3.2 Current parameter space surveyed

There are many free parameters that specify a BH-NS binary; the mass and spin of the BH, and the mass of the NS. Furthermore, the EOS of the NS is still unknown, and thus, it should also be regarded as one of the free parameters. Here, the internal motion of the NS is believed to be close to an irrotational velocity field [24, 104] because the viscosity of the NS matter is too small to realize a corotational velocity field, and, in addition, the typical spin angular momentum of the NS (Prot = 0.1 –1 s) is slow enough that we can safely assume that the NS spin is zero (because an orbital period of ≲ 10 ms for m0 ≲ 10M ⊙ is much shorter than Prot). However, we have to accept a wide possible range for other parameters and EOS of NS. In particular, the mass and spin of a BH in a realistic binary system are totally unknown, and hence, a survey for a wide parameter space is required to fully understand the nature of BH-NS coalescence and to derive a complete catalog of possible gravitational waveforms.

In the first phase, all groups employed the Γ-law EOS in the form P = (Γ − 1)ρ𝜀 with the special value of Γ = 2, for which the initial condition is prepared by using the polytropic EOS P = κρΓ. In this EOS, physical parameters are non-dimensional quantities such as mass ratio, Q = MBH ∕MNS, and compactness of the NS, 𝒞 = MNS ∕RNS, because κ can be freely chosen. Since 2009, several more plausible EOS have been employed by the KT and CCCW groups.

The early work of the KT group was done with the Γ-law EOS for a = 0 and for a wide range of Q and 𝒞; 1.5 ≤ Q ≤ 5 and 0.145 ≤ 𝒞 ≤ 0.178. Since 2009, the KT group has employed a piecewise polytropic EOS [167, 150] with a wide variety of EOS parameters (see Table 4). Simulations have been systematically performed employing this EOS for a wide range of (Q, a,𝒞) [107Jump To The Next Citation Point, 109Jump To The Next Citation Point]; 2 ≤ Q ≤ 5, − 0.5 ≤ a ≤ 0.75, and 0.12 ≲ 𝒞 ≲ 0.19. (Here, the negative value of the spin implies that the BH spin and orbital angular momentum vector are anti-parallel.)

The simulations of the UIUC group were performed employing the Γ-law EOS with Γ = 2. The UIUC group has chosen in total nine parameter sets for (Q, a, 𝒞) as follows; Q = 1,2,3, and 5, a = 0, − 0.5, and 0.75, and 𝒞 = 0.088 and 0.145. Simulations were performed for the relatively small value of NS compactness.

The CCCW group performed simulations employing two types of EOS; Γ-law EOS with Γ = 2 and 2.75, and Shen’s EOS, which is a tabulated EOS derived in a relativistic mean field theory [188, 189]. They focused on special parameter sets of (Q, a,𝒞) as Q = 1 and 3, a = 0 and 0.5, and 𝒞 ≈ 0.145 and 0.174. In their latest work, they focused on the case Q = 3 and a = 0.5, paying particular attention to the dependence of the merger process on the EOS and on the misalignment angle of the BH spin and orbital angular momentum axes.

The LBPLI group has performed one simulation to date, using the Γ-law EOS with Γ = 2, and with (Q, a, 𝒞) = (5, 0.5, 0.1) [41Jump To The Next Citation Point]. In their first work, the compactness was chosen to be small and not very realistic. In this simulation, magnetic fields in the ideal magnetohydrodynamics MHD approximation were incorporated, but they did not play an important role. The AEI group has performed simulations using Γ-law EOS with Γ = 2, and with Q = 5, a = 0, and 𝒞 = 0.1 –0.17 [154Jump To The Next Citation Point].

To summarize, the total number of simulations is still small, although a systematic survey is required to fully understand the complete picture of the coalescence of BH-NS binaries. In particular, many simulations were performed in a not very realistic setting using a simple Γ-law EOS and small values of 𝒞. As mentioned in Section 1, tidal disruption is more subject to the less compact NS, and hence, it should be in particular cautioned that a simulation with unphysically-small values of compactness may lead to an incorrect conclusion that tidal disruption and subsequent disk formation are easily achieved.

Nevertheless, the simulations performed so far have clarified a basic picture for the merger process of BH-NS binaries, the properties for the remnant, gravitational waveforms, and gravitational-wave spectrum. In the following, we summarize our current understanding of these topics based on work to date.


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