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

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4. Neutrino-antineutrino annihilation

Neutrino-antineutrino annihilation in the surroundings of the merger has been proposed to create a sufficiently energetic fireball of [FORMULA] -pairs and photons to explain gamma-ray bursts at cosmological distances. We attempt to put this idea to a quantitative test. With the given information about the fluxes and spectra of the neutrino emission of all grid cells (see Paper I for technical details), it is possible to evaluate our hydrodynamical models for the energy deposition by [FORMULA] -annihilation in a post-processing step. Since the neutrino luminosities become large only after the merging of the two neutron stars and in particular after the gas torus around the compact central body has formed, we consider the late stages of our simulated merger evolutions as the most interesting ones to perform the analyses. In the phase when quasi-stationary conditions have been established, the neutrino luminosities have reached their saturation levels and the annihilation rates have become maximal.

4.1. Numerical evaluation

Neglecting phase space blocking effects in the phase spaces of [FORMULA] and [FORMULA], the local energy deposition rate (energy [FORMULA]) at a position [FORMULA] by annihilation of [FORMULA] and [FORMULA] into [FORMULA] -pairs (which is the dominant reaction between neutrinos and antineutrinos) can be written in terms of the neutrino and antineutrino phase space distribution functions [FORMULA] and [FORMULA] as (Goodman et al. 1987, Cooperstein et al. 1987, Janka 1991)

[EQUATION]

When the energy integrations are absorbed into (energy-integrated) neutrino intensities [FORMULA] and [FORMULA],

[EQUATION]

Eq.  can be rewritten as

[EQUATION]

The integrals over [FORMULA] and [FORMULA] sum up neutrino and antineutrino radiation incident from all directions. [FORMULA] is the angle between neutrino and antineutrino beams and [FORMULA] and [FORMULA] are suitably defined average spectral energies of neutrinos and antineutrinos, respectively. The weak interaction cross section is [FORMULA], [FORMULA] is the electron rest-mass energy, c the speed of light, and the weak coupling constants are [FORMULA], [FORMULA] and [FORMULA], [FORMULA] for [FORMULA] and [FORMULA], [FORMULA] with [FORMULA]. The total energy deposition rate at the position [FORMULA] is given as the sum of the contributions from annihilation of [FORMULA] and [FORMULA], [FORMULA] and [FORMULA], and [FORMULA] and [FORMULA]:

[EQUATION]

When working with a discrete grid the integrals in Eq.  are replaced by sums over all cells k,

[EQUATION]

[FORMULA] is the solid angle with which cell k is seen from a position [FORMULA] at distance [FORMULA] when [FORMULA] is the location of the center of cell k. In order to avoid the need to take into account projection effects, we define an effective radius D associated with the cells of the cartesian grid used for the hydrodynamical modelling by setting the cell volume [FORMULA] equal to the volume of a sphere [FORMULA]:

[EQUATION]

With the projected area [FORMULA] we obtain

[EQUATION]

Using the simplifying assumption that a grid cell radiates neutrinos with isotropic intensity into the half space around the outward direction defined by the local density gradient [FORMULA], the flux [FORMULA] is related to the neutrino radiation intensity [FORMULA] by [FORMULA]. With an effective emissivity [FORMULA] (see Paper I) which represents the energy emission of cell k per [FORMULA] in a single neutrino species [FORMULA] or [FORMULA], the intensity [FORMULA] is therefore given by

[EQUATION]

Eq.  and Eq.  allow us to evaluate the sum of Eq.  sufficiently accurately by

[EQUATION]

With the average energy of neutrinos emitted from cell k, [FORMULA], (see Paper I) and the angle enclosed by the radiation from cells k and [FORMULA] at the position [FORMULA], [FORMULA], the integrals of Eq.  finally become sums over all combinations of grid cells k with all cells [FORMULA]. This double sum has to be evaluated at all positions [FORMULA] where the energy deposition rate by [FORMULA] -annihilation is to be determined. The corresponding computational load is appreciable but can be significantly reduced by taking into account only those grid cells which emit towards [FORMULA], i.e. whose emission into the outward directed half space around [FORMULA] is also pointing to position [FORMULA]. The criterion for this being fulfiled is [FORMULA].

4.2. Numerical results

The numerical post-processing procedure described in the previous section yields the total energy deposition rate [FORMULA] by neutrino-antineutrino annihilation as a function of the position  [FORMULA]. Fig. 16 shows the results, averaged over azimuthal angles, in a quadrant of the r -z -plane perpendicular to the orbital plane for the three models A64, B64, and C64. The evaluation was performed only in that region around the central part of the merger where the local energy loss rate by neutrino emission is less than [FORMULA] and the density is below [FORMULA]. Density levels are indicated by dashed contour lines in the plots.

[FIGURE] Fig. 16a-c. Maps of the total local energy deposition rates (in erg/cm3 /s) by neutrino-antineutrino annihilation into electron-positron pairs in the vicinity of the merger for the three models A64, B64, and C64 at time [FORMULA]  ms after the start of the simulations. In one quadrant of the r -z plane orthogonal to the orbital plane (at [FORMULA]), the plots show values obtained as averages of the energy deposition rate over azimuthal angles. The corresponding solid contour lines are logarithmically spaced in steps of 0.5 dex, the grey shading emphasizes the levels with dark grey meaning high energy deposition rate. The dashed contours indicate levels of the azimuthally averaged density, also logarithmically spaced with intervals of 0.5 dex. The energy deposition rate was evaluated only in that region around the merged object, where the mass density is below [FORMULA]  g/cm3 and the energy loss rate by neutrino emission is smaller than [FORMULA]  erg/cm3 /s. One can see that due to the closeness to the main neutrino radiating disk region, most annihilation energy is deposited in the "outer" (in r -direction) and "upper" (in z -direction) regions of the disk (dark grey areas)

One can see that in all three models the highest rate of energy deposition ([FORMULA]) occurs in the outer regions of the disk ([FORMULA] in the orbital plane) within about 25 km above and below the orbital plane. Here the energy deposition rate is much larger than the energy loss rate. Because this deposition transfers energy into gas layers with densities of still more than [FORMULA], one must suspect that a baryonic wind will be created very similar to the neutrino-driven wind caused by neutrino energy deposition in the surface layers of the nascent neutron star in a type-II supernova (for information about the neutrino-driven wind from forming neutron stars, see Duncan et al. 1986, Woosley & Baron 1992, Witti et al. 1994, Woosley et al. 1994; see also Hernanz et al. 1994). Instead of creating a fireball of a nearly pure relativistic electron-positron-photon plasma which might lead to an energetic gamma-ray burst, this energy is used up to lift baryons in the strong gravitational field of the massive central body. Consequently, the expansion velocities of this matter are nonrelativistic. If too much of this wind material gets mixed into the pair-photon plasma, the baryonic load [FORMULA] can become too high to allow for Lorentz factors [FORMULA] in the required range of [FORMULA].

The large energy deposition rates at radii between 30 km and 50 km in the equatorial plane and at moderate heights above and below the orbital plane can be explained by the closeness to the main neutrino emitting ring in the disk between 20 km and 30 km (see Figs. 5-10). According to Eqs. (3), (5), and (9) the annihilation rate decreases at least with the fourth power of the distance to the neutrino radiating grid cells k and [FORMULA]: [FORMULA]. Because of the influence of the geometrical factor [FORMULA] in Eq. (3) (this factor accounts for the dependence of the annihilation probability on the relative velocity of the interacting neutrinos and the kinematically allowed phase space for the reaction), [FORMULA] decreases even more steeply when one moves away from the source and large- [FORMULA] collisions between neutrinos become less and less frequent. At distances from the merged object [FORMULA] -60 km we find that along each radial beam the local energy deposition rate as a function of d becomes a power law with power law index between -6 and -8. Towards the rotation (z -) axis, the growing distance to the neutrino producing disk region cannot be compensated by the higher chance of head-on collisions between neutrinos. This is the reason why the contour levels in the plots of Fig. 16 bend towards the polar region of the merger near the z -axis.

The integral values of the energy deposition rate by [FORMULA] -annihilation in the surroundings of the merger out to equatorial distances r are shown in Fig. 17. For each of the models A64, B64, and C64 we have evaluated the spatial integral once for vertical heights [FORMULA] and another time for [FORMULA]. The maximum upper integration limit is determined by the largest distances where the local rates [FORMULA] were calculated. However, one can see from Fig. 17 that the curves tend to approach a saturation level for [FORMULA]  km from which we conclude that extending the integrations into the region [FORMULA]  km would not change the results significantly. From a comparison of both cases one can recognize that only a minor fraction of about [FORMULA] - [FORMULA] of the annihilation energy is converted into [FORMULA] -pairs in the region above and below the disk. Only at heights [FORMULA] is the baryon density low enough that most of the converted energy might end up in a relativistic fireball. However, in the whole region [FORMULA] the energy deposition rate is rather small, only about 2- [FORMULA] ; the "useful" fraction is probably less than [FORMULA] of that. The models were evaluated at times when the neutrino emission of the models had already achieved a maximum value and a quasi-stationary state. At earlier times the neutrino luminosities are much lower and therefore the integral values of the energy deposition rate are even smaller than those displayed in Fig. 17.

[FIGURE] Fig. 17. Cumulative total energy deposition rates by neutrino-antineutrino annihilation into electron-positron pairs for the three models A64, B64, and C64 as functions of the equatorial distance r from the center of the merger once for the case that the integration is done for polar distances [FORMULA] and another time for [FORMULA]. The integrals were performed over the values of the local energy deposition rates plotted in the three panels of Fig. 16

For the disk or torus geometry of our models the annihilation efficiency, defined as [FORMULA] where [FORMULA] is the total energy deposition rate by [FORMULA] -annihilation, can be found to be of the order of a few tenths of a per cent:

[EQUATION]

The factors in this equation can be deduced by simple analytical transformations and dimensional considerations of the volume integral of the annihilation rate (Eq. (10)). [FORMULA] is the inner radius of the disk, the radius which roughly separates the compact inner core from the more dilute gaseous cloud of matter around. The inserted numbers are representative values taken from Figs. 1, 3, and 4. The efficiency of Eq. (10) for converting emitted neutrino energy into electron-positron pairs by [FORMULA] -annihilation is in good agreement with what is obtained in supernova simulations or analytical estimates assuming emission in a spherical geometry instead of the disk-dominated emission of our numerical models.

From the results discussed here we conclude that there is no chance to obtain the energies needed for a cosmological gamma-ray burst by [FORMULA] -annihilation during the instant of the merging of binary neutron stars. Before the hot cloud of gas around the central, dense body has formed, the neutrino luminosities stay rather low and only very little energy is deposited by [FORMULA] -annihilation. Even later when a disk around the compact, merged body has formed and the neutrino emission has reached a high level, the energy deposition rates of 2- [FORMULA] lead to a total deposited energy of only a few [FORMULA] within the computed time of 10 ms. One would need the strong neutrino emission to be maintained for periods of about 10 s to pump an energy of more than [FORMULA] into a fireball of [FORMULA] -pairs and photons. Note that these statements are not changed if the observed gamma-ray bursts are beamed events. If one relies on the emission geometry of our models, a constrained solid angle [FORMULA] into which the fireball expands and becomes visible in a jet-like outburst would also (and by the same factor) reduce the spatial volume where [FORMULA] -annihilation in the surroundings of the merger could deposit the useful energy. Correspondingly, the numbers given in Fig. 17 would have to be scaled down by a factor [FORMULA]. In view of the fact discussed in Sect.  2.3 that the central, compact core of the merger has a mass that can neither be stabilized by internal pressure nor by rotation and should therefore collapse into a black hole on a time scale of milliseconds, we conclude that the energy available by pair annihilation of neutrinos is lower than the desired and required "canonical" value of [FORMULA] by more than a factor of 1000.

So far we have considered the phase of the merging of binary neutron stars and the evolution that follows immediately afterwards. This, however, leaves the question unanswered whether a disk around the most likely forming black hole could produce and emit neutrinos on a much longer time scale and could thus power a gamma-ray burst by the discussed mechanism of neutrino-pair annihilation? Since simulations of this scenario are beyond the present capabilities of the employed numerical code and, in addition, the results depend on the unknown viscosity in the disk, we attempt to develop a simple model of the behavior and properties of such a disk with respect to its neutrino emission and the strength of [FORMULA] -annihilation.

4.3. Simple model for the post-merging emission from the disk

Our simulations suggest that some material, possibly about [FORMULA], could remain in a disk around the central black hole. This disk will be heated by viscous dissipation and will emit neutrinos and antineutrinos until its matter is accreted into the central black hole. The efficiency of [FORMULA] -annihilation increases linearily with the [FORMULA] luminosity (Eq. (10)) and thus a short, very luminous neutrino burst is more efficient to create an [FORMULA] -pair fireball than the same energy emitted on a longer time scale with smaller neutrino fluxes. It has to be investigated whether enough energy can be provided in the pair-photon fireball by the neutrino emission from the disk to explain a [FORMULA] -ray burst at cosmological distances. Viscosity effects have a crucial influence on the disk evolution and on the neutrino emission. Viscous forces, on the one hand, transfer angular momentum between adjacent fluid elements and determine the accretion time scale and accretion rate. Viscous dissipation of rotational energy, on the other hand, heats the disk and is thus essential for the neutrino emission. Disk size, disk temperature, disk viscosity, and neutrino emission properties can therefore not be chosen independent of the accretor mass and disk mass. In the following we shall attempt to relate these quantities by simple considerations and conservation arguments.

The lifetime of the disk will decrease with larger dynamic viscosity [FORMULA] because the viscous force that generates a torque carrying angular momentum outward is increased. For a (Newtonian) Keplerian disk the viscous force (per unit area) in the angular ([FORMULA]) direction, [FORMULA], is simply expressed by the component [FORMULA] of the viscous stress tensor (see, e.g., Shapiro & Teukolsky 1983):

[EQUATION]

where [FORMULA] is the Keplerian angular velocity. The torque T exerted by the viscous stress is given by [FORMULA] when [FORMULA] is taken as the vertical diameter of the thick disk around the black hole with mass M and Schwarzschild radius [FORMULA]. The accretion rate [FORMULA] can be estimated (roughly) by setting the viscous torque equal to the rate [FORMULA] at which angular momentum is consumed by the black hole due to the accretion of matter from the disk:

[EQUATION]

From that, the accretion time scale of a disk of mass [FORMULA] is estimated to be

[EQUATION]

Thus, the lifetime of the disk is determined by the outward transport of angular momentum through the viscous torque. Eq. (13) shows that it decreases with the value of the dynamic viscosity as [FORMULA].

Viscous dissipation generates heat in the disk at a rate per unit volume of (see Shapiro & Teukolsky 1983)

[EQUATION]

At steady-state conditions the maximum dissipation rate occurs at a radius [FORMULA] when [FORMULA] is the inner radius of the disk which is taken to be the innermost stable circular orbit around the central accreting black hole, [FORMULA]. Using this in Eq. (14) one obtains for the maximum rate at which frictional heat is liberated,

[EQUATION]

Thus, the viscous heating rate increases linearly with [FORMULA].

For small viscosity [FORMULA] the viscous heating time scale is long and the disk remains rather cool, also because cool matter is comparatively transparent for neutrinos and therefore the neutrino cooling time scale is short. In that case the neutrino luminosity for a disk with volume [FORMULA] is [FORMULA] and the total energy radiated in neutrinos, [FORMULA], becomes independent of [FORMULA] because of [FORMULA]. With Eq. (10) one finds that the energy converted into [FORMULA] by [FORMULA] -annihilation increases proportional to [FORMULA]: [FORMULA]. In the optically thin case the mean energy of emitted neutrinos, [FORMULA], which enters the calculation of [FORMULA] will also increase with [FORMULA] and cause a slightly steeper than linear dependence of [FORMULA] on [FORMULA].

If [FORMULA] is large, the disk is heated rapidly and strongly and thus becomes opaque for neutrinos. With a neutrino diffusion time scale [FORMULA] that is much longer than the heating time scale the neutrino luminosity is [FORMULA] which is only indirectly dependent on [FORMULA] through [FORMULA] and thus the (viscosity dependent) gas temperature T. In that case [FORMULA] and [FORMULA]. Note that the average energy of emitted neutrinos, [FORMULA], which also determines [FORMULA], is only very weakly dependent on the viscosity of the disk in the optically thick case because it reflects the conditions at the neutrino decoupling sphere (see Eq. (21) below). The diffusion time scale increases with the disk temperature and thus with the disk viscosity due to the energy dependence of the weak interaction cross sections. This leads to a decrease of [FORMULA] with [FORMULA] that is steeper than [FORMULA].

The considerations above suggest that the annihilation energy [FORMULA] has a pronounced maximum at a particular value [FORMULA] of the dynamic viscosity. Because of [FORMULA] the annihilation of neutrinos and antineutrinos is more efficient when a certain energy is emitted in a short time with a high luminosity rather than over a long period with a moderate flux. If [FORMULA] is small, [FORMULA] stays low. If [FORMULA] is very large and the interior of the disk very hot and thus neutrino-opaque, the neutrino luminosity [FORMULA] scales with the inverse of the neutrino diffusion time scale and with the total energy [FORMULA] that can be emitted in neutrinos during the lifetime [FORMULA] of the disk. This energy [FORMULA] decreases in case of very large [FORMULA] because [FORMULA] becomes shorter and the internal energy cannot be completely radiated away in neutrinos before the neutrino-opaque matter is accreted into the black hole. The kinetic energy that is converted into internal energy by viscous friction is entirely transported away by neutrinos and the fluxes are largest, if the diffusion time scale is similar to the accretion time scale but not much longer. The optimum value [FORMULA] is therefore determined by the condition [FORMULA].

Let us assume that the part of the disk where most of the neutrinos are emitted has a mass [FORMULA] and is a homogeneous torus with center at [FORMULA] and radius [FORMULA] (inner radius [FORMULA], outer radius [FORMULA]) (Mochkovitch et al. 1993; Jaroszyski 1993). This is a fairly good picture in view of the shape and structure of the disk that we obtained in our numerical simulations 1. In terms of [FORMULA] the volume of the disk torus is [FORMULA] and its surface [FORMULA]. With the neutrino mean free path [FORMULA] and [FORMULA] the diffusion time scale is approximately given by

[EQUATION]

Here [FORMULA] is the atomic mass unit and the thermally averaged effective neutrino interaction cross section [FORMULA] is defined as the sum of the cross sections [FORMULA] times the number fractions [FORMULA] of the corresponding reaction targets for all neutrino processes in the medium, i.e., neutrino scattering off n, p, [FORMULA], [FORMULA], charged-current absorptions of [FORMULA] and [FORMULA] by n and p, respectively, and [FORMULA] -pair interactions. We find

[EQUATION]

the exact value depending on the neutrino type, the neutrino degeneracy and neutrino spectra, and the detailed composition of the medium. For the entropies, densities, and temperatures obtained in our simulations the gas in the disk is completely disintegrated into free nucleons; nucleon as well as lepton degeneracy plays a negligible role (see Sect.  3.2). Therefore fermion phase space blocking effects are unimportant. The thermal average of the neutrino cross section was evaluated by using a Fermi-Dirac distribution function with a vanishing neutrino chemical potential, [FORMULA]. In case of incomplete dissociation of the nuclei the neutrino opacity should still be within the uncertainty range associated with the cross section variation of Eq. (17). Setting [FORMULA] (Eq. (13)) with [FORMULA] equal to [FORMULA] (Eq. (16)), one determines the value of the shear viscosity in the disk, where [FORMULA] -annihilation yields the largest energy, as

[EQUATION]

The range of values accounts for the uncertainty in the effective neutrino interaction cross section. For the typical composition of the disk material the cross section is more likely near the upper limit of the given interval, in which case the lower value of [FORMULA] is favored.

Let us now consider a disk with this optimum value [FORMULA]. Making use of [FORMULA], the interior temperature of the disk can be estimated by setting the integral rate of viscous energy generation in the disk, [FORMULA], equal to the luminosity due to neutrino diffusion, [FORMULA]. Here [FORMULA] is an approximation to the gradient of the neutrino energy density in the disk and [FORMULA] is the sum of the energy densities of all three kinds of non-degenerate [FORMULA] -pairs with [FORMULA] being the radiation constant. One finds

[EQUATION]

where [FORMULA] and [FORMULA] is normalized to the Schwarzschild radius of a [FORMULA] black hole. Note that due to the dependence of [FORMULA] and of [FORMULA] on [FORMULA] (according to Eqs. (16) and (18), respectively) [FORMULA] does not depend on the neutrino interaction cross section and is therefore insensitive to its uncertainty.

Plugging the result of Eq. (19) for the interior disk temperature into Eq. (18) yields for the optimum disk viscosity

[EQUATION]

the interval of values again corresponding to the range of possible values of the effective neutrino interaction cross section. [FORMULA] from Eq. (20) can now be used in Eq. (15) to calculate [FORMULA], which, when set equal to the neutrino luminosity expressed in terms of temperature and surface area [FORMULA] of the neutrinosphere, [FORMULA], leads to an estimate of the neutrinospheric temperature

[EQUATION]

The temperature of the neutrino emitting disk surface is around [FORMULA] and rather insensitive to the exact value of the effective neutrino interaction cross section (slightly larger result for smaller cross section), to the disk mass [FORMULA], and to the inner disk radius [FORMULA].

The optimum value [FORMULA] for the dynamic viscosity as given in Eq. (20) corresponds to an effective [FORMULA] -parameter of [FORMULA] when [FORMULA], [FORMULA], and [FORMULA] are used, and the sound speed [FORMULA] is evaluated with [FORMULA]. For these values of density and temperature the gas pressure is dominated by relativistic particles, i.e., photons, electrons, positrons, and neutrinos. Neutrino shear viscosity does not contribute significantly to [FORMULA]. In the neutrino-opaque case it is estimated to be

[EQUATION]

where the interval of the numerical value is again associated with the uncertainty of the effective cross section [FORMULA]. For [FORMULA] and [FORMULA] one finds [FORMULA]. A temperature as high as [FORMULA] - [FORMULA] is required for the neutrino viscosity to become large enough to account for [FORMULA].

For the diffusion and accretion time scales one obtains by inserting Eq. (20) into Eq. (13)

[EQUATION]

This time is much longer than the dynamical time scale ([FORMULA])) and the neutrino equilibration time scale ([FORMULA]). Therefore our assumptions that neutrinos diffuse in the disk and are in equilibrium with the matter are confirmed a posteriori. The total neutrino luminosity is

[EQUATION]

In Eq. (23) the smaller values and in Eq. (24) the larger ones correspond to the case of larger viscosity [FORMULA] and thus smaller neutrino cross section [FORMULA] according to the postulated equality of Eqs. (13) and (16). The total energy [FORMULA] radiated away over the time [FORMULA] is independent of both and becomes

[EQUATION]

which is (approximately) equal to the Newtonian gravitational binding energy [FORMULA] of mass [FORMULA] at the inner disk radius [FORMULA] where the matter is swallowed by the black hole ([FORMULA] is the gravitational potential energy, [FORMULA] the rotational energy). Here it is assumed that no rotational kinetic energy is extracted from the black hole which is equivalent to a zero stress boundary condition at [FORMULA]. This requires that within [FORMULA] the gas spirals into the black hole rapidly without radiating, an idealization which is probably justified (see, e.g., Shapiro & Teukolsky 1983). The small discrepancy between the factors [FORMULA] and [FORMULA] of our calculation results from the fact that we consider a simple one-zone model of a homogeneous disk. We find that the radiation efficiency of the disk in our simplified treatment is [FORMULA] (exact value for a thin, Newtonian accretion disk: [FORMULA]). This result has to be compared with the radiation efficiency of about 5.7% for relativistic disk accretion onto a nonrotating black hole and with the radiation efficiency of 42.3% for a maximally rotating black hole with a prograde accretion disk (see Shapiro & Teukolsky 1983). Since our numerical simulations suggest the formation of a central black hole with a relativistic rotation parameter [FORMULA] that is clearly less than 1 (Ruffert et al. 1996), the reference value for the radiation efficiency for disk accretion onto a nonrotating black hole is relevant and our Newtonian disk evolution model most likely overestimates the amount of energy that can be carried away by neutrinos before the accreted mass finally plunges rapidly from [FORMULA] to the event horizon.

Using the results of Eqs. (21) and (24) to compute the [FORMULA] -annihilation efficiency [FORMULA] according to Eq. (10) and employing the integral energy [FORMULA] emitted in neutrinos as given from Eq. (25), one can obtain a result for the energy [FORMULA] deposited in an [FORMULA] -pair-photon fireball by the annihilation of [FORMULA] and [FORMULA] radiated from the disk. With [FORMULA], [FORMULA], and [FORMULA] (see Sect.  3.1), i.e, [FORMULA], we get

[EQUATION]

The upper and lower bounds of the interval for [FORMULA] correspond to the most extreme (maximum and minimum, respectively) choices for [FORMULA], [FORMULA], and [FORMULA]. Notice that the analytical estimates of the neutrino luminosity (Eq. (24)) and the mean energy of neutrinos emitted from the disk, [FORMULA] (Eq. (21)), agree well with our numerical results for the phase shortly after the merging. While the dynamical time scale of the merging is of the order of 1 ms and the post-merging evolution was followed by our numerical simulations for a period of about 10 ms, the disk emits neutrinos with similar luminosities for a much longer time of a few hundred milliseconds (cf. Eq. (23)). Therefore, Eq. (26) gives a number for the energy deposition by [FORMULA] -annihilation that is a factor of 10-100 larger than the [FORMULA] calculated in Sect.  4.2.

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Online publication: July 3, 1998
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