After having discussed in Sect. 2 the numerical method (including the initial setup) and all possible sensitivities and uncertainties, we have performed in total 11 numerical simulations (run A to K), to test all of these aspects. In this way we intend to analyze the robustness of the results from the (corotating) standard case and to understand what variation of these results can be expected for different system parameters and physics input. The specifics which characterize each of these runs are given in Table 1. The following discussion of the results will refer to these table entries. In all of these runs we follow the evolution of matter for 12.9 ms.
Table 1. Summary of the different runs: 1: corotation; 2: no initial spins; 3: spin against the orbit; LS: Lattimer and Swesty (1991), P: polytropic EOS; MG: Monaghan and Gingold (1983); MM: Morris and Monaghan (1997)
Figs. 1 to 4 show the particle positions of runs A, C, I and J. The different colors in Fig. 1 label those particles that end up (last dump at ms) in the central object (black), the disk (green), the tails (blue) and those that are unbound at the end of the simulation (red). Run A stands representative for all the corotating runs using the Lattimer/Swesty EOS (the particle plots of runs A, B, D and E are practically indistinguishable; runs F, G and H differ only slightly). The plots only show the first half of the simulation. In the second half practically no additional angular motion is visible, matter just expands radially up to km for all runs apart from the one without initial stellar spins. In this case the outermost layers only extend out to km from the center of mass.
In all corotating runs those parts having the largest distance from the common center of mass are driven into thick spiral arms. Those get "wrapped up" around the central object to form a ring-like structure (see Figs. 1 and 2). In the further evolution this ring contracts to form a thick disk, while the matter in the spiral arms continues to wind up around the coalesced, massive object in the center of mass.
The corotating runs show, apart from the "main" spiral arms, also a very narrow "side" spiral structure (see, for example, the second panel in Fig. 2; in outlines, this structure is also visible in run I), that emerges directly after the stars have come into contact. The side spiral arm matter comes from the "contact zone" (see panel one in Fig. 1), is then strongly compressed and finally, in a kind of "tube-of-toothpaste"-effect ejected into two narrow jets in opposite directions. While matter in the main arms is smoothly decompressed due to gravitational torques, matter in the side spiral arms gets compressed and heated up. The pressure responsible for the formation of the side spiral arms is of thermal nature. This is suggested since no such spiral arms emerge in run G, where neutrino emission leads to a very efficient cooling. In an additional test run, where the viscous pressure contribution to the energy equation was disregarded, also no side spiral arms occurred.
In the end three well-separated structures are visible: an oblate, massive central object, a thick surrounding disk, and long spiral arms. Going to lower total angular momentum, i.e. to runs I and J, the differences between these structures get washed out.
It is interesting to note the dependence of the spiral arms on the equation of state: while they are thick and pumped up in all corotating runs using the LS-EOS, they remain narrow and well-defined in the run with the stiff polytropic EOS (, run C; somewhat thicker for , run K). This expansion is a result of an increase of the adiabatic index and the energy release due to the recombination of the nucleons into heavy nuclei. Fig. 5 shows the total amount of nuclear binding energy present according to the LS-EOS:
where the summation ranges over all SPH-particles, denotes the particle mass, and the abundances of heavy nuclei and alpha particles, and and the nuclear binding energies. Around erg of nuclear binding energy are deposited (i.e. present in the last dump) in the spiral arms containing only (see Table 2).
Table 2. Masses of the different morphological regions, is the minimum gravitational mass (see Eq. (22)), a is the relativistic stability parameter (see text).
The degree of conservation of energy and angular momentum can be inferred from Fig. 6. Both are conserved to approximately .
3.2. Mass distribution
Density contours (cuts through the y-z-plane at the last dump, t=12.9 ms) of run E (representative for the corotating runs), run G (to see the influence of the cooling by neutrino emission), run I and run J (to see the influence of the initial spins) are shown in Figs. 7 and 8. The distribution of mass with density and radius is shown in Figs. 9 and 10. The sharp bends in Fig. 9 indicate the separations of the different morphological regions. The amounts of mass in the central object, the disk and the tails (collectively used for the low-density material outside the disk) can be inferred from this plot, the corresponding masses are listed in Table 2.
The runs A, B, D and E practically coincide in Fig. 9, suggesting that the mass distribution is insensitive to our approximation of initially spherical stars, to the details of artificial viscosity, and also shows that the resolution in our standard run is sufficient to describe the mass distribution properly. The temporal evolution of the masses and densities in the three regions (run A) can be inferred from Fig. 11 (Fig. 9 just corresponds to the last temporal slice of Fig. 11). In the beginning practically all the mass () has a density above (the visible oscillations result from the initially spherical stars). Around t ms a strong expansion sets in (less mass has a density above, say, ). Then, around t ms, the three regions (central object, disk and tails) become visible. At around t ms, when the wrapped-up spiral arms are contracted by gravity, a recompactification corresponding, to the "bump" in Fig. 11 sets in (to localize it in time and density the contour lines of 2.8, 2.9, 3.0 and 3.1 are projected on the log()-t-plane). This recompactification is accompanied by a reheating (this is also reflected in the neutrino emission of the disk, where a strong increase is visible, compare panel 2 in Fig. 22) that finally reexpands the disk. When our simulation stops, the disk has a density range from (compare also to the contour plots 7 and 8), densities above this range are associated with the central object, densities below with the tails.
3.2.1. The central object
Figs. 14 and 15 show density contours of runs E, C, I and J. The central objects (especially for runs A to H) consist of a core with densities above and a diffuse edge ranging from to forming a kind of a "hot skin" around the core (see the smooth transition from the central object to the disk in Fig. 9). In the run including neutrino energy losses (run G) there is a very sharp transition corresponding to a huge density gradient from the central object to the disk, suggesting that the edge diffuseness of the central object in the other runs emerges from thermal pressure (also visible in the temperature, Fig. 19). Because of the absence of this pressure, matter of the order , otherwise located in the disk, has fallen onto the central object increasing its mass to . (This hot skin might be an artifact of the viscosity scheme, since it less pronounced in run D). The masses of the final central objects range from to (see Table 2) and are thus well above the maximum precisely known neutron star (gravitational) mass of for the pulsar of the binary system PSR 1913+16 (Taylor 1994; there are, however, objects, whose mass error bars reach up to 2.5 ; see Prakash (1997b) and references therein). Theoretical calculations for maximally rotating neutron stars, using a large set of 17 (11 relativistic nuclear field theoretical and 6 non-relativistic potential models for the nucleon-nucleon-force) nuclear equations of state (Weber & Glendenning 1992) find maximum values for the gravitational masses of . Since these values refer to gravitational rather than to baryonic masses (which we are referring to due to the Newtonian character of our calculations) we have to estimate the gravitational masses of our central objects. The relation obtained by Lattimer and Yahil (1989) for reasonable uncertainties in the equation of state reads
This gives the minimal gravitational mass of the baryonic mass as
which is typically 0.4-0.5 smaller than the baryonic mass. The corresponding values for our central objects are shown as entry 3 in Table 2). These values are still very high, but there are equations of state for which fast rotating neutron stars in this mass range are stable. Thermal pressure, which is disregarded in the work of Weber and Glendenning since they assume old neutron stars, could further stabilize the central object on a cooling time scale.
The above quoted masses refer to neutron stars rotating with the maximum possible velocity. To estimate the influence of rotation for our calculations we plot in Fig. 17 the tangential velocities in the central objects of run E, I and J and compare them to the Kepler velocity , i.e. the velocity where the centrifugal forces balance the gravitational forces ( is the mass enclosed in the cylindrical radius r). In all cases the velocities are clearly below the Kepler velocity and we thus do not suppose rotation to play an important role for stabilization (in this Newtonian consideration). In their general relativistic analysis Weber and Glendenning find the maximum possible rotation frequency for the equations of state referred to above corresponding to a maximum stable mass of (Weber & Weigel 1989; Weber et al. 1991). This is larger than our rotation frequencies by a factor of about two.
In the general relativistic case the stability support from rotation is determined by the dimensionless parameter (Stark & Piran 1985). We give a for our central objects in column six in Table 2. We find values of in agreement with previous simulations (Shibata et al. 1992, Rasio & Shapiro 1992, Ruffert et al. 1996). This is well below the critical value , meaning that the central object cannot be stabilized against collapse by rotation.
Prakash et al. (1995) studied the influence of the composition on the maximum neutron star mass. Their general result is that trapped neutrinos together with nonleptonic negative charges (such as hyperons, or d and s quarks) lead to an increase of the maximum possible mass (in contrast to nucleons-only matter). Thus, the collapse of a star with almost the maximum mass could be delayed on a neutrino diffusion time scale. As an estimate of this time scale for our central objects, we use R km (center to pole distance, see Fig. 7) and a typical neutrino energy of MeV, where T MeV and has been used. This gives a mean free path of
where the baryon number density n corresponding to and a neutrino nucleon scattering cross section with cm2 (see Tubbs & Schramm 1975) has been used. An estimate for the delay time scale is then given by
Thus if there were nonleptonic negative charges present in the central object, the collapse could be delayed for a few seconds.
To summarize the stability question of the central object: if all the stabilizing effects mentioned above (EOS, rotation, thermal pressure and exotic matter) should conspire, the central object could (at least in some cases) be stabilized against gravitational collapse. However, we regard this possibility to be fairly unlikely.
If the central object collapses to a black hole, the question arises of how much mass has enough angular momentum to avoid being swallowed. For a simple estimate we assume that the specific angular momentum of a particle must be larger than the one of a test particle with Kepler velocity at the marginally stable circular direct orbit of a Kerr black hole (see Bardeen et al. 1972). Thus, , where
denotes the specific angular momentum of the central object. With this estimate we find that the central black hole will be surrounded by a disk of for initial corotation ( 0.1 and 0.06 for the cases of no initial stellar spins and spins against the orbit).
3.2.2. The disk
In all runs the central object is finally surrounded by a thick disk (see Figs. 7 and 8). In the case of initial corotation the disks contain around ( in run F; 0.25 in run G, where have fallen onto the central object; in run H, neglecting the backreaction force leads to an impact with a higher angular momentum and thus more mass is expelled into the tails; the disk, however, also contains ). In all cases, apart from run J, empty funnels form above the poles of the central object. These are strongly enlarged in run G, where the efficient cooling by neutrinos leads to strongly reduced thermal pressure contributions, i.e. to a softening of the EOS that makes matter more prone to the centrifugal forces thereby leading to a flattening of the central object as well as of the disk. These funnels have the appealing feature that here large gradients of radiation pressure can be built up which can lead to two well-collimated jets in opposite directions, pointing away from the poles. These funnels would be an ideal site for a fireball scenario (Goodman 1986; Shemi & Piran 1990; Paczyski 1990; Piran & Shemi 1993) since they are practically free of baryons. A baryonic load as small as could prevent the formation of a GRB. Our present resolution (the lightest SPH-particle has a mass of a few times ) is presently still too low to draw conclusions on this point.
3.2.3. The tails
Despite considerable morphological differences - exploding tails with the Lattimer/Swesty EOS, well-defined, narrow tails for the polytrope - the amount of mass in the tails is rather insensitive to the EOS (see Table 2). It is mainly dependent on the total amount of angular momentum during the impact, thus leading to the most massive tails in run H (no angular momentum lost in gravitational waves) with a decreasing tendency going from the standard run to runs I and J.
3.3. Temperatures and vortex structures
Fig. 18 shows the maximum temperatures during the different runs. The curves in the upper panel refer to the maximum temperature of a single particle, the ones below to the maximum smoothed temperature ,
where are the temperatures, the masses, the densities, W the spherical spline kernel (see e.g. Benz 1990) and the arithmetic mean of the smoothing lengths of particles i and j.
The smoothed (the particle) temperatures reach peak values of about 50 (80) MeV for run J where the most violent shear motion is present, around 45 (70) MeV in run I and about 30 (50) MeV in the corotating runs. The resolution and viscosity seem to be of minor importance for the temperature calculation. Starting with spherical stars leads to an increased temperature since oscillation energy is transformed into heat (see also Table 3). In the corotating runs the hot band that forms at the contact surface dissolves into two hot spots (see Fig. 19). This structure develops further to finally form a hot, s-shaped band through the central object. For an understanding of these structures we plotted in the right columns of Figs. 19 to 21 the projections of those particles that are contained within a thin slice ( km). The projected particle positions of star one and two are marked with different symbols. One sees the formation of two macroscopic vortices that can be identified with the hot spots. The panels in the second line of Fig. 19 show patterns that are typical for Kelvin-Helmholtz instabilities (see e.g. Drazin and Reid 1981). The basic properties of this process, the formation of a hot band in the contact region, separation into two (hot) vortices and the final s-shaped hot band, are unaffected by resolution and the change from the standard SPH-viscosity to the new scheme. The finger like structures that extend into the matter of each star (last panel Fig. 19) are just broader with lower resolution (run A). With the new viscosity scheme substructures form along the "fingers" leading to a more fractal appearance of the line separating the material of both stars. This is a result of a lower viscosity which leads to a dissipation of the energy of turbulent eddies on smaller scales (see e.g. Padmanabhan (1996)), where L is is a macroscopic scale and R the Reynolds number.
Table 3. Kinetic and thermal energies (in erg) in the different morphological regions.
In run I three macroscopic vortices, a large central one and two smaller ones to the left and right, form along the contact surface. The two smaller vortices get attracted by the central one and fuse on a time scale of approximately one millisecond. Approximately eight milliseconds after impact the material of both stars is well mixed and wrapped up around the central vortex.
In the case with maximum shear motion, run J, we find one large vortex in the centre and several smaller ones along the contact surface. The smaller ones on each side of the center merge so that there are in total three such macroscopic vortices. The outer vortices move towards the origin and finally merge with the central one. The material of both stars gets mixed turbulently on a time scale of a few milliseconds.
Ruffert et al. (1996) also find two macroscopic vortices for the corotating case. However, they find in their simulation that the vortices dissolve practically independently of the initial spin state into a ringlike structure, while our hot spots develop into an s-like shaped hot band along the line separating the matter of the different stars. Since Ruffert et al. do not find a particular influence of the shear motion on the growth time scale and since their results for different initial spins look very similar, they question on the Kelvin-Helmholtz picture and propose an alternative, macroscopic explanation for the flow pattern. A quantitative analysis of the incompressible, inviscid case in terms of linear normal mode analysis yields a growth rate for an unstable mode of wavelength (for the case ; see e.g. Padmanabhan (1996))
This implies that the shortest wavelengths will grow fastest and that the perturbation should grow faster with larger shear velocity.
The shortest wavelengths to grow are determined by our numerical resolution. Let us assume that this length scale corresponds to the typical distance over which neighbours can interact, i.e. , where the smoothing length is to be taken in the shear region. Then we find km. We then look at the shear velocities v. Here we find by looking at relative velocity projections along the shear interface values of for the corotation, for run I and for run J. Thus typical growth time scales in the different runs should be s, s and s, where the subscripts label the runs (A is representative for corotation). This means that the perturbations have enough time to grow into the macroscopic regime on a dynamical time scale of the system (milliseconds). The growth times scale approximately like 1: 2: 5 and this is approximately what is seen in our simulations. Since in our calculations a strong dependence of the growth time scale on the shear motion, consistent with the Kelvin-Helmholtz time scales, is encountered and the vortex structures for different initial spins are clearly different, we do not see the necessity of an alternative explanation.
3.4. Neutrino emission
The main reason for an additional run with a very simple inclusion of neutrinos is to investigate whether neutrino emission has a noticeable effect on the amount of ejected mass. To estimate the appropriateness of our neutrino treatment for the different regions we estimate typical neutrino diffusion time scales. A typical neutrino diffusion time scale for the central object is of the order of a few seconds (see discussion above). Thus our neutrino treatment is definitely inappropriate there (typical time scales are milliseconds). Applying the same formula (24) for the disk, using 10 MeV, and 60 km, we find s, which is comparable to the dynamical time scales. Thus at least in the outer regions of the disk, the free streaming approximation might be justified. In the tails free streaming neutrinos are clearly a good approximation (apart from, perhaps, the very first moments after impact). In Fig. 22 we compare the amount of nuclear binding energy present (see Eq. (20)) to the amount of energy radiated in neutrinos
where denotes the neutrino luminosity of process . The are given by
where is the energy emission rate of process and particle i (in erg s-1cm-3). Until the end of the simulation the energy lost in neutrinos exceeds that gained from nuclear processes by two orders of magnitude. This is mainly due to the central object, where unphysical amounts of neutrinos are emitted. In reality we expect the disk to be the dominant source of neutrino emission. When the disk reaches the above mentioned reheating phase the neutrino emission exceeds the nuclear energy by about one order of magnitude (see panel two in Fig. 22.)
The most important result is that in the spiral arms more energy is gained by nuclear processes than is lost by neutrino emission (in spite of our overestimate). Such conclusions were already reached in Davies et al. (1994) for low density regions. This is a crucial point since it shows that our results concerning mass ejection are independent of the neutrino treatment.
3.5. Ejected mass and nucleosynthesis
We regard a particle to be unbound if the sum of its energies - macroscopic as well as microscopic - is positive, i.e. if
For we count all kinds of internal energies apart from the (negative) nuclear binding energies, i.e.
where the indices and N denote photons, electrons and nucleons. All these terms are taken from the Lattimer/Swesty EOS (for the polytropic case we just use ). We do not consider nuclear binding energies, since we regard an isolated nucleus at (the only internal energy comes from nuclear binding) with as unbound. However, near the end of the evolution is negligible and does practically not influence the total amount of ejected material.
This criterion can be cross-checked by adopting a simple model, where we regard the particles as free point masses, i.e. we neglect hydrodynamic forces resulting from pressure gradients and disregard internal degrees of freedom (internal energies). The particles are supposed to move on Kepler orbits around a point mass in the origin with ( is the mass of the central object, see Table 2), which is large compared to the particle masses , . Under these assumptions the numerical eccentricities of the orbits are given by
where is the sum of the particle's kinetic and potential energy and its angular momentum. We generally find a good agreement of both criteria, with deviations lying in the range of .
All corotating runs using the LS-EOS eject around , about twice as much as the run using the stiff polytrope. This is caused by a variation of the adiabatic exponent and the formation of nuclei when matter is decompressed and thereby releases the gained nuclear binding energy. For reasons of illustration we plot in Fig. 23 the amount of unbound mass versus the time and compare this with the amount of nuclear binding energy present in the mass that ultimately escapes (see Fig. 24). The flatness of the curves in Fig. 23 indicates that not much more mass will be ejected during the further evolution. The rise in the adiabatic exponent and the deposition of a few times erg in (see Table 4) leads to an explosive expansion of the spiral arm tips, thereby supporting the ejection of mass. Tidal deformation (run B) leads to an increase of the ejected mass since the system contains more angular momentum due to its elongated shape. As expected, the 1.4 run ejects more mass since the gravitational potential to be overcome is shallower than in the 1.6 case. We suspect this to be true also in the general relativistic case where the less massive stars have larger radii and their outer parts therefore contain more angular momentum. The realistic reduction of viscosity also tends to increase the amount of ejecta. The inclusion of neutrinos does not alter the results concerning unbound mass (the emitted energy in neutrinos is approximately one order of magnitude lower than the released nuclear binding energy for the ejected matter, see panel three in Fig. 22). The basic intention of run H was to test the sensitivity of the amount of ejected mass on the details of the treatment of the gravitational radiation backreaction force. Here almost twice as much matter is ejected (since no angular momentum is lost in gravitational waves), indicating that our results could be influenced by our simplified treatment of this force (see Eqs. (3) and (4)). In the runs with lower initial angular momentum (run I and J) the amount of ejecta is substantially lower, indicating a very strong dependence on the initial spins. The sensitivity to the EOS is underlined by the fact that in the run with the soft polytrope () no resolvable amount of mass is ejected, in agreement with the result of Rasio and Shapiro (1992).
Table 4. Amount of mass that is unbound at the end of the simulation
The amount of r-process material that could be formed in this merging scenario is basically determined by and the entropies (and expansion time scales). In Fig. 25 we plot entropies (in per nucleon) and densities at the time of ejection for the three different neutron star spins (run E, I and J). It is very interesting to note that the densities at the moment of ejection are very high, the bulk of matter becomes unbound at densities from to g cm-3. This is well above the neutron drip ( g cm-3), where in spite of the high temperatures very large ( according to the LS-EOS) and very neutron rich () nuclei are present in appreciable amounts (). These nuclei are far from being experimentally well-known. Hence, to start r-process calculations from these initial conditions, very exotic nuclei (not in vacuo, but immersed in a dense neutron gas) have to be implemented in the corresponding reaction networks. In addition, the effects of the high Fermi-energies on the reaction rates including beta decays (Pauli-blocking) have to be accounted for.
The temperatures are strongly dependent on the ejection mechanism which is closely related to the stellar spins. In the corotating runs the ejecta are initially located on the front side of each star (with respect to the orbital velocity). Due to gravitational torques this matter gets smoothly stretched and thereby decompressed, no spikes in pressure, temperature and the time dependent viscosity parameter (see Eq. 12) are visible. In this case the temperatures are only slightly above the initial temperature ( K; see Appendix A). We suspect this temperature increase to be mainly due to (artificially high) viscosity. The material is ejected in a different way for the other spin configurations. In the case without stellar spins the ejecta can be separated into two groups according to their ejection mechanism. The material of the first group is found in the spiral arm structure, ejected in a way similar to corotation and is thus essentially cold. The second group comes from a region that gets strongly compressed and thereby heated up to temperatures around 6 MeV. In the following expansion, however, this material cools down quickly. In the case where the stars spin against their orbital motion all the material is ejected by the second mechanism thereby reaching even higher temperatures in the compression phase (around 9 MeV) for a short time.
Since the material that gets unbound in the coalescence of initially corotating systems stays essentially cold ( K, see Lai 1994), we expect the of this matter to be close to the initial values of the cold neutron stars, i.e. 0.01 0.05 with small contributions from the stellar crust. The cases with different stellar spins eject material that gets heated appreciably before being cooled by the expansion. In these cases might be different from the initial values, since temperatures are high enough for the charged current reactions (capture) to set in at non-negligible rates.
Owing to the problems in explaining the observed r-process abundances entirely by type II supernovae, there seems to be a need for at least one further astrophysical scenario that is able to produce r-process nuclei in appreciable amounts. Neutron star mergers are attractive candidates since they would in a natural way provide large neutron fluxes, low s and moderate entropies (which provides r-process matter more easily than high and entropy conditions). An r-process under such conditions should be very efficient and produce mostly elements in the high mass region. Thus, perhaps all of the r-process matter with , that can only be produced in the right amounts in supernova calculations if artificially high entropies are applied (see Freiburghaus et al. 1997; Takahashi et al. 1994), perhaps all of this matter could be synthesized in neutron star binary (or BH-NS) mergers.
Assuming a core collapse supernova rate of (year galaxy)-1 (Ratnatunga 1989), one needs to of ejected r-process material per supernova event to explain the observations if type II supernovae are assumed to be the only source. The rate of neutron star mergers, which is by far more uncertain, has recently been estimated to be (year galaxy)-1 (see van den Heuvel & Lorimer 1996). Taking these numbers, one would hence need to per event for an explanation of the observed r-process material exclusively by neutron star mergers. Thus our results for the ejected mass from to look promising (see Fig. 26).
Meyer (1989) calculated the decompression of initially cold neutron star material ( K) assuming the expansion to be given by multiples of the free-fall time scale. He found that the decompressed neutron star material gives always, i.e. regardless of the expansion rate, rise to r-process conditions. Thus even the initially cold ejecta from corotating configurations should heat up during the expansion and form r-process nuclei. If, as suggested by Meyer (1989), large parts of the ejected material should consist of r-process nuclei, neutron star mergers could account for the whole observed r-process material in the galaxy. However, whether the observed abundance patterns can be explained with this scenario remains an open question and is left to further investigations.
© European Southern Observatory (ESO) 1999
Online publication: December 4, 1998