Solving the Darwin problem in the first post-Newtonian approximation of general relativity: compressible model

TL;DR

This paper models corotating binary stars in the first post-Newtonian approximation of general relativity using an ellipsoidal density configuration, analyzing stability and effects of tidal forces.

Contribution

It introduces an analytical ellipsoidal model for binary stars in post-Newtonian gravity and compares results with numerical data, providing insights into stability and density changes.

Findings

Orbital angular velocity increases with star compactness.

Stability features align with previous numerical results.

Tidal effects slightly decrease the stars' central density.

Abstract

Using the ellipsoidal model for the density configuration, we calculate the equilibrium sequence of the corotating binary stars of the polytropic equation of state in the first post-Newtonian approximation of general relativity. After we calibrate this model by comparing with previous numerical results, we perform the stability analysis by calculating the energy and the angular momentum of the system as a function of the orbital separation. We find that the orbital angular velocity at the energy and/or momentum minimum increases with the increase of the compactness of each star, and this fact holds irrespective of the polytropic index. These features agree with those in previous numerical works. We also show that due to the influence of the tidal field from the companion star, the central density of each star slightly decreases.