed in the case of the ACES mission, i.e. a clock aboard the International Space Station and passing above a mid-latitude observing site. We obtain orders of magnitude for all interesting relativistic effects. We show that the influence on frequency shift of the mass quadrupole moment J2 of the Earth at the order 1/c3 has an amplitude around 10-18, below the expected sensitivity of ACES but close to the one of future missions such as RACE