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

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5. Discussion

In this paper we have reported about hydrodynamical calculations of the merging of equal-mass binary neutron stars with different initial spins. We have analysed the models for their neutrino emission, for neutrino-antineutrino annihilation in the surroundings of the merger, and for the thermodynamical conditions in the merged object.

5.1. Mass loss and nucleosynthesis

The dynamical merging proceeds within a few milliseconds after the simulations were started from an initial center-to-center distance of 42 km of the two 1.6  [FORMULA] neutron stars. Shortly after the two stars have fused into one compact object with a mass of about 3  [FORMULA] and an average density of more than [FORMULA], spun-off matter forms a less dense toroidal cloud ([FORMULA]) that is heated to temperatures of 5-10 MeV by friction in shock waves and strong pressure waves sent into the surrounding gas by the oscillations and periodic pulsations of the central high-density body. Roughly 0.1  [FORMULA] of material receive a large momentum and are pushed beyond the grid boundaries. However, only a small fraction of at most [FORMULA]... [FORMULA] of the matter has a total energy (internal plus kinetic plus gravitational) large enough to allow the gas to become unbound. In model A128 which has the best numerical resolution, in fact none of the matter can escape from the gravitational potential of the merger, in contrast to model A64 where it is several [FORMULA]. Notice, however, that energy released by the recombination of free nucleons into nuclei - an effect which is taken into account in our simulations by the use of the "realistic" equation of state of Lattimer & Swesty (1991) - and energy production by nuclear reactions proceeding in the cooling and expanding gas (Davies et al. 1994) can aid the mass ejection and could increase the unbound mass relative to our estimates.

The ejection of matter is also very sensitive to the amount of angular momentum that is present in the merging binary. The largest mass ejection was found in model B64 immediately after the merging, because the initial configuration of this model had a solid-body type rotation and thus the largest specific angular momentum in regions far away from the system axis. In contrast, model C64 showed the largest mass loss a few milliseconds later. The initial anti-spin setup of this model led to vigorous vibrations of the central body and to the outward acceleration of material some time after the two neutron stars had formed a single object. Whether and how much mass can be dynamically lost during the post-merging evolution, however, will also depend on the stability of the merged object. The central, compact core of the merger is so massive that it can be stabilized neither by internal pressure for the currently favored supranuclear equations of state, nor by its rapid rotation (for details, see Sect. 4.1.3 of Paper I). One therefore has to expect its collapse into a black hole within a few milliseconds. In case of model C64 not much matter would be expelled if the gravitational instability sets in before the large-amplitude post-merging oscillations have taken place.

The less dense cloud of gas that surrounds the massive, very dense central body is stabilized by internal pressure because its rotational velocities are significantly less than the Kepler velocity (see Fig. 12 of Paper I). When the massive core collapses into a black hole, the ambient matter will therefore be swallowed up by the black hole on a dynamical time scale of [FORMULA]... [FORMULA]  s. Only gas with a sufficiently large angular momentum will have a chance to remain in a disk or extended torus around the black hole. From there it will spiral into the black hole on the much longer time scale of viscous angular momentum transport. One can estimate that with typical orbital velocities of about [FORMULA] as found in our models, only gas at radii beyond about 44-107 km has an angular momentum that is large enough (for more information, see Sect. 4.1.3 of Paper I). This minimum orbital radius is outside of the grid boundaries of our simulations. We therefore conclude that essentially all the mass on the computational grid will disappear in a forming black hole more or less immediately, and only the [FORMULA] - [FORMULA] of material that have been swept off the grid might be able to end up in a toroidal "disk" around the central black hole. A disk mass of about [FORMULA] should be taken as an extreme upper limit for the considered scenario. Like the dynamically ejected mass, the amount of gas that ends up in a disk is sensitive to the initial angular momentum of the neutron star binary and to the relative times of black hole formation and mass spin-off. Last but not least, a quantitative answer seems to depend also on the numerical resolution, with the trend that better resolved models yield smaller estimates for the possible disk mass and mass ejection.

A mass of [FORMULA]... [FORMULA] that is dynamically ejected during the merging of binary neutron stars might have important implications for nucleosynthesis (Lattimer & Schramm 1974, 1976; Eichler et al. 1989). Dependent on the phase when the mass loss occurs, the expelled gas will start its expansion from different initial conditions of entropy and composition. The ejection of initially very cool, low-entropy material might be caused by the tidal interaction during the last stages of the inspiral and during the mass transfer phase of very close non-equal mass binaries. If the two components have "nearly equal" initial masses, the mass transfer is unstable and within a few orbital periods the lighter star can be completely dissipated into a thick, axially symmetric disk around the primary. Some fraction of the surface material might escape the system (Lattimer & Schramm 1974, 1976). If the initial mass ratio of the two stars is large, the binary is stable against dynamical-time scale mass transfer and there is the interesting possibility that the secondary (the less massive component) is stripped to the minimum mass of stable neutron stars, at which stage it will explode (Page 1982; Blinnikov et al. 1984, 1990; Eichler et al. 1989; Colpi et al. 1989, 1991, 1993; Colpi & Rasio 1994). However, recent investigations suggest that stable mass transfer is unlikely because the initial mass of the secondary must already be very small (below [FORMULA] ; Bildsten & Cutler 1992, Kochanek 1992, Rasio & Shapiro 1994, Lai et al. 1994). In addition, an unreasonably high value of the neutron star viscosity is needed to enforce corotation and to maintain tidal locking, because stable mass transfer requires a dynamically stable Roche limit configuration which can only exist in synchronized systems with extreme mass ratios. Such systems are essentially ruled out for neutron stars (Lai et al. 1994). If mass shedding in the discussed situations occurs, it would lead to the ejection of initially cold, very low-entropy and very neutron-rich material. Subsequent radioactive [FORMULA] -decays of unstable, neutron-rich heavy nuclei that are present in the decompressed matter will heat the expanding gas to temperatures around 0.1 MeV, which will give rise to r-process conditions (Lattimer et al. 1977, Meyer 1989, Eichler et al. 1989).

Gas spun off the exterior parts of the dilute toroidal cloud that surrounds the compact core of the merged binary, instead, has been heated to temperatures of [FORMULA]  MeV by friction during the merging and post-merging evolution. This heating has produced entropies of a few [FORMULA] per nucleon. Neutrino emission has already raised [FORMULA] from initial values [FORMULA] to slightly less neutron-rich conditions with [FORMULA] -0.2. The electron degeneracy is only moderate, [FORMULA]. All these parameters are very similar to the conditions found in the shocked outer layers of the collapsed stellar core in a type-II supernova where the site of the classical r-process has been suggested (see, e.g., Hillebrandt 1978). Compared with the supernova case, the range of [FORMULA] -values in the potentially ejected merger material is on the low side. An r-processing occurring under such neutron-rich conditions would be very efficient and should preferentially produce r-process nuclei with very high mass numbers (Fig. 18). The predominant production of high-mass r-process elements would be in concordance with the fact that one cannot expect the formation of all Galactic r-process material in the considered low-entropy ejecta. With our numerical estimates of a few [FORMULA] for the mass loss per merger event and with the possible event rate of NS-NS and NS-BH mergers of [FORMULA]... [FORMULA] per year per galaxy (Narayan et al. 1991, Phinney 1991, Tutukov et al. 1992 Tutukov & Yungelson 1993), which corresponds to about [FORMULA]... [FORMULA] events during the lifetime of the Galaxy, only 1- [FORMULA] of the [FORMULA] of Galactic r-process elements could be produced. But if the r-process were strong enough, all the Galactic actinides (e.g., about 40-50  [FORMULA] of Th) might be accounted for by the material shed during neutron star merging!

[FIGURE] Fig. 18. Expected average mass numbers of nuclei formed by an r-process starting with NSE conditions for low entropies and with the [FORMULA] -process for high entropies. The plot shows contours of constant average mass number in the [FORMULA] -s plane (s in [FORMULA] per nucleon). [FORMULA] and s define the thermodynamical conditions in the expanding gas at a temperature of about [FORMULA]  K, i.e., before the r-processing takes place. The dynamical evolution is characterized by the time scale [FORMULA] of the adiabatic expansion between [FORMULA]  K and [FORMULA]  K. Solid lines correspond to a time scale of 50 ms, dashed lines to 100 ms. Between the contours for mass numbers [FORMULA] and [FORMULA] the shaded area marks the region where a suitably mass-weighted combination of the r-process yields for different conditions will produce a solar-system like abundance pattern. For larger values of [FORMULA], no significant r-processing can take place, for lower values of [FORMULA] very strong r-processing will primarily lead to nuclei in the region of the actinides

However, the neutron-rich wind that is driven by neutrino energy deposition in the outer disk regions can also contribute to the nucleosynthetic input into the interstellar medium. With a total neutrino luminosity of about [FORMULA] the wind will have a mass outflow rate of the order of 0.001- [FORMULA] (Qian & Woosley 1996, Woosley 1993b), depending on the gravitating mass of the merger, the disk mass and geometry, and the neutrino emission as a function of time. For a duration of the outflow between some fractions of a second and a few seconds, one might therefore have another [FORMULA] up to several [FORMULA] of material that are expelled with very interesting thermodynamical properties.

Like the neutrino-driven wind from new-born neutron stars, the neutrino-heated material should have significantly higher entropies than the matter that is dynamically ejected from the disk or torus by momentum transfer during core pulsations. Owing to the particularities of geometry, gravitational potential, and neutrino emission in the merger situation, the expansion time scales as well as the degree of neutronization might be significantly different from the supernova case. Since the neutrino emission from the core and the disk of the merger is dominated by the [FORMULA] fluxes, absorptions of [FORMULA] in the wind material ([FORMULA]) will be more frequent than the absorption of less abundant [FORMULA] ([FORMULA]) and will keep the expanding material neutron-rich. Because of a larger luminosity ratio [FORMULA] -4.5 but similar mean [FORMULA] and [FORMULA] energies with [FORMULA] -1.8, the expanding wind will have a lower electron fraction than in the supernova case (see Qian & Woosley 1996 for a discussion of the electron fraction in neutrino-driven winds). Values as low as [FORMULA] -0.2 seem possible in the neutrino wind from the merger. For such low values of [FORMULA] a strong r-processing can happen even at modest entropies of [FORMULA] - [FORMULA] (see Sect.  5.2 for an estimate of the wind entropies) which are too low to allow for the formation of r-process nuclei in the neutrino-driven winds from protoneutron stars for the typical electron fractions of [FORMULA] -0.4 found there (Witti et al. 1994, Takahashi et al. 1994, Woosley et al. 1994, Qian & Woosley 1996). Fig. 18 visualizes this and shows that even for rather slow expansions with expansion time scales of more than 100 ms nuclei with mass numbers A between 150 and 210 can be formed.

Unfortunately our current models allow only rough estimates of the mass loss and the conditions in the ejected matter. Our simulations could not directly follow the ejection of mass from the merger because they suffered from the limitations due to the use of the computational grid and due to an insufficient numerical resolution, especially of matter at low densities. A detailed and meaningful analysis of the very interesting aspects of a possible r-processing in the dynamically ejected low-entropy material and in the high-entropy neutrino-driven winds from NS-NS or NS-BH mergers has to be postponed until models are available which yield more quantitative information about the long-time evolution of the merged object, the torus geometry, and the neutrino-matter interactions in the outer parts of the torus. Only such models can give evidence about the duration of the mass loss and the amount of material that is ejected with different entropies, different expansion time scales, and different degrees of neutronization, all of which determine the nucleosynthetic processes (see Fig. 18 and also Woosley & Hoffman 1992, Witti et al. 1994).

5.2. Neutrino emission and gamma-ray bursts

The luminosities and mean energies of the neutrinos emitted from merging neutron stars are very similar to those calculated for supernovae and protoneutron stars. After the two neutron stars have merged, luminosities up to several [FORMULA]  erg/s are reached for every neutrino species and the average energies of [FORMULA] leaking out of the merger are 10-13 MeV, of [FORMULA] they are 19-21 MeV, and of heavy-lepton neutrinos around 26-28 MeV. However, the neutrino emission exhibits characteristic differences from the supernova case, too.

The total neutrino luminosity from merging neutron stars does not increase to a value above [FORMULA]  erg/s before the hot, toroidal gas cloud around the dense and compact core of the merger starts to form. More than 90% of the peak neutrino emission of about [FORMULA]  erg/s stems from the "disk" region where the optical depths and thus the neutrino diffusion time scales are significantly smaller than in the core. Another pecularity is the fact that the very neutron-rich, decompressed and heated neutron star matter predominantly emits electron antineutrinos. This should hold on until the electron fraction in the medium has grown to a level where the increase of the lepton number by [FORMULA] emission is compensated by the [FORMULA] losses. If the merged configuration remained stable for a sufficiently long time, the deneutronization phase will be superseded by an extended period where the heated gas deleptonizes and cools again and thus evolves back to the state of cold neutron star matter. However, it is very likely that the merged object, which contains essentially the baryonic mass of two typical neutron stars (about [FORMULA]), does not remain gravitationally stable. For all currently favored nuclear equations of state it should collapse to a black hole long before the cooling is finished.

If the gravitational instability of the massive core of the merged object sets in before the hot gaseous torus has formed and if all the surrounding gas falls into the black hole immediately, the neutrino emission from the merger will stay fairly low with a total luminosity of less than [FORMULA]  erg/s. This is much too low to get sufficient energy for a cosmological gamma-ray burst from [FORMULA] -annihilation during the final stages of the inspiral of the two neutron stars and during the first 1-4 ms right after the merging. The annhilation efficiency of neutrinos and antineutrinos, [FORMULA], increases proportional to the neutrino luminosity (Eq. (10)) and the neutrino energy deposition rate [FORMULA] with the product of neutrino and antineutrino luminosities. At the time of maximum neutrino emission some 6-8 ms after the stars have merged, we calculate an annihilation efficiency of [FORMULA] - [FORMULA] and a neutrino energy deposition rate of 2- [FORMULA]  erg/s in the whole space outside the high-density regions of the compact core and the surrounding "disk". The integrated energy deposition during the simulated evolution of about 10 ms is therefore less than [FORMULA]  erg (assuming maximum neutrino fluxes during the whole considered times). Even with the unrealistic assumption that all the [FORMULA] -annihilation energy could be useful to power a relativistic pair-photon fireball, this energy would fail to account for the canonical [FORMULA]  erg/steradian of a typical gamma-ray burst at cosmological distances (e.g., Woods & Loeb 1994; Quashnock 1996) by nearly three orders of magnitude. Of course, these arguments are not conclusive if there is strong focussing of the expanding fireball into a narrow solid angle [FORMULA]. In this case an observer would deduce a largely overestimated value (by a factor [FORMULA]) for the energy in the fireball and in the gamma-ray burst if he assumed isotropy of the emission. However, for the considered merger scenario and the geometry of the post-merging configurations in our simulations, it is very hard to imagine how the required strong beaming of the fireball into a jet-like outflow could be achieved.

It is interesting to note that if the neutrino emission calculated for our merger models would continue for a few seconds, which is a typical duration of observed gamma-ray bursts (e.g., Norris et al. 1994, Kouveliotou 1995), a burst energy of about [FORMULA]  erg could well be accounted for by the annihilation of neutrinos and antineutrinos. An accretion disk or torus around the central black hole could provide a luminous neutrino source for the required period of time. This time span is much longer than the times covered by our hydrodynamical modelling. Since the current numerical simulations were neither able to mimic the effects of a central black hole nor to track the evolution of the merger for a sufficiently long time, we attempted to develop a simple analytical model in Sect.  4.3 to give us insight into the principal dependences of the energy deposition by the annihilation of neutrinos emitted from the disk or torus.

This torus model was based on Newtonian physics and did not determine the torus structure self-consistently. However, the analytic treatment took into account the effects of viscous angular momentum transport, viscous heating, neutrino cooling, and partial neutrino opaqueness. To first order and on a qualitative level, our considerations should also be valid for accretion disks around black holes in general relativity (see, e.g., Shapiro & Teukolsky 1983) and for tori around Schwarzschild black holes (see, e.g., Chakrabarti 1996). In fact, the employed assumptions about the torus geometry are supported by general relativistic investigations of neutron tori (Witt et al. 1994; Jaroszyski 1993, 1996) and the neutrino luminosities from our simple torus model are compatible with those obtained from the relativistic analyses. Moreover, the neutrino flux and neutrinospheric temperature calculated analytically are also in good agreement with the results of our numerical models for the post-merging phase when the neutrino emission has reached its saturation level.

Eq. (26) gives the estimate of the [FORMULA] -annihilation energy [FORMULA] for our analytical disk model. We find that [FORMULA] could at best lie between [FORMULA] and [FORMULA] for a disk of [FORMULA] around a [FORMULA] black hole. The lifetime of such a disk is determined by the time scale of the outward transport of angular momentum and is estimated to be several ten up to a few hundred milliseconds. The ranges of values account for the uncertainties in the neutrino opacity (which depends on the composition of the medium and on the neutrino spectra) and for the corresponding variation of the neutrino luminosity and mean energy of the emitted neutrinos. Further uncertainties due to the unknown disk viscosity are circumvented by deriving the result for a value [FORMULA] of the dynamic shear viscosity (Eq. (20)) which assures a maximum result for [FORMULA].

Unless focussing or beaming of the expanding pair-photon fireball towards the observer plays an important role, the annihilation energy of Eq. (26) is too low by more than a factor of 10 to explain powerful cosmological gamma-ray bursts. Weak bursts with an energy of about [FORMULA] and durations of less than or around a second, however, do not seem to be completely excluded on grounds of Eqs. (26) and (23). The result of Eq. (26) and in particular the upper value of [FORMULA] should be considered as a very optimistic maximum estimate for the energy deposited by [FORMULA] -annihilation in the surroundings of the disk. Eq. (26) was derived for the most favorable conditions and by making a whole sequence of most extreme assumptions, the combination of all of which appears rather unlikely.

In the first place it was assumed that the dynamic viscosity of the disk adopts the optimum value [FORMULA] of Eq. (18). For much smaller viscosities the toroidal disk should stay rather cool and the neutrino fluxes correspondingly low. For much larger values of the viscosity the lifetime of the disk will decrease because the mass accretion rate into the black hole increases with the rate of the outward transport of angular momentum mediated by viscous forces. In this case the neutrino luminosities will be bounded by the fact that the viscous friction will heat up the disk to very high temperatures and thus the neutrino absorption and scattering cross sections, which scale roughly with the square of the neutrino energy, will increase. Therefore the neutrino diffusion time scale will increase, too, and the neutrino cooling will become inefficient. As a consequence, most of the dissipated gravitational and rotational energy could be advected into the black hole when the matter, after having lost part of its angular momentum, spirals in through the innermost stable circular orbit (advection-dominated regime).

Eq. (26) also represents an optimistically high value of the energy deposition by [FORMULA] -annihilation because the radiation efficiency of about 8% obtained for our Newtonian model of a Keplerian accretion disk is an upper bound to the radiation efficiency of a relativistic disk around a nonrotating black hole where it is less than 6%. Moreover, it should be remembered that the annihilation efficiency [FORMULA] of Eq. (10) most likely overestimates the useful fraction of the annihilation energy by a considerable factor. Our hydrodynamical simulations show that the fraction of the neutrino energy deposited in the possibly baryon-poor region above and below the disk but not in the plane of the disk (where it will serve to drive a baryonic, nonrelativistic wind instead of creating a relativistically expanding pair-photon fireball) could be as small as some 20-25% of the number given in Eq. (26) (cf. Fig. 17). Even more, general relativistic effects were neglected in the numerical simulations and analytical considerations presented here. Jaroszyski (1993) showed that they lower the energy that can be transported to infinity significantly (by about 80%).

For the whole uncertainty range of the neutrino interaction cross section (Eq. (17)) and for the corresponding range of neutrino luminosities and mean energies of emitted neutrinos, we find that the result of Eq. (26) falls short of the desired value [FORMULA]  erg/steradian by at least an order of magnitude if the disk mass is of the order of [FORMULA] and the central black hole has a mass of about [FORMULA]. From our numerical models of NS-NS mergers, one concludes that a disk with a mass close to or even larger than [FORMULA] might be formed only under very special conditions. Even if there is a high angular momentum in the system due to neutron star spins as in our model B64, the amount of material that has a chance to form a disk is hardly as much as [FORMULA] (see Paper I). In models A64 and C64 the lower angular momentum allows only little material to possibly remain in a disk. Only matter outside of the boundary of our computational grid has enough angular momentum to be rotationally stabilized. Before some matter in models A64 and C64 has acquired a sufficiently large angular momentum to be lost off the computational grid - a process that is partly aided by pressure waves created by the wobbling and ringing of the central, compact object - the merger, however, has probably already collapsed to a black hole which swallows up most of the surrounding, pressure supported matter on a dynamical time scale.

Massive, self-gravitating tori around black holes may be subject to general relativistic global instabilities that lead to catastrophic runaway mass loss and may provide by far the shortest evolutionary time scale of such tori as recently argued for stationary polytropic tori by Nishida et al. (1996) and for stationary neutron tori by Nishida & Eriguchi (1996). Such an instability would have the same implications as the case of extremely large disk viscosity discussed above where the accretion time scale of the torus due to the rapid viscous angular momentum transport could be much smaller than the neutrino diffusion time scale. As a consequence of the rapid accretion of the torus material, most of the internal energy of the gas would be carried into the black hole along with the gas instead of being radiated away by neutrinos. Because the duration of the neutrino emission would be very short without a compensating increase of the neutrino luminosity, the total energy emitted in neutrinos would be much smaller than in the extreme and optimum situation considered in the derivation of Eq. (26) in Sect.  4.3. Therefore, global disk instabilities might be another threat to neutrino-powered gamma-ray bursts. However, it has still to be demonstrated whether global runaway instabilities develop in the non-stationary situation and how they behave and evolve in the presence of changes of the angular momentum distribution.

Even if the lifetime of the merger and of the neutrino radiating accretion torus is long enough and neutrino annihilation could provide a powerful "engine" for creating a fireball, yet another major concern for the viability of the considered [FORMULA] -ray burster scenario comes from the baryonic wind that is blown off the surface of the merger and accretion torus by neutrino heating. This neutrino-driven wind is unavoidable when large neutrino fluxes are emitted and a small fraction of these neutrinos annihilate or react with nucleons in the low-density gas in the outer layers of the merger. In order to obtain bulk Lorentz factors [FORMULA] one can only allow for mass loss rates [FORMULA] if the rate at which the pair-photon fireball is supplied with energy is [FORMULA]  erg/s. With about [FORMULA] or [FORMULA]  erg/s of this energy being transferred to the dilute outer regions of the accretion torus in our models (Sect.  4.2) by [FORMULA] -annihilation (neglecting additional heating by neutrino-electron scattering and neutrino absorption), we compute a mass loss rate of at least [FORMULA] for a black hole with mass [FORMULA] and Schwarzschild radius [FORMULA]  km. From this we get [FORMULA] and estimate an entropy [FORMULA] ([FORMULA] internal energy density, P pressure) of [FORMULA] -200  [FORMULA] when [FORMULA]  MeV. Therefore, unless the neutrino-driven wind can be hindered to penetrate into the pair-photon fireball, e.g., in a region along the system axis by centrifugal forces, there is no chance to obtain highly relativistic fireballs with [FORMULA].

These issues might also be critical when a neutron star merges with a black hole or when two non-equal mass neutron stars coalesce. Simulations indicate (Lee & Kluzniak 1995) that already after the dynamical interaction of a neutron star and a black hole a cloud of baryonic material might "pollute" the surroundings even near the system axis. Moreover, instead of an accretion torus, a stable binary system might form with a more massive black hole circulated by a low-mass neutron star companion which is possibly unstable to explosion (see also Sect.  5.1 and Blinnikov et al. 1984, Eichler et al. 1989). Future, better resolved computations covering a wider range of equations of state and masses of the interacting stars will have to show the conclusiveness of their results.

It is interesting to speculate whether there is a chance to get tori more massive than in the merging of two (nearly) equal-mass neutron stars when a neutron star merges with a black hole or when a small neutron star is tidally disrupted before merging with a companion neutron star which has a significantly larger mass (e.g., Eichler et al. 1989, Narayan et al. 1992, Mochkovitch et al. 1993, and references therein). On the one hand, a more massive disk can radiate neutrinos for a longer time than the accretion time scale of the innermost, most strongly neutrino radiating part near the last stable orbit around the black hole. With much more matter being at larger radii, the neutrino emitting torus region considered and normalized to a mass of [FORMULA] in the derivation of Eq. (26) will be continuously refed by material advected inward. On the other hand, Eq. (26) suggests that the energy that can be provided in the pair-photon fireball by [FORMULA] -annihilation increases steeply with the mass M of the central black hole, [FORMULA]. Larger gravitating masses at the center, e.g., a massive stellar black hole, might therefore allow for more powerful [FORMULA] -ray bursts, at least, if one relies on the simplified picture developed in the derivation of Eq. (26) which does not take into account the dynamics of the tidal interaction between neutron star and black hole and its implications for the disk formation and the disk mass.

For the reasons outlined above it appears to be extremely hard to account for the energies of cosmological [FORMULA] -ray bursts by [FORMULA] -annihilation, at least in the case of merging binary neutron stars if beaming or focussing of the fireball towards the observer does not play a crucial role. The same conclusion was arrived at by Jaroszyski (1993, 1996) who investigated models of relativistic tori around rotating stellar mass black holes and tested different values of specific angular momentum, viscosity, and entropy. He found that the neutrino emission and annihilation energy from these tori is insufficient to explain the energies of cosmological [FORMULA] -ray bursts except for tori around Kerr black holes with very high angular momentum, i.e., with relativistic rotation parameters [FORMULA]. This might indeed be realized in the collapsed cores of rapidly rotating Wolf-Rayet stars in the "failed supernova" or "collapsar" scenario (Woosley 1993a). However, models for the stellar evolution of Wolf-Rayet stars seem to indicate that the angular momentum loss during the mass-loss phases is too large to allow for the formation of very rapidly spinning black holes (Langer, personal communication). Moreover, with [FORMULA] -annihilation as the source of the energy of the pair-photon fireball, these models might come into additional trouble if, as claimed by Quashnock (1996), the homogeneity of the distribution of the bursts in the BATSE 3B Catalog and the non-association of the bursts with large-scale structures of luminous matter in our extragalactic neighbourhood implies such large distances of the [FORMULA] -ray burst sources that the required total energy output in [FORMULA] -rays is larger than [FORMULA].

Our investigations do not include effects due to possible convective overturn and instabilities in the disk that might occur as a result of specific entropy (or composition) inversions caused by neutrino effects or viscous heating. Such dynamical processes were recently found to be present in multi-dimensional models of type-II supernovae (Herant et al. 1994; Burrows et al. 1995; Janka & Müller 1995, 1996) and were computed for axisymmetric advection-dominated accretion flows with two-dimensional hydrodynamical simulations by Igumenshchev et al. (1996) (see also Chen 1996). Although the disk remains globally stable, shorter-wavelength modes may affect the flow dynamics and effective disk viscosity significantly. In addition, such instabilities might increase the neutrino fluxes and the average neutrino energies considerably and thus might help [FORMULA] -annihilation. Our three-dimensional hydrodynamical simulations were not followed for a sufficiently long time to see whether such overturn processes occur in the merging scenario, and the simplified one-zone torus model does not take into account an enhancement of the neutrino fluxes by possible convective transport. Also, magnetic fields in the merging neutron stars and in the torus were disregarded. Within a lifetime of a few tenths of a second, initial B -fields of [FORMULA] - [FORMULA] might be amplified by a factor of 100 or more in the rapidly rotating disk around the black hole (rotation periods [FORMULA]) and might become energetically important (Rees, personal communication). A relativistic magneto-hydrodynamical wind of extremly high luminosity, perhaps associated with a binary neutron star merger, was suggested to generate [FORMULA] -ray bursts by Thompson (1994). In this respect NS-BH mergers and neutron star collisions and coalescence need further theoretical investigation. Moreover, neutron star collisions were recently pointed out as potential origin of short cosmological [FORMULA] -ray bursts by Katz & Canel (1995a) who argued that hot matter fragments could be ejected and, when they become neutrino-transparent on dynamical time scales, could lead to the emission of neutrinos with very high luminosities.

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Online publication: July 3, 1998