Relative error on the change in elements C = (p, e, ι, Ω, ω, s, t, ψ), for different values of the refractivity at the surface (x-axis) and for different values of J₂ (see values in the inset panel). The relative error on the change in elements is computed as err (C) = |ΔC_num − ΔC_ana|∕|ΔC_num|, with ΔC being the total change in the value of the elements between the transmission and the reception of the signal. The subscripts “num” refers to the numerical predictions while the subscript “ana” refers to the analytical solutions (cf. Eqs. (B.21)). The change in s, t, and ψ is the total change including the hyperbolic and the non-hyperbolic contributions.