Extending the first-order post-Newtonian scheme in multiple systems to the second-order contributions to light propagation

In this paper, we extend the first-order post-Newtonian scheme in multiple systems presented by Damour-Soffel-Xu to the second-order contribution to light propagation without changing the virtueof the scheme on the linear partial differential equations of the potential and vector potential. The spatial components of the metric are extended to second order level both in a global coordinates ($q_{ij}\mathrm{/}{c}^4$) and a local coordinates ($Q_{ab}\mathrm{/}{c}^4$). The equations of $q_{ij}$ (or $Q_{ab}$) are obtained from the field equations.The relationship between $q_{ij}$ and $Q_{ab}$ are presented in this paper also. In special case of the solar system (isotropic condition is applied ($q_{ij} = \delta_{ij} q $)),weobtainthesolutionof$q$. Finally, a further extension of the second-order contributions in the parametrized post-Newtonian formalism is discussed.