Hydrodynamics of binary coalescence. II. Polytropes with Γ= 5/3

We present a new numerical study of the equilibrium and stability properties of close binary systems. We use the smoothed-particle hydrodynamics (SPH) technique both to construct accurate equilibrium configurations in three dimensions and to follow their hydrodynamic evolution. We adopt a simple polytropic equation of state p = Kρ^Γ with Γ = 5/3 and K = constant within each star, applicable to low-mass degenerate dwarfs as well as low-mass main-sequence stars. For degenerate configurations, we set the two polytropic constants equal, K = K′, independent of the mass ratio. For main-sequence stars, we adjust K and K′ so as to obtain a simple mass-radius relation of the form R/R′ = M/M′, where R′ and M′ are the radius and mass of the secondary. 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 ≲ 0.4 we find that the (synchronized) Roche limit configuration is secularly unstable. For q ≲ 0.25, a dynamical instability is encountered before the Roche limit. Degenerate binary configurations remain hydrodynamically stable all the way to the Roche limit for all mass ratios q ≠ 1. However, unstable mass transfer can occur beyond the Roche limit, and this is indeed observed in our numerical simulations. Dynamically unstable mass transfer also leads to the rapid coalescence of the binary system, although the details of the hydrodynamic evolution are quite different compared to that of an unstable equilibrium. We discuss the implications of our results for the evolution of double white-dwarf systems and W Ursae Majoris binaries.