Various features of quasiequilibrium sequences of binary neutron stars in general relativity

Quasiequilibrium sequences of binary neutron stars are numerically calculated in the framework of the Isenberg-Wilson-Mathews (IWM) approximation of general relativity. The results are presented for both rotation states of synchronized spins and irrotational motion, the latter being considered as the realistic one for binary neutron stars just prior to the merger. We assume a polytropic equation of state and compute several evolutionary sequences of binary systems composed of different-mass stars as well as identical-mass stars with adiabatic indices gamma=2.5, 2.25, 2, and 1.8. From our results, we propose as a conjecture that if the turning point of binding energy (and total angular momentum) locating the innermost stable circular orbit (ISCO) is found in Newtonian gravity for some value of the adiabatic index gamma_0, that of the ADM mass (and total angular momentum) should exist in the IWM approximation of general relativity for the same value of the adiabatic index.Comment: Text improved, some figures changed or deleted, new table, 38 pages, 31 figures, accepted for publication in Phys. Rev.

Possible explanation for star-crushing effect in binary neutron star simulations

A possible explanation is suggested for the controversial star-crushing effect seen in numerical simulations of inspiraling neutron star binaries by Wilson, Mathews and Marronetti (WMM). An apparently incorrect definition of momentum density in the momentum constraint equation used by WMM gives rise to a post-1-Newtonian error in the approximation scheme. We show by means of an analytic, post-1-Newtonian calculation that this error causes an increase of the stars' central densities which is of the order of several percent when the stars are separated by a few stellar radii, in agreement with what is seen in the simulations.Comment: 4 pages, 1 figure, uses revetx macros, minor revision

Tidal Interaction between a Fluid Star and a Kerr Black Hole in Circular Orbit

We present a semi-analytic study of the equilibrium models of close binary systems containing a fluid star (mass $m$ and radius $R_0$) and a Kerr black hole (mass $M$) in circular orbit. We consider the limit $M\gg m$ where spacetime is described by the Kerr metric. The tidally deformed star is approximated by an ellipsoid, and satisfies the polytropic equation of state. The models also include fluid motion in the stellar interior, allowing binary models with nonsynchronized stellar spin (as expected for coalescing neutron star-black hole binaries) to be constructed. Tidal disruption occurs at orbital radius $r_{\rm tide}\sim R_0(M/m)^{1/3}$, but the dimensionless ratio $\hat r_{\rm tide}=r_{\rm tide}/[R_0(M/m)^{1/3}]$ depends on the spin parameter of the black hole as well as on the equation of state and the internal rotation of the star. We find that the general relativistic tidal field disrupts the star at a larger $\hat r_{\rm tide}$ than the Newtonian tide; the difference is particularly prominent if the disruption occurs in the vicinity of the black hole's horizon. In general, $\hat r_{\rm tide}$ is smaller for a (prograde rotating) Kerr black hole than for a Schwarzschild black hole. We apply our results to coalescing black hole-neutron star and black hole-white dwarf binaries. The tidal disruption limit is important for characterizing the expected gravitational wave signals and is relevant for determining the energetics of gamma ray bursts which may result from such disruption.Comment: 29 pages including 8 figures. Minor changes and update. To appear in ApJ, March 20, 2000 (Vol.532, #1

A relativistic formalism for computation of irrotational binary stars in quasi equilibrium states

We present relativistic hydrostatic equations for obtaining irrotational binary neutron stars in quasi equilibrium states in 3+1 formalism. Equations derived here are different from those previously given by Bonazzola, Gourgoulhon, and Marck, and have a simpler and more tractable form for computation in numerical relativity. We also present hydrostatic equations for computation of equilibrium irrotational binary stars in first post-Newtonian order.Comment: 5 pages, corrected eqs.(2.10), (2.11) and (3.1

Equilibrium sequences of irrotational binary polytropic stars : The case of double polytropic stars

Solutions to equilibrium sequences of irrotational binary polytropic stars in Newtonian gravity are expanded in a power of $\epsilon=a_0/R$, where R and $a_0$ are the orbital separation of the binary system and the radius of each star for $R=\infty$. For each order of $\epsilon$, we should solve ordinary differential equations for arbitrary polytropic indices n. We show solutions for polytropic indices n= 0.5, 1, 1.5 and 2 up to $\epsilon^6$ orders. Our semi-analytic solutions can be used to check the validity of numerical solutions.Comment: 59 pages including 15 tables and 13 figures, revtex, accepted to Phys. Rev.

Merger of binary neutron stars of unequal mass in full general relativity

We present results of three dimensional numerical simulations of the merger of unequal-mass binary neutron stars in full general relativity. A $\Gamma$-law equation of state $P=(\Gamma-1)\rho\epsilon$ is adopted, where $P$, $\rho$, \varep, and $\Gamma$ are the pressure, rest mass density, specific internal energy, and the adiabatic constant, respectively. We take $\Gamma=2$ and the baryon rest-mass ratio $Q_M$ to be in the range 0.85--1. The typical grid size is $(633,633,317)$ for $(x,y,z)$ . We improve several implementations since the latest work. In the present code, the radiation reaction of gravitational waves is taken into account with a good accuracy. This fact enables us to follow the coalescence all the way from the late inspiral phase through the merger phase for which the transition is triggered by the radiation reaction. It is found that if the total rest-mass of the system is more than $\sim 1.7$ times of the maximum allowed rest-mass of spherical neutron stars, a black hole is formed after the merger irrespective of the mass ratios. The gravitational waveforms and outcomes in the merger of unequal-mass binaries are compared with those in equal-mass binaries. It is found that the disk mass around the so formed black holes increases with decreasing rest-mass ratios and decreases with increasing compactness of neutron stars. The merger process and the gravitational waveforms also depend strongly on the rest-mass ratios even for the range $Q_M= 0.85$--1.Comment: 32 pages, PRD68 to be publishe

A new numerical method for constructing quasi-equilibrium sequences of irrotational binary neutron stars in general relativity

We propose a new numerical method to compute quasi-equilibrium sequences of general relativistic irrotational binary neutron star systems. It is a good approximation to assume that (1) the binary star system is irrotational, i.e. the vorticity of the flow field inside component stars vanishes everywhere (irrotational flow), and (2) the binary star system is in quasi-equilibrium, for an inspiraling binary neutron star system just before the coalescence as a result of gravitational wave emission. We can introduce the velocity potential for such an irrotational flow field, which satisfies an elliptic partial differential equation (PDE) with a Neumann type boundary condition at the stellar surface. For a treatment of general relativistic gravity, we use the Wilson--Mathews formulation, which assumes conformal flatness for spatial components of metric. In this formulation, the basic equations are expressed by a system of elliptic PDEs. We have developed a method to solve these PDEs with appropriate boundary conditions. The method is based on the established prescription for computing equilibrium states of rapidly rotating axisymmetric neutron stars or Newtonian binary systems. We have checked the reliability of our new code by comparing our results with those of other computations available. We have also performed several convergence tests. By using this code, we have obtained quasi-equilibrium sequences of irrotational binary star systems with strong gravity as models for final states of real evolution of binary neutron star systems just before coalescence. Analysis of our quasi-equilibrium sequences of binary star systems shows that the systems may not suffer from dynamical instability of the orbital motion and that the maximum density does not increase as the binary separation decreases.Comment: 20 pages, 18 figures, more results of convergence tests are added, revised version accepted for publication in PR

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

We analytically calculate the equilibrium sequence of the corotating binary stars of incompressible fluid in the first post-Newtonian(PN) approximation of general relativity. By calculating the total energy and total angular momentum of the system as a function of the orbital separation, we investigate the innermost stable circular orbit for corotating binary(we call it ISCCO). It is found that by the first PN effect, the orbital separation of the binary at the ISCCO becomes small with increase of the compactness of each star, and as a result, the orbital angular velocity at the ISCCO increases. These behaviors agree with previous numerical works.Comment: 33 pages, revtex, 4 figures(eps), accepted for publication in Phys. Rev.

Gravitational radiation from corotating binary neutron stars of incompressible fluid in the first post-Newtonian approximation of general relativity

We analytically study gravitational radiation from corotating binary neutron stars composed of incompressible, homogeneous fluid in circular orbits. The energy and the angular momentum loss rates are derived up to the first post-Newtonian (1PN) order beyond the quadrupole approximation including effects of the finite size of each star of binary. It is found that the leading term of finite size effects in the 1PN order is only $O(GM_{\ast}/c^2 a_{\ast})$ smaller than that in the Newtonian order, where $GM_{\ast}/c^2 a_{\ast}$ means the ratio of the gravitational radius to the mean radius of each star of binary, and the 1PN term acts to decrease the Newtonian finite size effect in gravitational radiation.Comment: 26 pages, revtex, 9 figures(eps), accepted for publication in Phys. Rev.

Black Hole-Neutron Star Binaries in General Relativity: Quasiequilibrium Formulation

We present a new numerical method for the construction of quasiequilibrium models of black hole-neutron star binaries. We solve the constraint equations of general relativity, decomposed in the conformal thin-sandwich formalism, together with the Euler equation for the neutron star matter. We take the system to be stationary in a corotating frame and thereby assume the presence of a helical Killing vector. We solve these coupled equations in the background metric of a Kerr-Schild black hole, which accounts for the neutron star's black hole companion. In this paper we adopt a polytropic equation of state for the neutron star matter and assume large black hole--to--neutron star mass ratios. These simplifications allow us to focus on the construction of quasiequilibrium neutron star models in the presence of strong-field, black hole companions. We summarize the results of several code tests, compare with Newtonian models, and locate the onset of tidal disruption in a fully relativistic framework.Comment: 17 pages, 7 figures; added discussion, tables; PRD in pres