Hydrodynamics of Binary Coalescence. II. Polytropes with Gamma=5/3
  We present a new numerical study of the equilibrium and stability properties of close binary systems using the smoothed-particle hydrodynamics (SPH) technique. We adopt a simple polytropic equation of state $p=K\rho^\gam$ with $\gam=5/3$ and $K=\,$constant within each star, applicable to low-mass degenerate dwarfs as well as low-mass main-sequence stars. Along a sequence of binary equilibrium configurations for two identical stars, we demonstrate the existence of both secular and dynamical instabilities, confirming directly the results of recent analytic work. We use the SPH method to calculate the nonlinear development of the dynamical instability and to determine the final fate of the system. We find that the two stars merge together into a single, rapidly rotating object in just a few orbital periods. Equilibrium sequences are also constructed for systems containing two nonidentical stars. These sequences terminate at a Roche limit, which we can determine very accurately using SPH. For two low-mass main-sequence stars with mass ratio $q\lo0.4$ we find that the (synchronized) Roche limit configuration is secularly unstable. Degenerate binary configurations remain hydrodynamically stable all the way to the Roche limit for all mass ratios $q\ne1$. Dynamically unstable mass transfer can also lead to the rapid coalescence of a binary system, but the details of the hydrodynamic evolution are quite different. We discuss the implications of our results for double white-dwarf and W Ursae Majoris systems. 
F. A. Rasio and S. L. Shapiro