Enhanced term of order $G^3$ in the light travel time: discussion for some solar system experiments

It is generally believed that knowing the light travel time up to the post-post-Minkowskian level (terms in $G^2$) is sufficient for modelling the most accurate experiments designed to test general relativity in a foreseeable future. However, we have recently brought a rigorous justification of the existence of an enhanced term of order $G^3$ which becomes larger than some first-order contributions like the gravitomagnetic effect due to the rotation of the Sun or the solar quadrupole moment for light rays almost grazing the solar surface. We show that this enhanced term must be taken into account in solar system experiments aiming to reach an accuracy less than $10^{-7}$ in measuring the post-Newtonian parameter $\gamma$.