1.1 Overview of BH-NS binaries

Black hole–neutron star (BH-NS) binaries are believed to be formed as a result of two supernovae in a massive binary system (see [125] for an alternative possibility). After their formation, the orbital separation decreases gradually due to the longterm gravitational radiation reaction (i.e., two objects are in an adiabatic inspiral motion), and eventually, two objects merge to be a black hole system. The lifetime of a binary in quasi-circular orbit is approximately written by (see [156Jump To The Next Citation Point, 155Jump To The Next Citation Point] or Chapter 16 of [186Jump To The Next Citation Point])
5c5 r4 τGW = ------3----------------------- 256G (MBH + MNS )MBHMNS 10 -----r----- 4 MBH--− 1 -MNS---−1 --m0--- −1 ≈ 1.34 × 10 yrs(6 × 106 km ) (6M ⊙ ) (1.4M ⊙) (7.4M ⊙ ) , (1 )
where r, MBH, and MNS are the orbital separation, masses of the BH and NS, respectively, and m = M + M 0 BH NS. G is the gravitational constant and c the speed of light, respectively. The lifetime for a binary of elliptic orbits with the semi-major axis r is shorter than τGW [156, 155Jump To The Next Citation Point]. Thus, if the initial semi-major axis is smaller than ∼ 107 km, the BH and NS merge within the Hubble time scale after a substantial emission of gravitational waves [155]. In most of the inspiral phase during which the binary separation gradually decreases due to the gravitational radiation reaction, two compact objects are well approximated by two point masses in an adiabatic orbit, because their radii are much smaller than the orbital separation (finite-size effects, such as tidal deformation, are negligible) and also the gravitational-radiation-reaction time scale is much longer than the orbital period (cf. Equation (2View Equation) with r ≫ Gm0 ∕c2). The evolution through the inspiral phase is well understood within the post-Newtonian (PN) approximation [25Jump To The Next Citation Point]. On the other hand, throughout the late inspiral to the merger phases, the orbital evolution process depends significantly on their finite-size effects and the resulting modification on the interaction between the two objects. In addition, the adiabatic approximation for the orbital evolution becomes worse, because the gravitational-radiation-reaction time scale is as short as the orbital period; the ratio of τ GW to the orbital period, P orb, is approximately written as
τGW r 5∕2 MBH −1 MNS − 1 m0 2 -----≈ 1.8(--------2) (-----) (-------) (------) , (2 ) Porb 6Gm0 ∕c 6M ⊙ 1.4M ⊙ 7.4M ⊙
and thus, for the orbit close to the last one with r ∼ 6Gm0 ∕c2, τGW is comparable to Porb. In particular, in the merger phase and subsequent remnant-formation phase, the dynamics of the system depends strongly on the structure of the NS (the radius and density profile, or its equation of state; hereafter EOS) and the BH spin, as well as on general relativistic gravity. This implies that a numerical study in the framework of general relativity is required for precisely understanding the final evolution phase of BH-NS binaries.

BH-NS binaries have not been observed yet even in our galaxy in contrast to NS-NS binaries [205, 131]. However, many of statistical studies based on the stellar evolution synthesis suggest that the coalescence will occur by 1 – 10% as frequently as that of NS-NS binaries in our galaxy and hence in the normal spiral galaxies [144, 160, 223, 98, 97, 20, 21, 148] (every ∼ 106 – 107 years). In addition, coalescence in elliptic galaxies could contribute to the total coalescence rate of the universe by a significant fraction [147]. This implies that coalescence is likely to occur frequently in the Hubble volume, and therefore, the evolution process and the final fate of BH-NS binaries deserve a detailed theoretical study. In particular, the following two facts have recently enhanced the motivation for the study of BH-NS binaries: First, BH-NS binaries in close orbits are among the most promising sources for the large laser-interferometric gravitational-wave detectors, such as LIGO [126Jump To The Next Citation Point, 2Jump To The Next Citation Point, 1], VIRGO [222, 4, 3], LCGT [120], and Einstein Telescope [91Jump To The Next Citation Point, 92Jump To The Next Citation Point]: The frequency and amplitude of gravitational waves near the last orbit are estimated to give

Ω- 6Gm0-- 3∕2 -m0----−1 f ≈ π ≈ 594 Hz ( c2r ) (7.4M ) , (3 ) ⊙ h ≈ 4Gm0--μ-≈ 3.6 × 10 −22(MBH--)( -MNS--)(6Gm0--)(---D-----)−1, (4 ) c4Dr 6M ⊙ 1.4M ⊙ c2r 100 Mpc
where μ is the reduced mass of the binary defined by M M ∕m BH NS 0, and D is the distance to the source. The frequency for the late inspiral orbits is just within the frequency-band sensitivity for the advanced gravitational-wave detectors, ∼ 10 – 3000 kHz, and the amplitude of ∼ 10–22 is high enough that the signal of gravitational waves may be detected. The detection rate of BH-NS binaries will be ∼ 0.5 – 50 events per year for the advanced detectors such as advanced-LIGO [126]. To detect gravitational waves and to extract physical information from the gravitational-wave signal, theoretical templates must be prepared. This has motivated PN and numerical-relativity studies as well as two-body approximate general relativistic studies (e.g., [37, 50Jump To The Next Citation Point, 11Jump To The Next Citation Point, 12Jump To The Next Citation Point]) for the coalescing compact binaries. The second fact is that BH-NS binaries may be some of the progenitors of the central engine of gamma-ray bursts with short time duration ≲ 2 s (SGRB) [142Jump To The Next Citation Point], for which the source is still unknown. To elucidate whether the merger of BH-NS binaries could be a promising source for the progenitor of the central engine, numerical-relativity simulations are required (see also Section 1.3).

The final fate of BH-NS binaries is classified into two categories; a NS is tidally disrupted by its companion BH before it is swallowed by the BH or a NS is simply swallowed by its companion BH in the final phase. There is a third possibility, in which stable mass transfer occurs after the onset of mass shedding of the NS by the BH tidal field. Although this may be possible, numerical simulations performed so far have not shown this to be the case, as will be mentioned in Section 1.6.

The final fate of a NS depends primarily on the mass of its companion BH and the compactness of the NS. When the BH mass is small enough or the NS radius is large enough, the NS will be tidally disrupted before it is swallowed by the BH. A necessary (not sufficient) condition for this is semi-quantitatively derived from the following analysis. Mass shedding of a non-spinning NS occurs when the tidal force of its companion BH at the surface of the NS is stronger than the self-gravity of the NS. This condition is approximately (assuming Newtonian gravity) written as

2 GMBH--(cRRNS-)-≳ -GMNS----, (5 ) r3 (cRRNS )2
where R NS is the circumferential radius of the NS at r → ∞. c R is a function of r and an EOS-dependent parameter, which is larger than unity and denotes a degree of tidal deformation of the NS, i.e., the semi-major axis is assumed to be elongated as cRRNS. The left-hand side denotes the tidal force by the BH at the surface of the NS and the right-hand side is the self-gravitational force of the NS at the inner edge of its surface.

We emphasize here that Equation (5View Equation) is the necessary condition for the onset of mass shedding, strictly speaking. Tidal disruption occurs after substantial mass is stripped from the surface of the NS, during the decrease of the orbital separation due to gravitational-wave emission. Thus, tidal disruption should occur for a smaller orbital separation (larger orbital angular velocity) than that derived from Equation (5View Equation). We also note that the NS radius is assumed to depend weakly on the NS mass. If the radius quickly increases with mass loss, tidal disruption may occur soon after the onset of mass shedding.

Assuming that the binary is in a circular orbit with the Keplerian angular velocity, Equation (5View Equation) may be written in terms of the angular velocity ∘ -------- Ω(= Gm0 ∕r3 ) as

2 −3GMNS--- − 1 Ω ≥ 0.5cR R3 (1 + Q ), (6 ) NS
where Q denotes the mass ratio defined by MBH ∕MNS. The latest general-relativistic numerical study for quasi-equilibrium states of BH-NS binaries derives a more quantitative result [209Jump To The Next Citation Point, 210Jump To The Next Citation Point] (see Section 2),
2 2 GMNS −1 Ω ≥ C Ω(---3--)(1 + Q ), (7 ) R NS
where the value of the constant, C Ω, is ≲ 0.3 for a system composed of a NS of irrotational velocity field and a non-spinning BH. The small value of --- C (< √ 0.5) Ω indicates that mass shedding is enhanced by significant tidal deformation (i.e., cR > 1) and/or possibly by general relativistic gravity of the NS. Equation (7View Equation) indicates that the frequency of gravitational waves at mass shedding is
Ω- C-Ω -MNS--- 1∕2 -RNS---− 3∕2∘ -----−1- f = π ≥ 1.0 kHz (0.3)(1.4M ⊙ ) (12 km ) 1 + Q . (8 )
Thus, tidal disruption is likely to occur for a high frequency f ≳ 1 (0.7) kHz for a NS of MNS = 1.4M ⊙ and RNS = 12 (15) km, irrespective of the BH mass.

Up to now, we implicitly assume that orbits with arbitrary orbital separations may be possible for the binary system. However, BH-NS binaries always have the innermost stable circular orbit (ISCO) determined by the general relativistic effect, and hence, we should impose the condition that mass shedding (and tidal disruption) has to occur before the binary orbit reaches the ISCO in this analysis. According to PN analysis [25Jump To The Next Citation Point], a non-dimensional orbital compactness parameter, Gm Ω ∕c3 0, at the ISCO, is ∼ 0.10 for 1 ≤ Q ≤ 5 for a system composed of a non-spinning BH and a NS. (The tidal-deformation effect reduces this value by ∼ 10 – 20% [209Jump To The Next Citation Point, 210Jump To The Next Citation Point]). In the presence of the ISCO, the condition for the onset of mass shedding is written by

𝒞 ≡ Gm0-ΩISCO--≥ Gm0--Ω-≥ C (GMNS--)3∕2(1 + Q )3∕2Q −1∕2, (9 ) ISCO c3 c3 Ω RNSc2
where ΩISCO is the angular velocity at the ISCO, and 𝒞ISCO ∼ 0.1 for a system composed of non-spinning BH (for a spinning BH, this could be much larger than 0.1; see Section 1.2). Equation (9View Equation) is rewritten to give
1∕3 ( 𝒞ISCO-)2∕3 Q----≥ GMNS---≡ 𝒞, (10 ) CΩ 1 + Q RNSc2
where 𝒞 denotes the compactness of the NS, which will be, according to nuclear physics theories for high-density matter, in the range between ∼ 0.12 and ∼ 0.25 [115Jump To The Next Citation Point, 116Jump To The Next Citation Point] for the typical NS of mass 1.2– 1.5M ⊙. This estimate suggests that tidal disruption by a non-spinning BH could occur for a binary of a relatively low mass ratio of Q ≲ 5. Equation (10View Equation) also shows that the conditions for the onset of mass shedding and for tidal disruption depend strongly on the compactness of the NS.

In the above simple estimate, the tidal-deformation effect of the NS to the orbital motion is not taken into account. As a result of the tidal deformation, the gravitational force between two stars is modified, so are the orbital evolution and the criterion for tidal disruption. Lai, Rasio, and Shapiro [111Jump To The Next Citation Point, 113, 112] thoroughly investigated the effects associated with tidal deformation in the Newtonian framework with the ellipsoidal approximation. They found that the two-body attractive force is strengthen by the effect of the tidal deformation and, by this, the orbital separation of the ISCO is increased (see  [165, 166, 114Jump To The Next Citation Point, 190] for related issues) and that gravitational waveforms in the late inspiral phase are modified (see [66, 68, 67, 72] for related issues). Uryū and Eriguchi confirmed the fact that the tidal force modifies the orbital motion of BH-NS binaries by numerically computing equilibrium states of a binary composed of a point mass and a fluid star. The essence is that in addition to the Newtonian potential, which has the form −1 ∝ r, the correction term of the form −6 ∝ r appears when a NS is tidally deformed. The magnitude of this correction increases steeply with the decrease of the orbital separation, and modifies the location of the ISCO. Also, the tidal effect accelerates the orbital evolution, because the orbital velocity (and the centrifugal force) has to be increased to maintain circular orbits in an enhanced attractive force, and then, the emissivity of gravitational waves is enhanced, leading to a shorter inspiral time.

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