Through the discussions of this paper, we obtain the following conclusions: a) The 1PN gravitational potential can be expanded in terms of the time-slowly-changing multipole moments such as , , , which can be determined by observation. Its expansion is formally similar to that in terms of BD moments , and . b) For the multipole moments expansion of the 1PN potential, we suggest to adopt the standard PN gauge shown by Eq. (12). Under this gauge, the expansion of the 1PN scalar potential has the simplest form, and it is formally identical with the expansion of the Newtonian gravitational potential. c) The scalar potential under the aforesaid standard PN gauge is the simplest generalization of the Newtonian gravitational potential in the 1PN case. Then the conventional relation between the dynamical form-factor and the mass quadrupole moment of the Earth would be kept in the 1PN case, if the latter be identified as the projection of BD moments of the Earth in its co-rotating system. d) The vector potential under the aforesaid gauge can be also expanded in terms of spherical harmonics. Its expansion coefficients are related with the Cartesian time-slowly-changing multipole moments, , and radius r. Although the expansion of the vector potential is not simple under the standard PN gauge, this does not cause too much trouble at all, since in astronomical practice only few terms in the expansion of the vector potential would be needed. e) Under the standard PN gauge and rigidity approximation, the vector spherical harmonic coefficients, and , are determined by and , the projections of and in the Earth's co-rotating reference system.
© European Southern Observatory (ESO) 1998
Online publication: April 28, 1998