3.3 Merger process

3.3.1 Zero BH spin

As mentioned in Section 1, broadly speaking, the final fates of BH-NS binaries are divided into two classes. One in which the NS is tidally disrupted before it is swallowed by the companion BH and the other is that the NS is simply swallowed by the BH. Figures 11View Image and 12View Image display snapshots of the rest-mass density profiles and the location of the apparent horizon on the equatorial plane at selected time slices for two typical cases [107Jump To The Next Citation Point]. For these results, the NS are modeled by the piecewise polytropic EOS described in Table 4. Figure 11View Image illustrates the process in which the NS is tidally disrupted before the binary reaches the ISCO and then a disk is formed surrounding the companion BH. For this model, MBH = 2.7M ⊙, a = 0, MNS = 1.35M ⊙, and RNS = 15.2 km (EOS 2H); the mass ratio (Q = 2) is small and the NS radius is large. For this setting, the NS is significantly tidally deformed in close orbits, and eventually, mass shedding from an inner cusp of the NS sets in far outside the ISCO. After a substantial fluid element is removed from the inner cusp, the NS is tidally disrupted outside the ISCO. It should be emphasized that tidal disruption does not occur immediately after the onset of mass shedding in this case. Tidal disruption occurs for an orbital separation smaller than that for the onset of mass shedding.

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Figure 11: Evolution of the rest-mass density profile in units of 3 g∕cm and the location of the apparent horizon on the equatorial plane for a model with MBH = 2.7M ⊙, a = 0, MNS = 1.35M ⊙, and RNS = 15.2 km (EOS 2H). This simulation was performed by the KT group. The filled circle denotes the region inside the apparent horizon of the BH. The colored panel on the right-hand side of each figure shows log (ρ ) 10. This figure is taken from [107Jump To The Next Citation Point].
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Figure 12: The same as Figure 11View Image but for a model with M = 4.05M BH ⊙, a = 0, MNS = 1.35M ⊙, and RNS = 11.0 km (EOS B).

After tidal disruption occurs, the material of the NS forms a one-armed spiral. As a result of angular momentum transport in the arm, a large amount of material spreads outward, and after the spiral arm is wound from the differential rotation, a disk of approximately axisymmetric configuration is formed around the BH located approximately at the center. However, because of the presence of a non-axisymmetric structure at its formation, however, the disk does not completely relax to an axisymmetric state in the rotational period ∼ 10 ms. Rather, a one-armed spiral of small amplitude is present for a long time, and helps gradually transporting angular momentum outward, resulting in a gradual mass infall into the BH. However, the mass accretion time scale is much longer than the rotational period, and hence, the disk remains quasi-stationary for ≫ 10 ms. This evolution process agrees qualitatively with that found in the study of the longterm evolution of BH-disk systems [90, 103].

The tidal disruption process as illustrated in Figure 11View Image is qualitatively common for the model with a low-mass BH or a large-radius NS or a high-spin BH. However, quantitative details depend on the parameters of the binary. For a small mass ratio (Q ∼ 2) with a = 0, the typical size of the disk (the region with ρ > 1010 g∕cm3) is 50 – 100 km with maximum density ≳ 1012 g∕cm3 for a disk of mass ∼ 0.1M ⊙ as shown in Figure 11View Image. Thus, the disk is rather compact. The disk relaxes to a nearly axisymmetric configuration in a short time duration, approximately equal to the rotational period around the BH. We note that these properties depend on the parameter of the binary. For example, for a large mass ratio with a high BH spin (e.g., Q = 5 and a = 0.75), the typical size is also ∼ 100 km, but the maximum density is smaller than those for the smaller value of Q; the time scale until the disk relaxes to an axisymmetric configuration is relatively long. A remarkable point is that the tidal debris of relatively low density ∼ 1010 g/cm3 could be ejected to a distance of ≫ 100 km [63Jump To The Next Citation Point, 57Jump To The Next Citation Point, 41Jump To The Next Citation Point, 109Jump To The Next Citation Point], i.e., a wider but less dense disk is formed (see also Section 3.3.3).

Figure 12View Image illustrates the case in which the NS is not tidally disrupted before it is swallowed by the BH. For this model, MBH = 4.05M ⊙, a = 0, MNS = 1.35M ⊙, and RNS = 11.0 km (EOS B in Table 4). In this case, the NS is tidally deformed only in a close orbit. Then, mass shedding sets in for a too close orbit to induce subsequent tidal disruption outside the ISCO. As a result, most of the NS material falls into the BH approximately simultaneously. Also, the infall occurs from a narrow region of the BH horizon. These processes help exciting a non-axisymmetric fundamental quasi-normal mode (QNM) of the remnant BH. The mass of the disk formed after the onset of the merger is negligible (much smaller than 0.01M ⊙), because the BH simply engulfs the NS.

Generally speaking, the final fate depends on the location where mass shedding of the NS sets in. If the location is in the vicinity of or inside the ISCO, most of the NS material falls into the companion BH, and a BH with negligible surrounding material is the outcome. With the increase of the orbital separation at the onset of mass shedding, the mass of the material surrounding the BH increases. Note again that the mass shedding has to set in sufficiently outside the ISCO to induce tidal disruption, because tidal disruption occurs only after substantial mass is removed from the NS.

3.3.2 Nonzero BH spin

The effect of the BH spin significantly modifies the orbital evolution process in the late inspiral phase and merger dynamics, as first demonstrated by the UIUC group [63Jump To The Next Citation Point]. Figure 13View Image shows the trajectories of the BH and NS for models with Q = 3, 𝒞 = 0.145, and a = 0 (left) and a = 0.75 (right) [63Jump To The Next Citation Point]. The NS is modeled by the Γ-law EOS with Γ = 2. The initial orbital angular velocity is the same for two models. For the binary composed of a non-spinning (a = 0) BH and NS, the merger occurs after about 4 orbits, whereas for the case with a spinning BH (a = 0.75), it occurs after about 6 orbits. For the case with a spinning BH, the decreased rate of the orbital separation appears to be small. Qualitatively, these differences may be explained primarily by the presence of a spin-orbit coupling effect, which is accompanied by an additional repulsive force for a > 0 (and attractive force for a < 0), and thus, reduces the attractive force between two objects (see the equations of motion for two-body systems in the context of the PN approximation [102, 226, 100]). In the presence of this additional repulsive force, centrifugal forces should be reduced for a given orbital separation. This slows down the orbital velocity, and therefore, the luminosity of gravitational waves is decreased and orbital evolution due to gravitational radiation reaction is delayed (the lifetime of the binary becomes longer). In addition, the orbital radius at the ISCO around the BH is decreased (and the absolute value of the binding energy at the ISCO around the BH is increased) due to the spin-orbit coupling effect (e.g., [16]). This further helps to increase the lifetime of the binary because it evolves as a result of gravitational radiation reaction and hence has to emit more gravitational waves to reach the ISCO.

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Figure 13: Trajectories of the BH and NS coordinate centroids for models with Q = 3, 𝒞 = 0.145, and a = 0 (left) and a = 0.75 (right). This simulation was performed by the UIUC group. The NS is modeled by the Γ-law EOS with Γ = 2. The BH coordinate centroid corresponds to the centroid of the BH, and the NS coordinate centroid denotes a mass center. This figure is taken from [63Jump To The Next Citation Point].
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Figure 14: A remnant BH-disk system for models shown in Figure 13View Image. The simulation was performed by the UIUC group. The contour curves, velocity fields (arrows), and BH (solid circles) are plotted. This figure is taken from [63Jump To The Next Citation Point].

This longer lifetime for a binary with a spinning BH enhances the possibility of tidal disruption, and the final outcome is modified. Figure 14View Image displays the contour curves and the location of the remnant BH for the same models as in Figure 13View Image. For both models, the NS is tidally disrupted outside the ISCO and a disk is formed. For the spinning BH case (a = 0.75), a more extended, more massive, and denser disk is the outcome. For the non-spinning case (a = 0), the disk mass is only ≈ 4% of the total rest mass whereas for a = 0.75, it is ≈ 13% (see Section 3.4 for details of the remnant disk). This is probably due to the effect that the physical radius of the ISCO (or specific angular momentum of a particle orbiting the ISCO) around the spinning BH is smaller than that for the non-spinning BH and also that the radial approaching velocity at tidal disruption is smaller for a spinning BH because of the repulsive nature of the spin-orbit coupling effect.

The CCCW group subsequently studied the effects of BH spin on the final remnant with several EOS [57Jump To The Next Citation Point]. They reached a similar conclusion about the orbital evolution and final outcome to that of the UIUC group even for the case in which BH spin is slightly smaller, a = 0.5. This conclusion was reconfirmed also by the KT group [109Jump To The Next Citation Point] for a wide variety of piecewise polytropic EOS and mass ratios. Therefore, a moderately large BH spin, a = 0.5, is substantial enough for modifying the merger process and enhancing the disk formation. The CCCW group also performed a simulation with a = 0.9 and 𝒞 = 0.145 [74Jump To The Next Citation Point] and found that tidal disruption occurs far outside the ISCO and the resulting disk mass is very high, ∼ 36% of the total rest mass (see Section 3.4).

The KT group also found that [109Jump To The Next Citation Point] for binaries composed of a high-spin BH (a = 0.75) and NS, tidal disruption may occur for a large value of mass ratio, Q ∼ 5, for a wide variety of NS EOS as far as it gives 𝒞 ≲ 0.18. This implies that tidal disruption of a NS may be possible for a large BH mass over a wide area. In such case, the material of the tidally-elongated NS is swallowed from a relatively narrow region of the BH surface. As will be discussed in Section 3.6, this helps excite a non-axisymmetric fundamental QNM of the remnant BH. On the other hand, for the non-spinning BH case for which tidal disruption occurs only for a small BH, the material of a tidally-elongated NS is always swallowed by a wide region of the BH surface. This suppresses the excitation of non-axisymmetric QNM. This difference is reflected in gravitational waveforms and spectra, as predicted in [177, 178] (in which a BH perturbation study was performed).

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Figure 15: The same as Figure 11View Image but for a model with MBH = 4.05M ⊙, a = 0.75, MNS = 1.35M ⊙, and RNS = 11.6 km (EOS HB). The simulation was performed by the KT group. The figure is taken from [109Jump To The Next Citation Point].
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Figure 16: The same as Figure 15View Image but for a model with a = 0.5. This simulation was performed by the KT group. This figure is taken from [109Jump To The Next Citation Point].
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Figure 17: The same as Figure 16View Image but for a model with a = − 0.5. This simulation was performed by the KT group. This figure is taken from [109Jump To The Next Citation Point].

To clarify the fact described above, Kyutoku et al. [109Jump To The Next Citation Point] generated Figures 15View Image17View Image, which show the evolution of the rest-mass density profile for Q = 3, MNS = 1.35M ⊙, and EOS HB (cf. Table 4) with a = 0.75,0.5, and − 0.5, respectively. The evolution processes shown in Figures 15View Image and 17View Image are similar to those in Figures 11View Image and 12View Image, respectively. Figure 15View Image shows the case in which mass shedding of the NS occurs at an orbit sufficiently far from the BH. Subsequently, the NS is extensively elongated, a one-armed spiral is formed, and then the spiral arm composed of dense material is wound around the BH. The material located in the outer region of the spiral arm forms a disk, while that in the inner region falls into the BH. The infall of the dense material proceeds from a wide region of the BH surface as seen in the fourth panel of Figure 11View Image. By contrast, for a = − 0.5 (see Figure 17View Image), tidal disruption does not occur and more than 99.99% of the NS matter falls into the BH from a narrow region of the BH horizon and in a short time scale.

The type of merger process for a = 0.5 shown in Figure 16View Image is qualitatively new. Tidal disruption occurs in a relatively close orbit (in the vicinity of the BH ISCO). The subsequent evolution process is similar to that for a = 0.75, but the infall of dense NS material to the BH occurs from a relatively narrow region (see the second to fourth panels of Figure 16View Image). Eventually, the infall proceeds from a wide region of the BH surface, but the density of the infalling material seems to be too low to enhance a QNM oscillation of the BH significantly (see the fifth panel of Figure 16View Image). This feature is often found for a binary of high-mass ratio and high BH spin.

To date, three types of merger process have been found. Type-I: the BH mass is low or the BH spin is high, and the NS is tidally disrupted for an orbit far from the BH ISCO; Type-II: the BH mass is not low, the BH spin is small (or a < 0), and the NS is not tidally disrupted; Type-III: the BH mass is not low, the BH spin (a > 0) is high, and the NS is tidally disrupted for an orbit close to the BH ISCO. Their merger processes are schematically described in Figure 18View Image. These differences in the infall process are well reflected in gravitational waveforms and spectra.

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Figure 18: Schematic pictures for the three types of merger process that have been found to date. Left (type-I): the NS is tidally-disrupted and the extent of the tidally disrupted material is as large as or larger than the BH surface area. Middle (type-II): the NS is not tidally disrupted and simply swallowed by the BH. Right (type-III): the NS is tidally disrupted and the extent of the tidally disrupted material in the vicinity of the BH horizon is smaller than the BH surface area. The solid black sphere is the BH, the blue distorted ellipsoid is the NS, the solid red circle is the location of the BH ISCO, and the dashed circle is the location of the tidal-disruption limit. This figure is taken from [109Jump To The Next Citation Point].

In the latest work of the CCCW group [74Jump To The Next Citation Point], the effect of spin orientation, which is misaligned with that of the orbit rotation axis, was studied for the first time. They performed simulations for a = 0,0.5, and 0.9, and Q = 3 with Γ-law EOS (Γ = 2), and found that the remnant disk mass decreases sensitively with increase of the inclination (misalignment) angle for given values of a and Q (see Figure 20View Image). This is quite natural because the spin-orbit coupling force is proportional to S ⋅ L, where S and L denote BH spin and orbital angular momentum vectors, respectively, and the radius of the ISCO around the BH approaches that for a = 0 with increasing of the inclination angle. Thus, the BH spin effect becomes less important with the increase of the inclination angle. They also found that the inclination angle is significantly decreased after a substantial mass of the NS falls into the companion BH, implying that the angular momentum vector of the remnant disk misaligns only modestly with the BH spin vector (by ≲ 20∘). This is also quite natural because the orbital angular momentum is as large as or larger than the spin angular momentum of the BH for small mass ratios like Q = 3. However, for a high value of Q for which the fraction of the BH spin angular momentum in the total angular momentum is large, this conclusion will be modified. The initial inclination angle will not be significantly modified and an inclined disk, which subsequently precesses around the spinning BH, may be the outcome.

3.3.3 Extent of remnant disk

A tail of a one-armed spiral formed at tidal disruption often extends far away from the BH, in particular for the case in which the BH spin is high. The CCCW group reported that for Q = 3, a = 0.5, and 𝒞 = 0.144 – 0.173 (i.e., for realistic values of the compactness), a tidal tail of mass 0.01 – 0.1M ⊙ goes to a distance l = 200 – 2000 km away from the BH before falling back to the central region [57Jump To The Next Citation Point]. They also reported that the fall-back time scale was ∼ 200 ms for l = 2000 km assuming that the material obeys geodesic motion. Here, 200 ms agrees roughly with the dynamic infall time scale ∘ -------- l3∕Gm0. This indicates that the time scale of mass accretion from the disk onto the BH is much longer than the rotational period of the disk in the vicinity of the BH ∼ 10 ms. The typical duration of SGRB is 0.1 – 1 s [142]. Such a time scale may be explained by the time scale of the fall-back motion.

The LBPLI group also estimated the fall-back time for their simulation with Q = 5, a = 0.5, and 𝒞 = 0.1 [41]. In their simulation, the compactness of the NS was assumed to be smaller than that for a realistic NS, and thus, the formation of the tidal tail can be significantly enhanced. They reported that for a large fraction of the material ∼ 0.05M ⊙, the fall-back time may be longer than 1 s. However, they followed the motion of the tidal tail only for a short time duration (16.3 ms; i.e., they did not show that the material really went far away, l ≳ 104 km), and in addition, did not describe the detail of the method for estimating the fall-back time. (Note that a fall-back time longer than 1 s is realized only for an element, which reaches a distance from the BH of 4 l ≳ 10 km.) Hence, their conclusion should be confirmed by a longer-term simulation in the future. Their finding in the framework of numerical relativity that the disk can extend to a large distance ≫ 10Gm ∕c2 0 for a high BH spin was qualitatively new.

The KT group performed simulations for MNS = 1.35M ⊙, Q = 2– 5, a = 0.75 with several piecewise polytropic EOS [109Jump To The Next Citation Point]. They found that even for a binary with a NS of realistic compactness 𝒞 = 0.16 –0.18, the disk can extend to ≳ 500 km (which is approximately the location of the outer boundary of the computational domain in their simulations). This conclusion agrees qualitatively with the previous results by CCCW and LBPLI. Thus, for the merger of a rapidly-spinning BH and a NS, it may be concluded that a widely-spread disk is formed, and the lifetime of the accretion disk will be fairly long ≳ 0.1 s. The KT groups also found that the density of the disk decreases with increase of Q (or the BH mass); for a high-mass BH, a widely spread but less dense disk is formed.

3.3.4 Effects of EOS

The dependence of the merger process on the NS EOS comes primarily from the fact that the NS radius depends sensitively on the EOS. For a NS with a large radius, tidal disruption (and subsequent disk formation) is more likely. This fact was clearly shown in the works by the KT group [107Jump To The Next Citation Point, 109Jump To The Next Citation Point], performed employing a variety of the piecewise polytropic EOS.

The EOS also determines the density profile of a NS. Even if the radius is the same for a given mass, the density profiles for two NS may be different if the hypothetical EOS is different. The KT group showed that a NS with a small adiabatic index for the core region, with which the density is concentrated in the central region, is less subject to tidal disruption than that with a larger adiabatic index (with relatively uniform density profile), even if the radius and mass are identical; e.g., for the piecewise polytropic EOS listed in Table 4, a NS with a smaller value of Γ 2 is less subject to tidal disruption. The reason for this is that the star with a high degree of central density concentration is less subject to tidal deformation, as reviewed in Section 1.2.

The CCCW group performed a simulation incorporating a tabulated finite-temperature EOS for the first time [57Jump To The Next Citation Point]. The advantage of this approach is that one can determine the temperature and composition, such as electron fraction in the disk formed after tidal disruption occurs. This will be useful for discussing the possibility that the remnant BH-disk system could be a central engine of an SGRB. To avoid taking into account the effects of neutrino emission, they assumed that the system is in β-equilibrium or that the electron fraction is unchanged in the fluid-moving frame. In the former and latter, they assumed that the system is in either of the following two limiting cases; the weak interaction time scale is either much shorter or much longer than the dynamic time scale, respectively. They performed simulations focusing on the case a = 0.5 and Q = 3. Irrespective of the EOS, they found that the remnant disk is neutron rich with Ye ∼ 0.1 –0.2 and the temperature is only moderately high (maximum is ∼ 10 MeV with the average ∼ 3 MeV) with the maximum density ∼ 1012 g ∕cm3 and disk mass ∼ 0.1M ⊙. Perhaps, because of the relatively low mass and density of the disk, the temperature is not as high as that found in a Newtonian simulation with detailed microphysics [95].


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