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Astron. Astrophys. 319, 122-153 (1997)

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2. Computational procedure, initial conditions, and hydrodynamical evolution

In this section we summarize the numerical methods and the treatment of the input physics used for the presented simulations. In addition, we specify the initial conditions by which our different models are distinguished. Also, the results for the dynamical evolution as described in detail in Paper I are shortly reviewed.

2.1. Numerical treatment

The hydrodynamical simulations were done with a code based on the Piecewise Parabolic Method (PPM) developed by Colella & Woodward (1984). The code is basically Newtonian, but contains the terms necessary to describe gravitational wave emission and the corresponding back-reaction on the hydrodynamical flow (Blanchet et al. 1990). The modifications that follow from the gravitational potential are implemented as source terms in the PPM algorithm. The necessary spatial derivatives are evaluated as standard centered differences on the grid.

In order to describe the thermodynamics of the neutron star matter, we use the equation of state (EOS) of Lattimer & Swesty (1991) in a tabular form. The inversion for the temperature is done with a bisection iteration. Energy loss and changes of the electron abundance due to the emission of neutrinos and antineutrinos are taken into account by an elaborate "neutrino leakage scheme". The energy source terms contain the production of all types of neutrino pairs by thermal processes and of electron neutrinos and antineutrinos also by lepton captures onto baryons. The latter reactions act as sources or sinks of lepton number, too, and are included as source term in a continuity equation for the electron lepton number. When the neutron star matter is optically thin to neutrinos, the neutrino source terms are directly calculated from the reaction rates, while in case of optically thick conditions lepton number and energy are released on the corresponding neutrino diffusion time scales. The transition between both regimes is done by a smooth interpolation. Matter is rendered optically thick to neutrinos due to the main opacity producing reactions which are neutrino-nucleon scattering and absorption of neutrinos onto baryons.

More detailed information about the employed numerical procedures can be found in Paper I, in particular about the implementation of the gravitational wave radiation and back-reaction terms and the treatment of the neutrino lepton number and energy loss terms in the hydrodynamical code.

2.2. Initial conditions

We start our simulations with two identical Newtonian neutron stars with a baryonic mass of about 1.63  [FORMULA] and a radius of 15 km which are placed at a center-to-center distance of 42 km on a grid of 82 km side length. With the employed EOS of Lattimer & Swesty (1991), this baryonic mass corresponds to a (general relativistic) gravitational mass of approximately 1.5  [FORMULA] for a cool star with a radius as obtained from the general relativistic stellar structure equations of 11.2 km. Having a compressibility modulus of bulk nuclear matter of [FORMULA]  MeV (which is the "softest" of the three available cases), the Lattimer & Swesty EOS may overestimate the stiffness of supranuclear matter, in particular, since in its present form it neglects the possible occurrence of new hadronic states besides the neutron and proton at very high densities. For a softer supranuclear EOS, neutron stars would become more compact and their gravitational binding energy larger so that a baryonic mass of 1.63 [FORMULA] would more likely correspond to a gravitational mass between 1.4 and 1.45 [FORMULA].

The distributions of density [FORMULA] and electron fraction [FORMULA] are taken from a one-dimensional model of a cold, deleptonized neutron star in hydrostatic equilibrium. For numerical reasons the surroundings of the neutron stars cannot be assumed to be evacuated. The density of the ambient medium was set to [FORMULA]  g/cm3, more than five orders of magnitude smaller than the central densities of the stars. The internal energy density and electron fraction of this gas were taken to be equal to the values in the neutron stars at a density of [FORMULA]  g/cm3. In order to ensure sufficiently good numerical resolution, we artificially softened the extremely steep density decline towards the neutron star surfaces by not allowing for a density change of more than two orders of magnitude from zone to zone. From this prescription we obtain a thickness of the neutron star surface layers of about 3 zones.

The orbital velocities of the coalescing neutron stars were prescribed according to the motion of point masses spiralling towards each other due to the emission of gravitational waves. The tangential velocities of the neutron star centers were set equal to the Kepler velocities on circular orbits and radial velocity components were attributed as calculated from the quadrupole formula. In addition to the orbital angular momentum, spins around their respective centers were added to the neutron stars. The assumed spins and spin directions were varied between the calculated models. Table 1 lists the distinguishing model parameters, the number N of grid zones per spatial dimension (in the orbital plane) and the spin parameter S. The neutron stars in models A64 and A128 did not have any additional spins ([FORMULA]), in model B64 the neutron star spins were parallel to the orbital angular momentum vector ([FORMULA]), in model C64 the spins were in the opposite direction ([FORMULA]). In both models B64 and C64, the angular velocities of the rigid neutron star rotation and of the orbital motion were chosen to be equal. Model A128 had the same initial setup as model A64 but had twice the number of grid zones per spatial dimension and thus served as a check for the sufficiency of the numerical resolution of the computations with [FORMULA] zones.


Table 1. Parameters and some computed quantities for all models. N is the number of grid zones per dimension in the orbital plane, S defines the direction of the spins of the neutron stars relative to the direction of the orbital angular momentum, and [FORMULA] gives the maximum temperature (in energy units) reached on the grid during the simulation of a model. [FORMULA] is the electron neutrino luminosity after approaching a saturation level at about 8 ms, [FORMULA] is the corresponding electron antineutrino luminosity, and [FORMULA] the luminosity of each individual species of [FORMULA], [FORMULA], [FORMULA], [FORMULA]. [FORMULA] gives the total neutrino luminosity after a quasi-stationary state has been reached at [FORMULA] -8 ms and [FORMULA] denotes the integral rate of energy deposition by neutrino-antineutrino annihilation at that time

The rotational state of the neutron stars is determined by the action of viscosity. If the dynamic viscosity of neutron star matter were large enough, tidal forces could lead to tidal locking of the two stars and thus spin-up during inspiral. Kochanek (1992), Bildsten & Cutler (1992), and Lai (1994), however, showed that microscopic shear and bulk viscosities are probably orders of magnitude too small to achieve corotation. Moreover, they argued that for the same reason it is extremely unlikely that the stars are heated up to more than about [FORMULA]  K by tidal interaction prior to merging. In this sense, models A64 and A128 can be considered as reference case for two non-corotating neutron stars, while model B64 represents the case of rigid-body like rotation of the two stars, and model C64 the case where the spin directions of both stars were inverted.

Since in the case of degenerate matter the temperature is extremely sensitive to small variations of the internal energy, e.g. caused by small numerical errors, we did not start our simulations with cold ([FORMULA]) or "cool" ([FORMULA]  K) neutron stars as suggested by the investigations of Kochanek (1992), Bildsten & Cutler (1992), and Lai (1994). Instead, we constructed initial temperature distributions inside the neutron stars by assuming thermal energy densities of about 3% of the degeneracy energy density for a given density [FORMULA] and electron fraction [FORMULA]. The corresponding central temperature was around 7 MeV and the average temperature was a few MeV and thus of the order of the estimates obtained by Mészáros & Rees (1992b) for the phase just prior to the merging. Locally, these initial temperatures were much smaller than the temperatures produced by the compression and dissipative heating during coalescence (see Table 1 for the maximum temperatures). However, they are orders of magnitude larger than can be achieved by tidal dissipation with plausible values for the microscopic viscosity of neutron star matter. Even under the most extreme assumptions for viscous shear heating, the viscosity of neutron star matter turns out to be at least four orders of magnitude too small (see Janka & Ruffert 1996).

Models A64, B64, and C64 were computed on a Cray-YMP 4/64 where they needed about 16 MWords of main memory and took approximately 40 CPU-hours each. For the better resolved model A128 we employed a grid with [FORMULA] zones instead of the [FORMULA] grid of models A64, B64, and C64. Note that in the direction orthogonal to the orbital plane only half the number of grid zones was used but the spatial resolution was the same as in the orbital plane. Model A128 was run on a Cray-EL98 4/256 and required about 22 MWords of memory and 1700 CPU-hours.

2.3. Hydrodynamical evolution

The hydrodynamical evolution and corresponding gravitational wave emission were detailed in Paper I. Again, we only summarize the most essential aspects here.

The three-dimensional hydrodynamical simulations were started at a center-to-center distance of the two neutron stars of 2.8 neutron star radii. This was only slightly larger than the separation where the configurations become dynamically unstable which is at a distance of approximately 2.6 neutron star radii. Gravitational wave emission leads to the decay of the binary orbit, and already after about one quarter of a revolution, approximately 0.6 ms after the start of the computations, the neutron star surfaces touch because of the tidal deformation and stretching of the stars.

When the two stars begin to plunge into each other, compression and the shearing motion of the touching surfaces cause dissipation of kinetic energy and lead to a strong increase of the temperature. From initial values of a few MeV, the gas heats up to peak temperatures of several ten MeV. In case of model C64, a temperature of nearly 50 MeV is reached about 1 ms after the stars have started to merge. After one compact, massive body has formed from most of the mass of the neutron stars about 3 ms later, more than 50 MeV are found in two distinct, extremely hot off-center regions.

At the time of coalescence and shortly afterwards, spiral-arm or wing-like extensions are formed from material spun off the outward directed sides of the neutron stars by tidal and centrifugal forces. Due to the retained angular momentum, the central body performs large-amplitude swinging motions and violent oscillations. This wobbling of the central body of the merger creates strong pressure waves and small shocks that heat the exterior layers and lead to the dispersion of the spiral arms into a vertically extended "ring" or thick toroidal disk of gas that surrounds the massive central object. While the mass of the compact body is larger than 3  [FORMULA] and its mean density above [FORMULA], the surrounding cloud contains only 0.1-0.2  [FORMULA] and is more dilute with an average density of about [FORMULA].

The central body is too massive to be stabilized by gas pressure for essentially all currently discussed equations of state of nuclear and supranuclear matter. Moreover, it can be argued (see Paper I) that its angular momentum is not large enough to provide rotational support. Therefore, we expect that in a fully general relativistic simulation, the central object will collapse into a black hole on a time scale of only a few milliseconds after the neutron stars have merged. Dependent on the total angular momentum corresponding to the initial setup of the neutron star spins, some of the material in the outer regions of the disk obtains enough angular momentum (e.g., by momentum transfer via pressure waves) to flow across the boundary of the computational grid at a distance of about 40 km from the center. In model B64 which has the largest total angular momentum because of the assumed solid-body type rotation, 0.1-0.15  [FORMULA] are able to leave the grid between 2 ms and 4 ms after the start of the simulation. However, at most a few times [FORMULA]   [FORMULA] of this material have a total energy that might allow them to escape from the gravitational field of the merger. In model C64 the anti-spin setup leads to violent oscillations of the merged body which create a very extended and periodically contracting and reexpanding disk. In this model nearly 0.1  [FORMULA] are lost across the outer grid boundaries even during the later stages of the simulation ([FORMULA]  ms).

While by far the major fraction of the merger mass ([FORMULA]   [FORMULA]) will be swallowed up by the forming black hole almost immediately on a dynamical time scale, some matter with sufficiently high specific angular momentum might be able to remain in a disk around the slowly rotating black hole. Estimates show (Paper I) that with the typical rotation velocities obtained in our models, this matter must orbit around the central body at radii of at least between about 40 km and 100 km. This means that only matter that has been able to leave the computational grid used in our simulations has a chance to end up in a toroidal disk around the black hole. From these arguments we conclude that such a possible disk might contain a mass of at most 0.1-0.15  [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998