SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


Astron. Astrophys. 341, 499-526 (1999)

Previous Section Next Section Title Page Table of Contents

2. Numerical method and physical parameters of the system

2.1. The numerical method

We perform hydrodynamic calculations using a 3D Newtonian SPH code based on the one used in Davies et al. (1994). The basic equations of our SPH formulation can be found in the review of Benz (1990), the gravitational forces are calculated using a hierarchical binary tree (Benz et al. 1990). Since the method is well-known and comprehensive reviews exist (see e.g. Hernquist & Katz (1989); Benz (1990); Monaghan (1992); Müller (1998)), we do not go into the details of SPH here and refer the reader to the literature.

The Lagrangian nature of SPH is perfectly suited to the study of this intrinsically three dimensional problem, since it is not subject to spatial restrictions imposed by a computational grid. Thus, no matter can be lost from the computational domain (as for example in Eulerian methods). In addition no CPU resources are wasted for uninteresting empty regions. These zones are in other methods sometimes filled by low density material that can possibly lead to artifacts. Especially, the (tiny) amounts of ejecta could be influenced since they have to struggle against surrounding material. SPH also allows one to follow individual particles and therefore to keep track of the history of chosen blobs of matter.

The details of the initial SPH-particle setup are given in Appendix A.

2.2. The physical parameters of the binary system

In the following we want to discuss the physical parameters of the binary system together with the chosen approaches and their uncertainties that will be tested.

2.2.1. Masses

Present state of the art nuclear equations of state allow gravitational masses for maximally rotating neutron stars of [FORMULA] (see e.g. Weber & Glendenning (1992)). However, all observed neutron star masses are centered in a narrow band around 1.4 [FORMULA] (e.g. van den Heuvel (1994)), a mass that might be given rather by evolutionary constraints than by the stability limit of the EOS. The masses that are found in the five known neutron star binaries are very close to each other, the maximum deviation being [FORMULA] 4% in the case of PSR 1913+16 (see Thorsett 1996). Considering the observational facts we only investigate equal mass binary systems. We regard a baryonic mass of 1.6 [FORMULA], corresponding approximately to 1.4 [FORMULA] of gravitational mass, as most reasonable. The amount of mass that is ejected into the interstellar medium is directly related to the gravitational potential that has to be overcome by the neutron star debris. Hence we vary the neutron star masses, investigating also the case of [FORMULA] of baryonic matter per star. Starting with the initial separation [FORMULA] km, which is our standard value, the inspiral will take longer in this case than for the heavier stars. For an estimate one may apply the formula for the inspiral time of a point mass binary (see e.g. Misner et al. 1973)

[EQUATION]

giving an inspiral time that is longer by a factor of approximately [FORMULA] for the lighter case.

2.2.2. Gravitational radiation backreaction

Since the backreaction force, resulting from the emission of gravitational waves, tends to circularize elliptic orbits (see Peters & Mathews (1963, 1964)), we start with basically circular orbits. Starting on the x-axis of our coordinate system, we give each SPH-particle a tangential velocity corresponding to a circular Kepler motion with a fixed value [FORMULA], where M denotes the total mass of the system, and add the radial velocity of a point mass binary in quadrupole approximation (see e.g. Shapiro & Teukolsky 1983)

[EQUATION]

([FORMULA], [FORMULA] is the mass of each star). As in Davies et al. (1994) we apply in addition to the hydrodynamic and gravitational accelerations the backreaction of a point mass binary to each SPH-particle:

[EQUATION]

Here [FORMULA], E is the energy, J the angular momentum, µ the reduced mass and [FORMULA], [FORMULA] and [FORMULA] refer to the point mass binary (not to be confused with the SPH-particle properties). Since this acceleration is applied to each particle, the circulation of the fluid will be conserved. When the distances of the centers of mass of the stars from the origin are equal to one stellar radius (and thus the point mass approximation definitely breaks down) the backreaction force is switched off.

This treatment of the backreaction force is clearly a simplification of the relevant gravitational physics. To test the corresponding sensitivity of our results we will investigate also the case where this force is totally neglected. To push the system over the critical radius where it becomes dynamically unstable (for an estimate of this radius obtained by applying an energy variational method to compressible, triaxial ellipsoids obeying a polytropic equation of state see Lai et al. (1994a) and references therein), we have to begin in this case with a smaller initial separation (42 km).

2.2.3. Tidal deformation

For the standard case of our calculation we will start with spherical initial configurations. However, a neutron star binary system that has spiraled down to a center of mass distance of [FORMULA] km will show non-negligible tidal deformations. In a crude estimate one finds for the height of the tidal bulge in our equal mass system [FORMULA] km, i.e. parts of the surfaces of the neutron stars will be raised to up to half a kilometer above their spherical radius R. Thus starting with initially spherical stars will lead to oscillations. To estimate the impact of the approximation of spherical stars we compare the results with the case where the initial configuration has been relaxed in the mutual gravitational field.

In order to obtain a hydrostatic equilibrium for the tidally locked, corotating binary system, we consider the coordinate system in which the neutron stars are at rest. A particle i of mass [FORMULA] is accelerated not only by the forces in the inertial frame [FORMULA], but also by non-inertial forces (see e.g. Landau 1976), so that the total force reads:

[EQUATION]

where [FORMULA] is the rotational frequency of the non-inertial system, i.e. the Kepler frequency of the corresponding point mass binary, and [FORMULA] and [FORMULA] are position and velocity vectors of particle i in the corotating system. The first term in the brackets is the Coriolis-force [FORMULA], the second the centrifugal force [FORMULA] and the last term, [FORMULA], results from the change of [FORMULA] due to the gravitational radiation backreaction force. To obtain equilibrium we apply a velocity dependent damping force, so that the total force on a SPH-particle i reads:

[EQUATION]

where [FORMULA] and [FORMULA] are forces due to the presence of self-gravity and pressure gradients, both of which are evaluated with the SPH-code. We do not have to consider the backreaction force here, since it is applied (in our approximation) to each SPH-particle in the same way and hence does not influence the shape of the stars. The Coriolis force does not have to be considered here since we are interested in the construction of hydrostatic equilibrium where [FORMULA]. As mentioned by Rasio and Shapiro (1992) the relaxation time [FORMULA] should be chosen slightly overcritical to guarantee an efficient relaxation process (the typical oscillation time scale can be estimated from the sound crossing time [FORMULA]).

We first calculate the resulting forces on the centers of mass and then subtract them from the forces on individual particles in order to remove the force on the stellar centers of mass:

[EQUATION]

where the center of mass forces are given by

[EQUATION]

Here j labels the binary stars, with [FORMULA] their masses and the sum loops over the particle numbers in each object. Because of numerical noise the centers of mass drift very slightly and thus are reset each dump to keep the stars exactly at the desired positions.

Kepler's law [FORMULA], which we applied in the relaxation process, is strictly valid only for point masses. In their ellipsoidal approximation for polytropic equations of state Lai et al. (1994a) find a modified Kepler law for close equal mass binaries:

[EQUATION]

where

[EQUATION]

and [FORMULA]. The [FORMULA] are the semi-major axes of the ellipsoid and [FORMULA] is a constant depending on the polytropic index n ([FORMULA] in the incompressible case, [FORMULA]; numerical values are tabulated in Lai et al. (1993a)). We calculated [FORMULA] a posteriori using the parameters of the relaxed system. We found [FORMULA], which justifies using [FORMULA] instead of [FORMULA] in the relaxation process.

Starting with a binary system relaxed in the way described above no oscillations of the stars were observed.

2.2.4. Viscosity

Since the neutrons and protons are likely to be in superfluid states, the main contribution to the microscopic viscosity of the neutron star fluid is supposed to come from electron-electron scattering (Flowers & Itoh 1979). The simulation of the almost inviscid neutron star fluid poses a problem for numerical hydrodynamics since the latter is subject to numerical as well as to artificial viscosity.

The SPH standard tensor of artificial viscosity (Monaghan & Gingold 1983; we use the standard values [FORMULA] and [FORMULA]) is known to yield good results in shocks, but also to introduce spurious forces in pure shear flows, where a vanishing viscosity would be desirable. We thus decided to compare the results obtained using the standard viscosity with those found using a new scheme proposed by Morris & Monaghan (1997). The basic idea is to give each particle its own time dependent viscosity coefficient that is calculated by integrating a simple differential equation including a source term and a term describing the decay of this coefficient. The viscosity tensor then reads

[EQUATION]

where [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. The [FORMULA] denote the particle sound velocities, [FORMULA] the smoothing lengths, [FORMULA] and [FORMULA] positions and velocities, [FORMULA], [FORMULA] and [FORMULA]. [FORMULA] is calculated from

[EQUATION]

The first term on the right hand side causes the viscosity coefficient [FORMULA] to decay on a time scale [FORMULA] towards some minimum value [FORMULA] that keeps the particles well ordered in the absence of shocks. The source term [FORMULA], leading to an increase of [FORMULA] when a shock is detected, is chosen to be

[EQUATION]

We chose [FORMULA] with [FORMULA].

2.2.5. Spins

The degree of synchronization of a close binary system depends decisively on the viscous dissipation rate (compared to the inspiral time). Bildsten & Cutler (1992), Kochanek (1992) and Lai (1994) found any reasonable assumption on the microscopic viscosity of the neutron star matter to be orders of magnitude too low to lead to a tidal locking of a binary system within the inspiral time. Hence, it seems very unlikely that a corotating system is the generic case. We suspect the viscosity to be also too low to lead necessarily to an alignment of the stellar spins with the orbital angular momentum. However, for reasons of simplicity we will only treat aligned systems in this work.

For purely numerical reasons, i.e. to minimize the effects of viscosity during the inspiral and the shear motion at the contact surface of both stars during the merger, we focus here mainly on synchronized, "corotating" systems. This might also be useful as some kind of benchmark for future simulations with a more sophisticated incorporation of general relativistic effects (for a general relativistic simulation of a corotating inspiral see e.g. Baumgarte et al. 1997).

However, to explore the dependence of our results on the initial neutron star spins, we also investigate the cases where the stars have no initial spin (in a space-fixed frame) and the case where they are spinning against the orbital motion with a period equal to the orbital period. Starting with 45 km of initial separation, this corresponds to a spin period of 2.91 ms. We expect all mergers to lie in this range of initial spins.

2.2.6. Equation of state

The equation of state is a major uncertainty in all neutron star related calculations. Our code is coupled to the physical equation of state (EOS) for hot and dense matter of Lattimer and Swesty (1991). This EOS models the hadronic matter as consisting of an average heavy nucleus (representing all heavy nuclei), alpha particles (representative for an ensemble of light nuclei) and nucleons outside nuclei. Since further hadronic degrees of freedom, such as hyperons, pions, kaons and the quark-hadron phase transition are disregarded, we use the lowest available nuclear compressibility, [FORMULA] MeV, to mimic the possible softening of the EOS due to the appearance of "exotic" matter at higher densities. Since the chemical composition of matter is calculated for the thermodynamically most favourable state, effects of recombinations of nucleons into nuclei are taken into account automatically. For further details concerning the LS-EOS see Appendix B.

To test the dependence of our results on the underlying EOS we also use a polytropic EOS where the exponent [FORMULA] was fitted to the central part of our 1.6 [FORMULA] neutron star obtained with the LS-EOS. We obtained a good fit with [FORMULA]. The pressure p can be written in terms of density [FORMULA] and specific internal energy [FORMULA]

[EQUATION]

We assign the specific internal energy of our polytrope, [FORMULA], in a way that the pressure distribution of the LS-EOS star is reproduced:

[EQUATION]

where [FORMULA] and [FORMULA] are the values of pressure and density at radius r in the star. Using these profiles a spherical equilibrium configuration was constructed following the steps described in Appendix A.

To test the sensitivity on the adiabatic exponent we also considered a test case of a much softer EOS ([FORMULA]) where the polytropic constant K ([FORMULA]) was adjusted in a way that the radius of the LS-EOS star was reproduced.

2.2.7. Temperatures

When the neutron stars of a binary system have reached a separation of a few stellar radii (typical time scales are of the order of [FORMULA] years) they should be essentially cold ([FORMULA] K). When the stars come closer and tidal interactions become important, the stars might be heated up again by viscous processes acting in the tidal flow. However, these temperatures are estimated to be only of the order of [FORMULA] K (Lai 1994).

With respect to the thermodynamic properties, the accurate determination of low temperatures in the high density regime (say above [FORMULA]) is very difficult. Since the temperature is a very steeply rising function of the specific internal energy u (our independent variables are u, the electron fraction [FORMULA] and the density [FORMULA]), numerical noise in u in the degenerate regime may lead to substantial temperature fluctuations. However, as long as there are no additional physical processes involved that depend on temperature, this does not influence the dynamical evolution of matter.

2.2.8. Neutrino emission

The merger of two neutron stars will be accompanied by strong neutrino emission (see e.g. Ruffert et al. (1996, 1997)). To set a limit on the maximum influence of the cooling by neutrino emission on the ejection of mass, we also explore the extreme case where we assume that all locally produced neutrinos stream out freely without any interaction with overlying material (which also implies that a possible quenching of neutrino rates due to neutrino final-state blocking is ignored), thereby reducing the local thermal energy content. This scheme is clearly crude and will largely overestimate the emission from the hot, dense central regions. However, we are here only interested in the possible influence on the amount of ejected material and the free streaming limit will give us a secure upper limit of the effects resulting from neutrino emission.

Looking at typical densities and temperatures encountered in our scenario, one can conclude that the only processes giving possibly non-negligible contributions to the neutrino emission in addition to the electron and positron capture on nucleons, are the plasma and the pair processes (for a comparison of the importance of different neutrino processes as a function of [FORMULA] and T see e.g. Schinder et al. (1987), Itoh et al. (1989) and Haft et al. (1994)).

We thus include the following processes as sources of energy loss:

[EQUATION]

where the last process is the decay of a plasmon, i.e. the decay of both longitudinal and transverse electromagnetic excitations of the medium. Due to the very high temperatures, we ignore the electron mass [FORMULA] and the mass difference Q of neutron and proton for our treatment of electron and positron captures. The energy emission rates due to these processes are discussed in Appendix C. For the energy emission rates of pair and plasma neutrino processes we use the fit formulae of Itoh et al. (1989) and Haft et al. (1994). The neutrino emission then appears as a sink in the energy equation of each particle:

[EQUATION]

where [FORMULA] contains the usual terms from pdV-work and viscous heating (see e.g. Benz (1990)). The enormous dependence of the rates on temperature ([FORMULA]; [FORMULA]) gave rise to numerical instabilities in the degenerate regime, where temperature fluctuations can be appreciable due to noise in the specific internal energy. To overcome this problem the emission rates of particle i were calculated from mean temperatures [FORMULA], where [FORMULA] denotes the number of neighbours of particle i, rather than from the individual particle temperatures.

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 1999

Online publication: December 4, 1998
helpdesk.link@springer.de