Darwin-Riemann Problems in Newtonian Gravity

In this paper, we have reviewed the present status of the theory ofequilibrium configurations of compact binary star systems in Newtonian gravity.Evolutionary processes of compact binary star systems due to gravitational waveemission can be divided into three stages according to the time scales andconfigurations. The evolution is quasi-stationary until a merging processstarts, since the time scale of the orbital change due to gravitational waveemission is longer than the orbital period. In this stage, equilibriumsequences can be applied to evolution of compact binary star systems. Along theequilibrium sequences, there appear several critical states where someinstability sets in or configuration changes drastically. We have discussedrelations among these critical points and have stressed the importance of themass overflow as well as the dynamical instability of orbital motions.Concerning the equilibrium sequences of binary star systems, we have summarizedclassical results of incompressible ellipsoidal configurations. Recent resultsof compressible binary star systems obtained by the ellipsoidal approximationand by numerical computations have been shown and discussed. It is important tonote that numerical computational solutions to {\it exact equations} show thatcompressibility may lead realistic neutron star binary systems to massoverflows instead of dynamical disruptions for a wide range of parameters.Comment: 17 pages, 10 figures, PTPTeX style files are include