Time transfer and frequency shift to the order 1/c⁴ in the field of an axisymmetric rotating body

Within the weak-field, post-Newtonian approximation of the metric theories of gravity, we determine the one-way time transfer up to the order 1/c⁴, the unperturbed term being of order 1/c, and the frequency shift up to the order 1/c⁴. We adapt the method of the world function developed by Synge to the Nordvedt-Will parametrized post-Newtonian (PPN) formalism. We get an integral expression for the world function up to the order 1/c³ and we apply this result to the field of an isolated, axisymmetric rotating body. We give a new procedure enabling us to calculate the influence of the mass and spin multipole moments of the body on the time transfer and the frequency shift up to the order 1/c⁴. We obtain explicit formulas for the contributions of the mass, of the quadrupole moment and of the intrinsic angular momentum. In the case where the only PPN parameters different from zero are β and γ, we deduce from these results the complete expression of the frequency shift up to the order 1/c⁴. We briefly discuss the influence of the quadrupole moment and of the rotation of the Earth on the frequency shifts in the ESA’s Atomic Clock Ensemble in Space mission.