## 4. Neutrino-antineutrino annihilationNeutrino-antineutrino annihilation in the surroundings of the merger has been proposed to create a sufficiently energetic fireball of -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 -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 evaluationNeglecting phase space blocking effects in the phase spaces of and , the local energy deposition rate (energy ) at a position by annihilation of and into -pairs (which is the dominant reaction between neutrinos and antineutrinos) can be written in terms of the neutrino and antineutrino phase space distribution functions and as (Goodman et al. 1987, Cooperstein et al. 1987, Janka 1991) When the energy integrations are absorbed into (energy-integrated) neutrino intensities and , The integrals over and
sum up neutrino and antineutrino radiation
incident from all directions. is the angle
between neutrino and antineutrino beams and
and are suitably defined average spectral
energies of neutrinos and antineutrinos, respectively. The weak
interaction cross section is ,
is the electron rest-mass energy, When working with a discrete grid the integrals in Eq. are
replaced by sums over all cells is the solid angle with which cell With the projected area we obtain 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
, the flux is related
to the neutrino radiation intensity by
. With an effective emissivity
(see Paper I) which represents the energy
emission of cell Eq. and Eq. allow us to evaluate the sum of Eq. sufficiently accurately by With the average energy of neutrinos emitted from cell ## 4.2. Numerical resultsThe numerical post-processing procedure described in the previous
section yields the total energy deposition rate
by neutrino-antineutrino annihilation as a
function of the position . Fig. 16
shows the results, averaged over azimuthal angles, in a quadrant of
the
One can see that in all three models the highest rate of energy deposition () occurs in the outer regions of the disk ( 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 , 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 can become too high to allow for Lorentz factors in the required range of . 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 The integral values of the energy deposition rate by
-annihilation in the surroundings of the merger
out to equatorial distances
For the disk or torus geometry of our models the annihilation efficiency, defined as where is the total energy deposition rate by -annihilation, can be found to be of the order of a few tenths of a per cent: 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)). 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 -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
-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
-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- lead to a total
deposited energy of only a few 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 into a fireball of
-pairs and photons. Note that these statements
are 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 -annihilation. ## 4.3. Simple model for the post-merging emission from the diskOur simulations suggest that some material, possibly about , 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 -annihilation increases linearily with the luminosity (Eq. (10)) and thus a short, very luminous neutrino burst is more efficient to create an -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 -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 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 () direction, , is simply expressed by the component of the viscous stress tensor (see, e.g., Shapiro & Teukolsky 1983): where is the Keplerian angular velocity.
The torque From that, the accretion time scale of a disk of mass is estimated to be 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 . Viscous dissipation generates heat in the disk at a rate per unit volume of (see Shapiro & Teukolsky 1983) At steady-state conditions the maximum dissipation rate occurs at a radius when is the inner radius of the disk which is taken to be the innermost stable circular orbit around the central accreting black hole, . Using this in Eq. (14) one obtains for the maximum rate at which frictional heat is liberated, Thus, the viscous heating rate increases linearly with . For small viscosity 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 is and the total energy radiated in neutrinos, , becomes independent of because of . With Eq. (10) one finds that the energy converted into by -annihilation increases proportional to : . In the optically thin case the mean energy of emitted neutrinos, , which enters the calculation of will also increase with and cause a slightly steeper than linear dependence of on . If is large, the disk is heated rapidly and
strongly and thus becomes opaque for neutrinos. With a neutrino
diffusion time scale that is much longer than
the heating time scale the neutrino luminosity is
which is only indirectly dependent on
through and thus the
(viscosity dependent) gas temperature The considerations above suggest that the annihilation energy has a pronounced maximum at a particular value of the dynamic viscosity. Because of 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 is small, stays low. If is very large and the interior of the disk very hot and thus neutrino-opaque, the neutrino luminosity scales with the inverse of the neutrino diffusion time scale and with the total energy that can be emitted in neutrinos during the lifetime of the disk. This energy decreases in case of very large because 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 is therefore determined by the condition . Let us assume that the part of the disk where most of the neutrinos
are emitted has a mass and is a homogeneous
torus with center at and radius
(inner radius , outer
radius ) (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
Here is the atomic mass unit and the
thermally averaged effective neutrino interaction cross section
is defined as the sum of the cross sections
times the number fractions
of the corresponding reaction targets for all
neutrino processes in the medium, i.e., neutrino scattering off
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, . 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 (Eq. (13)) with equal to (Eq. (16)), one determines the value of the shear viscosity in the disk, where -annihilation yields the largest energy, as 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 is favored. Let us now consider a disk with this optimum value . Making use of , the interior temperature of the disk can be estimated by setting the integral rate of viscous energy generation in the disk, , equal to the luminosity due to neutrino diffusion, . Here is an approximation to the gradient of the neutrino energy density in the disk and is the sum of the energy densities of all three kinds of non-degenerate -pairs with being the radiation constant. One finds where and is normalized to the Schwarzschild radius of a black hole. Note that due to the dependence of and of on (according to Eqs. (16) and (18), respectively) 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 the interval of values again corresponding to the range of possible values of the effective neutrino interaction cross section. from Eq. (20) can now be used in Eq. (15) to calculate , which, when set equal to the neutrino luminosity expressed in terms of temperature and surface area of the neutrinosphere, , leads to an estimate of the neutrinospheric temperature The temperature of the neutrino emitting disk surface is around 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 , and to the inner disk radius . The optimum value for the dynamic viscosity as given in Eq. (20) corresponds to an effective -parameter of when , , and are used, and the sound speed is evaluated with . 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 . In the neutrino-opaque case it is estimated to be where the interval of the numerical value is again associated with the uncertainty of the effective cross section . For and one finds . A temperature as high as - is required for the neutrino viscosity to become large enough to account for . For the diffusion and accretion time scales one obtains by inserting Eq. (20) into Eq. (13) This time is much longer than the dynamical time scale ()) and the neutrino equilibration time scale (). 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 In Eq. (23) the smaller values and in Eq. (24) the larger ones correspond to the case of larger viscosity and thus smaller neutrino cross section according to the postulated equality of Eqs. (13) and (16). The total energy radiated away over the time is independent of both and becomes which is (approximately) equal to the Newtonian gravitational binding energy of mass at the inner disk radius where the matter is swallowed by the black hole ( is the gravitational potential energy, 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 . This requires that within 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 and 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 (exact value for a thin, Newtonian accretion disk: ). 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 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 to the event horizon. Using the results of Eqs. (21) and (24) to compute the -annihilation efficiency according to Eq. (10) and employing the integral energy emitted in neutrinos as given from Eq. (25), one can obtain a result for the energy deposited in an -pair-photon fireball by the annihilation of and radiated from the disk. With , , and (see Sect. 3.1), i.e, , we get The upper and lower bounds of the interval for correspond to the most extreme (maximum and minimum, respectively) choices for , , and . Notice that the analytical estimates of the neutrino luminosity (Eq. (24)) and the mean energy of neutrinos emitted from the disk, (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 -annihilation that is a factor of 10-100 larger than the calculated in Sect. 4.2. © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |