On the completion of the post-Newtonian gravitational two-body problem with spin

Previous work by the authors [B. M. Barker and R. F. O\u27Connell, Phys. Rev. D 12, 329 (1975); 14, 861 (1976); B. M. Barker, G. G. Byrd, and R. F. O\u27Connell, Astrophys. J. 305, 623 (1986); B. M. Barker and R. F. O\u27Connell, Gen. Relativ. Gravit. 18, 1055 (1986)] on the post-Newtonian (order c-2) gravitational two-body problem with spin and parametrized post-Newtonian parameters γ and β was concerned with the relative position r = r1 - r2. Here this work is completed by finding the individual positions r1 and r2, which is necessary for the interpretation of certain binary-system observations. First the center of inertia rCI is found. This makes it possible to obtain the positions r1 and r2 and the center of mass rCM as a function of the relative position r, relative velocity v, and spin angular momenta S(1) and S(2) of the two bodies. Thus, if a solution r = r(t) can be obtained, then solutions r1 = r 1(t) and r2 = r2(t) can also be obtained. The final results are given in a very general coordinate system specified by four arbitrary dimensionless parameters. In particular, the spin-orbit potential energy terms VS1 and VS2 are given without going to a frame of reference where the total momentum is zero. © 1987 American Institute of Physics