Binary-Induced Gravitational Collapse: A Trivial Example

We present a simple model illustrating how a highly relativistic, compact object which is stable in isolation can be driven dynamically unstable by the tidal field of a binary companion. Our compact object consists of a test-particle in a relativistic orbit about a black hole; the binary companion is a distant point mass. Our example is presented in light of mounting theoretical opposition to the possibility that sufficiently massive, binary neutron stars inspiraling from large distance can collapse to form black holes prior to merger. Our strong-field model suggests that first order post-Newtonian treatments of binaries, and stability analyses of binary equilibria based on orbit-averaged, mean gravitational fields, may not be adequate to rule out this possibility.Comment: 7 pages, 5 figures, RevTeX, to appear in Phys. Rev. D, Jan 15 199

Post-Newtonian Models of Binary Neutron Stars

Using an energy variational method, we calculate quasi-equilibrium configurations of binary neutron stars modeled as compressible triaxial ellipsoids obeying a polytropic equation of state. Our energy functional includes terms both for the internal hydrodynamics of the stars and for the external orbital motion. We add the leading post-Newtonian (PN) corrections to the internal and gravitational energies of the stars, and adopt hybrid orbital terms which are fully relativistic in the test-mass limit and always accurate to PN order. The total energy functional is varied to find quasi-equilibrium sequences for both corotating and irrotational binaries in circular orbits. We examine how the orbital frequency at the innermost stable circular orbit depends on the polytropic index n and the compactness parameter GM/Rc^2. We find that, for a given GM/Rc^2, the innermost stable circular orbit along an irrotational sequence is about 17% larger than the innermost secularly stable circular orbit along the corotating sequence when n=0.5, and 20% larger when n=1. We also examine the dependence of the maximum neutron star mass on the orbital frequency and find that, if PN tidal effects can be neglected, the maximum equilibrium mass increases as the orbital separation decreases.Comment: 53 pages, LaTex, 9 figures as 10 postscript files, accepted by Phys. Rev. D, replaced version contains updated reference

Innermost Stable Circular Orbit of Inspiraling Neutron-Star Binaries: Tidal Effects, Post-Newtonian Effects and the Neutron-Star Equation of State

We study how the neutron-star equation of state affects the onset of the dynamical instability in the equations of motion for inspiraling neutron-star binaries near coalescence. A combination of relativistic effects and Newtonian tidal effects cause the stars to begin their final, rapid, and dynamically-unstable plunge to merger when the stars are still well separated and the orbital frequency is $\approx$ 500 cycles/sec (i.e. the gravitational wave frequency is approximately 1000 Hz). The orbital frequency at which the dynamical instability occurs (i.e. the orbital frequency at the innermost stable circular orbit) shows modest sensitivity to the neutron-star equation of state (particularly the mass-radius ratio, $M/R_o$, of the stars). This suggests that information about the equation of state of nuclear matter is encoded in the gravitational waves emitted just prior to the merger.Comment: RevTeX, to appear in PRD, 8 pages, 4 figures include

The central density of a neutron star is unaffected by a binary companion at linear order in μ/R\mu/R

Recent numerical work by Wilson, Mathews, and Marronetti [J. R. Wilson, G. J. Mathews and P. Marronetti, Phys. Rev. D 54, 1317 (1996)] on the coalescence of massive binary neutron stars shows a striking instability as the stars come close together: Each star's central density increases by an amount proportional to 1/(orbital radius). This overwhelms any stabilizing effects of tidal coupling [which are proportional to 1/(orbital radius)^6] and causes the stars to collapse before they merge. Since the claimed increase of density scales with the stars' mass, it should also show up in a perturbation limit where a point particle of mass $\mu$ orbits a neutron star. We prove analytically that this does not happen; the neutron star's central density is unaffected by the companion's presence to linear order in $\mu/R$. We show, further, that the density increase observed by Wilson et. al. could arise as a consequence of not faithfully maintaining boundary conditions.Comment: 3 pages, REVTeX, no figures, submitted to Phys Rev D as a Rapid Communicatio

General-relativistic coupling between orbital motion and internal degrees of freedom for inspiraling binary neutron stars

We analyze the coupling between the internal degrees of freedom of neutron stars in a close binary, and the stars' orbital motion. Our analysis is based on the method of matched asymptotic expansions and is valid to all orders in the strength of internal gravity in each star, but is perturbative in the ``tidal expansion parameter'' (stellar radius)/(orbital separation). At first order in the tidal expansion parameter, we show that the internal structure of each star is unaffected by its companion, in agreement with post-1-Newtonian results of Wiseman (gr-qc/9704018). We also show that relativistic interactions that scale as higher powers of the tidal expansion parameter produce qualitatively similar effects to their Newtonian counterparts: there are corrections to the Newtonian tidal distortion of each star, both of which occur at third order in the tidal expansion parameter, and there are corrections to the Newtonian decrease in central density of each star (Newtonian ``tidal stabilization''), both of which are sixth order in the tidal expansion parameter. There are additional interactions with no Newtonian analogs, but these do not change the central density of each star up to sixth order in the tidal expansion parameter. These results, in combination with previous analyses of Newtonian tidal interactions, indicate that (i) there are no large general-relativistic crushing forces that could cause the stars to collapse to black holes prior to the dynamical orbital instability, and (ii) the conventional wisdom with respect to coalescing binary neutron stars as sources of gravitational-wave bursts is correct: namely, the finite-stellar-size corrections to the gravitational waveform will be unimportant for the purpose of detecting the coalescences.Comment: 22 pages, 2 figures. Replaced 13 July: proof corrected, result unchange

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.

Binary Neutron Stars in General Relativity: Quasi-Equilibrium Models

We perform fully relativistic calculations of binary neutron stars in quasi-equilibrium circular orbits. We integrate Einstein's equations together with the relativistic equation of hydrostatic equilibrium to solve the initial value problem for equal-mass binaries of arbitrary separation. We construct sequences of constant rest mass and identify the innermost stable circular orbit and its angular velocity. We find that the quasi-equilibrium maximum allowed mass of a neutron star in a close binary is slightly larger than in isolation.Comment: 4 pages, 3 figures, RevTe

Revised Relativistic Hydrodynamical Model for Neutron-Star Binaries

We report on numerical results from a revised hydrodynamic simulation of binary neutron-star orbits near merger. We find that the correction recently identified by Flanagan significantly reduces but does not eliminate the neutron-star compression effect. Although results of the revised simulations show that the compression is reduced for a given total orbital angular momentum, the inner most stable circular orbit moves to closer separation distances. At these closer orbits significant compression and even collapse is still possible prior to merger for a sufficiently soft EOS. The reduced compression in the corrected simulation is consistent with other recent studies of rigid irrotational binaries in quasiequilibrium in which the compression effect is observed to be small. Another significant effect of this correction is that the derived binary orbital frequencies are now in closer agreement with post-Newtonian expectations.Comment: Submitted to Phys. Rev.

Tidal Stabilization of Rigidly Rotating, Fully Relativistic Neutron Stars

It is shown analytically that an external tidal gravitational field increases the secular stability of a fully general relativistic, rigidly rotating neutron star that is near marginal stability, protecting it against gravitational collapse. This stabilization is shown to result from the simple fact that the energy $\delta M(Q,R)$ required to raise a tide on such a star, divided by the square of the tide's quadrupole moment $Q$, is a decreasing function of the star's radius $R$, $(d/dR)[\delta M(Q,R)/Q^2]<0$ (where, as $R$ changes, the star's structure is changed in accord with the star's fundamental mode of radial oscillation). If $(d/dR)[\delta M(Q,R)/Q^2]$ were positive, the tidal coupling would destabilize the star. As an application, a rigidly rotating, marginally secularly stable neutron star in an inspiraling binary system will be protected against secular collapse, and against dynamical collapse, by tidal interaction with its companion. The ``local-asymptotic-rest-frame'' tools used in the analysis are somewhat unusual and may be powerful in other studies of neutron stars and black holes interacting with an external environment. As a byproduct of the analysis, in an appendix the influence of tidal interactions on mass-energy conservation is elucidated.Comment: Revtex, 10 pages, 2 figures; accepted for publication in Physical Review D. Revisions: Appendix rewritten to clarify how, in Newtonian gravitation theory, ambiguity in localization of energy makes interaction energy ambiguous but leaves work done on star by tidal gravity unambiguous. New footnote 1 and Refs. [11] and [19

Stability of coalescing binary stars against gravitational collapse: hydrodynamical simulations

We perform simulations of relativistic binary stars in post-Newtonian gravity to investigate their dynamical stability prior to merger against gravitational collapse in a tidal field. In general, our equations are only strictly accurate to first post-Newtonian order, but they recover full general relativity for spherical, static stars. We study both corotational and irrotational binary configurations of identical stars in circular orbits. We adopt a soft, adiabatic equation of state with $\Gamma = 1.4$, for which the onset of instability occurs at a sufficiently small value of the compaction $M/R$ that a post-Newtonian approximation is quite accurate. For such a soft equation of state there is no innermost stable circular orbit, so that we can study arbitrarily close binaries. This choice still allows us to study all the qualitative features exhibited by any adiabatic equation of state regarding stability against gravitational collapse. We demonstrate that, independent of the internal stellar velocity profile, the tidal field from a binary companion stabilizes a star against gravitational collapse.Comment: 13 pages, 10 figures, RevTex, to appear in Phys. Rev.