un 2 00 3 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 virtue of 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 /c 4) and a local coordinates (Q ab /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 = δ ij q)), we obtain the solution of q. Finally, a further extension of the second-order contributions in the parametrized post-Newtonian formalism is discussed. Recently a series of space astrometric missions are proposed, e.g. LISA(Laser Interfer-ometer space Antenna), [1] GAIA (new generation of HIPPARCOS(HIgh Precision PARal-lax Collecting Satellite) for attaining angular accuracy of about 10 −6 arcsec in few hours of integration), [2] SIM(Space Interferometry Mission), [3] DIVA(Double Interferometer for Visual Astrometry), [4] FAME(Full-sky Astrometric Mapping Explorer), [5] ASTROD(Astrodynamical Space Test of Relativity using Optical Devices) [6] , Mini-ASTROD [7] and so on. LISA and GAIA are taken as cornerstones of ESA(European Space Agency). SIM and FAME have been planned by NASA(National Aeronautica and Space Administration) and DIVA is in the progress in Germany. Especially Mini-ASTROD is proposed by our Chinese colleagues and is under Phase A Study. [7] In all of above missions the light propagation has to be calculated to a more precise level, e.g. in Mini-ASTROD the precision is about 10 −13 , which is the second order post Newtonian(2PN) level (2PN precision for the sun is 10 −12 R ⊙ r 2 , where R ⊙ and r are the radius of the sun and the closest distance from light trajectory to the center of the sun). Therefore the second order contribution to light propagation is required to be considered. On other hand, when the light pass nearby the sun or any planet, the potential produced by 2 N multipole moment is [8] V mono (l/r) N , where V mono is the potential created by the mass monopole, l the length deviated to monopole and r the distance of the action. For the quadrupole of the sun (N = 2) (l/r) 2 < …