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

TL;DR

This paper derives 1PN hydrostatic equations for irrotational, incompressible binary systems, analyzing their equilibrium configurations, stability, and the accuracy of ellipsoidal approximations, revealing minimal changes in angular velocity at ISCO with increased compactness.

Contribution

It provides the first post-Newtonian analysis of irrotational, incompressible binary systems, including equilibrium configurations and the validity of ellipsoidal approximations.

Findings

ISCO separation decreases with higher compactness.

Angular velocity at ISCO remains nearly unchanged at 1PN order for increased compactness.

Ellipsoidal approximation accurately predicts energy, angular momentum, and angular velocity.

Abstract

The first post-Newtonian (PN) hydrostatic equations for an irrotational fluid are solved for an incompressible binary system. The equilibrium configuration of the binary system is given by a small deformation from the irrotational Darwin-Riemann ellipsoid which is the solution at Newtonian order. It is found that 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 at the 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 Newtonian order when one increases the compactness parameter. Also, we do not find the instability point driven by the deformation at 1PN order, where a new sequence bifurcates throughout the equilibrium sequence of the binary system until the ISCO. We also investigate the validity of an ellipsoidal approximation, in which a 1PN solution is obtained assuming an ellipsoidal figure and neglecting the deformation. It is found that the ellipsoidal approximation gives a fairly accurate result for the total energy, total angular momentum and angular velocity. However, if we neglect the velocity potential of 1PN order, we tend to overestimate the angular velocity at the ISCO regardless of the shape of the star (ellipsoidal figure or deformed figure).