Darwin-Riemann Problems in Newtonian Gravity

TL;DR

This paper reviews the equilibrium configurations and evolution of compact binary star systems in Newtonian gravity, highlighting the importance of critical points, mass overflow, and recent numerical findings on compressibility effects.

Contribution

It provides a comprehensive review of classical and recent results on binary star equilibrium sequences, including new insights from numerical solutions on compressibility effects.

Findings

Compressibility can lead to mass overflow rather than disruption.

Equilibrium sequences show critical points related to instabilities.

Numerical solutions reveal realistic neutron star binaries may undergo mass overflow.

Abstract

In this paper, we have reviewed the present status of the theory of equilibrium configurations of compact binary star systems in Newtonian gravity. Evolutionary processes of compact binary star systems due to gravitational wave emission can be divided into three stages according to the time scales and configurations. The evolution is quasi-stationary until a merging process starts, since the time scale of the orbital change due to gravitational wave emission is longer than the orbital period. In this stage, equilibrium sequences can be applied to evolution of compact binary star systems. Along the equilibrium sequences, there appear several critical states where some instability sets in or configuration changes drastically. We have discussed relations among these critical points and have stressed the importance of the mass overflow as well as the dynamical instability of orbital motions. Concerning the equilibrium sequences of binary star systems, we have summarized classical results of incompressible ellipsoidal configurations. Recent results of compressible binary star systems obtained by the ellipsoidal approximation and by numerical computations have been shown and discussed. It is important to note that numerical computational solutions to {\it exact equations} show that compressibility may lead realistic neutron star binary systems to mass overflows instead of dynamical disruptions for a wide range of parameters.