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

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3. Neutrino emission and thermodynamical evolution

3.1. Neutrino emission

The local energy and lepton number losses due to neutrino emission are included via source terms in our code as described in the appendix of Paper I. We treated the neutrino effects in terms of an elaborate leakage scheme that was calibrated by comparison with results from diffusion calculations in one-dimensional situations. By adding up the local source terms over the whole computational grid, one obtains the neutrino luminosities of all individual neutrino types, the sum of which gives the total neutrino luminosity. In the same way number fluxes of electron neutrinos ([FORMULA]), electron antineutrinos ([FORMULA]), and heavy-lepton neutrinos ([FORMULA], [FORMULA], [FORMULA], and [FORMULA], which will be referred to as [FORMULA] in the following) can be calculated. The mean energies of the emitted neutrinos result from the ratios of neutrino luminosities to neutrino number fluxes.

Figs. 1-4 display the results for the neutrino fluxes and mean energies of the emitted neutrinos for our models A64, B64, C64, and A128. In Fig. 1 one can see that the total neutrino luminosities start to increase above [FORMULA] at about 2.5-3.5 ms after the start of the simulations. This is about the time when the spiral arm structures become dispersed into a spread-out cloud of material that surrounds the merger and is heated by the interaction with compression waves. At the same time the temperatures in the interior of the massive central body reach their peak values. Since the neutrino luminosities are by far dominated by the contributions from the disk emission (see below), the heating of the cloud and torus material is reflected in a continuous increase of the neutrino fluxes. Also, the dynamical expansions and contractions caused by the oscillations and wobbling of the central body impose fluctuations on the light curves. The periodic expansions and contractions have particularly large amplitudes in model C64 because of the anti-spin setup of the neutron star rotations which leads to very strong internal shearing and turbulent motions after merging as well as to higher temperatures than in the other models (see Table 1). The light curve fluctuations in model C64 are as large as 25-30% of the average luminosity and proceed with a period of 0.5-1 ms which is about the dynamical time scale of the merger. When the oscillating central body enters an expansion phase, very hot matter that is located in a shell around the central core of the merger, is swept to larger radii. In course of the expansion the neutrino optical depth decreases and the very hot material releases enhanced neutrino fluxes. The neutrino outburst is terminated, when the surface-near matter has cooled by adiabatic expansion or when re-contraction sets in and the neutrino optical depth increases again. At times later than about 6-8 ms quasi-stationary values of the fluxes are reached which are between about [FORMULA] in case of model A128 and about [FORMULA] for C64.

[FIGURE] Fig. 1. Total neutrino luminosities as functions of time for all four models
[FIGURE] Fig. 2. Comparison of the cumulative energies emitted in gravitational waves and in neutrinos as functions of time for all four models
[FIGURE] Fig. 3. Luminosities of individual neutrino types ([FORMULA], [FORMULA], and the sum of all [FORMULA]) as functions of time for the three models A64, B64, and C64
[FIGURE] Fig. 4. Average energies of emitted neutrinos [FORMULA], [FORMULA], and [FORMULA] as functions of time for the three models A64, B64, and C64

Fig. 2 shows that the energy lost in gravitational waves during the merging is about two orders of magnitude larger than the energy radiated away in neutrinos during the simulated period of approximately 10 ms. While the gravitational wave luminosity peaks around the time when the dynamical instability of the orbit sets in and the two neutron stars start to interact dynamically and fuse into a single object (t between 0.5 ms and 1.5 ms), the energy emitted in neutrinos becomes sizable only after the extended toroidal cloud of matter has formed around the merged stars.

The heated torus or "disk" consists of decompressed neutron star matter with an initially very low electron number fraction [FORMULA] between about 0.01 and 0.04. The neutrino emission of the disk is therefore clearly dominated by the loss of electron antineutrinos which are primarily produced in the process [FORMULA], because positrons are rather abundant in the hot and only moderately degenerate, neutron-rich matter (see Sect.  3.2). In Fig. 3 one sees that the [FORMULA] luminosity [FORMULA] is a factor 3-4.5 larger than [FORMULA] and between [FORMULA] for model B64 and about [FORMULA] for C64 when quasi-stationary conditions have been established after about 6 ms from the start of the simulations (Table 1). The sum of all heavy-lepton neutrino fluxes which is four times the individual luminosities of [FORMULA], [FORMULA], [FORMULA] or [FORMULA], is around [FORMULA]. The better resolved model A128 yields slightly smaller values for all luminosities than A64 (Table 1).

The mean energies of the emitted neutrinos are displayed in Fig. 4 for models A64, B64, and C64. The differences between the different models are smaller than in case of the neutrino luminosities. This means that the effective temperatures in the neutrinospheric regions are very similar in all models. Electron neutrinos are emitted with an average energy of 12-13 MeV, electron antineutrinos have 19-21 MeV, and heavy lepton neutrinos leave the merger with mean energies of 26-28 MeV during the stationary phase. Except for the dominance of [FORMULA] relative to [FORMULA], both the mean neutrino energies and the neutrino luminosities are in good overall agreement with typical numbers obtained in stellar core collapse and supernova simulations at a stage some time after the prompt [FORMULA] burst has been emitted and after the collapsed stellar core has heated up along with the post-shock settling (see, e.g., Mayle et al. 1987, Myra & Burrows 1990, Bruenn 1993, Bruenn et al. 1995).

Just like the neutrino emission from the less opaque, hot mantle region of the protoneutron star dominates the supernova neutrino fluxes for several ten to some hundred milliseconds after core bounce, more than 90% of the neutrino emission of the merger comes from the extended, hot torus and only a minor fraction of less than 10% originates from the very opaque and dense central core of the merged object. Figs. 5, 6, 8, and 9 display the local emission rates of [FORMULA], [FORMULA], the sum of [FORMULA], and the sum of all neutrino types in the orbital plane for models A64, A128, B64, and C64, respectively, at a time when quasi-stationary conditions have been established. Figs. 7 and 10 give the corresponding information in two orthogonal cut planes vertical to the equatorial plane for the representative models A128 and C64. One can clearly see that the surface-near regions, in particular the toroidal cloud of matter surrounding the central, compact body, emits neutrinos at much higher rates.

[FIGURE] Fig. 5a-d. Energy emission rates (in erg/cm3 /s) of electron neutrinos (panel a), electron antineutrinos (panel b), the sum of all heavy-lepton neutrinos (panel c), and the total neutrino energy loss rate in the orbital plane of model A64 at the end of the simulation (time in the top right corner of the panels). The contours are logarithmically spaced in intervals of 0.5 dex, bold contours are labeled with their respective values. The grey shading emphasizes the emission levels, dark grey corresponding to the strongest energy loss by neutrino emission
[FIGURE] Fig. 6a-d. Same as Fig. 5 but for model A128 at time [FORMULA]  ms. The higher resolution of this simulation allows more fine structure to be visible

[FIGURE] Fig. 7a-d. Energy loss rates by neutrino emission in two orthogonal planes vertical to the orbital plane for model A128 at time [FORMULA]  ms. The displayed information is the same as in Fig. 5

[FIGURE] Fig. 8a-d. Same as Fig. 5 but for model B64 at time [FORMULA]  ms

[FIGURE] Fig. 9a-d. Same as Fig. 5 but for model C64 at time [FORMULA]  ms
[FIGURE] Fig. 10a-d. Same as Fig. 7 but for model C64 at time [FORMULA]  ms

Using the values from these figures, one can estimate the relative importance of core and disk emission. The dense core regions ([FORMULA]) inside a radius of [FORMULA]  km lose neutrino energy with a typical rate of [FORMULA] and account for a luminosity of about [FORMULA], while the emission from the surrounding torus-shaped disk region with outer radius [FORMULA]  km, height [FORMULA]  km, and typical total neutrino energy loss rate [FORMULA] emits a luminosity of roughly [FORMULA]. The sum of [FORMULA] and [FORMULA] is of the size of the results displayed in Fig. 1.

Figs. 5-10 also demonstrate that the disk emits [FORMULA] fluxes that are larger than the [FORMULA] fluxes. Electron neutrinos are most strongly emitted from surface-near regions where the optical depth to [FORMULA] by absorption on neutrons ([FORMULA]) in the neutron-rich matter is smallest. In contrast, the production of electron antineutrinos by positron captures on neutrons and of heavy-lepton neutrinos via electron-positron pair annihilation ([FORMULA]) requires the presence of large numbers of positrons and therefore occurs predominantly in those parts of the disk which have been heated most strongly and have a low electron degeneracy. Since [FORMULA] are absorbed on the less abundant protons in the inverse [FORMULA] process and the opacity to [FORMULA] is primarily caused by neutrino-nucleon scatterings only, electron antineutrinos and heavy-lepton neutrinos can escape on average from deeper and hotter layers than electron neutrinos (see Figs. 5-10). This explains the higher mean energies of the emitted [FORMULA] and [FORMULA].

As expected from the very similar neutrino luminosities and nearly equal mean energies of emitted neutrinos, the neutrino emission maps do not reveal major differences between the models A64, B64, and C64. Most neutrino emission comes from the disk region where the neutrino optical depths are lower, and in Figs. 5, 8, and 9 one can clearly recognize the high-density, very opaque inner part of the core of the merger from its roughly two orders of magnitude smaller energy loss rates. The cuts perpendicular to the orbital plane (see Fig. 10) show the ring-like main emission region that surrounds the central core and has a banana- or dumb-bell shaped cross section. Model A128 has significantly more fine structure but the overall features and characteristics of the neutrino emission do not change with the much better numerical resolution of this simulation.

3.2. Thermodynamics and composition

Fig. 11 displays contour levels in the temperature-density plane of the electron neutrino chemical potential (measured in MeV) for an electron fraction of [FORMULA]. In Fig. 12 the contours corresponding to vanishing electron neutrino chemical potential [FORMULA] are plotted for different values of [FORMULA]. Figs. 11 and 12 provide information about how chemical equilibrium is shifted according to the equation of state of Lattimer & Swesty (1991) when neutron star matter changes its temperature-density state.

[FIGURE] Fig. 11. Contours of the neutrino chemical potential (measured in MeV) in the temperature-density plane for fixed electron fraction [FORMULA]. The contours are spaced with increments of 10 and are labeled with their respective values
[FIGURE] Fig. 12. Contours of vanishing neutrino chemical potential ([FORMULA]) in the temperature-density plane for different values of the electron fraction [FORMULA] as indicated by the line labels

Cold neutron star matter at neutrinoless [FORMULA] -equilibrium is in a state with [FORMULA]. When such gas of the outermost [FORMULA] - [FORMULA] of the neutron star with density [FORMULA] and electron fraction [FORMULA] (see Fig. 2 in Paper I) is expanded and heated while the lepton fraction [FORMULA] stays roughly constant as it is the case for a fast change where neutrino losses are too slow to compete, the [FORMULA] -equilibrium is shifted into the region of negative [FORMULA] values. This usually implies that the electron degeneracy of the matter is drastically decreased, too, because the electron chemical potential [FORMULA] also drops when [FORMULA] attains negative values. For hot gas (i.e., gas at conditions lying above the nose-like feature of the curves in Figs. 11 and 12) the state with [FORMULA] corresponds to a higher value of [FORMULA] which is seen in Fig. 12 by moving along lines parallel to the ordinate. On the lepton-number loss time scale associated with neutrino emission, the gas will tend to evolve again towards the [FORMULA] -equilibrium with [FORMULA] by an enhanced production and emission of [FORMULA] relative to [FORMULA]. Notice that from Figs. 11 and 12 one infers that the same arguments are true for the case that neutron star matter with densities above [FORMULA] is strongly compressed. From the properties of the high-density equation of state we therefore deduce that during the coalescence of neutron star binaries the hot gas in the compact, compressed core region of the merger as well as the heated, decompressed disk matter will radiate [FORMULA] more copiously than [FORMULA].

This explains the relative sizes of electron neutrino and antineutrino luminosities as discussed in Sect.  3.1. Driven by this imbalance of the emission of [FORMULA] and [FORMULA], the initially very neutron-rich matter ([FORMULA] everywhere in the neutron star, see Fig. 2 in Paper I) gains electron lepton number and becomes more proton-rich again. Fig. 13 shows this evolution for model A128 from the start of the simulation until its end at 8.80 ms. The snapshots of the [FORMULA] distribution in the orbital plane visualize how, as a consequence of the rapid neutrino loss from the disk region, [FORMULA] in this region climbs from initial values of 0.02-0.06 to values of more than 0.18 in some parts. In the core region the neutrino emission proceeds much more slowly so that [FORMULA] changes only slightly during the simulated time. If we continued our computations for a long enough time to see the matter cooling again by neutrino losses (provided the configuration is stable for a sufficiently long period), this process of [FORMULA] increase would again be inverted and the gas would evolve towards the cold, deleptonized, very neutron-rich state again.

[FIGURE] Fig. 13a-f. Time evolution of the spatial distribution of the electron fraction [FORMULA] in the orbital plane of model A128. The times of the snapshots are given in the upper right corners of the panels. The contours are linearly spaced with intervalls of 0.02, bold lines are labeled with their respective values, and the grey shading emphasizes the contrasts, higher values of [FORMULA] being associated with darker grey

Fig. 14 displays the final situations ([FORMULA] -11 ms) in the models A64, B64, and C64 to be compared with panel f of Fig. 13. As in case of the neutrino emission, one notices very similar properties of all four models. In model A64 the peak [FORMULA] values in the disk region are as high as 0.22. Note that in all four models very neutron-rich matter is swept off the grid, a tiny fraction of which might potentially become unbound (see Sect.  2.3 and Paper I). It is also interesting to see the still very elongated and deformed neutron-rich inner region of models A64 and B64 which indicates ongoing strong dynamical and pulsational activity of the massive core. This is not so pronounced in model C64 where the anti-spin setup of the initial model has caused the dissipation of a large fraction of the rotational energy during the coalescence of the neutron stars. Model A128 has also a much more circular core region because, as described in Paper I, the better resolution allows for a much more fine-granular flow pattern which contains a large fraction of the initial vorticity and kinetic energy in small vortex structures.

[FIGURE] Fig. 14a-c. The spatial distributions of the electron fraction [FORMULA] in the orbital plane for models A64, B64, and C64 at the end of the calculations (times given in the top right corners). The contours are linearly spaced with intervalls of 0.02 and the bold contours are labeled with their respective values. The grey shading emphasizes the levels with darker shading indicating higher values of [FORMULA]. The plots have to be compared with panel f of Fig. 13 which shows the same information for model A128 at time [FORMULA]  ms

Fig. 15 presents a collection of plots of parameters that give information about the thermodynamical state in the orbital plane of model A128 and about the nuclear composition of the gas at the end of the simulation. Panels a and b show the electron degeneracy parameter [FORMULA] ([FORMULA] is the Boltzmann constant, [FORMULA] the electron chemical potential) and the electron neutrino degeneracy parameter [FORMULA], respectively. Concordant with the discussion above, the [FORMULA] -equilibrium conditions in the whole star are characterized by [FORMULA] (panel b). In the core values between -3 and -6 can be found, while in the disk moderately negative values are present (around -1) and in some regions the medium has evolved back to a state close to [FORMULA]. Comparison with panels a and b of Fig. 6 shows that in these regions the emission rates of [FORMULA] and [FORMULA] are already very similar again whereas the production of electron antineutrinos is clearly dominant in those parts of the disk with the most negative values of [FORMULA]. The electron degeneracy is moderate ([FORMULA] -3) in the disk but climbs to numbers around [FORMULA] near the center.

[FIGURE] Fig. 15a-e. Cuts in the orbital plane of model A128 showing different thermodynamical and composition parameters at the end of the simulated evolution (time [FORMULA]  ms). Panel a gives the electron degeneracy parameter [FORMULA] (contours linearly spaced with steps of one unit), panel b shows the electron neutrino degeneracy parameter [FORMULA] (contours linearly spaced with steps of 0.5 units), panel c displays the temperature distribution (contours linearly spaced with steps of 2 MeV), panel d is a plot of the entropy per nucleon (contours linearly spaced with steps of 1  [FORMULA] /nucleon), and panel e informs about the mass fraction of [FORMULA] particles in the medium (contours logarithmically spaced in steps of one dex)

The temperature [FORMULA] and entropy are displayed in panels c and d, respectively, of Fig. 15. Detailed information about the evolution of the temperature in all models was given in Paper I (Figs. 4-7, 14-17, and 20, 21). At the end of the simulation model A128 has the highest temperatures of [FORMULA]  MeV in hot spots located in a shell around the central high-density core where [FORMULA]  MeV. The disk has been heated up to [FORMULA]  MeV in regions of density [FORMULA] - [FORMULA], [FORMULA]  MeV where [FORMULA], and [FORMULA] -2 MeV for [FORMULA]. The corresponding entropies are less than 1  [FORMULA] /nucleon in the core region and between 3 and slightly more than 7  [FORMULA] /nucleon in the disk (panel d). In model C64 similar disk entropies are found while in model A64 specific entropies up to about 9  [FORMULA] /nucleon and in model B64 up to even 10  [FORMULA] /nucleon develop towards the end of the simulated evolution.

Such high entropies allow only minor contributions of nuclei to be present in the gas in nuclear statistical equilibrium at the densities found for the disk matter on our computational grid. Most of the nuclei are completely disintegrated into free nucleons (the mass fraction of heavy nuclei is below the lower limit of [FORMULA] returned from the equation of state of Lattimer & Swesty (1991)), and only small admixtures of [FORMULA] particles are possible. Panel e of Fig. 15 shows that the mass fraction [FORMULA] of [FORMULA] particles is typically less than about [FORMULA]. Only in the outermost parts of the disk [FORMULA] because there the temperatures are low enough, [FORMULA] -2 MeV, that some of the free nucleons can recombine.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998