Irrotational and Incompressible Binary Systems in the First Post-Newtonian Approximation of General Relativity

The first post-Newtonian (PN) hydrostatic equations for an irrotational fluidare solved for an incompressible binary system. The equilibrium configurationof the binary system is given by a small deformation from the irrotationalDarwin-Riemann ellipsoid which is the solution at Newtonian order. It is foundthat the orbital separation at the innermost stable circular orbit (ISCO)decreases when one increases the compactness parameter $M_{\ast}/c^2 a_{\ast}$,in which $M_{\ast}$ and $a_{\ast}$ denote the mass and the radius of a star,respectively. If we compare the 1PN angular velocity of the binary system atthe ISCO in units of $\sqrt{M_{\ast}/a_{\ast}^3}$ with that of Newtonian order,the angular velocity at the ISCO is almost the same value as that at Newtonianorder when one increases the compactness parameter. Also, we do not find theinstability point driven by the deformation at 1PN order, where a new sequencebifurcates throughout the equilibrium sequence of the binary system until theISCO. We also investigate the validity of an ellipsoidal approximation, in which a1PN solution is obtained assuming an ellipsoidal figure and neglecting thedeformation. It is found that the ellipsoidal approximation gives a fairlyaccurate result for the total energy, total angular momentum and angularvelocity. However, if we neglect the velocity potential of 1PN order, we tendto overestimate the angular velocity at the ISCO regardless of the shape of thestar (ellipsoidal figure or deformed figure).Comment: 36 pages with 5 figures, revtex, Prog. Theor. Phys. in pres