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

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6. Summary and conclusions

The neutrino emission and neutrino-antineutrino annihilation during the coalescence of binary neutron stars were investigated. To this end the three-dimensional Newtonian equations of hydrodynamics were integrated by the Riemann-solver based "Piecewise Parabolic Method" on an equidistant Cartesian grid with a resolution of [FORMULA] or [FORMULA] zones. The properties of neutron star matter were described by the equation of state of Lattimer & Swesty (1991). Energy loss and changes of the electron abundance due to the emission of neutrinos were taken into account by an elaborate "neutrino leakage scheme". We have simulated the coalescence of two identical, cool (initially [FORMULA]) neutron stars with a baryonic mass of about [FORMULA], a radius of 15 km, and an initial center-to-center distance of 42 km for three different cases of initial neutron star spins.

The total neutrino luminosity prior to and during the dynamical phase of the coalescence is very small ([FORMULA]  erg/s), becomes about 1- [FORMULA]  erg/s when the stars have merged into one rapidly spinning massive body, and climbs to 1- [FORMULA]  erg/s after spun off material has formed a hot toroidal cloud with a mass of 0.1- [FORMULA] around the wobbling and pulsating central object. The neutrino fluxes are clearly dominated ([FORMULA] 90-95%) by the emission from this "disk". Since the disk matter is neutron-rich, [FORMULA] are radiated with a luminosity that is a factor 3-6 higher than the (individual) luminosities of [FORMULA] and [FORMULA] ([FORMULA]). The mean energies of the emitted neutrinos are very similar to those of supernova neutrinos, [FORMULA], [FORMULA], and [FORMULA].

When the neutrino luminosities are highest, only about 0.2-0.3% of the energy emitted in neutrinos is deposited in the immediate neighborhood of the merger by [FORMULA] -annihilation, and the maximum integral energy deposition rate is found to be about 3- [FORMULA]  erg/s. Thus, to pump an energy of the order of [FORMULA]  erg/steradian into a fireball of [FORMULA] -pairs and photons, the strong neutrino emission would have to continue for several seconds. Since a collapse of the central core of the merger with a mass of [FORMULA] into a black hole within milliseconds seems unavoidable, we conclude that the available energy is insufficient by a factor of about 1000 to explain gamma-ray bursts at cosmological distances. However, it appears possible that an accretion torus with a mass of [FORMULA] - [FORMULA] remains around the central black hole and is accreted on the time scale of viscous angular momentum transport. Analytical estimates suggest that even under the most favorable conditions in this torus and with an optimum value of the disk viscosity, annihilation of [FORMULA] pairs emitted from this torus provides an energy that is still more than a factor of 10 too small to account for powerful cosmological gamma-ray bursts, unless focussing of the fireball expansion plays an important role.

A few [FORMULA] of very neutron-rich, low-entropy matter may be dynamically ejected shortly after the neutron stars have merged, and another [FORMULA] up to a few [FORMULA] of strongly neutronized, high-entropy material might be carried away from the accretion torus in a neutrino-driven wind on a time scale between a fraction of a second and a few seconds. The contamination with these baryons is a severe threat to a relativistic fireball. Aspects of nucleosynthesis in these ejecta were discussed. Because of the neutron-richness of the ejected material and the dominance of the [FORMULA] luminosity from the merged object and its accretion torus, conditions suitable for the formation of r-process elements might be realized more easily than in the neutrino wind from newly formed neutron stars.

It seems to be very difficult to fulfil the energetic requirements of cosmological gamma-ray bursts with the annihilation of [FORMULA] pairs emitted from an accretion disk or torus around a stellar mass black hole. If [FORMULA] -annihilation is nevertheless to be saved as energy source for relativistic pair-photon fireballs - despite of the problems exposed by our numerical and analytical results and the critical issues addressed in the discussion of Sect.  5- then one is forced to consider the following possibilities.

The neutrino luminosities from the accretion torus could be considerably higher than obtained in our models, but the mechanism to achieve this has yet to be identified, e.g., it is possible that the neutrino transport in the torus is enhanced by convective instabilities. Because of the quadratic dependence on the neutrino luminosities, an increase of the neutrino fluxes would affect the [FORMULA] -annihilation sensitively. Alternatively, still relying on the simple picture described in Sects.  4.2 and 4.3, one might feel tempted to interpret the estimates of the annihilation energy towards the optimistic side, although they were derived by employing a number of very generous and favorable assumptions. In this case some interesting implications for the hypothesis of stellar mass accretion disks around black holes as sources of gamma-ray bursts (Paczyski 1991; Narayan et al. 1992; Woosley 1993a, 1996) can be inferred from combining the theoretical results with information about measured burst time scales (Meegan et al. 1995a, Kouveliotou 1995) and energies of cosmological bursts (e.g., Woods & Loeb 1994; Quashnock 1996).

Short gamma-ray bursts have a typical duration [FORMULA] of several tenths of a second and a typical total energy that is a factor of [FORMULA] smaller than that of long bursts (Mao et al. 1994). They require the release of neutrino energy from accretion tori with masses [FORMULA]  few  [FORMULA] and a beaming of the expanding fireball into a solid angle [FORMULA], when [FORMULA] is the typical energy of a cosmological gamma-ray burst if the gamma-rays were radiated isotropically. [FORMULA] becomes smaller if the useful energy from [FORMULA] -annihilation is less than [FORMULA]  erg. For [FORMULA]  erg the focussing of the fireball is noticable, [FORMULA], whereas for [FORMULA]  erg it is essentially absent. The required accretion mass might suggest merging events of compact binaries as the origin of the bursts in this case. The disk masses and a possible focussing of the fireball expansion towards the observer have to be explained by theoretical modelling.

For long bursts, most of which have durations [FORMULA] between some 10 s and about 100 s, the accretion of [FORMULA]  few  [FORMULA] is needed and could provide an energy [FORMULA]  erg which is of the order of [FORMULA]  erg without significant focussing of the relativistic pair-photon plasma being necessary. For more energetic bursts some beaming of the fireball expansion would have to be invoked. The large accretion mass favors the "failed supernova" or "collapsar" scenario (Woosley 1993a). In this model several solar masses of material surround the most strongly neutrino radiating region of the accretion torus close to the innermost stable orbit around the black hole. This should lead to mixing of baryons with a significant fraction of the pair-photon plasma and will confine the volume where [FORMULA] -annihilation might create a relativistic fireball to a baryon-poor region along the system axis. The fireball will expand into a limited solid angle which should compensate for the reduction of the useful fraction of the [FORMULA] -annihilation energy.

This interpretation of the bimodal distribution of burst durations (Kouveliotou et al. 1993) employs two different kinds of astrophysical events. In contrast, Wang (1996) attempted to explain the bimodality by a superposition of two distinct time scales in the temporal structure of individual bursts corresponding to peak widths and separations between adjacent peaks. Katz & Canel (1995a, b) have recently suggested the association of short and long bursts with two different classes of models and have hypothesized that short bursts are produced by neutron star collisions and long bursts originate from accretion-induced collapse of bare degenerate white dwarfs ( Dar et al. 1992). In both types of models the burst energy would be provided by the annihilation of emitted [FORMULA] pairs. Accretion induced collapse, however, was ruled out as a source of gamma-ray bursts situated at cosmological distances by Woosley & Baron (1992) on grounds of the unacceptably large baryonic pollution of the surroundings of the collapsed star caused by a nonrelativistic neutrino-driven wind. The same worries also hold for collisions of neutron stars where explosions of ejected low-mass fragments might create an envelope of baryonic material around the collision site. Moreover, one has to be suspicious whether the neutrino emission will be luminous and long enough that [FORMULA] -annihilation can provide an energy [FORMULA]  erg for a short and most likely unbeamed gamma-ray burst.

Neutrino emission and [FORMULA] -annihilation would be the energy source for the gamma-ray bursts also in the two classes of models that could lead to accretion tori around black holes, i.e., the merging of binary neutron stars or of neutron star black hole systems in case of short bursts, and collapsing, very massive stars which do not succeed to explode as type-II supernovae in case of long bursts. The bimodal distribution of burst durations would reflect the two distinct mass ranges of the accretion tori around the accreting stellar mass black holes, some 0.1  [FORMULA] or a few [FORMULA], respectively. The similar peak luminosities of both short and long bursts (Mao et al. 1994) could be explained by the same underlying energy source for the gamma-ray bursts. The short-time variability of the gamma-ray signal could be a consequence of jet precession (Hartmann & Woosley 1995) or of inhomogeneities and instabilities in the accretion torus that give rise to fluctuations of the accretion rate. And the individual characteristics of burst events might be associated with different masses of accretion tori and accreting black holes, different torus structures due to different angular momentum distributions, and different accretion time scales because of variations of the angular momentum transport, e.g., caused by magnetic fields or viscosity producing dissipative processes in the torus.

Movies in mpeg format of the dynamical evolution of all models are available in the WWW at http://www.mpa-garching.mpg.de/~mor/nsgrb.html

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998