gravitational waves, neutrino emission, and gamma-ray...

Astron. Astrophys. 338, 535–555 (1998) ASTRONOMYAND

ASTROPHYSICS

Colliding neutron stars

Gravitational waves, neutrino emission, and gamma-ray bursts

M. Ruffert 1? and H.-Th. Janka2??

1 Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK2 Max-Planck-Institut fur Astrophysik, Postfach 1523, D-85740 Garching, Germany

Received 15 April 1998 / Accepted 21 July 1998

Abstract. Three-dimensional hydrodynamical simulations arepresented for the direct head-on or off-center collision of twoneutron stars, employing a basically Newtonian PPM code butincluding the emission of gravitational waves and their back-reaction on the hydrodynamical flow. A physical nuclear equa-tion of state is used that allows us to follow the thermodynami-cal evolution of the stellar matter and to compute the emissionof neutrinos. Predicted gravitational wave signals, luminositiesand waveforms, are presented. The models are evaluated fortheir implications for gamma-ray burst scenarios. We find anextremely luminous outburst of neutrinos with a peak luminos-ity of more than4 · 1054 erg/s for several milliseconds. Thisleads to an efficiency of about 1% for the annihilation of neu-trinos with antineutrinos, corresponding to an average energydeposition rate of more than1052 erg/s and a total energy ofabout1050 erg deposited in electron-positron pairs around thecollision site within 10 ms. Although these numbers seem veryfavorable for gamma-ray burst scenarios, the pollution of thee±

pair-plasma cloud with nearly10−1 M of dynamically ejectedbaryons is 5 orders of magnitude too large. Therefore the for-mation of a relativistically expanding fireball that leads to agamma-ray burst powered by neutrino emission from collidingneutron stars is definitely ruled out.

Key words: gamma rays: bursts – gravitational waves – elemen-tary particles – hydrodynamics – stars: binaries: close – stars:neutron

1. Introduction

Themergingof two neutron stars that make up a binary systemis of interest both because it is a powerful source of gravitationalwaves and because it might be the central engine of gamma-raybursts. These mergings have been studied intensely during thepast few years, since the occurrence rate is high enough (10−6

to 10−5 per year per galaxy, e.g. Narayan et al. 1991, Phinney1991, Tutukov et al. 1992, Tutukov & Yungelson 1993, Lipunov

Send offprint requests to: H.-Th. Janka? e-mail: [email protected]

?? e-mail: [email protected]

et al. 1995, van den Heuvel & Lorimer 1996, Lipunov et al. 1997,Prokhorov et al. 1997, Bethe & Brown 1998) that one might beable to measure the observable consequences. However, directcollisionsof two neutron stars are much rarer and thus have notreceived much attention. Considering dense star clusters andtaking into account encounters between single neutron stars andbinaries that contain at least one neutron star, Portegies Zwartet al. (1997) estimate a rate of∼ 10−8 per year in the galaxy,which is about two orders of magnitude larger than the roughestimates by Centrella & McMillan (1993).

Two previously published sets of hydrodynamical models ofcolliding neutron stars were computed by Centrella & McMil-lan (1993) and by Rasio & Shapiro (1992). Both used an SPHcode to simulate the dynamics of two polytropic neutron starsof equal masses with adiabatic indexγ = 2 and started at an ini-tial distance of about four neutron star radii on parabolic orbits.Using a relatively small number of particles (2048), Centrella& McMillan (1993) were able to produce a catalog of gravita-tional wave luminosities and gravitational waveforms for dif-ferent impact parameters. Rasio & Shapiro (1992) restrictedthemselves to fewer simulations with higher resolution (16000particles) of neutron star coalescence as well as head-on colli-sions. They found that up to 5% of the total mass can escapefrom the systems and that strong shocks occur. Their results forthe gravitational waveforms and luminosities reveal a smallerfirst maximum (caused by the initial free fall of the two stars),followed by the main peak associated with the rapid decelera-tion of the colliding matter during the propagation of the recoilshocks.

Neutron star collisions have repeatedly been suggested inthe literature as possible sources of gamma-ray bursts (e.g., Katz& Canel 1996, Dokuchaev& Eroshenko 1996; also Dokuchaevet al. 1998), powered either by neutrino-antineutrino annihila-tion which produces ane± pair-photon fireball, or by highlyrelativistic shocks which are formed during the collision (or co-alescence) and eject matter at relativistic velocities (Shaviv &Dar 1995). Katz & Canel (1996), in particular, developed theidea that long gamma-ray bursts might be explained by accre-tion induced collapse of bare degenerate white dwarfs, whileshort bursts might originate from neutron starcollisions. Theyestimated the post-collision temperature to bekT ≈ 100 MeV,

536 M. Ruffert & H.-Th. Janka: Colliding neutron stars

and expected a neutrino pulse with a width of 2.5 ms, and atotal number of escaping neutrinos of1058 (refering to Dar etal. 1992). With a mean neutrino energy of about6 MeV thiscorresponds to a total energy of1053 erg emitted in neutrinos.They sketched the picture that shortly after the collision, denselumps of hot matter expand and become optically thin to neu-trinos which are released in a powerful outburst that is lumi-nous enough to produce the desired gamma-ray burst energy byneutrino-antineutrino annihilation if one assumes an efficiencyof 1%. In order to obtain the observed gamma-ray burst ratesthey had to invoke a total of109 hypothetical clusters withinz ≈ 1 with 108 neutron stars each. They stated that “we cannotexclude the possibility that such clusters are commonly foundat the centers of galaxies”.

Neutron star collisions might also be considered as interest-ing, because they could be viewed at as more violent variants ofthe merging process with a more extreme dynamical evolution.This might support the formation of relativistic shocks and pos-sibly lead to a larger burst of gravitational waves and neutrinos.More violence of the merging process can be expected from gen-eral relativistic effects which were so far neglected in the largemajority of simulations (see, however, Wilson et al. 1996 foran interesting step towards fully general relativistic modeling).Oohara & Nakamura (1997), using a grid-based TVD scheme,performed preliminary simulations to compare the coalescencewith a Newtonian potential to the case with post-Newtonianphysics. Their neutron stars were modeled as two polytropeswith γ = 2, M = 0.62 M, andR = 15 km. The principaldifference between the two cases was that the post-Newtonianmerging indeed turned out to be more violent: the impact ismore central and shocks develop, because “general relativity ef-fectively increases the gravitational force”. Although the strongshock itself has little immediate effect on the gravitational wave-form, the dynamics can be changed because of the higher tem-peratures and densities. Wex (1995) and Ogawaguchi & Kojima(1996) showed that both spin-orbit and spin-spin interactions ap-pear formally to befirst order post-Newtonian corrections, justas gravitational potential corrections are (gravitational wavesare 2.5 PN), but the inferred magnitude of these corrections forknown compact binary systems is actually smaller.

Our project of simulating neutron star collisions with a New-tonian PPM code was motivated by the aspects described in thelast two paragraphs. On the one hand, we intended to put po-tential gamma-ray burst scenarios to a test, on the other handwe wanted to study a situation that mimics themergingof twoneutron stars with extreme violence and maximal parameterslike pre-merging kinetic energy and angular momentum in thesystem. Thus we hoped not only to obtain an upper bound onthe gravitational wave emission to be expected from the merg-ing of two neutron stars, even with general relativistic effectsincluded. We also wanted to see whether the most extreme con-ditions during the collision of the two stars lead to sufficientlylarge neutrino emission to explain the gamma-ray burst ener-getics by the annihilation of neutrinos and antineutrinos emittedduring the dynamical event. The latter seems impossible in caseof the final stages of the coalescence of binary neutron stars be-

cause the prompt neutrino burst, although very luminous, failsby several orders of magnitude to produce about1051 erg ofgamma-rays within the short time of only a few millisecondsthat it takes the two neutron stars to merge into one massivebody that is most likely going to collapse into a black hole on adynamical timescale (Ruffert et al. 1996).

The paper is organized as follows. In Sect. 2 the basic aspectsand new elements of our computational method (in extensionof Ruffert et al. 1996) and the chosen initial conditions for oursimulations are given. Our results are presented in the followingsections, where we describe the hydrodynamical and thermo-dynamical evolution (Sect. 3), the gravitational wave emission(Sect. 4), and the neutrino production (Sect. 5) in the collidingstars together with the evaluation of our models for the effi-ciency of neutrino-antineutrino annihilation (Sect. 6). Sect. 7contains a summary and conclusions.

2. Computational procedure and initial conditions

In this section we summarize the numerical methods and thetreatment of the input physics used for the presented simula-tions. In addition, we specify the initial conditions by whichour different models are distinguished. More detailed informa-tion about the employed numerical procedures can be found inRJS (Ruffert et al. 1996) and RJTS (Ruffert et al. 1997).

2.1. Hydrodynamical code

The hydrodynamical simulations were done with a code basedon the Piecewise Parabolic Method (PPM) developed by Colella& Woodward (1984). The code is basically Newtonian, but con-tains the terms necessary to describe gravitational-wave emis-sion and the corresponding back-reaction on the hydrodynam-ical flow (Blanchet et al. 1990). The modifications that followfrom the gravitational potential are implemented as source termsin the PPM algorithm. The necessary spatial derivatives are eval-uated as standard centered differences on the grid.

In order to describe the thermodynamics of the neutron starmatter, we use the equation of state (EOS) of Lattimer & Swesty(1991) for a compressibility modulus of bulk nuclear matter ofK = 180 MeV in tabular form. Energy loss and changes of theelectron abundance due to the emission of neutrinos and antineu-trinos are taken into account by an elaborate “neutrino leakagescheme”. The energy source terms contain the production of alltypes of neutrino pairs by thermal processes and additionallyof electron neutrinos and antineutrinos by lepton captures ontobaryons. The latter reactions act as sources or sinks of leptonnumber, too, and are included as source terms in a continu-ity equation for the electron lepton number. Matter is renderedoptically thick to neutrinos due to the main opacity producingreactions which are neutrino-nucleon scattering and absorptionof electron-type neutrinos onto nucleons.

More detailed information about the employed numericalprocedures can be found in RJS, in particular about the imple-mentation of the gravitational-wave radiation and back-reaction

M. Ruffert & H.-Th. Janka: Colliding neutron stars 537

Table 1. Characterizing parameters and some computed quantities for all models.N is the number of grid zones per dimension,L the size of thelargest grid,l the size of the smallest zone,Mρ<11 the gas mass with density below1011 g/cm3 at the end of the simulation, andMd the masswith specific angular momentum larger thanj∗ ≡ vKepler(3Rs)3Rs whereRs is the event horizon of a∼ 3 M black hole probably formingfrom the colliding neutron stars.Mg denotes the mass swept off the grid,Mu is the mass leaving the grid unbound, andTex the maximumtemperature (in energy units) reached on the grid during the simulation of a model.

Model impact spin N L l Mρ<11 Md Mg Mu Tex

direction km km 10−2M 10−2M 10−2M 10−2M MeV

h head-on none 32 328 1.28 4.0 0.0 5.2 1.5 96.H head-on none 64 328 0.64 4.1 0.0 6.6 1.5 96.H head-on none 128 328 0.32 — — — — —

o off-center none 32 328 1.28 9.0 0.03 1.5 0.16 57.O off-center none 64 328 0.64 11.0 0.03 1.8 0.19 58.

Table 2. Gravitational-wave and neutrino emission properties for all models.L is the maximum gravitational-wave luminosity,E the totalenergy emitted in gravitational waves,rh the maximum amplitude of the gravitational waves as observed from a distancer, Lνe the stationaryvalue of the electron neutrino luminosity which is reached at about 6–10 ms after the start of the simulations,Lνe the corresponding electronantineutrino luminosity, andLνx the luminosity of each individual species ofνx (= νµ, νµ, ντ , ντ ). Lν represents the total neutrino luminosityat the end of the simulation,〈ενe〉, 〈ενe〉 and〈ενx〉 are the mean energies of the different neutrino and antineutrino flavors.Eνν denotes theintegral rate of energy deposition by neutrino-antineutrino annihilation, averaged over the simulation time of 10 ms.

Model impact L E rh Lνe Lνe Lνx Lν 〈ενe〉〈ενe〉〈ενx〉 Eνν

direction 1055 ergs 1052 erg 104cm 1053 erg

s 1053 ergs 1053 erg

s 1053 ergs MeV MeV MeV 1050 erg

s

h head-on 4.2 0.42 6.3 2.4 5.0 1.4 13. 14. 20. 26. —H head-on 3.61 0.39 6.2 2.0 4.0 1.0 10. 13. 20. 25. 100H head-on 3.55 — 6.2 — — — — — — — —

o off-center 6.6 4.1 12.3 1.3 2.8 0.8 7.3 11. 18. 25. —O off-center 6.1 3.6 11.9 1.2 3.1 0.8 7.5 11. 18. 25. —

terms and the treatment of the neutrino lepton number and en-ergy loss terms in the hydrodynamical code.

We have extended and improved the numerical treatment ascompared to RJS in several aspects (a comparison of publishedresults for coalescing neutron stars obtained with the old codeagainst results from the improved one will be given in a separate,forthcoming paper):

(a) Numerical resolution: The presented simulations weredone on multiply nested and refined grids. With an only modestincrease in CPU time, the nested grids allow one to simulatea substantially larger computational volume while at the sametime they permit a higher local spatial resolution of the mergedobject. The former is important to follow the fate of matterthat is flung out to distances far away from the collision siteeither to become unbound or to eventually fall back. The latteris necessary to adequately resolve the strong shock fronts andsteep discontinuities of the plasma flow that develop during thecollision. The procedures used here are based on the algorithmsthat can be found in Berger & Colella (1989), Berger (1987)and Berger & Oliger (1984). Our version is described in detail inSect. 4 of Ruffert (1992) so only the most important features areto be summarized here. The individual grids are equidistant andCartesian with each finer grid having a factor of two smaller zonesize and extent than the next coarser one. Hereby the number

of zones remains the same for all grids, typically323 for low-resolution test calculations,643 for our “standard” simulations,and1283 for models of special interest where high resolutionseems desirable.

(b) Physics input: The table for the Lattimer & Swesty(1991) equation of state was extended to higher and lowertemperatures and now spans0.01 MeV ≤ T ≤ 100 MeV,and also the lower density bound was moved down to nowρmin = 5 · 107 g cm−3 so that the density range now coveredby the table is5 · 107 g cm−3 ≤ ρ ≤ 2.9 · 1015 g cm−3.

2.2. Evaluation of neutrino-antineutrino annihilation

In a post-processing step, performed after the hydrodynami-cal evolution had been calculated, we evaluated our models forneutrino-antineutrino (νν) annihilation in the surroundings ofthe collided stars in order to construct a map showing the lo-cal energy deposition rates per unit volume. Spatial integrationfinally yields the total rate of energy deposition outside the neu-trino emitting high-density regions. The “brute force” approach,however, which was applied by RJTS, is not feasible any longerbecause it involved explicit summation of the contributions ofneutrino and antineutrino loss terms of all grid cells and at everylocation where the local annihilation rate was to be determined.


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Fig. 1a–d.Contour plots of ModelH (left panels) and H (right panels) showing cuts in a plane containing thex-axis which is the symmetryaxis of the initial model. The displayed physical quantities are the density together with the velocity field (panelsa andb) and the temperature(panelsc andd). The density is measured ing cm−3, the temperature in MeV. The density contours are spaced logarithmically with intervalsof 0.5 dex, while the temperature contours are linearly spaced, starting with 1 MeV, 3 MeV, 5 MeV, and then continuing with an increment of5 MeV. The bold contours are labeled with their corresponding values (1010, 1012, and1014 g cm−3, and 10, 30, and 50 MeV, respectively).In the box in the upper right corner of each panel, the velocity vectors and the time elapsed since the beginning of the simulation are given.The mirror symmetry relative to the planex = 0 and the rotational symmetry around thex-axis are broken during the evolution (panelsb andd) because of an instability of the contact layer of the two neutron stars against shear motions by which numerical fluctuations (panelc) areamplified.

The computational load for this procedure increases roughlywith the third power of the number of grid zones if the anni-hilation map has about the same spatial resolution as the gridfor the hydrodynamical simulation. With the larger number ofzones on several levels of the nested grid, such a strategy iscurrently computationally impossible.

Therefore we resort to a different approach which involvesfive distinct steps. (1) First, the relevant physical quantities aremapped from only that fractional volume of the nested gridswhere most of the neutrino emission (and neutrino opacity)comes from, to an equidistant Cartesian grid of fairly high reso-lution (typically1403). This mapping is done by tri-linear inter-polation. (2) Second, on this grid, the neutrinosphere for each


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Fig. 2a–d.Contour plots of ModelH (left panels) and H (right panels) showing cuts in a plane containing thex-axis which is the symmetryaxis of the initial model. The displayed physical quantities are the electron fractionYe (panelsa andb) and the entropy (panelsc andd), thelatter measured in units of Boltzmann’s constantk per nucleon. The contours of the electron fraction are linearly spaced with intervals of 0.02below 0.1 and with intervals of 0.05 above, the entropy contours are given in steps of1 k/nucleon from 1 to 6, in steps of2 k/nucleon between6 and 16, and then for values of 20, 25, 30, and 40k/nucleon. The bold contours are labeled with their corresponding values (0.02, 0.06, 0.10,0.20, 0.30, and 0.40 forYe, and 6, 10, and 20 for the entropy). Maximum values ofYe are above 0.4, of the entropy near30 k/nucleon. In thebox in the upper right corner of each panel, the time elapsed since the beginning of the simulation is given.

flavor of neutrino or antineutrinoνi is determined. We definethis two-dimensional hypersurface forνi by the set of thosetriples (x, y, z) where, for each chosen pair of coordinatesxandy, the vertical optical depthτνi

(z) satisfies the conditionτνi

(z) = ∆z · ∑∞j=z κi(j) = 1 where∆z is the cell size of

the Cartesian grid andκi(j) the local opacity of neutrinoνi atpositionj. (3) Thirdly, the local neutrino number and energyloss terms of neutrinoνi are added up along thez-direction(for each fixed pair(x, y)), and the total neutrino emissivity

and the corresponding average energy of the emitted neutrinosare projected to originate from the neutrinosphere of neutrinoνi determined in step (2). (4) Fourthly, and most importantly,the energy deposition rates byνν annihilation are calculatedby integrating (summing) only over the two-dimensional neu-trinospheres as neutrino and antineutrino sources instead of thethree-dimensional stellar volume as done in RJTS. Additionalconditions imposed on the construction of the annihilation mapby RJTS are also used here, i.e., the neutrino emission is as-


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Fig. 3a–d.Contour plots of Model O showing cuts in the orbital plane for the density together with the velocity field (panelsa andb) and forthe temperature (panelsc andd). The density is measured ing cm−3, the temperature in MeV. The density contours are spaced logarithmicallywith intervals of 0.5 dex, while the temperature contours are linearly spaced, starting with 1 MeV, 3 MeV, 5 MeV, and then continuing with anincrement of 5 MeV. The bold contours are labeled with their corresponding values (1010, 1012, and1014 g cm−3, and 10, 30, and 50 MeV,respectively). In the box in the upper right corner of each panel, the velocity vectors and the time elapsed since the beginning of the simulationare given.

sumed to occur isotropically around theoutward pointinglo-cal density gradients at the neutrinospheres, and the energy de-position byνν annihilation is evaluated only in those regionswhere the baryon density is below a certain threshold, typicallyρ < 1011 g cm−3, and on a cylindrical grid with coarser res-olution than the Cartesian grid used to represent the neutrinosources, in order to limit the costs of the numerically intensecalculations. (5) Finally, the local energy deposition rates per

unit volume,Eann($, φ, z), are averaged over theφ directionof the cylindrical grid:

Eann($, z) =12π

∫ 2π

0Eann($, φ, z) dφ . (1)

From these average values two-dimensional maps like the oneshown in Sect. 6 are plotted and integral numbers can be ob-tained by summation along radial or vertical directions in thecylindrical grid.


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Fig. 4a–d.Contour plots of Model O showing cuts in the orbital plane for the electron fractionYe (panelsa andb) and the entropy (panelsc andd), the latter quantity measured in units of Boltzmann’s constantk per nucleon. The contours are linearly spaced with intervals of 0.02 for theelectron fraction and1 k/nucleon for the entropy. The bold contours are labeled with their corresponding values (0.02, 0.06, 0.10, 0.20 forYe

and 5 and 10 for the entropy). Maximum values ofYe are around 0.16, of the entropy about10 k/nucleon. In the box in the upper right cornerof each panel, the time elapsed since the beginning of the simulation is given.

The total energy deposition rate (“annihilation luminosity”)Lann is obtained from the local values of the energy depositionrate per unit volume,Eann($, z), by integration over the wholespace outside the neutrino emitting stellar source:

Lann =∫

Eann($, φ, z) dV (2)

with dV = $ d$ dφdz. Given the time dependent functionLann(t) one can then calculate the cumulative energy depositionby neutrino-antineutrino annihilation according to

Eann =∫

Lann(t) dt . (3)

Because the computation is so expensive, however,Lann(t) can-not be evaluated on a fine temporal grid, but only at a few discretepoints in time,ti. With the valuesLann(ti) we therefore makethe following approximation forEann:

Eann ≈ 1N

N∑i=1

(Lann(ti)F(ti)

)·∫

F(t) dt (4)

whereF(t) is defined by

F(t) ≡ 1R(t)

Lνe

(t) Lνe(t) [〈ενe

(t)〉 + 〈ενe(t)〉] +


density

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Fig. 5.Cut through Model O showing the density contours together withthe velocity field in the planey = 0 perpendicular to the orbital planenear the end of the simulation (t = 9.27ms). The contours are spacedlogarithmically with intervals of 0.5 dex, the bold contours correspondto density values of1010, 1012, and1014 g cm−3, respectively. Thevelocity vectors are normalized as in Fig. 3b.

2 · Lνµ(t) Lνµ

(t)[〈ενµ

(t)〉 + 〈ενµ(t)〉] . (5)

Here the term multiplied by the factor 2 accounts for the equalcontributions fromνµνµ annihilation andντ ντ annihilation. Theform of F(t) in Eq. (5) reflects the main dependences of theννannihilation rate: The energy deposition rate increases propor-tional to the product of the neutrino and antineutrino luminosi-ties times the sum of the mean energies of the annihilating neu-trinos and antineutrinos; in the denominator the characteristicradial extentR(t) of the neutrino source comes from the vol-ume integral of Eq. (2) when the latter is performed in sphericalcoordinates (compare Eqs. (3) and (10) in RJTS and referencestherein). The ratio appearing in Eq. (4) in the sum in front ofthe time integral then contains geometrical effects which resultfrom the dependence of theνν annihilation rate on the angulardistributions of neutrinos and antineutrinos. From the hydro-dynamical models, the neutrino luminosities for the individualneutrino flavors,Lνe

(t), Lνe(t) and the corresponding values

for muon and tau neutrinos and antineutrinos, are available asfunctions of time as well as the average energies of the emittedneutrinos,〈ενe(t)〉, 〈ενe(t)〉, and〈ενx(t)〉 for νµ, νµ, ντ , andντ . We found that the typical radial sizeR(t) of the neutrinoemitting object during the phase where by far most of theννannihilation happens is not very strongly time-dependent be-cause the wobblings and oscillations change the shape and sizeof the collision remnant only on smaller scales but not globally.Therefore instead ofF(t) from Eq. (5) we use in Eq. (4) thesimpler expression

F∗(t) ≡ Lνe(t) Lνe

(t) [〈ενe(t)〉 + 〈ενe

(t)〉] +2 · Lνµ

(t) Lνµ(t)

[〈ενµ(t)〉 + 〈ενµ

(t)〉] . (6)

ComputingEann from Eq. (4) instead of Eq. (3) involves theapproximation that the term abbreviated byF(t) or F∗(t) con-tains the main time dependence of the integral in Eq. (3). Ideally,the ratioLann(t)/F∗(t) would have to be constant. Since this isnot the case, we decided to employ an average value for a smallnumberN of time points where the spatial integral of Eq. (2)was evaluated. It turned out that the variation ofLann(t)/F∗(t)during the most interesting phase of the evolution is less than afactor 2.

2.3. Initial conditions

We started our simulations with two identical Newtonian neu-tron stars, each having a baryonic mass of about 1.63M and aradius of 15 km, which were placed at a center-to-center distanceof 42 km. The distributions of densityρ and electron fractionYe ≡ ne/nb (with ne being the number density of electrons mi-nus that of positrons, andnb the baryon number density) weretaken from a one-dimensional model of a cold, deleptonizedneutron star in hydrostatic equilibrium and were the same asin RJS. For numerical reasons the surroundings of the neutronstars cannot be treated as completely evacuated. The densityof the ambient medium was set to less than108 g/cm3, morethan six orders of magnitude smaller than the central densitiesof the stars. The total mass on the whole grid, associated withthis finite density is less than10−3 M.

The neutron stars were given the free-fall velocity at theirrespective initial positions(x, y, z) = (−21 km, 0, 0) and(x, y, z) = (21 km, 0, 0). The angle between the velocity vec-tors and the vector connecting the stellar centers was varied toproduce a head-on collision for Models h, H, andH, and anoff-center collision for Models o and O. The impact parameterof the latter was chosen to be one neutron star radius. This im-pact parameter is the minimum distance that two point massesreach along their orbits. A compilation of all models togetherwith their characterizing grid parameters, initial parameter set-tings, and some results of the numerical simulations is given inTables 1 and 2.

In degenerate matter variations of the temperature lead onlyto minor changes of the internal energy and pressure (both aredominated by degeneracy effects) or, inversely, the temperatureis extremely sensitive to small variations of the total internalenergy. Therefore any small fluctuation caused for example bysmall numerical errors in the calculation of the energy den-sity, will be amplified and reflected in temperature fluctuations.Subsequently, the neutrino emission, which scales with a highpower of the temperatureT , will be very noisy. For this reasonwe did not start our simulations with cold (T = 0) or “cool”(T <∼ 108 K) neutron stars as suggested by the investigationsof Kochanek (1992), Bildsten & Cutler (1992), and Lai (1994).Instead, we constructed initial temperature distributions insidethe neutron stars by assuming thermal energy densities of about3% of the degeneracy energy density for a given densityρ andelectron fractionYe. The corresponding central temperature wasaround 7 MeV, the surface temperature less than half an MeV,and the average temperature was a few MeV. Because of the


density separation

0 2 4 6 8 10time [ms]

0

10

20

30

40

50

60

70di

stan

ce [k

m]

model H

model O

model o

Fig. 6. The separation of the density maxima of the two neutron starsas a function of time for the three Models H, O, and o.

orbits of maximum density

-30 -20 -10 0 10 20 30x axis [km]

-30

-20

-10

0

10

20

30

y ax

is [k

m] 0ms<t<0.15ms

0.15ms<t<2.4ms

2.4ms<t<3.3ms

3.3ms<t<3.6ms

Fig. 7. Trajectory described by the density maximum of one of theneutron stars in a Eulerian frame for the off-center collision, Model O.Different line styles denote different orbits or phases between closestapproaches. The kinks are numerical and due to the fact that the positionof the maximum density is given by the integers corresponding to theindices of the numerical grid cells.

small contribution of thermal effects to the pressure, these tem-peratures are unimportant for the neutron star structure, andthe rapid and violent hydrodynamical evolution ensures that theresults are essentially unaffected by the assumed finite initialtemperatures.

The simulations were performed on a Cray-YMP 4/64. Themodels with 64 zones needed about 24 MWords of main memoryand took approximately 160 CPU-hours each, models with 32zones roughly a factor of 10 less. Movies were generated forevery model.

maximum density

0 2 4 6 8 10time [ms]

0

2

4

6

8

10

12

rho

[1014

g/cm

3 ]

model H

model O

Fig. 8. The maximum density on the grid as a function of time for thetwo Models H and O.

maximum temperature

0 2 4 6 8 10time [ms]

0

20

40

60

80

100

T [M

eV]

model H

model O

Fig. 9.The maximum temperature on the grid as a function of time forthe two Models H and O.

3. Hydrodynamical and thermodynamical evolution

3.1. Head-on collision

Figs. 1a–d and 2a–d show the densityρ, temperatureT , elec-tron fractionYe, and entropys at two different moments ofthe collision: Att = 0.23 ms after the start of the simulationthe largest compression is reached with a maximum density of1.1 ·1015 g cm−3 (see Fig. 8), associated with a prominent peakof the gravitational wave luminosity (see Figs. 15 and 17), andat t = 2.87 ms when the collision remnant has performed sev-eral cycles of oscillatory motions before it begins to settle intoa more quiet state, and the neutrino emission starts to decreasefrom its most powerful phase (Figs. 20 and 22). For the ear-lier time, data from the high-resolution ModelH are plotted inFigs. 1a–d and 2a–d, whereas for the later moment only datafrom the 643 Model H are available.


Shortly after the surfaces of the neutron stars have touchedduring their head-on collision (Figs. 1a and c), a strong shockwave is generated at the common surface by the abrupt deceler-ation of the matter. This shock propagates back into the as yetpractically undisturbed neutron star matter. The temperatures di-rectly behind this shock reach values of more than 45 MeV, whileahead of the shock they were around 5 MeV. The entropy in theinitially cool neutron star (s 1 k/nucleon) is increased to val-ues between 1k/nucleon and 2k/nucleon (Fig. 2c). At the sametime, matter is being squeezed out perpendicularly to the col-lision axis and expands behind a very strong shock (postshockentropies near 30k/nucleon). This shock-heated matter emitselectron antineutrinos in large numbers (cf. Fig. 22) and quicklydevelops from initially neutron-rich conditions to a much moresymmetric nuclear state characterized by an electron fractionYe >∼ 0.4 (Fig. 2a). In contrast, in the interior of the collidingbodies the composition remains essentially unchanged becauseof the long neutrino diffusion timescales in the hot neutron starmatter.

At the collision interface a thin “pancake” like layer withvery high temperatures up to about 70 MeV (Figs. 1c and 9) oc-curs. This sheet is dynamically unstable due to shear motions.First indications of a break-down of the mirror symmetry rel-ative to they-z-plane can be already seen att = 0.23 ms inFigs. 1c and 2a. Only a short moment later, when the mergedbodies bounce back and the oblate shape changes into a pro-late form, this flat pancake-like layer folds asymmetrically andbreaks up on a millisecond timescale (Fig. 1d). Within 3 ms thedensity distribution has already smoothed into a nearly spheri-cal shape (Fig. 1b) and most of the kinetic energy of the impacthas been dissipated by shocks into thermal energy or is carriedaway by ejected matter (see Fig. 12). The collision has leadto an increase of the entropy to values near 2k/nucleon in themerged object (Fig. 2c), whereas the shock heated gas that formsa very extended, nearly spherical cloud around the dense cen-tral body, has entropies between 6k/nucleon and 10k/nucleon(Fig. 2d). Ejected clumps of matter with even higher entropy(s ∼ 20 k/nucleon) can be identified, and positron capturesonto neutrons andνe production in the hot gas (T ∼ severalMeV) leads to a rapid increase of the electron fraction to valuesYe ∼ 0.3–0.4 and higher in the expanding debris.

The formation of shock waves at the moment of the im-pact in Model H (Fig. 1a–d) clearly indicates that the head-oncollision is strongly inelastic and the dissipation of kinetic en-ergy happens very efficiently. Therefore the two neutron starsare not able to separate again after the first compression andreexpansion, however, it takes several (4–6) violent oscillationsuntil all the kinetic energy is dissipated into heat. The reexpan-sions produce peaks of the separation of the density maxima inFig. 6, while the compression phases are reflected in a sequenceof very large density maxima in Fig. 8 and temperature maximain Fig. 9. The steady decrease of the density maxima in Fig. 8indicates that the oscillations come to a rest within about 4 ms.In contrast, the maximum temperature increases from one com-pression to the next (Fig. 9) because of the dissipative heatingof the stellar plasma. The most extreme temperatures that are

mass loss off grid

0 2 4 6 8 10time [ms]

0

1

2

3

4

5

6

7

M [%

Mso

l]

model H

model O

32 zones

64 zones

Fig. 10. Cumulative amount of matter flowing off the numerical grid(measured in10−2 M) as a function of time from the beginning ofthe simulations.

unbound mass

0 2 4 6 8 10time [ms]

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0lg

M [M

sol]

model H

model O

32 zones

64 zones

Fig. 11.Cumulative amount of matter that is unbound when it leaves thenumerical grid (in units ofM, plotted logarithmically) as a functionof time for Models o, O, h, and H. The unbound mass is determinedby the criterion that the sum of the specific gravitational, kinetic, andinternal energies is positive.

reached in Model H during the dynamical phase,0 < t <∼ 4 ms,are close to 100 MeV. After settling into a static statet > 4 ms,the maximum temperatures in the collision remnant is around40–50 MeV.

3.2. Off-center collision

The motion of the two neutron stars is rather complicated incase of theoff-centercollision, Model O. The first phase of theinfall (t <∼ 0.1 ms) proceeds essentially along point-mass binaryorbits (Fig. 7), until the stars start to touch and orbital energy andangular momentum are converted into neutron star spin and areconsumed by the acceleration of matter which is flung off the


energy, model H

0 2 4 6 8 10time [ms]

-10

-5

0

5E

[1053

erg]

internal

kineticgravi waves

total

potential

Fig. 12.The internal, kinetic, and potential energies of the matter on thegrid and the sum of these energies as functions of time for Model H.Also plotted is the cumulative energy carried away in gravitationalwaves.

energy, model O

0 2 4 6 8 10time [ms]

-10

-8

-6

-4

-2

0

2

4

E [1

053er

g]

internal

kinetic

gravi waves

total

potential

Fig. 13.Same as Fig. 12 but for Model O. The thin solid line marks theE = 0 level.

neutron star surfaces. The corresponding loss of orbital angularmomentum and kinetic energy leads to a transformation of theinitially parabolic orbits into elliptic ones. Even more orbitalenergy is transfered to internal energy when the neutron starscome into contact and inelastic interaction sets in.

After the first closest approach, visible in Fig. 6 as minimumdistancedmin ≈ 3 km of the density maxima of the two neu-tron stars att ≈ 0.15 ms, the positions of the density maximadescribe three nearly elliptic orbits between moments of clos-est approach att ≈ 0.15 ms, 2.4 ms, 3.3 ms, and3.6 ms (seealso Fig. 6). In Fig. 7 these three orbits, represented by thex-y-trajectory of the density maximum of one of the neutron stars,are discerned by different line styles. The diameters of the or-bits (and thus the maximum separation of the density maxima)become successively smaller due to the inelasticity of the con-

tact between the stars during their close encounters. The majoraxis of the first ellipse has a length of more than 30 km, corre-sponding to an apastron separation of the density maxima onthe grid of about 65 km att ≈ 1.2 ms (Fig. 6), much larger thanthe initial distance of the neutron star centers which was42 km.This means that the neutron stars separate again after their firstencounter, but fall back towards each other again. Even duringthe second apastron att ≈ 2.9 ms, the density maxima of thetwo stars have a distance of about 35 km, which is larger thanthe sum of their initial radii2Rns,i = 30 km. During the thirdand the subsequent quasi-elliptic orbits the neutron stars are notable to separate again. While the first bound orbit has a periodof about 2.2 ms, the second, smaller one has only∼ 0.9 ms, andthe following are even shorter (Figs 6 and 7).

In Figs. 3a–d and 4a–d the density distribution, tempera-ture, electron fraction, and entropy per nucleon are plotted forModel O in the orbital plane at two different stages during theoff-center collision. The left panels show the results for a timeclose to the apastron of the first orbital ellipse, i.e., a little morethan a millisecond after the first closest approach. The neutronstars are tidally strongly deformed and gas has been swept intothe surroundings during the first direct contact and interaction(Fig. 3a). There is a dense gas bridge (ρ ∼ 1012–1013 g cm−3)between the stars which continue to wobble and oscillate alongtheir orbits. The temperature has climbed to nearly 40 MeV indistinct hot spots where the gas bridge hits the denser cores ofthe stars (Figs. 3c and 9), whereas the extended cloud of gas sur-rounding the orbiting bodies has a temperature of 1–5 MeV. Inthis ambient gas,νe production by positron capture onto neu-trons has raised the electron fraction from initially less thanabout 0.05 to maximum values around 0.2 (Fig. 4a). The max-imum entropy values ofs ∼ 11 k per nucleon are produced bybow shocks in front of the rather dilute (ρ ∼ 109–1010 g cm−3)clouds reaching outward from the two neutron stars. Clumpsof gas with entropys ∼ 5–6 k/nucleon are scattered in thesurroundings of the collision site. Note that in contrast to thehead-on collision, there is no shock heating of the interior of theneutron stars. Up to the end of the simulation the high-densitycores of the neutron stars retain their low initial entropy, evenafter they have merged into one body.

The right panels in Figs. 3a–d and 4a–d show snapshotsat a time near the end of the simulation (t = 9.27 ms). Thedistributions of density, entropy, and electron fraction have be-come roughly circular in thex-y-plane: A compact, dense cen-tral body (ρ > 1012 g cm−3) with T ∼ 5–10 MeV (outside oftwo distinct hot spots whereT ∼ 40 MeV), Ye ∼ 0.04–0.1,ands <∼ 7 k/nucleon is surrounded by an extended envelopewith somewhat largerYe ands (but lower temperature), whichis rapidly rotating and which is stabilized by centrifugal forces.Therefore the vertical extension of the gas envelope is signifi-cantly smaller than its diameter in the orbital plane; the densitycontour corresponding toρ = 1010 g cm−3 extends to a ra-dius of about 130 km in the orbital plane, whereas its butterflyshape has a maximum vertical height of roughly 70 km (Fig. 5).Even the compact core is rotationally deformed with an axisratio of 1:1.5. Nevertheless, there is a considerable amount of


gas at large heights|z| above and below the orbital plane. Byspatial integration we find3.8 × 10−2 M for |z| ≥ 40 km,2.2 × 10−2 M for |z| ≥ 50 km, and still6.0 × 10−3 M for|z| ≥ 80 km.

3.3. Comparison of head-on and off-center collisions

The different dynamical evolutions of the head-on collision,Model H, and the off-center collision, Model O, are reflected inthe different time histories of the maximum density (Fig. 8) andthe maximum temperature (Fig. 9) on the grid. Model H showsvery large amplitudes of the maximum density at the momentsof strongest compression with up to a factor of∼ 10 larger val-ues compared to the return points of expansion phases. Thesedensity fluctuations are damped in a sequence of 5–6 strong os-cillations, and a stationary value is reached after about 4 ms. Thetemperature evolution reveals spikes and valleys, respectively,at the same moments, but there is a general trend of the tempera-ture to increase during the dissipative oscillatory compressionsand reexpansions. Although in Model O the variations of theseparation of the density maxima are much more pronouncedthan in Model H (Fig. 6), the maximum density and tempera-ture on the grid show much less extreme fluctuations becausethe neutrons stars describe orbits around each other and do notcrash violently into each other. One can recognize peaks of themaximum density correlated with the moments of closest ap-proaches,t ≈ 0.15 ms, 2.4 ms, 3.3 ms, and 3.6 ms (compareFig. 8 and Fig. 7). The whole evolution of Model O is much lessviolent than that of Model H. Nevertheless, on a longer timescalet >∼ 4 ms, both models settle to roughly the same maximumtemperature of 40–50 MeV. The maximum density in Model Obecomes even somewhat larger than in Model H, because thelatter has been heated to higher entropies and therefore ther-mal pressure causes an expansion of the collision remnant. Inaddition, the compact core of the remnant of Model H is lessmassive as a result of the larger mass loss during the collision.

The more violent collision and therefore higher tempera-tures in Model H push more matter off the grid than in Model O(Fig. 10). Also a larger fraction of this matter gets unbound(Fig. 11) which is the case when the total specific energy of thegas, defined as the sum of the specific kinetic, internal, and grav-itational potential energies, becomes positive. In Model H about1.5 × 10−2 M are able to escape the gravitational potential ofthe collision remnant, whereas it is little more than one tenth ofthis amount in Model O (cf. Table 1). Obviously, the angular mo-mentum of Model O (see Fig. 14) and the associated centrifugalshredding of material can hardly compete with the ejection ofgas in the strong shock waves occurring in Model H. We note inpassing that Rasio & Shapiro (1992) (RS) found a significantlylarger amount of mass loss (up to about 5% of the total mass)in their simulations of head-on collisions between two identicalγ = 2 polytropes, compared to only∼ 0.46% that can escapefrom the system in our Model H. The difference is presumablycaused by a combination of reasons, the use of different equa-tions of state and correspondingly different stellar structure andmass (Lattimer & Swesty nuclear EOS here vs.γ = 2 adiabatic

angular momentum

0 2 4 6 8 10time [ms]

0.0

0.2

0.4

0.6

0.8

angu

lar

mom

entu

m

[1050

cm2 g/

s]

model O

model o

total on grid

gravitational wave loss

Fig. 14.Total angular momentum (z-component) of the matter on thegrid (upper curves) and cumulative angular momentum loss by gravita-tional wave emission (lower curves) as functions of time for Models Oand o. Gravitational waves carry away about 13% of the initial angularmomentum of the colliding neutron stars and another non-negligibleamount (∼ 3.5%) is taken away by the matter flowing off the numericalgrid.

EOS by RS), the inclusion of gravitational wave back-reactionon the hydrodynamical flow in our simulations, and last but notleast the use of different numerical schemes (Eulerian PPM herevs. SPH by RS) in combination with possibly different criteriato determine the unbound mass.

The temporal evolutions of the internal, kinetic, and grav-itational potential energies (Figs. 12 and 13) show structuresthat correspond to the stages of the dynamical interactions ofthe colliding stars. The internal energy of Model H oscillatesstrongly with maxima at the moments of strongest compression(compare Figs. 8 and 12) which coincide with maxima of thekinetic and minima of the potential energy. There is a generaltrend for the internal energy to increase with time. This corre-sponds to a decrease of the kinetic energy (while the potentialenergy fluctuates around an essentially constant level) and thusreflects the action of dissipative forces. In the off-center col-lision of Model O the kinetic energy is much less efficientlyconverted into internal energy. The latter exhibits a continu-ous increase with superimposed, but much less dramatic, localpeaks at the instants of closest approach, also coinciding withmaxima of the kinetic and minima of the potential energy. Thisreflects the dynamical transformation between the different en-ergy forms. Near the end of the computed evolution, practi-cally all of the kinetic energy of Model O is rotational energy,Ekin = Erot. Using the values from Fig. 13, we determine thefollowing ratio of kinetic energy to the gravitational bindingenergy:β = Ekin/Erot ≈ 0.08.

The total energies of Model H and Model O (defined as thesum of kinetic, internal, and potential energies of all gas on thegrid) are also shown in Figs. 12 and 13, respectively. In Model Hminor variations of the total energy betweent ≈ 0.2 ms andt ≈


2.2 ms are numerical because of the extremely violent collision.Since the energy carried away by gravitational waves and massloss off the numerical grid is negligible, the total energy at thebeginning and end of the simulation are nearly equal. In contrast,in Model O the energy emitted in gravitational waves leads to agradual decrease of the total energy of the gas on the grid.

Fig. 14 shows the total angular momentum (z-component)of the gas on the grid and the cumulative value of the an-gular momentum carried away by the emitted gravitationalwaves as a function of time for Models O and o. There isvery good agreement of both calculations concerning the angu-lar momentum loss in gravitational waves. This indicates thatthe overall mass distribution in the neutron stars (which en-ters the calculation of the mass quadrupole moment needed forthe evaluation of the gravitational-wave source terms) is suffi-ciently well represented even on the coarser grid of Model o.Most of the decrease of the gas angular momentum is ex-plained by the gravitational wave emission. An additional ef-fect comes from the mass loss off the computational grid at4 ms <∼ t <∼ 8 ms (Fig. 10) which removes an angular momen-tum of about∆Jz ≈ Mgrgvg ≈ 2 × 1048 g cm2 s−1 (with Mgtaken from Table 1 andrg ≈ 160 km being the grid radius andvg ≈ 3.5 × 109 cm s−1 the nearly tangential velocity of the gaswhen it leaves the grid) or 3.5% of the initial angular momen-tum. However, although the gravitational wave loss and the massflowing off the grid are very similar in both models, Model oexhibits a steeper decrease of the total angular momentum att >∼ 3 ms than Model O. This difference is purely numericaland caused by the coarser grid resolution of Model o. Even inthe better resolved calculation, Model O, about 7% of the initialangular momentum are destroyed by numerical effects at theend of the simulation.

The relativistic rotation parameter is defined asa ≡Jc/(GM2

tot)whereMtot is the total mass of the system (Mtot =2M initially with M being the mass of one of the neutron stars)andJ is the total angular momentum relative to the center ofmass. We have an initial value ofa = 0.60 and find a valueof a >∼ 0.47 (in Model O) at the end of the simulation af-ter angular momentum has been removed from the system bygravitational waves and ejected matter. Sincea < 1 rotationseems unable to prevent the collapse of the collision remnant toa black hole if the remnant mass exceeds the maximum stablemass of the employed equation of state (see also RJS and Rasio& Shapiro, 1992, and references therein). Thermal pressure canincrease this stable mass limit only insignificantly (cf. Goussardet al. 1997) and, if so, only during the transient period of neu-trino cooling (note that the interior of the neutron stars retains itsinitial low entropy in the off-center collision, see Fig. 4c and d,and thus the temperature remains fairly low), and also rotationleads to an increase of the upper stability limit on the baryonmass by only<∼ 20% (Friedman et al. 1986; Friedman & Ipser1987).

gravitational waves, model H

0 2 4 6 8 10time [ms]

48

50

52

54

56

lg lu

min

osity

[erg

/s]

64 zones32 zones

0.00

0.10

0.20

0.30

0.40

0.50

ener

gy [%

Mso

lc2 ]

Fig. 15.Gravitational-wave luminosity and cumulative energy emittedin gravitational waves (measured in units of10−2 Mc2) as functionsof time for Models H and h.

gravitational waves, model O

0 2 4 6 8 10time [ms]

48

50

52

54

56

lg lu

min

osity

[erg

/s]

64 zones32 zones

0

1

2

3

4

ener

gy [%

Mso

lc2 ]

Fig. 16.Same as Fig. 15 but for Models O and o.

4. Gravitational waves

The gravitational-wave luminosities and the cumulative energyloss in gravitational waves as functions of time for Models H andh and Models O and o are shown in Figs. 15 and 16, respectively,and the corresponding gravitational waveformsh+ andh× forModels H and O are plotted in Fig. 19.

In Model H the most prominent luminosity spike is createdat the moment when the two neutron stars crash into each otherand the gas flow is abruptly decelerated and redirected by therecoil shocks (t ≈ 0.22 ms, cf. Fig. 1a–d) which leads to a rapidchange of the mass quadrupole moment. The peak luminosityreaches about3.7 × 1055 erg/s (see also Fig. 17). A precursorwith about 1/4 of the maximum luminosity is caused by the in-creasing tidal deformation of the neutron stars as they approacheach other. After this initial outburst the gravitational-wave lu-minosity continues to oscillate regularly with a period betweentwo maxima of roughly half a millisecond but with peaks atleast one order of magnitude below the maximum luminosity.


gravitational wave energy

0.00 0.10 0.20 0.30 0.40 0.50time [ms]

0

1

2

3

4

5lu

min

osity

[1055

erg/

s]

128 zones

64 zones

32 zones

paper CM

paper RS

Fig. 17. Gravitational-wave luminosity as a function of time for thehead-oncollision ModelsH, H and h, compared with the results fromCentrella & McMillan (CM, 1993, Fig. 8) and from Rasio & Shapiro(RS, 1992, Fig. 9).

gravitational wave energy

0.0 0.2 0.4 0.6 0.8time [ms]

0

2

4

6

8

lum

inos

ity [1

055er

g/s] 64 zones

32 zonespaper CM

Fig. 18. Gravitational-wave luminosity as a function of time for theoff-centercollision Models O and o, compared with the result fromCentrella & McMillan (CM, 1993, Fig. 11).

This indicates that the bulk of the matter quickly adopts a moreor less spherical distribution. Within about 8 ms the luminosityfalls by more than 5 orders of magnitude to less than1050 erg/s.This dramatic drop is reflected in the gravitational waveforms ofModel H which indicate that after∼ 4 ms the activity has essen-tially ceased. This coincides with the complete dissipation of thekinetic energy at that time (see Fig. 12). More than 50% of thetotal energy emitted in gravitational waves are contained in theluminosity spike and after about3 ms only insignificant furthercontributions are added (cf. cumulative energy loss in Fig. 15).From Figs. 15 and 17 one learns that the coarser resolution ofModel h leads only to minor differences compared to Models HandH with the tendency to overestimate the gravitational-wave

luminosity and emitted energy. Models H andH are hardlydistinguishable in Fig. 17 and the computations seem to be con-verged.

Model O is an approximately ten times more energeticsource of gravitational waves than Model H and emits a to-tal energy of2 × 10−2 Mc2. On the one hand the first lu-minosity maximum att ≈ 0.22 ms is nearly twice as high(∼ 6 × 1055 erg/s) as in Model H and nearly four times longer(half width about0.2 ms compared to0.05 ms) (Fig. 18) On theother hand Model O continues its strong emission of gravita-tional waves for the whole computation period of 10 ms duringwhich the luminosity on average stays around1054 erg/s andhardly ever drops below1053 erg/s. This can be explained bythe rapid change of the quadrupole moment of the system asthe two neutron stars repeatedly come close and separate againon their quasi-elliptic orbits around each other (cf. Fig. 7) andby the large kinetic energy retained as rotational energy of thecollision remnant at the end of the simulation (Fig. 13). Severalpeaks of the gravitational-wave luminosity can be correlatedwith the moments of closest approach of the density maxima ofthe two neutron stars (compare Figs. 16 and 6), and two strong,short minima of the luminosity (att ≈ 1.2 ms andt ≈ 4.1 ms)coincide with instants of maximal separation. In Model O onlyabout 30% of the total energy emitted in gravitational wavesis contained in the first luminosity spike, another∼ 50% areadded during the second and third periastrons (t ≈ 2.4 ms andt ≈ 3.3 ms), and a non-negligible fraction (∼ 20%) comesat later times. The waveforms of Model O (Fig. 19) exhibit arather irregular structure during the first 4–5 ms of the evolutionwith a peak amplitude att ≈ 0.22 ms which is about twice ashigh as in Model H. They continue with a very regular, slowlydamped sinusoidal modulation until the end of the simulationat t ≈ 10 ms.

Finally, in Figs. 17 and 18 we compare the first large spikeof the gravitational-wave luminosity with the corresponding re-sults obtained by Centrella & McMillan (1993) and Rasio &Shapiro (1992). In order to do that we rescale and renormalizetheir dimensionless quantities to physical units for our chosenneutron star parameters by using the dynamical timescale

tD ≡(

R3

GM

)1/2

≈ 0.125 ms (7)

with M = 1.63 M as the mass andR = 15 km as the radiusof the neutron stars, and by the scaling the luminosity with thefactor

L0 ≡ 1G

(GM

cR

)5

≈ 3.86 · 1055 erg/s . (8)

Moreover, there is a time shift between their calculations andours because of the different initial center-to-center distances ofthe neutron stars. While it wasa0 = 42 km in our models, Rasio& Shapiro (1992) and Centrella & McMillan (1993) assumeda1 = 4R = 60 km for their head-on collisions, and Centrella &McMillan (1993) tooka1 ≈ 4.53R ≈ 68 km for the off-centercase. The time lag for the parabolic infall trajectories of the


waveform, model H

-4

-2

0

2

4r

h + [1

0-23 G

pc]

0 2 4 6 8 10time [ms]

-4

-2

0

2

4

r h x

[10-2

3 Gpc

]

waveform, model O

-4

-2

0

2

4

r h +

[10-2

3 Gpc

]

0 2 4 6 8 10time [ms]

-4

-2

0

2

4

r h x

[10-2

3 Gpc

]

Fig. 19. Gravitational waveforms,h+ andh×, for Models H and O,respectively, as functions of time. Note that the gravitational-wave fieldh× with the cross polarization is practically zero in case of the head-oncollision, Model H, because there are only very small deviations fromthe mirror symmetry relative to they-z plane when the hot contactlayer of the two neutron stars becomes unstable against shear motions(compare Figs. 1a–d and 2a–d).

two masses with different initial separations can be determinedas the difference of the times to reach the minimum distance(periastron), to be (Roy, 1982, Eqs. (4.82) and (4.85))

∆t =√

2(A1 − A0)3√

GMt(9)

with Ai ≡ (ai+2Rp)√

ai − Rp (i = 0, 1). Mt = M1+M2 =2M is the total mass of the system and the periastron distance

neutrino radiation

0 2 4 6 8 10time [ms]

0

10

20

30

40

lum

inos

ity [1

053er

g/s]

model H

model O

32 zones

64 zones

Fig. 20.Total neutrino luminosities (sums of the contributions from allneutrino and antineutrino flavors) as functions of time for Models Hand h and Models O and o.

0 2 4 6 8 10time [ms]

0

10

20

30

40

50

mea

n ne

utrin

o en

ergy

[MeV

]

model H

model O

nu

anti nu

mu, tau

Fig. 21.Average energies of emitted neutrinosνe, νe, andνx (≡ νµ,νµ, ντ , ντ ) as functions of time for Models H and O.

Rp is the separation of two point particles of massM at closestapproach, which is set to zero for the head-on collision and toRp = 1R for the off-center case.

Taking into account these aspects in Figs. 17 and 18, we findgood overall agreement between the gravitational-wave lumi-nosities from the different calculations, both in shape and mag-nitude. The remaining minor discrepancies can be attributed tothe use of the Lattimer & Swesty (1991) EOS in our calcula-tions instead of an adiabatic EOS with constant indexγ = 2,the possible influence from the inclusion of gravitational-waveback-reactions in our models, and the effects resulting from dif-ferent numerical schemes and different resolution.


neutrino radiation, model H

0 2 4 6 8 10time [ms]

0

5

10

15

20

25lu

min

osity

[1053

erg/

s]

anti nu e

4 * mu,tau

nu e

Fig. 22.Luminosities of the individual neutrino and antineutrino flavors(νe, νe, and the sum of allνx) as functions of time for Model H.

neutrino radiation, model O

0 2 4 6 8 10time [ms]

0

1

2

3

4

lum

inos

ity [1

053er

g/s] anti nu e

4 * mu,tau

nu e

Fig. 23.Same as Fig. 22 but for Model O.

5. Neutrino emission

Because of the different dynamical evolution, the head-on andoff-center collisions also show distinctive differences in the neu-trino emission. The total neutrino luminosities for Models H andh and Models O and o as functions of time are plotted in Fig. 20.

One can see that in the head-on collision (Models H andh) a first very luminous burst of neutrinos with a peak flux ofmore than5 × 1053 erg/s is produced at the moment when thetwo stars crash into each other and the neutron star matter isshock heated and hot gas is squeezed out perpendicular to thecollision axis (see Fig. 1a and c). A second and third luminositymaximum, however, with about 8 times larger peak fluxes ofmore than4×1054 erg/s are present in the time interval1.5 ms <∼t <∼ 3.5 ms which is the time of maximum temperatures in thecollision remnant (Fig. 9) when the hot matter starts to expandand to spread out over a larger volume (Figs. 1b and 1d). A

luminosity of Lν = 4 × 1054 erg/s ≈ 4πR2ν

c4 (3 · 7

8aradT 4ν )

corresponds to a neutrinosphere with radiusRν ≈ 50 km whichradiates neutrinos and antineutrinos of all flavors as black bodywith temperatureTν ≈ 8.5 MeV. Figs. 1b and d confirm that themassive, nearly spherical central body of the collision remnant,which is embedded in a cloud of less dense and cooler gas,has a radial size and “surface” temperature in this estimatedrange. The duration of this extremely luminous burst is onlyabout 2 ms, after which the total luminosity settles down to amuch lower but still sizable value around1054 erg/s. During the∼ 10 ms of simulation time, Models H and h emit an energy of6.7 × 10−3 Mc2 or 1.2 · 1052 erg in neutrinos, of which 50%are radiated during the double peak of the luminosity. Model His a stronger source of neutrinos than of gravitational waves;the latter carry away only about2.2 × 10−3 Mc2 (Fig. 24).Nevertheless, despite of the extremely high luminosity the totalenergy radiated in neutrinos by Model H is still one order ofmagnitude below the estimates by Katz & Canel (1996).

During the break-out of the bounce shock att ≈ 0.2 msneutrinos with average energies in excess of50 MeV are emit-ted from Model H (Fig. 21). A second phase of very high meanenergies coincides with the two luminosity spikes and thereforethe phase of highest temperatures in Model H. As is well knownfrom type II supernovae (see, e.g., Janka 1993), neutron-rich,hot neutron star matter is more opaque toνe than toνe becauseof frequent captures of theνe on the abundant free neutrons.Heavy-lepton neutrinos (νx ≡ νµ, νµ, ντ , ντ ) are even lessstrongly coupled to the stellar medium since their opacity isdominated by neutral-current neutrino-nucleon scatterings butthey do not interact with nucleons via charged-current reactions.For these reasonsνe decouple energetically from the hot plasmaat higher densities and thus usually higher temperatures thanνe,andνx at even higher densities and temperatures. This explainswhy typically the mean energy of the emittedνe is lower thanthat of νe which in turn is below the average energy ofνx (seeFig. 21). Aftert >∼ 4 ms we obtain mean energies of〈ενe

〉 ≈ 10–13 MeV, 〈ενe

〉 ≈ 15–20 MeV, and〈ενx〉 ≈ 20–25 MeV which

is in the range of values found during the neutrino cooling phaseof newly formed neutron stars in type II supernovae. Despite ofthis generic ranking of the mean energies, theνe luminosity ofthe collision remnant is larger than the luminosity in each indi-vidual type ofνx, and theνe luminosity of the neutron-rich, hotneutron star matter dominates theνe luminosity (Fig. 22). Thedifference in ranking between neutrino luminosities and meanenergies of emitted neutrinos reflects the fact that the emissionis not like an ideal black-body, but non-equilibrium effects playa role. Moreover, Fig. 25 shows that there is an extended region(with a broad range of temperatures and densities) where theneutrino fluxes are fed by local neutrino energy losses. Thereforethe neutrino emission can hardly be characterized by the condi-tions of thermodynamical equilibrium at a well defined neutrinoemitting surface (“neutrinosphere”) associated with each typeof neutrino or antineutrino.

In the off-center collision, Models O and o, the neutrinoluminosity reveals a steady increase and does not have suchpronounced maxima as seen in Model H (Fig. 20), although


emitted radiation

0 2 4 6 8 10time [ms]

-5

-4

-3

-2

-1lg

ene

rgy

[Mso

lc2 ]

32 zones

64 zones

gravitational waves

neutrinos

model O

model H

model O

Fig. 24. Comparison of the cumulative energies emitted in neutrinosand gravitational waves (in units ofMc2) as functions of time for thefour Models H, h, O, and o.

some fluctuations up to a factor of 2 are present. At the endof our simulations (t ≈ 10 ms), Model O has a total neu-trino luminosity of about7 × 1053 erg/s which is only 30% lessthan in Model H. Because of the lack of a phase of extremelyhigh neutrino emission and the strong production of gravita-tional waves during its whole evolution, Model O loses onlyhalf the energy in neutrinos as Model H but is a much strongergravitational-wave source (Fig. 24). A comparison of the twoplots in Fig. 25 shows that in Model O the neutrino emittinggas cloud (att ≈ 10 ms) is more extended than in Model H (att ≈ 2.5 ms) but the peak values of the local energy loss rate inneutrinos are only around3 × 1032 erg cm−3 s−1 in Model Owhereas they are above1034 erg cm−3 s−1 in case of Model H.The mean energies of the emitted neutrinos (Fig. 21) are verysimilar in Models O and H, and also the relative contributionsof the different neutrino types to the energy loss are roughlysimilar, as can be seen from a comparison of the relative sizesof the individual neutrino luminosities in Figs. 23 and 22.

6. Neutrino-antineutrino annihilation

The rate of energy deposition by neutrino-antineutrino anni-hilation increases, roughly, with the product of the local neu-trino and antineutrino energy densities times the mean energyof these neutrinos times a factor that accounts for the angulardistribution of the neutrinos (the process is very sensitive to theangle at which neutrinos and antineutrinos collide, for details,see RJST). In the spherically symmetric situation this can beconverted into a product of the neutrino and antineutrino lumi-nosities times the sum of the mean neutrino and antineutrinoenergies times a normalized factor which results from the phasespace integration over the local neutrino distribution functionsand which depends on the geometry of the considered problem.This was used to arrive at the approximate description sum-marized in Eqs. (1)–(6) of Sect. 2.2 employed here to evaluate

Model H for the energy deposition byνν annihilation in thesurroundings of the collision remnant.

Neutrino-antineutrino annihilation is considered as a mech-anism to pump energy into a fireball consisting ofe+, e−, pho-tons, and a small number of baryons. This fireball was suggestedto be a possible source of gamma-ray bursts from neutron starcollisions at cosmological distances (see, e.g., Katz & Canel1996) if the energy in the fireball is sufficiently large,Efb ≈Eγ ≈ Eann >∼ 1051δΩ/(4π) erg, and if the baryon loading ofthe fireball is sufficiently small,Mfb <∼ 10−5 M (for a canon-ical energyEfb ∼ 1051 erg) so that the fireball can expand rela-tivistically with a Lorentz factorΓfb = 1+Efb/(Mfbc2) >∼ 100.Two questions arise from this suggestion. First, is the conver-sion of neutrino-antineutrino energy into electron-positron pairsenough efficient to provide the desired energy, and, second, howlarge is the baryon mass contained in the fireball created throughνν annihilation?

In the following we attempt to give answers to these twoquestions on grounds of our hydrodynamical collision models.Here we only report on the evaluation of the head-on collisionModel H. We concentrate on this model for two reasons. On theone hand, the efficiency ofνν annihilation increases stronglywith the neutrino luminosities and with the mean energies ofthe emitted neutrinos. Therefore Model H with its larger neu-trino emission appeared to us as the more interesting one. Onthe other hand, our simulations have demonstrated that even inthe off-center collision the interaction of the two neutron starsis so dramatic that a lot of matter is ejected perpendicularly tothe orbital plane. Therefore, despite of the large angular mo-mentum in the system, the axis region is polluted with baryons(see Fig. 5), and both the remnants of the head-on and off-centercollisions adopt more or less spherical shapes after the dynamicinteractions, with a central massive object being surrounded bya less dense, extended cloud of hot baryonic gas (Figs. 1a–d and3a–d). From this point of view Model O did not seem to offerbetter perspectives for the emergence of a relativistic fireballfrom a baryon-depleted region near the collided neutron stars.

Fig. 26 gives a map of the energy deposition rate byννannihilation intoe+e− pairs (averaged over the azimuthal anglearound thez-axis according to Eq. (1)) in the surroundings ofthe collision remnant in Model H at timet = 3 ms which isinside the double peak structure of the neutrino luminosity ofFig. 20. One can see that the highest energy deposition ratesof the order of3 × 1030 erg cm−3 s−1 to 1031 erg cm−3 s−1

occur immediately outside the neutrinospheres but in layers withdensities still above and around1010 g cm−3. At the displayedtime, the integral energy deposition rate in matter with densitybelow 1011 g cm−3 (evaluated according to Eq. (2)) is3.4 ×1052 erg s−1.

Since the neutrino luminosities and mean energies of theemitted neutrinos show significant variation during the com-putation time (see Figs. 20, 21, and 22), we compute the timeintegral of the energy deposition rate, Eq. (3), by employing theapproximate treatment summarized in Eqs. (4)–(6). The phasebetweent ≈ 1.5 ms andt ≈ 3.5 ms has by far the highest neu-trino luminosity and therefore yields the largest contribution to


neutrino energy loss rates, H

-50 0 50x axis [ km ]

-50

0

50

y ax

is [

km ]

30.0

30.0

30.0

30.0

32.0

32.0

32.0

32.0

32.0

e

ee

a

aa

x

xx

-50 0 50

-50

0

50

neutrino energy loss rates, O

-150 -100 -50 0 50 100 150x axis [ km ]

-150

-100

-50

0

50

100

150

y ax

is [

km ]

30.0

30.0

30.0

32.0

32.0

e

e

a

a

x

x

-150 -100 -50 0 50 100 150

-150

-100

-50

0

50

100

150

Fig. 25.Contour plots of the local energy loss rates due to the emission of neutrinos and antineutrinos of all flavors. The left plot shows a cut inthez = 0 plane for Model H at timet = 2.47 ms, the right plot displays a cut in the orbital plane of Model O at timet = 9.94 ms. Note thatthe visualized region is larger in the latter figure. The contours are spaced logarithmically in steps of 0.5 dex, the bold lines are labeled withtheir corresponding values in units erg cm−3 s−1. The dashed lines indicate the approximate positions of the neutrinospheres (defined where theoptical depths forνe, νe, andνx, respectively, are approximately unity).

the time integral. For this reason we calculate the temporal av-erage of the termLann(t)/F∗(t) in Eq. (4) by summing overN = 3 time points in this interval:t1 = 2.47 ms,t2 = 3.01 ms,andt3 = 3.59 ms. We obtain

1N

N∑i=1

(Lann(ti)F∗(ti)

)≈ 2 × 10−57 MeV−1 erg−1 s . (10)

Because the three terms of the sum are different by less than afactor 2, we think that the splitting of the time integral of Eq. (3)which led to the approximate form of Eq. (4) was justified.Taking the data for the individual neutrino luminositiesLνe

(t),Lνe

(t), andLνx(t) (Fig. 22) and the average neutrino energies

〈ενe(t)〉, 〈ενe(t)〉, and〈ενx(t)〉 (Fig. 21), we further find∫ 10 ms

0F∗(t) dt ≈ 5 × 10106 MeV erg2 s−1 . (11)

Multiplying the results of Eqs. (10) and (11) we end up with a to-tal energy deposition ofEann ≈ 1050 erg within our simulationinterval of 10 ms for Model H (see also Fig. 27). The corre-sponding average energy deposition rate byνν annihilation of∼ 1052 erg s−1 is very large and means a conversion efficiencyof νν energy toe+e− pairs of the order of 1%. Most of thisenergy is liberated during the 2 ms interval betweent ≈ 1.5 msandt ≈ 3.5 ms after the start of the simulation because of theenormous neutrino luminosity shortly after the collision of theneutron stars.

The energy of approximately1050 erg in e+e− pairs andphotons is released nearly isotropically (Fig. 26) and is onlyone order of magnitude below the canonical fireball energy

Efb ≈ Eγ ∼ 1051δΩ/(4π) erg. It may therefore be sufficientto account for the shorter and weaker bursts whose energy isestimated to be typically more than a factor of 10 below themean energy of the longer and more powerful bursts (Mao etal. 1994). However, most of theνν energy deposition happensvery close to the neutrinospheres and thus in a high-densityregion (Fig. 26). Because of this, the baryon loading of thee+e−-photon fireball is a serious problem. This is obvious fromFig. 27 where we give the energy fromνν-annihilation and thebaryonic mass, integrated from outside inward to the radial po-sition R given on the abscissa. In the region where1050 ergare deposited, one has a baryon mass ofM >∼ 5 × 10−2 Mwhich is about 5 orders of magnitude too large to allow forhighly relativistic expansion. The Lorentz factors which can beestimated asΓ(R) − 1 ≡ Eνν(r ≥ R)/

[M(r ≥ R)c2

]are

therefore around10−3 instead of 100. For this reason, the hugeand nearly isotropic baryon pollution of theνν energy deposi-tion region seems to rule out the possibility that neutrinos fromcolliding neutron stars produce gamma-ray bursts.

7. Summary and discussion

We have reported results of three-dimensional Newtonian hy-drodynamical simulations of the collision along parabolic orbitsof two identical, non-rotating neutron stars with a baryonic massof about1.6 M. The simulations were done with a EulerianPPM code, employing nested grids, using a physical nuclearEOS, and taking into account gravitational-wave emission andits back-reaction on the hydrodynamical flow as well as theemission of neutrinos from the heated neutron star matter. We


annihilation rate

-150 -100 -50 0 50 100 150d / km

-150

-100

-50

0

50

100

150z

/ km

29.0

29.0

29.0

29.0

30.0

30.0

30.0

30.0

30.0

29.0

29.0

28.0

28.0

10.0

10.0

10.0

12.0

Fig. 26. Map of the local energy deposition rates (in erg cm−3 s−1)by νν annihilation intoe+e− pairs in the vicinity of the merger forModel H at timet = 3 ms after the start of the simulation. The valuesare obtained as averages over the azimuthal angle around thez-axis,dmeasures the distance from the grid center in thex-y-plane. The corre-sponding solid contour lines are logarithmically spaced in steps of 0.5dex, the grey shading emphasizes the levels with dark grey meaninghigh energy deposition rate. The dashed lines mark the (approximate)positions of the neutrinospheres ofνe, νe, andνx (from outside in-ward), defined by the requirement that the optical depths inz-directionareτz,νi = 1. The dotted contours indicate levels of the azimuthallyaveraged density, also logarithmically spaced with intervals of 0.5 dex.The energy deposition rate was evaluated only in that region aroundthe merged object, where the mass density is below1011 g cm−3. Theintegral value of the energy deposition rate at the displayed time is3.4 × 1052 erg s−1.

have studied two different cases, each with varied resolution onthe finest grid, a head-on collision and an off-center collisionwith a periastron distance of one stellar radius (≈ 15 km). Oursimulations allow us to draw the following main conclusions:1.)Dynamical evolution:During the head-on collision of the twoneutron stars, the collision remnant is heated by recoil shocksand the initial kinetic energy is efficiently shock-dissipated tothermal energy within a few violent pulsations on a timescaleof about 3–4 ms, after which the collision remnant has an es-sentially spherical mass distribution. The neutron star matteris transiently heated to peak temperatures close to100 MeV.Average temperatures are around 40–50 MeV, correspondingto entropies between 2k/nucleon and 10k/nucleon. The vio-lent crash of the stars into each other leads to the dynamicalejection of about 0.5% of the system mass. In contrast, theoff-center collision is much “milder” and the ejected mass isabout a factor of 10 smaller, although the collision remnant re-tains a large fraction (about 30%) of the initial kinetic energyas rotational energy even after 10 ms at the end of our simu-

baryon contamination

0 50 100 150radius [km]

-3

-2

-1

0

1

lg m

ass

[Mso

l]

mass

energy

gamma-1

48.6

48.8

49.0

49.2

49.4

49.6

49.8

50.0

lg e

nerg

y [e

rg]

0.0000

0.0005

0.0010

0.0015

0.0020

gam

ma-

1

Fig. 27. Cumulative massM(r ≥ R) and annihilation energyEνν(r ≥ R) outside of radiusR, as functions ofR for Model Hat time 3 ms (same time as in Fig. 26). The corresponding relativisticLorentz factorΓ(R) − 1 ≡ Eνν(r ≥ R)/

[M(r ≥ R)c2

]is also

plotted.

rotation parameter

0 20 40 60 80 100 120 140radius [km]

0.00

0.10

0.20

0.30

0.40

0.50

rota

tion

para

met

er gas mass / 8 Msol

rot par

Fig. 28. The thick solid line gives the relativistic rotation parametera(r) ≡ J(r)c/(GM2

gas(r)) for Model O as function of radiusr. J(r)is the angular momentum (inz-direction) insider, Mgas(r) is the gasmass insider which is plotted as thin solid line, given in units ofMand scaled down by a factor 8.

lations. Only for extreme assumptions about the nuclear EOSwill the >∼ 3 M remnant of the head-on collision not form ablack hole on a dynamical timescale. We also believe that theremnant of the off-center collision will probably not be able toescape the collapse to a black hole within a few milliseconds.A mass of3 M is distributed within a radius of 30 km and2.8 M are within 20 km (Fig. 28) which is only about twicethe Schwarzschild radius of a3 M black hole. Thermal pres-sure can only insignificantly raise the maximum stable massof neutron stars (Goussard et al. 1997) and rotation is able toincrease the stable mass limit by only<∼ 20% (Friedman etal. 1986). The interior∼ 2.8 M of the remnant rotate nearlyuniformly and the relativistic rotation parameter of this mass isonly a(r = 20 km) =

[Jc/(GM2)

]20 km ≈ 0.3 (cf. Fig. 28),

which is smaller than that of the maximally rotating, maximum-mass models constructed by Friedman et al. (1986). For all but


one extreme EOS tested by Friedman et al. (1986) such a config-uration is unstable. Therefore we conclude that it is very likelythat gravitational instability will set in as soon as the two starshave merged into one massive body withint >∼ 3–4 ms. Generalrelativistic effects will certainly influence the transformation ofthe infall orbit of the neutron stars into a bound one; this needsto be studied with relativistic simulations.2.) Gravitational waves and neutrinos:The gravitational-wavesignal will certainly depend on general relativistic effects andour basically Newtonian models have only a limited ability tomake predictions of a possibly measurable pulse. Moreover,the duration of the gravitational-wave and neutrino emissionfrom the hot collision remnant will depend on the timescaleof the delay until black hole formation. Our simulations yielda maximum amplitude of the gravitational waveform that ishmax ≈ 2–4 × 10−23 for neutron star collisions happeningat a distance of1 Gpc. The off-center collision is the strongergravitational-wave source due to the larger quadrupole momentof the rotating system and the longer duration of the emissionwhich might last for 10–20 wave periods in the 1000–2000 Hzrange. The gravitational-wave strain would be close to the lowersensitivity limit of the new generation of gravitational-wave in-terferometers which are currently under construction and willstart operation within the next years. Of course, neutron starcollisions are very rare and very short events and therefore thechance to catch a signal is rather small. The head-on collisionis the more powerful neutrino source of the two investigatedcases and emits an energy about 3 times larger in neutrinos thanin gravitational waves. The peak neutrino luminosity reaches4×1054 erg s−1 and the total energy radiated in neutrinos withina few milliseconds is around1052 erg.3.) Gamma-ray bursts:Because of the larger energy outputin neutrinos and the very high neutrino luminosity (Lν >∼1054 erg s−1) as well as high mean energies of the emitted neu-trinos (〈εν〉 <∼ 40 MeV), the head-on collision provides more fa-vorable conditions for producing gamma-ray bursts frome+e−-photon fireballs created byνν annihilation. We calculate a con-version efficiency of neutrino energy intoe+e−-pairs of about1% and find an integral value for the energy deposited in thevicinity of the collision remnant of1050 erg within only 10 ms.However, most of this energy is deposited in the immediateneighbourhood of the neutrinospheres where the density is stillhigher than about1010 g cm−3. Therefore the baryon loadingof the e+e−-photon fireball is at least 5 orders of magnitudetoo high and instead of having values above 100 the relativis-tic Lorentz factorΓ − 1 is estimated to be only around10−3.Dynamically ejected material together with a flow of baryonicmatter driven by neutrino-energy transfer to the surface layers ofthe collision remnant are therefore a harmful poisonous combi-nation which prevents relativistic expansion of the pair-plasmafireball even though the latter seems to obtain an interestingamount of energy fromνν-annihilation.

However, if the collision remnant collapses to a black holevery quickly, this neutrino-driven mass flow will be quenched,and the density in the cloud of ejected matter will rapidly de-crease as the gas is sucked into the newly formed black hole. If

the gas has too little angular momentum to be stabilized on orbitsaround the black hole (as in the head-on collision of Model H),the accretion will happen on a dynamical infall timescale and theneutrino emission and corresponding conversion of energy intoelectron-positron pairs will cease within only a few millisec-onds. With sufficiently large angular momentum in the system(favored by off-center collisions like Model O), some mattermay form an accretion torus in the orbital plane of the collidingstars, whereas the axis region should become evacuated as thegas falls into the black hole. This situation might allow for muchmore favorable conditions for the creation of a pair-plasma fire-ball with low baryon loading, powered by the annihilation ofneutrinos and antineutrinos that are emitted from the accretiontorus (see, e.g., Mochkovitch et al. 1993, 1995 and paper RJTS).Hydrodynamical simulations of the formation of such accretiontori in the context of merging or colliding neutron stars andblack holes and results on the associated neutrino emission andνν annihilation will be published in a separate paper (Ruffert &Janka, in preparation) where also the implications for gamma-ray burst scenarios will be addressed in detail.

Strong shock heating of the neutron stars during their vi-olent collision (or during their merging as suggested by post-Newtonian simulations, see Oohara & Nakamura 1997) can in-deed raise the neutrino luminosities significantly and can thusenhance the energy deposition byνν annihilation. The associ-ated dynamical ejection of gas and the neutrino-driven windcaused by the intense neutrino fluxes, however, impede theemergence of gamma-rays from this scenario. Fireballs pow-ered by neutrinos from colliding neutron stars should thereforebe ruled out as possible sources of cosmological gamma-raybursts. Also, we do not see the formation of highly relativis-tic shocks during the collisions by which a fraction of 0.1–1%of the kinetic energy at impact (>∼ 1053 erg) might be hydro-dynamically focused into a small amount of mass (∼ 10−6 to10−5 M). The expansion velocity of the collision shocks is atmost a few tenths of the speed of light and the masses ejected inthe presented models are more than 3 orders of magnitude toolarge to allow for ultrarelativistic motion.

Acknowledgements.MR thanks Sabine Schindler for her patiencein their common office. We would also like to thank M. Camen-zind, W. Hillebrandt, F. Meyer, M. Rees, S. Woosley and S.F. Porte-gies Zwart for educative discussions, and appreciate valuable com-ments by the referee, R. Mochkovitch. MR is grateful for supportby a PPARC Advanced Fellowship, HTJ acknowledges support bythe “Sonderforschungsbereich 375-95 fur Astro-Teilchenphysik” derDeutschen Forschungsgemeinschaft. The calculations were performedat the Rechenzentrum Garching on a Cray-YMP 4/64.

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