3.4 Properties of the remnant black hole and disk

During merger, NS material falls into its companion BH, and then, the mass and spin of the BH vary. Because a large fraction of the NS material falls into the BH for most of the numerical experiments performed so far and the total energy of gravitational waves emitted (EGW) is much smaller than the total mass energy of the system, the final BH mass becomes roughly MBH + MNS − Mdisk − EGW ≲ MBH + MNS = m0. (The disk mass Mdisk and EGW are less than 10% of the initial total mass m0.) Numerical results indeed show that the final BH mass is larger than 0.9m 0.

The final BH spin depends sensitively on the mass ratio and initial BH spin. This can be understood by the following simple analysis. In Newtonian gravity, the total orbital angular momentum for two point masses in a circular orbit with an angular velocity Ω is

G2-∕3MBHMNS--- Jorb = (Ωm0 )1∕3 . (122 )
Thus, the non-dimensional spin parameter of the system is approximately written as
−1 2 − 3 −1∕3 2 cG---Jorb +-M-BHa- (Gc---Ωm0-)----Q-+-aQ--- af = m20 = (1 + Q )2 , (123 )
where we assume that the BH spin aligns with the orbital angular momentum vector. If the mass and angular momentum of the remnant disk, and loss by gravitational waves may be neglected, af will be equal to the spin of the remnant BH. At the onset of the merger or at tidal disruption, the angular velocity becomes Gc− 3Ωm0 ∼ 0.05– 0.1, and thus, (Gc −3Ωm0 )− 1∕3 is in a narrow range of ∼ 2.1 – 2.7. This implies that af is primarily determined by Q and a.

Equation (123View Equation) gives rather qualitative estimate for the spin of the remnant BH. Nevertheless, it still gives a good approximate value of the final spin with the choice of − 3 −1∕3 (Gc Ωm0 ) = 3 as large as the remnant disk mass is small. With this choice, af = 0.67,0.56,0.48, and 0.42 for Q = 2,3, 4, and 5 and for a = 0. These values agree with the results derived by the CCCW [74Jump To The Next Citation Point], KT [194Jump To The Next Citation Point, 107Jump To The Next Citation Point, 108Jump To The Next Citation Point], and UIUC [63Jump To The Next Citation Point] groups within the error of Δa = 0.01 –0.02.

For a large BH spin with a ≳ 0.5, a disk of a large mass (≳ 0.1M ⊙) is often formed even for Q ∼ 3 –5 (see below). In such cases, Equation (123View Equation) overestimates the final BH spin. However, this equation still captures the qualitative tendency of the final spin; e.g., for small BH spin, the final spin is determined by the value of Q and the larger values of Q results in smaller final BH spin; for larger values of Q with a large BH spin a ≳ 0.5, the final BH spin is primarily determined by the initial BH spin.

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Figure 19: Left: Disk mass at 10 ms after the onset of merger as a function of NS compactness 𝒞 with various piecewise polytropic EOS and with various values of a for Q = 2. Right: The same as the left panel but for Q = 3. The simulations were performed by the KT group and the figure is taken from [109Jump To The Next Citation Point].
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Figure 20: Left: Summary of the remnant disk mass as a function of the BH spin for a fixed EOS (Γ = 2 EOS), NS compactness (𝒞 ≈ 0.145), and mass ratio (Q = 3), computed by the UIUC, CCCW, and KT groups. The vertical axis shows the fraction of the disk mass Mr>rAH ∕MB where MB is the baryon rest mass of the NS. “CCCWin” shows the results by the CCCW group with inclination angle of the BH spin, 40, 60, and 80 degrees (from upper to lower points). Right: The same as the left panel but for the disk mass in the solar mass unit for more compact NS (𝒞 ≈ 0.172) with a piecewise polytropic EOS (HB). The simulation was performed by the KT group [109Jump To The Next Citation Point]. For both panels, the disk mass is measured at t ≈ 10 ms after the onset of the merger (for the Γ-law EOS, MNS is assumed to be 1.4M ⊙).

The mass and characteristic density of the remnant disk surrounding the BH depend sensitively on the mass ratio (Q), the BH spin (a), and the EOS (or the compactness) of the NS. Figures 19View Image21View Image illustrate this fact. Figure 19View Image displays a result of the disk mass as a function of the NS compactness for Q = 2 (left) and Q = 3 (right) for various piecewise polytropic EOS and for various values of a, reported by the KT group [109Jump To The Next Citation Point]. This shows that the disk mass decreases steeply and systematically with the increase of the compactness irrespective of a and Q.

The left panel of Figure 20View Image plots together the results obtained by the UIUC, CCCW, and KT groups for Γ = 2 EOS with 𝒞 ≈ 0.145 and Q = 3 (the revised result by the KT group is plotted here; results in early work by the KT group do not agree with the result shown here [228Jump To The Next Citation Point, 194Jump To The Next Citation Point]; see below for the reason). This shows that the disk mass increases steeply with BH spin (a) for given values of 𝒞 and Q. The results by these three groups agree approximately with each other for a = 0. The CCCW group showed for the first time that the disk mass decreases with the increase of the inclination angle of the BH spin, and toward the limit to 90 degree, the disk mass approaches to that of a = 0. The right panel of Figure 20View Image plots the results by the KT group for different compactness (using EOS HB with MNS = 1.35M ⊙; see Table 4). This shows again that the disk mass increases with the increase of the BH spin, and also that for high BH spin (e.g., a = 0.75), the disk mass is larger than 0.1M ⊙ even for Q = 5 with 𝒞 ∼ 0.17.

Figure 21View Image shows disk mass as a function of NS compactness for a = 0.75 and a = 0.5 as performed by the KT group [109Jump To The Next Citation Point]. A steep decrease in disk mass with increasing compactness is found irrespective of the values of a and Q. Simulations for BH-NS binaries with a spinning BH for particular values of the compactness or mass ratio were also performed by the CCCW and UIUC groups for a = 0.5 and 0.75, respectively. The results of the CCCW group for a = 0.5 and Q = 3 with various EOS agree with those of Figure 21View Image within ∼ 10 – 20%. We note that the disk mass may be different in the different EOS even with the same compactness of the NS, because the density profile of the NS, and the resulting susceptibility to the BH tidal force is different. Thus, this disagreement is reasonable. Also, the results of the UIUC group for a = 0.75, Q = 3, and 𝒞 = 0.145 with the Γ-law EOS (the disk mass ∼ 0.15MB) agree with the relation expected from Figure 21View Image within ∼ 20% error.

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Figure 21: Left: Disk mass as a function of NS compactness for various values of Q with a = 0.75. Right: The same as the left panel but for a = 0.5. These simulations were performed by the KT group and this figure is taken from [109Jump To The Next Citation Point].

To clarify the dependence of the disk mass on relevant parameters, we consider three types of comparisons. First, we consider the case in which a, 𝒞, and the EOS are fixed, but Q is varied. In this case, the disk mass monotonically decreases with the increase of Q for many cases. For example, for a = 0 and 𝒞 = 0.145 with Γ = 2 EOS, the disk mass is larger (smaller) than 0.01MB for Q ≲ 4 (Q ≳ 4[63Jump To The Next Citation Point, 74Jump To The Next Citation Point, 154Jump To The Next Citation Point]. However, we should point out the exception to this rule, because for the case in which a large-mass disk is formed, this rule may not hold. For example, the comparison between the left and right panels of Figure 19View Image shows that for relatively small compactness (𝒞 ≲ 0.16), a large-mass disk is formed for high BH spins and the disk mass depends only weakly on the value of Q for given values of a and 𝒞.

Second, we consider the case in which Q and 𝒞 are fixed, but a is varied. The UIUC group compared the results for a = − 0.5,0, and 0.75 for Q = 3 and 𝒞 = 0.145 with the Γ-law EOS (Γ = 2), and the resulting disk mass at ∼ 10 ms after the onset of the merger is 0.008, 0.039, and 0.15MB for a = − 0.5,0, and 0.75, respectively [63Jump To The Next Citation Point]. The CCCW group performed a similar study for a = 0,0.5, and 0.9 for Q = 3 and 𝒞 = 0.144 with the Γ-law EOS (Γ = 2), and found that the disk mass at 10 ms after the onset of the merger is 0.034, 0.126, and 0.360M B for a = 0, 0.5, and 0.9, respectively [74Jump To The Next Citation Point]. Both groups found the systematic steep increase of the disk mass with the increase of the BH spin (cf. the left panel of Figure 20View Image). This was also reconfirmed by the KT group [109Jump To The Next Citation Point] (see the right panel of Figure 19View Image and Figure 21View Image).

Third, we consider the case in which a and Q are fixed, but the compactness, 𝒞, is varied. Systematic work was recently performed by the KT group [107Jump To The Next Citation Point, 108Jump To The Next Citation Point, 109Jump To The Next Citation Point], employing a piecewise polytropic EOS. Figures 19View Image and 21View Image show that the disk mass decreases monotonically with the increase of 𝒞 for given values of a and Q. For producing a disk of mass larger than 0.01M ⊙ for a = 0, 𝒞 should be smaller than ∼ 0.18 for Q = 2 and ∼ 0.16 for Q = 3 according to their results. For a = 0.75, the condition is significantly relaxed.

Finally, the KT group [107Jump To The Next Citation Point, 108] found that the disk mass depends not only on the compactness but weakly on the density profile. For the NS with a more centrally concentrated density profile, the disk mass is smaller. The reason for this is that if the degree of the central mass concentration is smaller, tidal deformation is enhanced and it encourages earlier tidal disruption after the onset of mass shedding.

It is interesting to note that for a = 0.75 and MNS ∼ 1.4M ⊙, the mass of a disk surrounding a BH can be larger than 0.1M ⊙ for Q ≲ 4 with 𝒞 = 0.18 and for Q ≲ 5 with 𝒞 = 0.17. For Q = 2, the disk mass will be ≳ 0.1M ⊙ for any realistic NS with 𝒞 ≲ 0.20. For a BH of higher spin, the disk mass will be even larger (cf. the left panel of Figure 20View Image). In addition, the disk mass does not decrease steeply with the increase of Q for such a high spin. In particular, for a small value of 𝒞, the disk mass depends very weakly on Q. All these results indicate that the disk mass is likely to be large even for a higher value of Q (≥ 5 ) with a high BH spin (a ≥ 0.75). The maximum density of the disk increases monotonically with the disk mass. For a disk mass larger than 0.1M ⊙, the maximum density is larger than 1012 g∕cm3 for Q ≲ 4 and ∼ 1011 g∕cm3 for Q = 5 [109Jump To The Next Citation Point]. Hence, a high-mass disk with a relatively small value of Q ≲ 4 is likely to be universally opaque against thermal neutrinos for the typical geometrical thickness of the disk; a neutrino-dominated accretion disk is the outcome and this is favorable for copious neutrino emission. This leads to the conclusion that the coalescence of BH-NS binaries with a high-spin BH with a ≳ 0.75 and Q ≲ 4 is a promising progenitor for forming a BH plus a massive disk system; that is the candidate for a central-engine of SGRB.

Before closing this subsection, we should note that different groups have reported different quantitative results for the disk mass in their earlier work, which has been improved upon. For example, the earlier work of the KT group presented a small disk mass [228, 194Jump To The Next Citation Point]. This is mainly due to an unsuitable choice of computational domain and partly due to a spurious numerical effect associated with insufficient resolution and an unsuitable choice of AMR grid. The first work of the UIUC group also underestimated the disk mass [62]. This is due to an unsuitable prescription for handling the atmosphere. However, these have been subsequently fixed.

Generally speaking, the quantitative disagreement is due to the numerics. First, the fluid elements in the disk have to acquire a sufficiently large specific angular momentum which is larger than that at the ISCO of the remnant BH. The material that forms a disk obtains angular momentum by a hydrodynamic angular momentum transport process from the inner part of the material. This implies that such a transport process has to be accurately computed in a numerical simulation. However, it is well known that this is one of the challenging tasks in computational astrophysics. Second, to avoid spurious loss and transport of the angular momentum, a high-resolution computation is required. However, the disk material is located in a relatively distant orbit around the central BH. In the AMR scheme, which is employed in all the groups, the resolution in this region is usually poorer than that in the central region. This might induce a spurious loss of angular momentum and resulting decrease of disk mass, even by a factor of ∼ 2. However, these issues are being resolved with the improvement of computational resources, the efficiency of the numerical code, and the skill for computation with the AMR algorithm. The left panel of Figure 20View Image shows as much.


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