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Coalescence of black hole–neutron star binaries

The Original Version of this article was published on 29 August 2011

Abstract

We review the current status of general relativistic studies for coalescences of black hole–neutron star binaries. First, high-precision computations of black hole–neutron star binaries in quasiequilibrium circular orbits are summarized, focusing on the quasiequilibrium sequences and the mass-shedding limit. Next, the current status of numerical-relativity simulations for the merger of black hole–neutron star binaries is described. We summarize our understanding for the merger process, tidal disruption and its criterion, properties of the merger remnant and ejected material, gravitational waveforms, and gravitational-wave spectra. We also discuss expected electromagnetic counterparts to black hole–neutron star coalescences.

Introduction

Why is the black hole–neutron star binary merger important?

After the first release of this review article in 2011 (Shibata and Taniguchi 2011), the research environment for compact binary coalescences has changed completely. The turning point was the first gravitational-wave event GW150914 from a binary-black-hole merger detected by the LIGO and Virgo Collaboration (Abbott et al. 2016b). We obtained the strongest evidence for the existence of black holes and mergers of their binaries within the Hubble time. We learned that some stellar-mass black holes are significantly more massive than those found in our Galaxy by X-ray observations (Abbott et al. 2016a). Furthermore, we confirmed that general relativity is consistent with observations even for this dynamical and strongly gravitating phenomenon (Abbott et al. 2016c; Yunes et al. 2016, see also Abbott et al. 2019b, 2021c for the update). After further observations, the number of stellar-mass black holes detected by gravitational waves has already exceeded those by electromagnetic radiation (see Abbott et al. 2019a, 2021a for reported detections as of 2020). Gravitational-wave astronomy of binary black holes is rapidly becoming an established branch of astrophysics.

Subsequently, the first binary-neutron-star merger, GW170817, was detected with not only gravitational waves (Abbott et al. 2017d, 2019c) but also electromagnetic waves by multiband instruments all over the world (Abbott et al. 2017c, e). Tidal deformability of the neutron star was constrained from gravitational-wave data analysis, and extremely stiff equations of state are no longer favored (Abbott et al. 2017d; De et al. 2018; Abbott et al. 2018, 2019c; Narikawa et al. 2020). Binary-neutron-star mergers were strongly suggested to be central engines of short gamma-ray bursts by the detection of a weak GRB 170817A (Abbott et al. 2017c; Goldstein et al. 2017; Savchenko et al. 2017) and by longterm observations of its off-axis afterglow (Mooley et al. 2018a, b; Alexander et al. 2018; Lamb et al. 2019). The host galaxy NGC 4993 was identified by the kilonova/macronova AT 2017gfo (Coulter et al. 2017; Arcavi et al. 2017; Lipunov et al. 2017; Soares-Santos et al. 2017; Tanvir et al. 2017; Valenti et al. 2017), and Hubble’s constant was inferred in a novel manner by combining the cosmological redshift of NGC 4993 and the luminosity distance estimated from GW170817 (Abbott et al. 2017b). Furthermore, binary-neutron-star mergers were indicated to be a site of r-process nucleosynthesis (Tanaka et al. 2017; Kasen et al. 2017; Watson et al. 2019). This event heralded a new era of multimessenger astronomy with gravitational and electromagnetic radiation.

Finally, during the review process of this article, detections of black hole–neutron star binaries, GW200105 and GW200115, are reported (Abbott et al. 2021b). Together with another candidate of a black hole–neutron star binary merger GW190426_152155 (Abbott et al. 2021a), we may now safely consider that black hole–neutron star binaries are actually merging in our Universe. The merger rate is currently inferred to be \(\approx 12\)\({240}\,{\text{Gpc}}^{-3}\,{\text{yr}}^{-1}\), which is largely consistent with previous theoretical estimation (Dominik et al. 2015; Kruckow et al. 2018; Neijssel et al. 2019; Zevin et al. 2020; Santoliquido et al. 2021, see also Narayan et al. 1991; Phinney 1991 for pioneering rate estimation). Unfortunately, despite the presence of neutron stars, no associated electromagnetic counterpart was detected. This is consistent with the current theoretical understanding reviewed throughout this article, because the most likely mass ratios of these binaries are as high as 4–5 and the spin of the black holes are likely to be zero or retrograde. Tidal disruption do not occur for these parameters, and thus neither mass ejection nor electromagnetic emission is expected to occur. We also note that some other gravitational-wave events reported in LIGO-Virgo O3 are also consistent with black hole–neutron star binaries under generous assumptions on the mass of compact objects (Abbott et al. 2020a, c),Footnote 1 partly because no electromagnetic counterpart was detected (Kyutoku et al. 2020; Han et al. 2020; Kawaguchi et al. 2020a).

One of the remaining issues for ground-based gravitational-wave detectors is to discover coalescences of black hole–neutron star binaries accompanied by tidal disruption and hence electromagnetic emission. Indeed, among the mergers of black hole–neutron star binaries, those resulting in tidal disruption of the neutron star by the black hole are of physical and astrophysical interest and deserve detailed investigations. Specifically, the tidal disruption is required to occur outside the innermost stable circular orbit of the black hole for inducing astrophysically interesting outcomes. If the neutron star is not disrupted, as is likely the case of GW200105 and GW200115, it behaves like a point particle throughout the coalescence, and the merger process will be indistinguishable from that of (highly asymmetric) binary black holes (Foucart et al. 2013a) except for possible electromagnetic emission associated with crust shattering (Tsang et al. 2012), magnetospheric activities (Hansen and Lyutikov 2001; McWilliams and Levin 2011; Lai 2012; Paschalidis et al. 2013; D’Orazio et al. 2016; Carrasco and Shibata 2020; Wada et al. 2020; East et al. 2021; Carrasco et al. 2021, see also Ioka and Taniguchi 2000 for earlier work on binary neutron stars), or charged black holes (Levin et al. 2018; Zhang 2019; Dai 2019; Pan and Yang 2019; Zhong et al. 2019). These two possibilities for the fate of merger are summarized schematically in Fig. 1.Footnote 2

Fig. 1
figure 1

Image adapted from Shibata and Hotokezaka (2019), copyright by Annual Reviews

Summary for the merger and postmerger evolution of black hole–neutron star binaries. The fate is classified into two categories according to whether the neutron star is tidally disrupted (right) or not (left).

Focusing on the cases in which tidal disruption occurs, many researchers have vigorously studied the following three aspects. Accordingly, most parts of this review will be devoted to their detailed discussions.

  • Gravitational waves will enable us to study the finite-size properties and hence the equation of state of neutron stars. First, tidal deformability, \(\varLambda \) (see also Sect. 1.3), of neutron stars will be extracted from the phase evolution in the inspiral phase (Flanagan and Hinderer 2008) along with the masses and the spins of binary components (Finn and Chernoff 1993; Jaranowski and Krolak 1994; Cutler and Flanagan 1994; Poisson and Will 1995). Although the tidal deformability could be inferred even if tidal disruption does not occur, realistic measurements will be possible only when the finite-size effect is so sizable that the neutron star is disrupted (Lackey et al. 2012, 2014). Second, the orbital frequency at tidal disruption depends on the compactness of the neutron star, \({\mathscr{C}}\) (Vallisneri 2000; Shibata et al. 2009). Because the mass can be extracted or constrained from inspiral signals along with the spin as stated above, gravitational waveforms from tidal disruption of a neutron star may bring us invaluable information about its radius, which is strongly but not perfectly correlated with the tidal deformability (Hotokezaka et al. 2016a; De et al. 2018). The measurement of these quantities with black hole–neutron star binaries could serve as an additional tool for exploring supranuclear-density matter (Lindblom 1992; Harada 2001). For this purpose, it is crucial to understand the dependence of gravitational waveforms, including characteristic observable features associated with tidal disruption, on possible equations of state by theoretical calculations.

  • The remnant disk formed from the disrupted neutron star is a promising central engine of short-hard gamma-ray bursts (Paczynski 1991; Narayan et al. 1992; Mochkovitch et al. 1993, see also Blinnikov et al. 1984 for an earlier idea and Paczynski 1986; Goodman 1986; Eichler et al. 1989 for binary-neutron-star scenarios). A typical beaming-corrected energy of the jet, \(\sim {10^{50}}\,{\text{erg}}\) (Fong et al. 2015), can be explained if, for example, \(\sim 0.1\%\) of the rest-mass energy is converted from a \(\sim 0.1 \,M_\odot \) accretion disk. This could be realized via neutrino pair annihilation (Rees and Meszaros 1992), which is effective when the disk is sufficiently hot and dense to cool via neutrino radiation, called the neutrino-dominated accretion flow (Popham et al. 1999; Narayan et al. 2001; Kohri and Mineshige 2002; Di Matteo et al. 2002; Kohri et al. 2005; Chen and Beloborodov 2007; Kawanaka and Mineshige 2007). Another possible energy source is the rotational energy of a spinning black hole extracted by magnetic fields, i.e., the Blandford–Znajek mechanism (Blandford and Znajek 1977; Mészáros and Rees 1997). For this mechanism to work, magnetic-field strength in the disk needs to be amplified by turbulent motion resulting from magnetohydrodynamic instabilities such as the magnetorotational instability (Balbus and Hawley 1991), and subsequently, strong magnetic fields threading the spinning black hole need to be developed to form a surrounding magnetosphere. One of the ultimate goals for numerical simulations of compact binary coalescences may be to clarify how, if possible, the ultrarelativistic jet is launched from the merger remnant. Theoretical investigations should also clarify whether longterm activity of short-hard gamma-ray bursts, e.g., the extended and plateau emission (Norris and Bonnell 2006; Rowlinson et al. 2013; Gompertz et al. 2013; Kisaka et al. 2017), can really be explained by the merger remnant of black hole–neutron star binaries. Because of the diversity associated with stellar-mass black holes, black hole–neutron star binaries might naturally explain the variety observed in short-hard gamma-ray bursts (see Nakar 2007; Berger 2014 for reviews).

  • A substantial amount of neutron-rich material will be ejected and synthesize r-process elements (Lattimer and Schramm 1974, see also Lattimer 2019 for retrospection by an originator), i.e., about half of the elements heavier than iron in the universe, whose origin has not yet been fully understood (Burbidge et al. 1957; Cameron 1957). Subsequently, radioactive decays of unstable nuclei will heat up the ejecta, resulting in quasithermal emission in UV-optical-IR bands on a time scale of \({\mathscr {O}}\)(10) days (Li and Paczyński 1998). This transient, called the kilonova (Metzger et al. 2010b) or macronova (Kulkarni 2005), serves as the most promising omnidirectional electromagnetic counterparts to gravitational waves (see Metzger 2019 for reviews). The ejecta are eventually mixed with the interstellar medium and contribute to the chemical evolution of galaxies, and this interaction may drive another electromagnetic counterpart such as synchrotron radiation from nonthermal electrons (Nakar and Piran 2011) and possibly inverse Compton emission (Takami et al. 2014). To derive nucleosynthetic yields and characteristics of electromagnetic counterparts, we need to understand properties of the ejecta such as the mass, the velocity, and the electron fraction that characterizes the neutron richness. In particular, the electron fraction primarily determines the nucleosynthetic yield, which controls features of the kilonova/macronova via the opacity (Kasen et al. 2013; Tanaka and Hotokezaka 2013; Tanaka et al. 2018, 2020; Banerjee et al. 2020) and the heating rate (Hotokezaka et al. 2016b; Barnes et al. 2016; Kasen and Barnes 2019; Waxman et al. 2019; Hotokezaka and Nakar 2020). If a significant fraction of the ejecta keeps extreme neutron richness of the neutron star, ultraheavy elements may be produced in abundance, and the associated fission and/or \(\alpha \)-decay will power the kilonova/macronova at late times (Wanajo et al. 2014; Zhu et al. 2018; Wu et al. 2019). They could also be the origin of exceptionally r-process enhanced metal-poor stars, so-called actinide-boost stars (see, e.g., Mashonkina et al. 2014). Last but not least, the geometrical shape of the ejecta could be important for understanding the diversity of electromagnetic counterparts to black hole–neutron star binaries (Kyutoku et al. 2013; Tanaka et al. 2014).

Life of black hole–neutron star binaries

We first overview the entire evolution of black hole–neutron star binaries from their birth. Binaries consisting of a black hole and/or a neutron star, hereafter collectively called compact object binaries, are generally born after evolution of isolated massive binaries (see, e.g., Postnov and Yungelson 2014 for reviews) or via dynamical processes in dense environments (see, e.g., Benacquista and Downing 2013 for reviews). Relative contributions of these two paths to black hole–neutron star binaries have not been understood yet, as well as for compact object binaries of other types. We do not go into details of the formation path in this article, commenting only that the evolution of isolated binaries is usually regarded as the dominant channel for black hole–neutron star binaries (see, e.g., discussions in Abbott et al. 2021b).

After the formation of black hole–neutron star binaries, their orbital separation decreases gradually due to longterm gravitational radiation reaction. If we would like to observe their coalescences, the binaries are required to merge within the Hubble time of \(\approx {1.4 \times 10^{10}}\,{\text{yr}}\). This condition is also a prerequisite for them to drive short-hard gamma-ray bursts and to produce r-process elements. The lifetime of a black hole–neutron star binary in a circular orbit for a given orbital separation r is given by

$$\begin{aligned} t_{\mathrm {GW}}&= \frac{5c^5}{256G^3} \frac{r^4}{( M_{\mathrm {BH}} + M_{\mathrm {NS}} ) M_{\mathrm {BH}} M_{\mathrm {NS}}} \nonumber \\&= {1.01\times 10^{10}}\, {\hbox {yr}} \left( \frac{r}{{6\times 10^{6}}\, {\hbox {km}}} \right) ^4 \left( \frac{M_{\mathrm {BH}}}{7\,M_\odot } \right) ^{-1} \left( \frac{M_{\mathrm {NS}}}{1.4\,M_\odot } \right) ^{-1} \left( \frac{m_0}{8.4\,M_\odot } \right) ^{-1} \end{aligned}$$
(1)

in the adiabatic approximation, which is appropriate when the radiation reaction time scale is much longer than the orbital period. Here, G, c, \(M_{\mathrm {BH}}\), \(M_{\mathrm {NS}}\), and \(m_0\) are the gravitational constant, the speed of light, the gravitational mass of the black hole, the gravitational mass of the neutron star, and the total mass of the binary \(m_0 := M_{\mathrm {BH}} + M_{\mathrm {NS}}\), respectively (cf., Table 1). The orbital eccentricity only reduces the time to merger for a given value of the semimajor axis (Peters and Mathews 1963; Peters 1964). Thus, a black hole–neutron star binary merges within the Hubble time if its initial semimajor axis is smaller than \(\sim {10^{7}}\, {\hbox {km}}\) with the precise value depending on the masses of the objects and the initial eccentricity. Because the spin and finite-size properties of the objects come into play only as higher-order corrections in terms of the orbital velocity or other appropriate parameters (see, e.g., Blanchet 2014 for reviews of the post-Newtonian formalism), Eq. (1) with eccentricity corrections is adequate for judging whether a binary merges within the Hubble time.

Two remarks should be made regarding the longterm evolution. First, the orbital eccentricity decreases rapidly, specifically \(e \propto a^{-19/12}\) in an asymptotically circular regime with a being the semimajor axis, due to gravitational radiation reaction (Peters 1964). Accordingly, black hole–neutron star binaries right before merger (e.g., when gravitational waves are detected by ground-based detectors) may safely be approximated as circular. Second, the neutron star is unlikely to be tidally-locked, because the effects of viscosity are likely to be insufficient (Kochanek 1992; Bildsten and Cutler 1992). Thus, the spin of the neutron star can affect the merger dynamics significantly only if the rotational period is extremely short at the outset and the spin-down is not severe. Quantitatively, the dimensionless spin parameter of the neutron star is approximately written as

$$\begin{aligned} \chi _{\mathrm {NS}}&= \frac{c I_{\mathrm {NS}} \varOmega _{\mathrm {rot}}}{G M^2_{\mathrm {NS}}} = {4.9 \times 10^{-4}} \left( \frac{I_{\mathrm {NS}} / ( M_{\mathrm {NS}} R_{\mathrm {NS}}^2 )}{1/3} \right) \left( \frac{M_{\mathrm {NS}}}{1.4\,M_\odot } \right) ^{-1} \left( \frac{R_{\mathrm {NS}}}{{12}\, {\hbox {km}}} \right) ^2 \left( \frac{P_{\mathrm {rot}}}{{1}\, {\hbox {s}}} \right) ^{-1} , \end{aligned}$$
(2)

where \(I_{\mathrm {NS}}\), \(R_{\mathrm {NS}}\), and \(P_{\mathrm {rot}}\) are the moment of inertia, the radius, and the rotational period, respectively, of the neutron star. Observationally, the shortest rotational period of known pulsars in Galactic binary neutron stars that merge within the Hubble time is \(\approx {17}\, {\hbox {ms}}\), which is equivalent to only \(\chi _{\mathrm {NS}} \approx 0.03\) (Stovall et al. 2018). Moreover, black hole–neutron star binaries are unlikely to harbor recycled pulsars, because the neutron star is expected to be formed after the black hole, having no chance for mass accretion. Hence, it is reasonable to approximate neutron stars as nonspinning in the merger of black hole–neutron star binaries. Exceptions to these remarks might arise from dynamical formation in dense environments such as galactic centers and globular clusters, e.g., exchange interactions involving recycled pulsars (see, e.g., Fragione et al. 2019; Ye et al. 2020), and/or black-hole formation from the secondary caused by mass transfer in isolated massive binaries (see, e.g., Kruckow et al. 2018).

The late inspiral and merger phases of black hole–neutron star binaries are promising targets of gravitational waves for ground-based detectors irrespective of the degree of tidal disruption. The frequency f and the amplitude h of gravitational waves from black hole–neutron star binaries with the orbital separation r at the luminosity distance D are estimated in the quadrupole approximation for two point particles as

$$\begin{aligned} f&\approx \frac{\varOmega }{\pi } = {523}\, {\hbox {Hz}} \left( \frac{r}{6Gm_0 / c^2} \right) ^{-3/2} \left( \frac{m_0}{8.4\,M_\odot } \right) ^{-1} , \end{aligned}$$
(3)
$$\begin{aligned} h&\approx \frac{4G^2 M_{\mathrm {BH}} M_{\mathrm {NS}}}{c^4 r D} = {3.7 \times 10^{-22}} \left( \frac{\mu }{1.17\,M_\odot } \right) \left( \frac{r}{6Gm_0 / c^2} \right) ^{-1} \left( \frac{D}{{100}\, {\hbox {Mpc}}} \right) ^{-1} , \end{aligned}$$
(4)

where \(\mu := M_{\mathrm {BH}} M_{\mathrm {NS}} / m_0\) is the reduced mass. Here, the most favorable direction and orientation are assumed for evaluating h. These values indicate that black hole–neutron star binaries near the end of their lives fall within the observable window of ground-based gravitational-wave detectors as far as the distance is sufficiently close.

However, the quadrupole approximation for point particles is not sufficiently accurate for describing the evolution of black hole–neutron star binaries in the late inspiral, merger, and postmerger phases. As the orbital separation gradually approaches the radius of the object, spins and finite-size effects such as tidal deformation begin to modify the gravitational interaction between the binary in a noticeable manner. The adiabatic approximation also breaks down for the very close orbit, because the radiation reaction time scale and the orbital period become comparable near an approximate innermost stable orbitFootnote 3 as

$$\begin{aligned} \frac{t_{\mathrm {GW}}}{P_{\mathrm {orb}}} = 2.0 \left( \frac{r}{6Gm_0 / c^2} \right) ^{5/2} \left( \frac{M_{\mathrm {BH}}}{7\,M_\odot } \right) ^{-1} \left( \frac{M_{\mathrm {NS}}}{1.4\,M_\odot } \right) ^{-1} \left( \frac{m_0}{8.4\,M_\odot } \right) ^2 . \end{aligned}$$
(5)

Thus, dynamics in the late inspiral and merger phases depends crucially on complicated hydrodynamics associated with neutron stars, whose properties are controlled by the equation of state, and on nonlinear gravity of general relativity. Furthermore, the evolution of the remnant disk in the postmerger phase is governed by neutrino emission triggered by shock-induced heating and turbulence associated with magnetohydrodynamic instabilities (Lee et al. 2004; Setiawan et al. 2004; Lee et al. 2005; Setiawan et al. 2006; Shibata et al. 2007). All these facts make fully general-relativistic numerical studies the unique tool to clarify the final evolution of black hole–neutron star coalescences in a quantitative manner.

Tidal problem around a black hole

As we stated in Sect. 1.1, this review will focus primarily on numerical studies of black hole–neutron star binaries for which finite-size effects play a significant role. To set the stage for understanding numerical results, in this Sect. 1.3, we will discuss requirement for the binary to cause significant tidal disruption, which starts with the mass shedding from the inner edge of the neutron star.

Mass-shedding condition

The orbital separation at which the mass shedding sets in is determined primarily by the mass ratio of the binary and the radius of the neutron star. The orbit at which the mass shedding sets in, the so-called mass-shedding limit, can be estimated semiquantitatively by Newtonian calculations as follows. Mass shedding from the neutron star occurs when the tidal force exerted by the black hole overcomes the self-gravity of the neutron star at the inner edge of the stellar surface. This condition is approximately given by

$$\begin{aligned} \frac{2G M_{\mathrm {BH}} ( c_{\mathrm {R}} R_{\mathrm {NS}} )}{r^3} \gtrsim \frac{G M_{\mathrm {NS}}}{( c_{\mathrm {R}} R_{\mathrm {NS}} )^2} , \end{aligned}$$
(6)

where the factor \(c_{\mathrm {R}} \ge 1\) represents the degree of tidal (and rotational if the neutron star is rapidly spinning) elongation of the stellar radius. The precise value of this factor depends on the neutron-star properties and the orbital separation. The mass-shedding limit may be defined as the orbit at which this inequality is approximately saturated,

$$\begin{aligned} r_{\mathrm {ms}} := 2^{1/3} c_{\mathrm {R}} \left( \frac{M_{\mathrm {BH}}}{M_{\mathrm {NS}}} \right) ^{1/3} R_{\mathrm {NS}} . \end{aligned}$$
(7)

We emphasize here that Eq. (6) is a necessary condition for the onset of mass shedding. Tidal disruption occurs only after substantial mass is stripped from the surface of the neutron star, while the orbital separation decreases continuously due to gravitational radiation reaction during this process. Thus, the tidal disruption should occur at a smaller orbital separation than Eq. (7). We also note that the neutron star will be disrupted immediately after the onset of mass shedding if its radius increases rapidly in response to the mass loss, although typical equations of state predict that the radius in equilibrium depends only weakly on the mass (see, e.g., Lattimer and Prakash 2016; Özel and Freire 2016; Oertel et al. 2017 for reviews).

Tidal disruption induces observable astrophysical consequences only if it occurs outside the innermost stable circular orbit of the black hole, inside which stable circular motion is prohibited by strong gravity of general relativity; If the tidal disruption fails to occur outside this orbit, the material is rapidly swallowed by the black hole and does not leave a remnant disk or unbound ejecta in an appreciable manner. This implies that observable tidal disruption requires, at least, the mass-shedding limit to be located outside the innermost stable circular orbit. The radius of the innermost stable circular orbit depends sensitively on the dimensionless spin parameter of the black hole, \(\chi \). Specifically, it is given in terms of a dimensionless decreasing function \({\hat{r}}_{\mathrm {ISCO}} ( \chi )\) of \(\chi \) for an orbit on the equatorial plane of the black hole by (Bardeen et al. 1972)

$$\begin{aligned} r_{\mathrm {ISCO}} = {\hat{r}}_{\mathrm {ISCO}} ( \chi ) \frac{GM_{\mathrm {BH}}}{c^2} . \end{aligned}$$
(8)

Here, we adopt the convention that the positive and negative values of \(\chi \) indicate the prograde and retrograde orbits, i.e., the orbits with their angular momenta aligned and anti-aligned with the black-hole spin, respectively. Specifically, the value of \({\hat{r}}_{\mathrm {ISCO}}\) is 9 for a retrograde orbit around an extremally-spinning black hole (\(\chi = -1\)), 6 for an orbit around a nonspinning black hole (\(\chi = 0\)), and 1 for a prograde orbit around an extremally-spinning black hole (\(\chi = 1\)). If the spin of the black hole is inclined with respect to the orbital angular momentum, the spin effect described here is not determined by the magnitude of the spin angular momentum but by that of the component parallel to the orbital angular momentum. Thus, even if the black-hole spin is high, its effect can be minor in the presence of spin misalignment.

To sum up, the final fate of a black hole–neutron star binary is determined primarily by the mass ratio of the binary, the compactness of the neutron star, and the dimensionless spin parameter of the black hole. The ratio of the radius of the mass-shedding limit and that of the innermost stable circular orbit is given by

$$\begin{aligned} \frac{r_{\mathrm {ms}}}{r_{\mathrm {ISCO}}} = \frac{2^{1/3} c_{\mathrm {R}}}{{\hat{r}}_{\mathrm {ISCO}} ( \chi )} \left( \frac{M_{\mathrm {BH}}}{M_{\mathrm {NS}}} \right) ^{-2/3} \left( \frac{GM_{\mathrm {NS}}}{c^2 R_{\mathrm {NS}}} \right) ^{-1} . \end{aligned}$$
(9)

This semiquantitative estimate suggests that tidal disruption of a neutron star could occur if one or more of the following conditions are satisfied:

  1. 1.

    the mass ratio of the binary, \(Q := M_{\mathrm {BH}} / M_{\mathrm {NS}}\), is low,

  2. 2.

    the compactness of the neutron star, \({\mathscr {C}} := GM_{\mathrm {NS}} / ( c^2 R_{\mathrm {NS}} )\), is small,

  3. 3.

    the dimensionless spin parameter of the black hole, \(\chi \), is high with the definition of signature stated above.

If we presume that the neutron-star mass is fixed, the conditions 1 and 2 may be restated as

  1. 1’.

    the black-hole mass is small,

  2. 2’.

    the neutron-star radius is large,

respectively.

Quantitative discussions have to take the general-relativistic nature of black hole–neutron star binaries into account. For this purpose, it is advantageous to rewrite Eq. (6) in terms of the orbital angular velocity as

$$\begin{aligned} \varOmega ^2 \ge \frac{1}{2c_{\mathrm {R}}^3} \frac{GM_{\mathrm {NS}}}{R_{\mathrm {NS}}^3} \left( 1 + Q^{-1} \right) , \end{aligned}$$
(10)

because \(\varOmega \) can be defined in a gauge-invariant manner even for a comparable-mass binary in general relativity. It should be remarked that the orbital frequency at the onset of mass shedding is determined primarily by the average density of the neutron star, \(\propto \sqrt{M_{\mathrm {NS}} / R_{\mathrm {NS}}^3}\). According to the results of fully general-relativistic numerical studies for quasiequilibrium states (Taniguchi et al. 2007, 2008, see Sect. 2 for the details), the mass-shedding condition is given by

$$\begin{aligned} \varOmega ^2 \ge \varOmega _{\mathrm {ms}}^2 := C_\varOmega ^2 \frac{G M_{\mathrm {NS}}}{R_{\mathrm {NS}}^3} \left( 1 + Q^{-1} \right) , \end{aligned}$$
(11)

where \(C_\varOmega \lesssim 0.3\) for binaries of a nonspinning black hole and a neutron star with the irrotational velocity field. The smallness of \(C_\varOmega < 1/\sqrt{2}\) indicates that the mass shedding is helped by significant tidal deformation, i.e., \(c_{\mathrm {R}}>1\), and/or by relativistic gravity. This condition also indicates that the gravitational-wave frequency at the onset of mass shedding is given by

$$\begin{aligned} f_{\mathrm {ms}} = \frac{\varOmega _{\mathrm {ms}}}{\pi } \gtrsim {1.0}\, {\hbox {kHz}} \left( \frac{C_\varOmega }{0.3} \right) \left( \frac{M_{\mathrm {NS}}}{1.4\,M_\odot } \right) ^{1/2} \left( \frac{R_{\mathrm {NS}}}{{12}\, {\hbox {km}}} \right) ^{-3/2} \left( 1 + Q^{-1} \right) ^{1/2} . \end{aligned}$$
(12)

This value might be encouraging for ground-based gravitational-wave detectors, which have high sensitivity up to \(\sim {1}\, {\hbox {kHz}}\). However, we again caution that the mass shedding is merely a necessary condition for tidal disruption, and thus the frequency at tidal disruption should be higher than this value.

Tidal interaction in the orbital evolution

The discussion in Sect. 1.3.1 did not take the effect of tidal deformation of a neutron star on the orbital motion into account except for a fudge factor \(c_{\mathrm {R}}\). Tidally-induced higher multipole moments of the neutron star modify the gravitational interaction between the binary components (see, e.g., Poisson and Will 2014), so are the orbital evolution and the criterion for tidal disruption. This problem has thoroughly been investigated in Newtonian gravity with the ellipsoidal approximation, in which the isodensity contours are assumed to be self-similar ellipsoids (Lai et al. 1993a, b, 1994a, b). They find that the tidal interaction acts as additional attractive force and accordingly the radius of the innermost stable circular orbit is increased (see also Rasio and Shapiro 1992, 1994; Lai and Wiseman 1996; Shibata 1996). Because (i) the tidally-deformed neutron star develops a reduced quadrupole moment with the magnitude of components being \(\propto r^{-3}\) associated with the tidal field of the black hole and (ii) the reduced quadrupole moment produces potential of the form \(\propto r^{-3}\), the gravitational potential in the binary develops an \(r^{-6}\) term in addition to the usual \(r^{-1}\) term of the monopolar (i.e., mass) interaction. The reason that this interaction works as the attraction is that the neutron star is stretched along the line connecting the binary components and the enhancement of the pull at the inner edge dominates over the reduction at the outer edge. The steep dependence of the potential on the orbital separation indicates that the tidal interaction is especially important for determining properties of the close orbit.

These discussions about the tidal effects on the orbital motion have been revived in the context of gravitational-wave modeling and data analysis (Flanagan and Hinderer 2008). Specifically, it has been pointed out that the finite-size effect of a star on the orbital evolution and hence gravitational waveforms are characterized quantitatively by the tidal deformability as far as the deformation is perturbative (Hinderer 2008; Binnington and Poisson 2009; Damour and Nagar 2009). Because the additional attractive force increases the orbital angular velocity required to maintain a circular orbit for a given orbital separation, the gravitational-wave luminosity is also increased. In addition, the coupling of the quadrupole moments between the binary and the deformed star also enhances the luminosity. These effects accelerate the orbital decay particularly in the late inspiral phase to the extent that the difference of gravitational waveforms may be used to extract tidal deformability of neutron stars. This extraction has been realized in GW170817 (Abbott et al. 2017d, 2018; De et al. 2018; Abbott et al. 2019c; Narikawa et al. 2020) and GW190425 (Abbott et al. 2020a), whereas the statistical errors are large. It should also be cautioned that the effect of tidal deformability is not very large compared to various other effects, e.g., the spin and the eccentricity (Yagi and Yunes 2014; Favata 2014; Wade et al. 2014). In particular, the tidal effect comes into play effectively at the fifth post-Newtonian order (\(r^{-6} / r^{-1} = r^{-5}\)), but the point-particle terms at this order have not yet been derived in the post-Newtonian approximation. Thus, accurate extraction of tidal deformability requires sophistication not only in the description of tidal effects but also in the higher-order post-Newtonian corrections to point-particle, monopolar interactions. This fact has motivated gravitational-wave modeling in the effective-one-body formalism (Buonanno and Damour 1999, 2000) and numerical relativity.

Tidal interaction and criteria for mass shedding in general relativity have long been explored for a circular orbit of a “test” Newtonian fluid star around a Kerr (or Schwarzschild) black hole as follows (Fishbone 1973; Mashhoon 1975; Lattimer and Schramm 1976; Shibata 1996; Wiggins and Lai 2000; Ishii et al. 2005). The center of mass of the star is assumed to obey the geodesic equation in the background spacetime, and the stellar structure is computed with a model based on the Newtonian Euler’s equation of the form

$$\begin{aligned} \frac{\hbox {d} u_i}{\hbox {d} \tau } = - \frac{1}{\rho } \frac{\partial P}{\partial x^i} - \frac{\partial \phi }{\partial x^i} - C_{ij} x^j , \end{aligned}$$
(13)

where \(\tau \), \(x^i\), \(u_i\), \(\rho \), P, \(\phi \), and \(C_{ij}\) denote the proper time of the stellar center, spatial coordinates orthogonal to the geodesic, the internal velocity, the rest-mass density, the pressure, the gravitational potential associated with the star itself, and the tidal tensor associated with the black hole, respectively. The self-gravity of the fluid star is computed in a Newtonian manner from Poisson’s equation sourced by \(4\pi G\rho \). The tidal force of the black hole is incorporated up to the quadrupole order via the tidal tensor derived from the fully relativistic Riemann tensor (Marck 1983, see also van de Meent 2020). Because the gravity of the fluid star is assumed not to affect the orbital motion and general relativity is not taken into account for describing its self-gravity, the analysis based on this model is valid quantitatively only for the cases in which the black hole is much heavier than the fluid star (\(Q \gg 1\)) and the fluid star is not compact (\({\mathscr {C}} \ll 1\)). In addition, the tidal force of the black hole beyond the quadrupole order, \(C_{ij}\), is neglected (Marck 1983), and this model is valid only if the stellar radius is much smaller than the curvature scale of the background spacetime (again, \(Q \gg 1\) is assumed). Regarding this point, higher-order tidal interactions have also been incorporated (Ishii et al. 2005) via the tidal potential computed in the Fermi normal coordinates (Manasse and Misner 1963).

A series of analysis described above confirms the qualitative dependence of the mass-shedding and tidal-disruption conditions inferred from Eq. (9) on binary parameters in a semiquantitative manner. Specifically, the mass shedding from an incompressible star is found to occur for the mass and the spin of the black hole satisfying

$$\begin{aligned} M_{\mathrm {BH}} \lesssim C_M ( \chi ) \,M_\odot \left( \frac{M_{\mathrm {NS}}}{1.4\,M_\odot } \right) ^{-1/2} \left( \frac{R_{\mathrm {NS}}}{{10}\, {\hbox {km}}} \right) ^{3/2} , \end{aligned}$$
(14)

where \(C_M (0) \approx 4.6\), \(C_M (0.5) \approx 7.8\), \(C_M (0.75) \approx 12\), \(C_M (0.9) \approx 19\), and \(C_M (1) \approx 68\) (Shibata 1996). This condition tells us that tidal disruption of a neutron star by a nonspinning black hole is possible only if the black-hole mass is small compared to astrophysically typical values (see, e.g., Özel et al. 2010; Kreidberg et al. 2012; Abbott et al. 2019a, 2021a). At the same time, the increase in the threshold mass by a factor of \(\approx 15\) for extremal black holes is impressive particularly in light of many massive black holes discovered by gravitational-wave observations.

The threshold mass of the black hole for mass shedding and thus tidal disruption also depends on the neutron-star equation of state even if the mass and the radius are identical (Wiggins and Lai 2000; Ishii et al. 2005). If we focus on polytropes, stiffer equations of state characterized by a larger adiabatic index are more susceptible to tidal deformation due to the flatter, less centrally condensed density profile. Conversely, neutron stars with a soft equation of state are less subject to tidal disruption than those with a stiff one. These features are also reflected in the tidal Love number and tidal deformability (Hinderer 2008). Note that the incompressible model corresponds to the stiffest possible equation of state. According to the computations performed adopting compressible stellar models (Wiggins and Lai 2000; Ishii et al. 2005), the threshold mass of the black hole may be reduced by 10%–20% for a soft equation of state characterized by a small adiabatic index.

In reality, the self-gravity of the neutron star needs to be treated in a general-relativistic manner. General-relativistic effects associated with the neutron star have been investigated by a series of work in a phenomenological manner based on the ellipsoidal approximation (Ferrari et al. 2009, 2010; Pannarale et al. 2011; Ferrari et al. 2012; Maselli et al. 2012). However, quantitative understanding of the mass shedding and tidal disruption ultimately requires numerical computations of quasiequilibrium states and dynamical simulations of the merger process in full general relativity.

Brief history of studies on black hole–neutron star binaries

Here, we briefly review studies on black hole–neutron star binaries from the historical perspectives in an approximate chronological order. We also introduce pioneering studies that are not fully relativistic, e.g., Newtonian computations of equilibrium states and partially-relativistic simulations of the coalescences. In the main part of this article, Sects. 2 and 3, we will review fully general-relativistic results, i.e., quasiequilibrium states satisfying the Einstein constraint equations and dynamical evolution derived by solving the full Einstein equation, from the physical perspectives.

Nonrelativistic equilibrium computation

Equilibrium configurations of a neutron star governed by Newtonian self-gravity in general-relativistic gravitational fields of a background black hole was first studied in Fishbone (1973) for incompressible fluids in the corotational motion (i.e., the fluid is at rest in the corotating frame of the binary). The criterion for mass shedding was investigated and qualitative results were obtained. This type of studies has been generalized to accommodate irrotational velocity fields (i.e., the vorticity is absent; Shibata 1996), compressible, polytropic equations of state (Wiggins and Lai 2000), and higher-order tidal potential of the black hole (Ishii et al. 2005). Another direction of extension was to remove the assumption of the extreme mass ratio of the binary. This extension was done in Taniguchi and Nakamura (1996) by adopting modified pseudo-Newtonian potential for the black hole based on the so-called Paczyński–Wiita potential (Paczyńsky and Wiita 1980) to determine the location of the innermost stable circular orbit.

However, all these studies have limitation even if we accept the Newtonian self-gravity of neutron stars. The ellipsoidal approximation is strictly valid only if the fluid is incompressible and the tidal field beyond the quadrupole order can be neglected (Chandrasekhar 1969). Thus, the internal structure of compressible neutron stars in a close orbit can be investigated only qualitatively.

The hydrostationary equilibrium of black hole–neutron star binaries was derived in Uryū and Eriguchi (1998) assuming that the black hole was a point source of Newtonian gravity and that the neutron star with irrotational velocity fields obeyed a polytropic equation of state (irrotational Roche–Riemann problem). The center-of-mass motion of the neutron star was computed fully accounting for its self-gravity, and the tidal field of the Newtonian point source was incorporated to the full order in the ratio of the stellar radius to the orbital separation. Their subsequent work, Uryū and Eriguchi (1999), considered both the corotational and irrotational velocity fields, and differences from the ellipsoidal approximation have been analyzed.

Relativistic quasiequilibrium computation

One of the essential features of general relativity is the existence of gravitational radiation, whose reaction prohibits exactly stationary equilibria of binaries. Still, an approximately stationary solution to the Einstein equation may be obtained by solving the constraint equations, quasiequilibrium conditions derived by some of the evolution equations, and hydrostationary equations. Such solutions are called quasiequilibrium states, and Eq. (5) suggests that they are reasonable approximations to inspiraling binaries except near merger (see also Blackburn and Detweiler 1992; Detweiler 1994). The quasiequilibrium states are important not only by their own but also as initial data of realistic numerical-relativity simulations.

Quasiequilibrium states and sequences of black hole–neutron star binaries in full general relativity were first studied in Miller (2001) with preliminary formulation. Approximate quasiequilibrium states in the extreme mass ratio limit were obtained for the corotational velocity field in Baumgarte et al. (2004) and later for the irrotational velocity field in Taniguchi et al. (2005). Because gravitational fields around the black hole are not required to be solved in the extreme mass ratio limit, these computations were performed only around (relativistic) neutron stars.

General-relativistic quasiequilibrium states for comparable-mass binaries were obtained in 2006 by various groups both in the excision (Grandclément 2006; Taniguchi et al. 2006) and the puncture frameworks (Shibata and Uryū 2006, 2007). A general issue in the numerical computation of black-hole spacetimes is how to handle the associated physical or coordinate singularity. The excision framework handles the black hole by removing the interior of a suitably-defined horizon (see, e.g., Dreyer et al. 2003; Ashtekar and Krishnan 2004; Gourgoulhon and Jaramillo 2006) from the computational domains and by imposing appropriate boundary conditions (Cook 2002; Cook and Pfeiffer 2004). The puncture framework separates the singular and regular components in an analytic manner so that only the latter terms are solved numerically (Bowen and York 1980; Brandt and Brügmann 1997). The details are presented in Appendix A.

Taniguchi et al. (2007) derived accurate quasiequilibrium sequences in the excision framework by adopting the conformally-flat background and investigated properties of close black hole–irrotational neutron star binaries such as the mass-shedding limit (see also Grandclément 2007). Taniguchi et al. (2008) further improved the sequences by enforcing nonspinning conditions for the black hole in a sophisticated manner via the boundary condition at the horizon. Quasiequilibrium states with spinning black holes were computed with the same code as initial data for numerical simulations (Etienne et al. 2009).

Quasiequilibrium states in the puncture framework were also derived for irrotational velocity fields in Shibata and Taniguchi (2008) by extending the formulation for corotating neutron stars (Shibata and Uryū 2006, 2007). Kyutoku et al. (2009) obtained quasiequilibrium sequences of nonspinning black holes with varying the method for determining the center of mass of the binary, which is not uniquely defined in the puncture framework. Quasiequilibrium states in the puncture framework were extended to black holes with aligned and inclined spins (Kyutoku et al. 2011a; Kawaguchi et al. 2015), and the same formulation has also been adopted to perform merger simulations in the conformal-flatness approximation (Just et al. 2015). Recently, the eccentricity reduction method has been implemented in this framework (Kyutoku et al. 2021, see below for preceding work in the excision framework).

Except for early work in the puncture framework (Shibata and Uryū 2006, 2007), all the computations described above were performed with the spectral-method library, LORENE, which enables us to achieve very high precision (see Grandclément and Novak 2009 for reviews). Note that Grandclément (Grandclément 2006) and Taniguchi (Taniguchi et al. 2006) are two of the main developers of LORENE.

Foucart et al. (2008) also computed quasiequilibrium sequences in the excision framework with an independent code, SPELLS (Pfeiffer et al. 2003). This code implemented a modified Kerr-Schild background metric for computing highly-spinning black holes (Lovelace et al. 2008) and the eccentricity reduction method for performing realistic inspiral simulations (Pfeiffer et al. 2007). Initial data with inclined black-hole spins were also derived (Foucart et al. 2011). The computations of quasiequilibrium states have now been extended to high-compactness (Henriksson et al. 2016) and/or spinning neutron stars (Tacik et al. 2016). Papenfort et al. (2021) have also derived quasiequilibrium sequences by using another spectral-method library, KADATH (Grandclément 2010).

Non/partially-relativistic merger simulations

The merger process of black hole–neutron star binaries was first studied in Newtonian gravity primarily with the aim of assessing the potentiality for the central engine of gamma-ray bursts. In the early work, the black hole was modeled by a point source of Newtonian gravity with (artificial) absorbing boundary conditions. A series of simulations with a smoothed-particle-hydrodynamics code explored influences of the rotational states of the fluids and stiffness of the (polytropic) equations of state for neutron-star matter (Kluźniak and Lee 1998; Lee and Kluźniak 1999a, b; Lee 2000, 2001). They studied the process of tidal disruption, subsequent formation of a black hole–disk system and mass ejection, properties of the remnant disk and unbound material, and gravitational waveforms emitted during merger.

Around the same time, Janka et al. (1999) performed simulations incorporating detailed microphysics with a mesh-based hydrodynamics code. Specifically, their code had implemented a temperature- and composition-dependent equation of state (Lattimer and Swesty 1991) and neutrino emission modeled in terms of the leakage scheme (Ruffert et al. 1996). By performing simulations for various configurations of binaries, a hot and massive remnant disk with \(\gtrsim {10}\, {\hbox {MeV}}\) and 0.2–\(0.7 \,M_\odot \) was suggested to be formed, and the neutrino luminosity was found to reach \({10^{53}}{-}{10^{54}}\, {\hbox {erg s}^{-1}}\) during \(10{-}{20}\, {\hbox {ms}}\) after formation of a massive disk. Pair annihilation of neutrinos was also investigated by post-process calculations (Ruffert et al. 1997) and was found to be capable of explaining the total energy of gamma-ray bursts. Although these Newtonian results were still highly qualitative and both the disk mass and temperature can be overestimated for given binary parameters (see below), it was first suggested by dynamical simulations that black hole–neutron star binary coalescences could be promising candidates for the central engine of short-hard gamma-ray bursts if the massive accretion disk was indeed formed. Rosswog et al. (2004) also performed simulations in similar setups with a smoothed-particle-hydrodynamics code adopting a different equation of state (Shen et al. 1998).

The gravity in the vicinity of a black hole modeled by a Newtonian point source is qualitatively different from that in reality. In particular, the innermost stable circular orbit is absent in the Newtonian point-particle model. Because of this difference, early Newtonian work concluded that the neutron star might be disrupted without an immediate plunge even in an orbit closer to the black hole than \(\lesssim 6GM_{\mathrm {BH}} / c^2\). Consequently, they indicated that a massive remnant disk with \(\gtrsim 0.1\,M_\odot \) might be formed around the black hole irrespective of the mass ratio and the rotational states of fluids. It should also be mentioned that the orbital evolution within Newtonian gravity frequently exhibited episodic mass transfer (see Clark and Eardley 1977; Cameron and Iben 1986; Benz et al. 1990 for relevant discussions). That is, the neutron star is only partially disrupted via the stable mass transfer during the close encounter with the black hole, becomes a “mini-neutron star” (Rosswog 2005) with increasing the binary separation, and continues the orbital motion. This has never been found in fully relativistic simulations (although not completely rejected throughout the possible parameter space) and may be regarded as another qualitative difference associated with the realism of gravitation (see also Appendix C.1).

To overcome these drawbacks, Rosswog (2005) performed smoothed-particle-hydrodynamics simulations by modeling the black-hole gravity in terms of a pseudo-Newtonian potential (Paczyńsky and Wiita 1980). A potential for modeling the gravity of a spinning black hole (Artemova et al. 1996) was also adopted in later mesh-based simulations (Ruffert and Janka 2010), and the episodic mass transfer was still observed for some parameters of binaries. These work found that the massive disk with \(\gtrsim 0.1\,M_\odot \) was formed only for binaries with low-mass and/or spinning black holes. Because this feature agrees qualitatively with the fully relativistic results, simulations with a pseudo-Newtonian potential might be helpful to understand the nature of black hole–neutron star binary mergers qualitatively or even semiquantitatively.Footnote 4

While numerical-relativity simulations have been feasible since 2006 (see Sect. 1.4.4), approximately general-relativistic simulations have also been performed without simplifying the black holes by point sources of gravity. This is particularly the case of smoothed-particle-hydrodynamics codes, which are especially useful to track the motion of the material ejected from the system but have not been available in numerical relativity (see Rosswog and Diener 2021 for recent development of smooth-particle hydrodynamics in numerical relativity). One of the popular approaches to incorporate general relativity is the conformal-flatness approximation (Faber et al. 2006a, b; Just et al. 2015). In these work, the gravity of neutron stars was also treated in a general-relativistic manner. Another work adopted the Kerr background for modeling the black hole, while the neutron star is modeled as a Newtonian self-gravitating object, and studied dependence of the merger process on the inclination angle of the black-hole spin with respect to the orbital angular momentum (Rantsiou et al. 2008). Results of these simulations agree qualitatively with those from pseudo-Newtonian and numerical-relativity simulations.

Fully-relativistic merger simulations

General-relativistic effects play a crucial role in the dynamics of close black hole–neutron star binaries. First of all, the inspiral is driven by gravitational radiation reaction. Dynamics right before merger is affected crucially by further general-relativistic effects, which include strong attractive force between two bodies, associated presence of the innermost stable circular orbit, spin-orbit coupling, and relativistic self-gravity of neutron stars. Accordingly, the orbital evolution, the merger process, the criterion for tidal disruption, and the evolution of disrupted material are all affected substantially by the general-relativistic effects. Although non-general-relativistic work has provided qualitative insights, numerical simulations in full general relativity are obviously required for accurately and quantitatively understanding the nature of black hole–neutron star binary coalescences. This is particularly the case in development of an accurate gravitational-wave template for the data analysis.

Soon after the breakthrough success in simulating binary-black-hole mergers (Pretorius 2005; Campanelli et al. 2006; Baker et al. 2006), Shibata and Uryū (2006, 2007); Shibata and Taniguchi (2008) started exploration of black hole–neutron star binary coalescences in full general relativity extending their early work on binary neutron stars (Shibata 1999; Shibata and Uryū 2000, 2002; Shibata et al. 2003, 2005; Shibata and Taniguchi 2006). While these work adopted initial data computed in the puncture framework for moving-puncture simulations, Etienne et al. (2008) independently performed moving-puncture simulations with excision-based initial data by extending their early work on binary neutron stars (Duez et al. 2003, see also Löffler et al. 2006 for early work on a head-on collision in a similar setup). Duez et al. (2008) also performed simulations for black hole–neutron star binaries based on the excision method by introducing hydrodynamics equation solvers to a spectral-method code, SpEC, for binary black holes (Boyle et al. 2007, 2008; Scheel et al. 2009). All these studies were performed for nonspinning black holes and neutron stars modeled by a polytropic equation of state with \(\varGamma = 2\).

To derive accurate gravitational waveforms, longterm simulations need to be performed. Effort in this direction was made with the aid of an adaptive-mesh-refinement code (see Appendix B.3), SACRA (Yamamoto et al. 2008), by the authors (Shibata et al. 2009, 2012). Around the same time, Etienne et al. (2009) independently studied the effect of black-hole spins by another adaptive-mesh-refinement code. Systematic longterm studies were started employing nuclear-theory based equations of state approximated by piecewise-polytropic equations of state (Read et al. 2009a) for both nonspinning (Kyutoku et al. 2010, 2011b) and spinning black holes (Kyutoku et al. 2011a). A tabulated, temperature- and composition-dependent equation of state (Shen et al. 1998) was also incorporated in simulations with SpEC around the same time (Duez et al. 2010), while neutrino reactions were not considered in this early work. SpEC was also used to simulate systems with inclined black-hole spins (Foucart et al. 2011) or an increased mass ratio of \(Q=7\) (Foucart et al. 2012). These simulations clarified quantitatively the criterion for tidal disruption, properties of the remnant disk and black hole, and emitted gravitational waves in the merger phase.

Mass ejection from black hole–neutron star binaries and associated fallback of material began to be explored in numerical relativity at the beginning of 2010’s (Chawla et al. 2010; Kyutoku et al. 2011a). Actually, these studies predate the corresponding investigations for binary neutron stars in numerical relativity (Hotokezaka et al. 2013b). Although the authors of this article suggested that \(\ge 0.01\,M_\odot \) may be ejected dynamically from black hole–neutron star binaries in this early work (Kyutoku et al. 2011a), they were unable to show the ejection of material with confidence, partly because the artificial atmosphere was not tenuous enough and the computational domains were not large enough. By contrast, the mass ejection from hyperbolic encounters was discussed clearly (Stephens et al. 2011; East et al. 2012, 2015).

Serious investigations of the dynamical mass ejection were initiated in 2013 (Foucart et al. 2013b; Lovelace et al. 2013; Kyutoku et al. 2013), right after the first version of this review article was released, stimulated by the importance of electromagnetic counterparts to gravitational waves (Abbott et al. 2020b, the preprint version of which appeared on arXiv in 2013). Kyutoku et al. (2015) systematically studied kinematic properties such as the mass and the velocity as well as the morphology of dynamical ejecta with reducing the density of artificial atmospheres and enlarging the computational domains. Kawaguchi et al. (2015) also investigated the impact of the inclination angle of the black-hole spin for both the remnant disk and the dynamical ejecta (see also Foucart et al. 2013b). The study of mass ejection has now become routine in numerical-relativity simulations of black hole–neutron star binary coalescences. Accordingly, a lot of discussions about mass ejection are newly added in this update.

The current frontier of numerical relativity for neutron-star mergers is the incorporation of magnetohydrodynamics and neutrino-radiation hydrodynamics as accurately as possible. These physical processes play essentially no role during the inspiral phase (Chawla et al. 2010; Etienne et al. 2012a, c; Deaton et al. 2013), and thus gravitational radiation, disk formation, and dynamical mass ejection are safely studied by pure hydrodynamics simulations except for the chemical composition of the dynamical ejecta. However, both neutrinos and magnetic fields are key agents for driving postmerger evolution including disk outflows and ultrarelativistic jets. In addition, the electron fraction of the ejected material, either dynamical or postmerger, can be quantified only by simulations implementing composition-dependent equations of state with appropriate schemes for neutrino transport.

Although numerical-relativity codes for magnetohydrodynamics were already available in the late 2000’s and a preliminary simulation was performed by Chawla et al. (2010), magnetohydrodynamics simulations are destined to struggle with the need to resolve short-wavelength modes of instability (see, e.g., Balbus and Hawley 1998 for reviews). Various simulations have been performed aiming at clarifying the launch of an ultrarelativistic jet, generally finding magnetic-field amplification difficult to resolve (Etienne et al. 2012a, c). The situation was much improved by high-resolution simulations performed in Kiuchi et al. (2015b, see also Wan 2017 for a follow-up). At around the same time, magnetohydrodynamics simulations with a presumed strong dipolar field have been performed to clarify the potentiality of black hole–neutron star binaries as a central engine of short-hard gamma-ray bursts (Paschalidis et al. 2015; Ruiz et al. 2018, 2020). Simulations beyond ideal magnetohydrodynamics have recently been performed aiming at clarifying magnetospheric activities right before merger (East et al. 2021).

As the electron fraction of the ejected material is crucial to determine the nucleosynthetic yield and the feature of associated kilonovae/macronovae, numerical-relativity simulations with neutrino transport have been performed extensively following those for binary-neutron-star mergers (Sekiguchi et al. 2011a, b). Neutrino-radiation-hydrodynamics simulations of black hole–neutron star binaries were first performed in Deaton et al. (2013); Foucart et al. (2014, 2017); Brege et al. (2018) with neutrino emission based on a leakage scheme (see Ruffert et al. 1996; Rosswog and Liebendörfer 2003, for the description in Newtonian cases) and with composition-dependent equations of state. These simulations are also capable of predicting neutrino emission from the postmerger system. Kyutoku et al. (2018) incorporated neutrino absorption by the material based on the two moment formalism (Thorne 1981; Shibata et al. 2011) again following work on binary neutron stars (Wanajo et al. 2014; Sekiguchi et al. 2015).

The longterm postmerger evolution of the remnant accretion disk also requires numerical investigations. These simulations need to be performed with sophisticated microphysics, because the evolution of the disk is governed by weak interaction processes such as the neutrino emission and absorption and magnetohydrodynamical processes. The liberated gravitational binding energy may eventually be tapped to launch a postmerger wind as well as an ultrarelativistic jet. Simulations focusing on the postmerger evolution with detailed neutrino transport are initially performed without incorporating sources of viscosity for a short term (Foucart et al. 2015). This work has been extended in the Cowling approximation to incorporate magnetic fields that provide effective viscosity and induce magnetohydrodynamical effects (Hossein Nouri et al. 2018). Fujibayashi et al. (2020a, b) performed fully-general-relativistic viscous-hydrodynamics simulations for postmerger systems with detailed neutrino transport. Finally, Most et al. (2021b) reported results of postmerger simulations for nearly-equal-mass black hole–neutron star binaries with neutrino transport and magnetohydrodynamics in full general relativity. Still, it is not feasible to perform fully-general-relativistic simulations incorporating both detailed neutrino transport and well-resolved magnetohydrodynamics. Because the postmerger mass ejection is now widely recognized as an essential source of nucleosynthesis and electromagnetic emission for binary-neutron-star mergers (Shibata et al. 2017a), sophisticated simulations for this problem will remain the central topic in the future study of black hole–neutron star binary coalescences.

After the detection of gravitational waves from binary neutron stars GW170817 (Abbott et al. 2017d) and GW190425 (Abbott et al. 2020a), it became apparent that robust characterization of source properties requires us to distinguish binary neutron stars from black hole–neutron star binaries (see Sect. 4.2.2). This situation motivated studies on mergers of very-low-mass black hole–neutron star binaries to clarify disk formation, mass ejection, and gravitational-wave emission in a more precise manner than what was done before (Foucart et al. 2019b; Hayashi et al. 2021; Most et al. 2021a). Here, “very low mass” means that it is consistent with being a neutron star and that observationally distinguishing binary types is not straightforward. Interestingly, these studies are revealing overlooked features of very-low-mass ratio systems. Finally, longterm simulations with \(\gtrsim 10\)–15 inspiral orbits have recently been performed aiming at deriving accurate gravitational waveforms (Foucart et al. 2019a, 2021).

Outline and notation

This review article is organized as follows. In Sect. 2, we review the current status of the study on quasiequilibrium states of black hole–neutron star binaries in general relativity. First, we summarize physical parameters characterizing a binary in Sect. 2.1. Next, the parameter space surveyed is summarized in Sect. 2.2. Results and implications are reviewed in Sects. 2.3 and 2.4, respectively. In Sect. 3, we review the current status of the study on coalescences of black hole–neutron star binaries in numerical relativity. Methods for numerical simulations are briefly described in Sect. 3.1, and the parameter space investigated is summarized in Sect. 3.2. The remainder of Sect. 3 is devoted to reviewing numerical results, and readers interested only in them should jump into Sect. 3.3, in which we start from reviewing the overall process of the binary coalescence and tidal disruption in the late inspiral and merger phases. Properties of the remnant disk, remnant black hole, fallback material, and dynamical ejecta are summarized in Sect. 3.4. Postmerger evolution of the remnant disk is reviewed in Sect. 3.5. Gravitational waveforms and spectra are reviewed in Sect. 3.6. Finally in Sect. 4, we discuss implications of numerical results to electromagnetic emission and characterization of observed astrophysical sources. Formalisms to derive quasiequilibrium states and to simulate dynamical evolution are reviewed in Appendices A and B, respectively. Appendix C presents analytic estimates related to discussions made in this article.

Table 1 List of symbols

The notation adopted in this article is summarized in Table 1. Among the parameters shown in this table, \(M_{\mathrm {BH}}\), \(\chi \), \(\iota \), \(M_{\mathrm {NS}}\), and \(R_{\mathrm {NS}}\) are frequently used to characterize binary systems in this article. The negative value of \(\chi \) is allowed for describing anti-aligned spins, meaning that the dimensionless spin parameter and the inclination angle are given by \(\left| \chi \right| \) and \(\iota = {180}^{\circ }\), respectively. Hereafter, the dimensionless spin parameter is referred to also by the spin parameter for simplicity. Greek and Latin indices denote the spacetime and space components, respectively. We adopt geometrical units in which \(G=c=1\) in Sects. 2, 3.6, Appendices A, and B.

This article focuses on fully general-relativistic studies of black hole–neutron star binaries, and other types of compact object binaries are not covered in a comprehensive manner. Numerical-relativity simulations of compact object binaries in general are reviewed in, e.g., Lehner and Pretorius (2014); Duez and Zlochower (2019). Simulations of compact object binaries involving neutron stars and their implications for electromagnetic counterparts are reviewed in, e.g., Paschalidis (2017); Baiotti and Rezzolla (2017); Shibata and Hotokezaka (2019). Black hole–neutron star binaries are also reviewed briefly in Foucart (2020).

Quasiequilibrium state and sequence

Quasiequilibrium states of compact object binaries in close orbits are important from two perspectives. First, they enable us to understand deeply the tidal interaction of comparable-mass binaries in general relativity. Second, they serve as realistic initial conditions for dynamical simulations in numerical relativity.

For the purpose of the former, a sequence of quasiequilibrium states parametrized by the orbital separation or angular velocity, i.e., quasiequilibrium sequences, should be investigated as an approximate model for the evolution path of the binary. In this section, we review representative numerical results of quasiequilibrium sequences derived to date. The formulation to construct black hole–neutron star binaries in quasiequilibrium is summarized in Appendix A. Because the differential equations to be solved are typically of elliptic type (see also the end of this section), most numerical computations adopt spectral methods for achieving high precision (see Grandclément and Novak 2009 for reviews). In this section, geometrical units in which \(G=c=1\) is adopted.

Physical parameters of the binary

In this Sect. 2.1, we present physical quantities required for quantitative analysis of quasiequilibrium sequences. Each sequence is specified by physical quantities conserved at least approximately along the sequences, and these quantities also serve as labels of binary models in dynamical simulations. We also need physical quantities that characterize each quasiequilibrium state to study its property.

To begin with, we introduce a helical Killing vector used in modeling quasiequilibrium states of black hole–neutron star binaries (see also Appendix A). Because the time scale of gravitational radiation reaction is much longer than the orbital period except for binaries in a very close orbit as we discussed in Sect. 1.2, the binary system appears approximately stationary in the comoving frame. Such a system is considered to be in quasiequilibrium and is usually modeled by assuming the existence of a helical Killing vector with the form of

$$\begin{aligned} \xi ^\mu = ( \partial _t )^\mu + \varOmega ( \partial _\varphi )^\mu , \end{aligned}$$
(15)

where \(\varOmega \) denotes the orbital angular velocity of the system. The helical Killing vector is timelike and spacelike in the near zone of \(\varpi \lesssim c/\varOmega \) and the far zone of \(\varpi \gtrsim c/\varOmega \), respectively, where \(\varpi \) denotes the distance from the rotational axis and we inserted c for clarity. Thus, if we focus only on quasiequilibrium configurations in the near zone, we may assume the existence of a timelike Killing vector, which allows us to define several physical quantities in a meaningful manner.

Here, it is necessary to keep the following two (not independent) caveats in mind if we consider helically symmetric spacetimes. First, spacetimes of binaries cannot be completely helically symmetric in full general relativity. That is, it is not realistic to assume that a helical Killing vector exists in the entire spacetime. The reason is that the helical symmetry holds throughout the spacetime only if standing gravitational waves are present everywhere. However, the total energy of the system diverges for such a case. Thus, the helical Killing vector can be supposed to exist only in a limited region of the spacetime, e.g., in the local wave zone. A simpler strategy for studying quasiequilibrium states of a binary is to neglect the presence of gravitational waves. Although this is an overly simplified assumption, this strategy has been employed in the study of quasiequilibrium states of compact object binaries. The results introduced in this section are derived by assuming that gravitational waves are absent. Moreover, the induced metric is assumed to be conformally flat (see Appendix A for the details.)

Second, gravitational radiation reaction violates the helical symmetry in full general relativity. To compute realistic quasiequilibrium states of compact object binaries, we need to take radiation reaction into account. Procedures for this are described in Appendix A.5.2.

Parameters of the black hole

It is reasonable to assume that the irreducible mass (i.e., the area of the event horizon) and the magnitude of the spin angular momentum of the black hole are conserved along a quasiequilibrium sequence, because the absorption of gravitational waves by the black hole is only a tiny effect (Alvi 2001; Chatziioannou et al. 2013, see also Poisson and Sasaki 1995; Poisson 2004 for relevant work in black-hole perturbation theory). The irreducible mass of the black hole is defined by (Christodoulou 1970)

$$\begin{aligned} M_{\mathrm {irr}} := \sqrt{\frac{A_{\mathrm {EH}}}{16\pi }} , \end{aligned}$$
(16)

where \(A_{\mathrm {EH}}\) is the proper area of the event horizon. Because the event horizon cannot be identified in quasiequilibrium configurations, its area, \(A_{\mathrm {EH}}\), is usually approximated by that of the apparent horizon, \(A_{\mathrm {AH}}\). It is reasonable to consider that \(A_{\mathrm {AH}}\) agrees at least approximately with \(A_{\mathrm {EH}}\) in the current context, because a timelike Killing vector is assumed to exist in the vicinity of the black hole (Hawking and Ellis 1973, Chap. 9). The magnitude of the spin angular momentum is determined in terms of an approximate Killing vector \(\xi _{\mathscr {S}}^i\) for axisymmetry on the horizon as (Dreyer et al. 2003; Caudill et al. 2006)

$$\begin{aligned} S_{\mathrm {BH}} := \frac{1}{8\pi } \oint _{\mathscr {S}} \left( K_{ij} - K \gamma _{ij} \right) \xi _{\mathscr {S}}^j \hbox {d}{S}^i . \end{aligned}$$
(17)

An approximate Killing vector, \(\xi _{\mathscr {S}}^i\), may be determined on the horizon by requiring some properties satisfied by genuine Killing vectors to hold (see Appendix A.1.1).

The angle of the black-hole spin angular momentum is usually evaluated in terms of coordinate-dependent quantities specific to individual formulations, because no geometric definition is known. It should be cautioned that the direction of the black-hole spin seen from a distant observer changes for the case in which the precession motion occurs (Apostolatos et al. 1994). Still, the angle between the black-hole spin and the orbital angular momentum of the binary is approximately conserved during the evolution, and hence, can be employed to characterize the system.

The gravitational mass (sometimes called the Christodoulou mass; Christodoulou 1970) of the black hole in isolation is given by

$$\begin{aligned} M_{\mathrm {BH}}^2 = M_{\mathrm {irr}}^2 + \frac{S_{\mathrm {BH}}^2}{4 M_{\mathrm {irr}}^2} . \end{aligned}$$
(18)

By introducing a dimensionless spin parameter of the black hole,

$$\begin{aligned} \chi := \frac{S_{\mathrm {BH}}}{M_{\mathrm {BH}}^2} , \end{aligned}$$
(19)

the gravitational mass is also written by

$$\begin{aligned} M_{\mathrm {BH}} = M_{\mathrm {irr}} \left[ \frac{2}{1 + ( 1 - \chi ^2 )^{1/2}} \right] ^{1/2} . \end{aligned}$$
(20)

Because \(M_{\mathrm {BH}}\) is directly measured in actual observations, this quantity rather than \(M_{\mathrm {irr}}\) is usually adopted to label quasiequilibrium sequences and the models of binary systems in dynamical simulations for spinning black holes, along with the spin parameter, \(\chi \).

Parameters of the neutron star

The baryon rest mass of the neutron star given by

$$\begin{aligned} M_{\mathrm {B}} = \int \rho u^t \sqrt{-g} \hbox {d}^{3}{x} = \int \rho \alpha u^t \sqrt{\gamma } \hbox {d}^{3}{x} \end{aligned}$$
(21)

is conserved along quasiequilibrium sequences assuming that the continuity equation holds and that mass ejection from the neutron star does not occur prior to merger. The spin angular momentum of the neutron star may be evaluated on the stellar surface by Eq. (17) (Tacik et al. 2016), although it is not conserved on a long time scale due to the spin-down. The orientation of the spin is also affected by the precession motion.

In contrast to black holes, an equation of state needs to be provided to specify finite-size properties of neutron stars such as the radius and the tidal deformability, although the realistic equation of state at supranuclear density is still uncertain (see, e.g., Lattimer and Prakash 2016; Oertel et al. 2017; Baym et al. 2018 for reviews). Because of rapid cooling by neutrino emission in the initial stage and subsequent photon emission (see, e.g., Potekhin et al. 2015 for reviews), temperature of not-so-young neutron stars relevant to coalescing compact object binaries is likely to be much lower than the Fermi energy of constituent particles. Thus, it is reasonable to adopt a fixed zero-temperature equation of state throughout the quasiequilibrium sequence. The zero-temperature equation of state allows us to express all the thermodynamic quantities as functions of a single variable, e.g., the rest-mass density.

As a qualitative model, the polytrope of the form

$$\begin{aligned} P ( \rho ) = \kappa \rho ^{\varGamma } , \end{aligned}$$
(22)

where \(\kappa \) and \(\varGamma \) are the polytropic constant and the adiabatic index, respectively, has often been adopted in the study of quasiequilibrium sequences. The neutron-star matter is frequently approximated by a polytrope with \(\varGamma \approx 2\)–3. More sophisticated models include piecewise polytropes (Read et al. 2009a) and generalization thereof (O’Boyle et al. 2020), spectral representations (Lindblom 2010), and various nuclear-theory-based tabulated equations of state. We will come back to this topic later in Sect. 3.1.2.

Once a hypothetical equation of state is given, the gravitational mass of a neutron star in isolation, \(M_{\mathrm {NS}}\), is determined for a given value of the baryon rest mass, \(M_{\mathrm {B}}\), via the Tolman-Oppenheimer-Volkoff equation (Tolman 1939; Oppenheimer and Volkoff 1939) [if the neutron star is spinning, the magnitude of the spin also comes into play (Hartle 1967; Friedman and Stergioulas 2013)]. The gravitational mass rather than the baryon rest mass is usually adopted to label the models of dynamical simulations, primarily because the gravitational mass is directly measured in actual observations. The equation of state also determines the radius and the compactness for a given mass of the neutron star. By imposing perturbative tidal fields on a background spherical configuration, the tidal deformability as a function of the neutron-star mass is computed from the ratio of the tidally-induced multipole moment and the exerted tidal field (Hinderer 2008; Binnington and Poisson 2009; Damour and Nagar 2009).

Parameters of the binary system

The Arnowitt–Deser–Misner (ADM) mass of the system (Arnowitt et al. 2008) is evaluated in isotropic Cartesian coordinates (see, e.g., York 1979; Gourgoulhon 2012, Chap. 8 for further details) as

$$\begin{aligned} M_{\mathrm {ADM}} = - \frac{1}{2\pi } \oint _{r \rightarrow \infty } \partial _i \psi \hbox {d}{S}^i , \end{aligned}$$
(23)

where \(\psi \) is the conformal factor, which is given by \(\psi = \gamma ^{1/12}\) in Cartesian coordinates for a conformally-flat case (see Appendix A). This quantity should decrease as the orbital separation decreases along a quasiequilibrium sequence because of the strengthening of gravitational binding. The binding energy of a binary system is often defined by

$$\begin{aligned} E_{\mathrm {b}} := M_{\mathrm {ADM}} - m_0 , \end{aligned}$$
(24)

where the total mass \(m_0 := M_{\mathrm {BH}} + M_{\mathrm {NS}}\) corresponds to the ADM mass of the binary system at infinite orbital separation.

The Komar mass is originally defined as a charge associated with a timelike Killing vector (Komar 1959) and is evaluated in the \(3+1\) formulation by (see, e.g., Shibata 2016, Sect. 5)

$$\begin{aligned} M_{\mathrm {K}} = \frac{1}{4\pi } \oint _{r \rightarrow \infty } \left( \partial _i \alpha - K_{ij} \beta ^j \right) \hbox {d}{S}^i . \end{aligned}$$
(25)

Since quasiequilibrium states of black hole–neutron star binaries are computed assuming the existence of a helical Killing vector which is timelike in the near zone, the Komar mass may be considered as a physical quantity, at least approximately. If the second term in the integral falls off sufficiently rapidly, as is typical for the case in which the linear momentum of the system vanishes, the Komar mass may be evaluated only from the first term, i.e., the derivative of the lapse function. Because the ADM and Komar masses should agree if a timelike Killing vector exists (Friedman et al. 2002; Shibata et al. 2004, see also Beig 1978; Ashtekar and Magnon-Ashtekar 1979), their fractional difference,

$$\begin{aligned} \delta M := \left| \frac{M_{\mathrm {ADM}} - M_{\mathrm {K}}}{M_{\mathrm {ADM}}} \right| , \end{aligned}$$
(26)

measures the global error in the numerical computation. This quantity is sometimes called the virial error.

An ADM-like angular momentum of the system may be defined by (York 1979)

$$\begin{aligned} J_i := \frac{1}{16\pi } {\underline{\epsilon }}_{ijk} \oint _{r \rightarrow \infty } \left( X^j K^{kl} - X^k K^{jl} \right) \hbox {d}{S}_l , \end{aligned}$$
(27)

where \(X^i\) and \({\underline{\epsilon }}_{ijk}\) denote, respectively, Cartesian coordinates relative to the center of mass of the binary and the Levi-Civita tensor for the flat space. It should be cautioned that this quantity is well-defined only in restricted coordinate systems (see, e.g., York 1979; Gourgoulhon 2012, Chap. 8). This subtlety is irrelevant to the results reviewed in this article. For binary systems with the reflection symmetry about the orbital plane, only the component normal to the plane is nonvanishing and will be denoted by J.

Current parameter space surveyed

Although a decade has passed since the release of the first version of this article, the parameter space surveyed for the study of quasiequilibrium sequences remains narrow. The main progress achieved during this period may be the computations of sequences involving high-compactness neutron stars with a tabulated equation of state (Henriksson et al. 2016). We classify the current study shown in Table 2 according to the following seven items (see Appendix A for the details).

  1. 1.

    Metric: Choice of the spatial background metric \({\hat{\gamma }}_{ij}\) and the extrinsic curvature K. The abbreviations “CFMS,” “KS,” and “MKS” mean, respectively, the conformally-flat and maximal-slicing condition (\(K=0\); see Appendix A.1), Kerr–Schild, and modified Kerr–Schild.

  2. 2.

    Hole: Method to handle the singularity associated with the black hole. The abbreviations “Ex” and “Pu” indicate the excision and the puncture approaches, respectively.

  3. 3.

    Spin: Dimensionless spin parameter of the black hole \(\chi \). Note that the spin is zero or anti-aligned with the orbital angular momentum in these work.

  4. 4.

    Flow: State of the fluid flow in the neutron star. The abbreviations “Ir” and “Co” indicate irrotational and corotational flows, respectively. Arbitrary spins of neutron stars are not considered in these computations.

  5. 5.

    EOS: Equation of state for neutron-star matter. Here, “Poly” means the polytrope (specifically, that with \(\varGamma = 2\)) and “Tab” means a tabulated equation of state. Specifically, the SLy equation of state (Douchin and Haensel 2001) is adopted for computing quasiequilibrium sequences in Henriksson et al. (2016).

  6. 6.

    Compactness: Compactness of the neutron star \({\mathscr {C}}\).

  7. 7.

    Mass Ratio: Mass ratio Q.

Quasiequilibrium sequences with high and/or misaligned spins of either component have not been derived. Meanwhile, computations of individual quasiequilibrium states have been extended to a wide range of the parameter space including spin vectors for both black holes and neutron stars (Tacik et al. 2016). Additionally, many quasiequilibrium configurations, including those with a radial approaching velocity to reduce the orbital eccentricity (Foucart et al. 2008; Kyutoku et al. 2021), have been computed as initial conditions of dynamical simulations, whose results will be discussed in Sect. 3.

Table 2 Summary of the study on quasiequilibrium sequences

Numerical results

Hereafter in this section, we focus on the results reported in Taniguchi et al. (2008), because a systematic survey for a wide range of the parameter space was performed only there. Accordingly, all the results are derived in the excision approach, the conformal-flatness and maximal-slicing condition, nonspinning black holes, the irrotational flow, and the \(\varGamma = 2\) polytrope. Still, this work reasonably captures the properties of quasiequilibrium sequences. As the polytropic equation of state has only a single dimensional parameter, \(\kappa \), we may normalize various quantities such as the length, mass, and time by the polytropic length scale,

$$\begin{aligned} R_{\mathrm {poly}} := \kappa ^{1/(2\varGamma - 2)} \left[ = \kappa ^{1/(2\varGamma - 2)} G^{-1/2} c^{(\varGamma - 2)/(\varGamma - 1)} \right] . \end{aligned}$$
(28)

We will put an overbar above a symbol for indicating quantities in the polytropic unit, e.g., \({\bar{M}}_{\mathrm {B}} := M_{\mathrm {B}} / R_{\mathrm {Poly}} [= G M_{\mathrm {B}}/ (c^2 R_{\mathrm {poly}}) ]\).

Figure 2 displays contours of the conformal factor, \(\psi \), for a black hole–neutron star binary with the mass ratio \(Q=3\) and the baryon rest mass of the neutron star \({\bar{M}}_{{\mathrm {B}}}=0.15\) (\({\mathscr {C}} = 0.145\)). This contour plot shows the configuration at the smallest orbital separation for which the code used in Taniguchi et al. (2008) successfully achieved a converged solution. The thick solid circle for \(X<0\) (left) denotes the excised surface, i.e., the apparent horizon, while that for \(X>0\) (right) denotes the surface of the neutron star. A saddle point exists between the black hole and the neutron star, and for this close orbit, it is located in the vicinity of the inner edge of the neutron star. This fact suggests that the orbit of the binary is close to the mass-shedding limit. The value of \(\psi \) on the excised surface is not constant, because a Neumann boundary condition is imposed (see Appendix A.1.1).

Fig. 2
figure 2

Contour of the conformal factor, \(\psi \), on the orbital plane for the innermost configuration with the mass ratio \(Q=3\) and the baryon rest mass of the neutron star \({\bar{M}}_{{\mathrm {B}}}=0.15\) (\({\mathscr {C}} = 0.145\)). The cross symbol indicates the position of the rotational axis. Image reproduced with permission from Taniguchi et al. (2008), copyright by APS

Binding energy and total angular momentum

Figure 3 shows the binding energy, \(E_{{\mathrm {b}}}/m_0 [= E_{\mathrm {b}} / (m_0 c^2)]\), and the total angular momentum, \(J/m_0^2 [= cJ/(Gm_0^2)]\), as functions of the orbital angular velocity, \(\varOmega m_0 (= G \varOmega m_0 /c^3)\), for a binary with \(Q=3\) and \({\bar{M}}_{{\mathrm {B}}}=0.15\) (\({\mathscr {C}} = 0.145\)). All the quantities of the binary are expressed as dimensionless quantities normalized by the total mass, \(m_0\). This figure shows that the numerical results agree quantitatively with the third-order post-Newtonian approximation (Blanchet 2002). The results also agree with the up-to-date, fourth-order post-Newtonian approximations (Blanchet 2014), which differ only by \(< 1\%\) and \(< 3\%\) for the binding energy and the total angular momentum, respectively, from the third-order ones in the range of \(\varOmega m_0\) considered here. For this parameter set, the numerical sequence terminates at the mass-shedding limit, i.e., at an orbit for which a cusp is formed at the inner edge of the neutron star and the material begins to flow out, before the innermost stable circular orbit is encountered, i.e., before the minimum of the binding energy appears.

Fig. 3
figure 3

Binding energy \(E_{{\mathrm {b}}}/m_0 [=E_{\mathrm {b}}/(m_0 c^2)]\) (left) and total angular momentum \(J/m_0^2 [=cJ/(Gm_0^2)]\) (right) as functions of \(\varOmega m_0 (=G \varOmega m_0 /c^3)\) for binaries with \(Q=3\) and \({\bar{M}}_{{\mathrm {B}}}=0.15\) (\({\mathscr {C}} = 0.145\)). The solid curve with the filled circles shows numerical results, and the dashed curve denotes the results derived in the third-order post-Newtonian approximation for point particles (Blanchet 2002). Image reproduced with permission from Taniguchi et al. (2008), copyright by APS

If the binary separation at the mass-shedding limit is substantially larger than the radius of the innermost stable circular orbit, the neutron star is expected not only to start shedding mass but also to be tidally disrupted before being swallowed by the black hole. Equation (9) presented in Sect. 1.3.1 suggests that the ratio of the binary separation at the mass-shedding limit to the radius of the innermost stable circular orbit decreases with increasing the mass ratio, Q, and/or the compactness of the neutron star, \({\mathscr {C}}\). Thus, it is naturally expected that quasiequilibrium sequences encounter minima in the binding energy and the total angular momentum as in the two-point-particle problem in general relativity, if the mass ratio is sufficiently high and/or the compactness is sufficiently large.

Figure 4 shows the binding energy, \(E_{\mathrm {b}}/m_0\), and the total angular momentum, \(J/m_0^2\), as functions of \(\varOmega m_0\) for a binary with \(Q=5\) and \({\bar{M}}_{\mathrm {B}}=0.15\) (\({\mathscr {C}}=0.145\)). While the compactness of the neutron star is the same as that shown in Fig. 3, the mass ratio is higher. In this sequence, an innermost stable circular orbit is encountered before the onset of mass shedding, i.e., we see minima in the binding energy and the angular momentum just before the end of the sequence.

Fig. 4
figure 4

Same as Fig. 3 but for the sequence with \(Q=5\). Image reproduced with permission from Taniguchi et al. (2008), copyright by APS

Figure 4 shows that the turning points in the binding energy and the total angular momentum appear simultaneously within the numerical accuracy as described by theory of binary thermodynamics (Friedman et al. 2002). A simultaneous turning point appears as a cusp in the curve representing the relation between the binding energy and the total angular momentum. This fact is clearly seen in Fig. 5, which shows the relations for sequences with \(Q=5\) but with different compactnesses of neutron stars. As suggested by Eq. (9), turning points are not found for small compactnesses such as \({\mathscr {C}}=0.120\), since the sequences terminate at mass shedding before encountering an innermost stable circular orbit. By contrast, for large compactnesses, these curves indeed form a cusp. Note that sequences derived in post-Newtonian approximations for point particles cannot identify mass shedding and therefore always exhibit turning points. The difference of \(E_{\mathrm {b}} / m_0\) as a function of \(J / m_0^2\) between the third- and fourth-order post-Newtonian approximations is less than 0.1% before the binary reaches an innermost stable circular orbit for the binary parameters chosen here. Although it may appear from Fig. 5 that the numerical results deviate from the post-Newtonian approximation as the compactness increases, i.e., approaching a point-particle limit, quasiequilibrium sequences of binary black holes also deviate from the analytic computation based on the post-Newtonian approximation in a similar manner to the high-compactness sequence of black hole–neutron star binaries (Cook and Pfeiffer 2004; Caudill et al. 2006).

Fig. 5
figure 5

Binding energy as a function of the total angular momentum for binaries with \(Q=5\) and different compactnesses of neutron stars. The solid curve denotes the results derived in the third-order post-Newtonian approximation for point particles (Blanchet 2002). Image reproduced with permission from Taniguchi et al. (2008), copyright by APS

Endpoint of the sequence

One of the most important questions in the study of black hole–neutron star binaries is whether the coalescence leads to mass shedding of the neutron star before reaching the innermost stable circular orbit or to the plunge of the neutron star into the black hole before the onset of mass shedding. The answer to this question is essential for the topics raised in Sect. 1.1: Orbital dynamics and gravitational waves in the late inspiral and merger phases are affected strongly by this issue, and hence, its precise understanding is necessary for developing theoretical templates. For driving a short-hard gamma-ray burst, formation of an accretion disk surrounding the black hole is the most promising model. The r-process nucleosynthesis and electromagnetic emission occur if the material of the neutron star is ejected from the system. Both the disk formation and the mass ejection can result only if the neutron star is disrupted prior to reaching the innermost stable circular orbit.

Quantitative exploration of this issue ultimately requires dynamical simulations, and their results will be reviewed in Sect. 3. However, in-depth studies of quasiequilibrium sequences also provide a guide to the binary parameters that separate mass shedding and the dynamical plunge. In the following, we summarize quantitative insights obtained from the study of quasiequilibrium sequences. Specifically, we will review semiquantitative expressions that may be used to predict whether a black hole–neutron star binary of arbitrary values of Q and \({\mathscr {C}}\) encounters an innermost stable circular orbit before the onset of mass shedding or not. Since the detailed study was performed only for nonspinning black holes, we do not consider the effect of black-hole spins in the following. However, as discussed in Sect. 1.3, we have to keep in mind that the spin of the black hole is crucial for determining the fate of general black hole–neutron star binaries.

Mass-shedding limit

First, we summarize the results for the orbital angular velocity at which mass shedding sets in, i.e., the mass-shedding limit. For this purpose, a mass-shedding indicatorFootnote 5 introduced in studies of binary neutron stars (Gourgoulhon et al. 2001; Taniguchi and Gourgoulhon 2002, 2003)

$$\begin{aligned} {\mathscr {X}} := \frac{\left. \partial {(\ln h)}/\partial {r} \right| _{\mathrm {eq}}}{\left. \partial {(\ln h)}/\partial {r} \right| _{\mathrm {pole}}} \end{aligned}$$
(29)

is used to determine the mass-shedding limit (Taniguchi et al. 2006, 2007, 2008). The numerator is the radial derivative of the log-enthalpy at the surface toward the black hole on the orbital plane. The denominator is that on the pole. Here, the radial coordinate is defined with respect to the center of the neutron star. The mass-shedding indicator, \({\mathscr {X}}\), is unity for a spherical neutron star at infinite orbital separation, while \({\mathscr {X}} = 0\) indicates the formation of a cusp on the stellar surface, and hence, the onset of mass shedding. Note that the quasiequilibrium sequences have been analyzed only for binary systems with the reflection symmetry about the orbital plane. Thus, “pole” and “eq” are well-defined.

In Newtonian gravity and partially-relativistic approaches, simple formulae may be introduced to fit the effective radius of a Roche lobe (Paczyński 1971; Eggleton 1983; Wiggins and Lai 2000; Ishii et al. 2005). In Shibata and Uryū (2006, 2007), a fitting formula is introduced for binaries composed of a nonspinning black hole and a corotating neutron star in general relativity. In this Sect. 2.4.1, we review how to derive a fitting formula from data of Taniguchi et al. (2008) for a nonspinning black hole and an irrotational neutron star.

To derive the fitting formula, we need to determine the orbital angular velocity at the mass-shedding limit. However, stellar configurations with cusps cannot be constructed by the numerical code used in Taniguchi et al. (2008), because it is based on a spectral method and accompanied by the Gibbs phenomena in the presence of a nonsmooth stellar surface (but see also Ansorg et al. 2003; Grandclément 2010). This is also the case for a configuration with smaller values of \({\mathscr {X}} \le 0.5\), even though a configuration with a cusp does not appear. Thus, data for the mass-shedding limits have to be determined by extrapolation.

Taniguchi et al. (2008) tabulated the values of \({\mathscr {X}}\) as a function of the orbital angular velocity and their sequence is extrapolated to \({\mathscr {X}} = 0\) by using polynomial functions to find the orbits at the onset of mass shedding. Figure 6 shows an example of such extrapolations for sequences with \({\mathscr {C}} = 0.145\) and \(Q=1,2,3,\) and 5. By extrapolating results toward \({\mathscr {X}} = 0\), the orbital angular velocity at the mass-shedding limit, \(\varOmega _{\mathrm {ms}}\), is approximately determined for each set of Q and \({\mathscr {C}}\).

Fig. 6
figure 6

Extrapolation of sequences with \({\mathscr {C}} = 0.145\) to the mass-shedding limit, \({\mathscr {X}} = 0\). The thick curves are sequences constructed using numerical data, and the thin curves are their extrapolations. The horizontal axis is the orbital angular velocity in the polytropic unit, \({\bar{\varOmega }} = \varOmega R_{{\mathrm {poly}}} [= \varOmega R_{\mathrm {poly}} / c]\). Image reproduced with permission from Taniguchi et al. (2008), copyright by APS

To derive a fitting formula for \(\varOmega _{\mathrm {ms}}\) as a function of Q and \({\mathscr {C}}\), the Newtonian expression of Eq. (11) is useful, although it is semiquantitative. By fitting the sequence of data with respect to this expression, Taniguchi et al. (2008) determined the value of \(C_{\varOmega }\) for the \(\varGamma = 2\) polytrope to be 0.270, i.e.,

$$\begin{aligned} {\bar{\varOmega }}_{\mathrm {ms}} = 0.270 \frac{{\mathscr {C}}^{3/2}}{{\bar{M}}_{{\mathrm {NS}}}} \left( 1 + Q^{-1} \right) ^{1/2} , \end{aligned}$$
(30)

or equivalently,

$$\begin{aligned} \varOmega _{{\mathrm {ms}}} m_0 = 0.270 {\mathscr {C}}^{3/2} (1 +Q) \left( 1 + Q^{-1} \right) ^{1/2}. \end{aligned}$$
(31)

Figure 7 shows the results of the fitting for the mass-shedding limit. The agreement is not perfect but fairly good for \(Q \ge 2\).

Fig. 7
figure 7

Fit of the mass-shedding limit by the analytic expression, Eq. (30). The mass-shedding limit for each compactness of the neutron star and mass ratio is computed by extrapolating the numerical data. Image reproduced with permission from Taniguchi et al. (2008), copyright by APS

It may be interesting to note that the value of \(C_{\varOmega } = 0.270\) is the same as that found for quasiequilibrium sequences in general relativity of binary neutron stars (Taniguchi and Shibata 2010) and of black hole–corotating neutron star binaries (Shibata and Uryū 2006, 2007). Thus, the value of \(C_{\varOmega } = 0.270\) could be widely used for an estimation of the orbital angular velocity at the mass-shedding limit of a neutron star in a relativistic binary system with \(\varGamma = 2\). We also note that \(C_\varOmega = 0.270\) corresponds to \(c_{\mathrm {R}} = 1.90\), which is defined in Sect. 1.3.1.

Innermost stable circular orbit

Next, we summarize the results for the orbital angular velocity at which the minimum of the binding energy, i.e., the innermost stable circular orbit appears. Because the numerical data are discrete and do not necessarily give the exact minimum, the minimum point may be located approximately by fitting three nearby points of the sequence to a second-order polynomial.

A simple empirical fitting that predicts the angular velocity \(\varOmega _{\mathrm {ISCO}}\) at the innermost stable circular orbit for an arbitrary companion orbiting a black hole may be derived in the manner of Taniguchi et al. (2008). They search for an expression of \(\varOmega _{\mathrm {ISCO}}\) as a function of the mass ratio Q and the compactness \({\mathscr {C}}\) of the companion. Specifically, they assume a functional form of

$$\begin{aligned} \varOmega _{\mathrm {ISCO}} m_0 = c_1 \left[ 1 - \frac{c_2}{Q^{p_1}} \left( 1 - c_3 {\mathscr {C}}^{p_2} \right) \right] , \end{aligned}$$
(32)

where \(c_1\), \(c_2\), \(c_3\), \(p_1\), and \(p_2\) are parameters to be determined in the following manner. The coefficients \(c_1\), \(c_2\), and \(c_3\) are determined for given values of \(p_1\) and \(p_2\) by requiring that three known values of \(\varOmega _{\mathrm {ISCO}}\) are recovered: (1) that of a test particle orbiting a Schwarzschild black hole, \(\varOmega _{\mathrm {ISCO}} m_0 =6^{-3/2}\) (for \(Q \rightarrow \infty \)), (2) that of an equal-mass binary-black-hole system, \(\varOmega _{\mathrm {ISCO}} m_0 =0.1227\) (for \(Q=1\) and \({\mathscr {C}} = 0.5\); Caudill et al. 2006), and finally (3) that of a black hole–neutron star configuration with \(Q=5\) and \({\mathscr {C}}=0.1452\), \(\varOmega _{\mathrm {ISCO}} m_0 = 0.0854\) (Taniguchi et al. 2008). The exponents \(p_1\) and \(p_2\) are determined by requiring the fitted curves to lie near the data points for all the systems.

As demonstrated in Fig. 8, the numerical data are fitted nicely by a function

$$\begin{aligned} \varOmega _{\mathrm {ISCO}} m_0 = 0.0680 \left[ 1 - \frac{0.444}{Q^{0.25}} \left( 1 - 3.54 {\mathscr {C}}^{1/3} \right) \right] . \end{aligned}$$
(33)

The agreement is sufficient for finding the orbital angular velocity at the innermost stable circular orbit within the error of \(\sim 10\%\).

Fig. 8
figure 8

Fit of the minimum point of the binding energy curve by Eq. (33). Image reproduced with permission from Taniguchi et al. (2008), copyright by APS

Critical mass ratio

Combining Eqs. (31) and (33), we can identify the critical binary parameters which separate two final fates that the binary encounters an innermost stable circular orbit before initiating mass shedding or that the neutron star reaches the mass-shedding limit before plunging into the black hole. Figure 9 illustrates the final fate of black hole–neutron star binaries with \({\mathscr {C}} = 0.145\). Because the orbital angular velocities at the mass-shedding limit [Eq. (31)] and the innermost stable circular orbit [Eq. (33)] depend differently on the mass ratio, Q, they intersect in this figure. An inspiraling binary evolves along horizontal lines toward increasing \({\bar{\varOmega }}\), i.e., from the left to the right, until it reaches either the innermost stable circular orbit or the mass-shedding limit. For a sufficiently high mass ratio, the binary reaches an innermost stable circular orbit. Quasiequilibrium sequences cannot predict the fate of the neutron star after that, because it enters a dynamical plunge phase (see Sect. 3). Thus, the mass-shedding limit for unstable quasiequilibrium sequences included in Fig. 9 should be regarded as only indicative. As shown in Fig. 9, the sequence with \(Q=6\) (dot-dashed line) encounters the innermost stable circular orbit, while that with \(Q=3\) (dot-dot-dashed line) ends at the mass-shedding limit. The intersection of the curve for the mass-shedding limit and that for the innermost stable circular orbit marks a critical point that separates the two distinct fates of the binary inspiral. Specifically, the critical mass ratio is found to be \(Q \approx 4.2\) for this case, i.e., binaries of a nonspinning black hole and an irrotational neutron star with \({\mathscr {C}} = 0.145\) modeled by a \(\varGamma = 2\) polytrope.

Fig. 9
figure 9

Example of the boundary between the mass-shedding limit and the innermost stable circular orbit for \({\mathscr {C}} = 0.145\). The solid and dashed curves denote the mass ratio that gives the mass-shedding limit and the innermost stable circular orbit (ISCO), respectively, for a given value of the orbital angular velocity in the polytropic unit. The thin dotted curve denotes the mass-shedding limit for unstable quasiequilibrium sequences. Image reproduced with permission from Taniguchi et al. (2008), copyright by APS

The critical mass ratio which separates black hole–neutron star binaries that encounter an innermost stable circular orbit before initiating mass shedding and those reach the mass-shedding limit before the plunge is obtained as a function of the compactness of the neutron star by equating Eqs. (31) and (33). Specifically, the critical mass ratio is determined by

$$\begin{aligned} 0.270 {\mathscr {C}}^{3/2} (1 + Q) \left( 1 + Q^{-1} \right) ^{1/2} = 0.0680 \left[ 1 - \frac{0.444}{Q^{0.25}} \left( 1 - 3.54 {\mathscr {C}}^{1/3} \right) \right] , \end{aligned}$$
(34)

and the curve that separates those two regions on the Q-\({\mathscr {C}}\) plane is shown in Fig. 10. If the mass ratio of a black hole–neutron star binary is higher than the critical value, the quasiequilibrium sequence terminates by encountering the innermost stable circular orbit, while if lower, it ends at the mass-shedding limit. We emphasize that the mass shedding is only a necessary condition for tidal disruption. In Sect. 3, we will see that dynamical simulations tend to predict that tidal disruption occurs for a more restricted range of parameters than that shown in Fig. 10.

Fig. 10
figure 10

Critical mass ratio which separates black hole–neutron star binaries that encounter an innermost stable circular orbit before initiating mass shedding and those undergoing tidal disruption, as a function of the compactness of the neutron star modeled by the \(\varGamma = 2\) polytrope. As we discuss in Sect. 3, while configurations in “ISCO” nearly certainly end up in the plunge without tidal disruption, only a part of configurations in “Mass-shedding” leads to appreciable tidal disruption. Image reproduced with permission from Taniguchi et al. (2008), copyright by APS

We again caution that the classification shown in Fig. 10 is appropriate only for nonspinning black hole–neutron star binaries. The spin of the black hole significantly modifies the critical mass ratio. For example, if the spin of the black hole is high and prograde, i.e., aligned with the orbital angular momentum of the binary, the region of “Mass-shedding” will be enlarged significantly. Systematic studies on the effect of black-hole spins have not been performed yet, and further investigations are awaited.

Summary and issues for the future

In this section, we have reviewed the current status of the studies on quasiequilibrium sequences of black hole–neutron star binaries in general relativity, focusing mainly on Taniguchi et al. (2008). In particular, we highlighted a curve of the critical mass ratio, which separates black hole–neutron star binaries that encounter an innermost stable circular orbit before initiating mass shedding and vice versa, as a function of the compactness of the neutron star. The result is shown in Eq. (34) and in Fig. 10. Such a critical curve clearly classifies the possible final fate of black hole–neutron star binaries, which depends on the mass ratio and the compactness of the neutron star for a given equation of state. The final fate depends also on the spin of the black hole (see Sect. 3), although only nonspinning configurations are discussed here.

As seen in Table 2, the parameter space surveyed is still quite narrow, partly because the community has been devoting effort to dynamical simulations. A systematic study of quasiequilibrium sequences has been done only for binaries composed of a nonspinning black hole and an irrotational neutron star with the \(\varGamma = 2\) polytrope. It may be useful to survey the remaining parameter space in a systematic manner in the future. Systematic numerical results for such a study will be helpful for predicting the final fate of black hole–neutron star binaries and for checking the results derived in numerical simulations in light of the high computational precision of quasiequilibrium states. Specifically, quasiequilibrium sequences of binaries composed of a spinning black hole and a neutron star with an equation of state other than the \(\varGamma = 2\) polytrope remain to be studied in detail.

The formulation for quasiequilibrium states should be improved further for more rigorous studies. To date, all the computations adopted formulations which solve only five out of ten components of the Einstein equation: constraint equations and the slicing condition (see Appendix A). Employing an improved formulation in which full components of the Einstein equation are solved is left for the future (see Shibata et al. 2004; Cook and Baumgarte 2008 for proposed formulations and Uryū et al. 2006, 2009 for studies of binary neutron stars).

Merger and postmerger simulation

Dynamical simulations for coalescences of black hole–neutron star binaries have been performed in full general relativity by several groups since Shibata and Uryū (2006). These studies have explored the merger process, the criterion for tidal disruption of a neutron star, the mass and the spin parameter of the remnant black hole, properties of the remnant disk and the ejected material, and gravitational waveforms. Results from different groups agree with each other quantitatively when they can be compared. Longterm simulations of the accretion disks around black holes for \(\sim {10}\, {\hbox {s}}\) are also becoming available in recent years. These studies begin to clarify the evolution of the dense and hot accretion disk, neutrino luminosity, the role of neutrino emission, properties of the disk outflow, and possible launch of an ultrarelativistic jet. In this section, we review our current understanding about these topics.

Numerical method for coalescence simulations

Numerical-relativity simulations for black hole–neutron star binary coalescences are performed by solving the Einstein evolution equations with appropriate gauge conditions and hydrodynamics equations, which may involve neutrino-radiation transfer and magnetohydrodynamics. General formulation and numerical techniques are summarized in Appendix B. In this Sect. 3.1, we summarize general aspects of initial data and equations of state adopted in numerical simulations of black hole–neutron star binary coalescences.

Numerical-relativity simulations for black hole–neutron star binaries throughout the inspiral-merger-postmerger phases have typically been performed only for \(\lesssim {100}\, {\hbox {ms}}\) and are still in the early stage for studying the postmerger evolution. To explore the longterm evolution of the merger remnant, simulations of black hole–accretion disk systems have also been performed in full general relativity for \(\gtrsim {1}\, {\hbox {s}}\) (Fujibayashi et al. 2020a, b, see also Most et al. 2021b). The setup of these simulations will be described separately in Sect. 3.5.

Initial condition

Realistic simulations of black hole–neutron star binary coalescences always adopt quasiequilibrium states reviewed in Sect. 2 as their initial conditions. Typically, simulations based on the generalized harmonic formalism with the excision method adopt quasiequilibrium states computed in the excision framework. While many simulations based on the BSSN formalism (or its extension) with the moving-puncture method adopt quasiequilibrium states computed in the puncture framework, quasiequilibrium states computed in the excision framework have also been adopted (Etienne et al. 2008, 2009, 2012a, c; Paschalidis et al. 2015; Ruiz et al. 2018, 2020; Most et al. 2021a). Because the moving-puncture method needs data of the metric inside the excision surface in the initial configurations, the interior needs to be filled artificially by extrapolating the data outside the excision surface. This extrapolation generally produces constraint-violating initial data, and care must be taken so that this violation does not affect significantly the evolution outside the excision surface (Etienne et al. 2007, see also Brown et al. 2007).

Strictly speaking, quasiequilibrium states derived under the assumption of the helical symmetry cannot be realistic, because the radial approaching velocity induced by gravitational radiation reaction is not taken into account. This drawback gives rise to the inspiral motion with the residual eccentricity of \(e \gtrsim 0.01\) for typical initial data. Because the eccentricity of \(e \approx 0.01\) introduces a phase shift larger than the tidal effect (Favata 2014), numerical simulations used for developing theoretical templates are required to adopt initial data with the eccentricity as low as \(e \lesssim {10^{-3}}\) in order not to bias estimation of tidal deformability in the analysis of gravitational waves from realistic circular binaries. Although the residual eccentricity may be reduced if we could start simulations from a distant orbit at which radiation reaction is sufficiently weak, this is not practical with current and near-future computational resources. To obtain low-eccentricity inspirals with a reasonable initial separation, iterative eccentricity reduction is routinely applied in the excision-based simulations (Foucart et al. 2008, see Appendix A.5.2 for details). Essentially the same technique has recently been developed for and applied to puncture-based initial data (Kyutoku et al. 2021).

Because all the quasiequilibrium states have been computed in the framework of pure ideal hydrodynamics assuming that the neutron star is cold, additional variables need to be specified if we perform simulations with detailed microphysics. For evolving neutron stars with composition-dependent equations of state, we need to give the electron fraction in the initial condition. These variables are usually determined by the condition of a(n approximate) zero-temperature \(\beta \)-equilibrium (see, e.g., Duez et al. 2010). This step is in particular necessary for neutrino-radiation-hydrodynamics simulations (Deaton et al. 2013; Foucart et al. 2014, 2017; Brege et al. 2018; Kyutoku et al. 2018; Foucart et al. 2019b; Most et al. 2021a).

For magnetohydrodynamics simulations, magnetic fields are superposed on the initial configuration with arbitrary strength and arbitrary geometry. Their magnitude should not be very large, because too strong magnetic fields destroy the hydrostationary equilibrium. Still, this condition admits an astrophysically strong magnetic fields of \(\lesssim {10^{17}}\, {\hbox {G}}\) in the neutron star, for which the gravitational binding energy is larger by orders of magnitude. To resolve short-wavelength modes associated with the magnetorotational instability in the postmerger phase (Balbus and Hawley 1991), it is customary to impose magnetar-level magnetic fields inside neutron stars. This may be justified, because the magnetic fields are likely to be amplified on a dynamical time scale in the accretion disk formed after merger in the real world. However, it has not been clarified yet whether a strong and coherent poloidal field is really established by some mechanism, e.g., the dynamo process. While many magnetohydrodynamics simulations have adopted poloidal magnetic fields initially confined in the neutron star to avoid difficulty in handling force-free magnetospheres (Chawla et al. 2010; Etienne et al. 2012a, c; Kiuchi et al. 2015b; Most et al. 2021a), pulsar-like dipolar magnetic fields are also adopted with a non-tenuous artificial atmosphere outside the neutron star in simulations that focus on the possible jet launch (Paschalidis et al. 2015; Ruiz et al. 2018, 2020). We note that an artificial atmosphere itself is always required by hydrodynamics simulations performed in a conservative scheme (see also Appendix B.2.2).

Equation of state

The equation of state for neutron-star matter is a key ingredient for deriving realistic outcomes of and multimessenger signals from black hole–neutron star binary coalescences. The primary reason for this is that the equation of state determines the density distribution and hence the radius of the neutron star for a given value of the mass. Whether tidal disruption occurs or not during the coalescence and, if it occurs, its degree are determined primarily by the radius or the compactness of the neutron star for given masses and spins of binary components [see Eq. (34)]. Thus, the gravitational waveform, the properties of the remnant disk, and the properties of the ejecta are governed crucially by the equation of state. The equation of state also determines the tidal deformability of the neutron star, which affects the late inspiral phase of compact binary coalescences.

However, as mentioned in Sect. 2.1.2, the equation of state for supranuclear-density matter is still uncertain (see, e.g., Lattimer and Prakash 2016; Oertel et al. 2017; Baym et al. 2018 for reviews). In the study of compact binary coalescences involving neutron stars, it is more beneficial to explore the possibility of determining the equation of state via gravitational-wave observations (Lindblom 1992; Vallisneri 2000; Read et al. 2009b; Ferrari et al. 2010; Lackey et al. 2012; Maselli et al. 2013; Lackey et al. 2014; Pannarale et al. 2015b) than to derive observable signals relying on a single candidate of the realistic equation of state. For this purpose, it is necessary to prepare theoretical templates of gravitational waveforms by performing simulations systematically over the parameter space of black hole–neutron star binaries with a wide variety of hypothetical equations of state. Such a systematic survey is also indispensable for predicting electromagnetic counterparts (see Sect. 3.4 for the dependence of the disk and ejecta properties on the equation of state). Thus, all the equations of state adopted in simulations reviewed in this article should be understood as hypothetical.

Due to the reason described in Sect. 2.1.2, we may safely adopt zero-temperature equations of state during the inspiral and early merger phases before the shock heating begins to play a role. Moreover, the zero-temperature equations of state are sufficient for simulating black hole–neutron star binary coalescences which do not result in tidal disruption of neutron stars, because essentially no heating process is involved. However, because polytropes are not quantitative models of neutron stars, it is desirable to adopt nuclear-theory-based equations of state for the purpose of investigating realistic black hole–neutron star binary coalescences.

Sophisticated zero-temperature equations of state are implemented in numerical simulations by various means. A straightforward method is to adopt numerical tables calculated based on hypothetical models of nuclear physics (see, e.g., Glendenning and Moszkowski 1991; Müller and Serot 1996; Akmal et al. 1998; Douchin and Haensel 2001; Alford et al. 2005). The drawback of this method is that the capability of systematic studies is limited by available tables. A popular tool for conducting a systematic study is an analytic, piecewise-polytropic equation of state, with which the pressure is given by a broken power-law function of the rest-mass density as

$$\begin{aligned} P ( \rho ) = \kappa _i \rho ^{\varGamma _i} \; ( \rho _i \le \rho < \rho _{i+1} ) , \end{aligned}$$
(35)

where \(i \in [0:n-1]\), \(\rho _0 = 0\), and \(\rho _n \rightarrow \infty \) (Read et al. 2009a, b; Özel and Psaltis 2009, see also O’Boyle et al. 2020 for generalization and Haensel and Potekhin 2004; Lindblom 2010; Potekhin et al. 2013 for other analytic approaches). It has been shown that most of the nuclear-theory-based equations of state for neutron-star matter can be approximated to reasonable accuracy up to the rest-mass density of \(\gtrsim 10^{15} \hbox {gcm}^{-3}\) by piecewise polytropes consisting of one for the crust region and three for the core region if we choose \(\rho _2 = {10^{14.7}} \hbox {gcm}^{-3}\) and \(\rho _3 = {10^{15}} \hbox {gcm}^{-3}\) (Read et al. 2009a).

It is remarkable that the maximum density in the system only decreases in time (except for possible minor fluctuations) during the black hole–neutron star binary coalescences. This is a striking difference from the binary-neutron-star merger, after which a massive or collapsing neutron star with increased central density is formed (see, e.g., Hotokezaka et al. 2011; Takami et al. 2015; Dietrich et al. 2015; Foucart et al. 2016a). This property indicates that black hole–neutron star binaries are not influenced by the equation of state at very high density, e.g., several times the nuclear saturation density, unless the neutron star is close to the maximum-mass configuration. Thus, numerical simulations for binaries with plausibly canonical \(\sim 1.4\,M_\odot \) neutron stars may safely adopt simplified models of nuclear-matter equations of state. For example, if the central density is lower than \({10^{15}} \hbox {gcm}^{-3}\), we may adopt piecewise polytropes with a reduced number of pieces for the core focusing only on its low-density part (Read et al. 2009b; Kyutoku et al. 2010, 2011a). This feature also indicates a weak point that we will not be able to investigate properties of ultrahigh-density matter from observations of black hole–neutron star binaries without extrapolation relying on theoretical models (see, e.g., Abbott et al. 2018; Raaijmakers et al. 2019).

Once the heating process is activated in the merger phase, particularly via the shock associated with self-crossing of the tidal tail, the zero-temperature approximation is no longer valid. Finite-temperature effects become increasingly important in the remnant disk, because temperature increases due to the shock interaction and presumably to subsequent viscous heating associated with magnetohydrodynamical turbulence, while the Fermi energy decreases due to the decreased rest-mass density compared to neutron stars.

One popular and qualitative approach for incorporating finite-temperature effects is to supplement zero-temperature equations of state with an approximate correction. A simple prescription for this purpose is to add an ideal-gas-like term (Janka et al. 1993),

$$\begin{aligned} P_{\mathrm {th}} = ( \varGamma _{\mathrm {th}} - 1 ) \rho \varepsilon _{\mathrm {th}} , \end{aligned}$$
(36)

where \(\varepsilon _{\mathrm {th}} ( \varepsilon , \rho ) := \varepsilon - \varepsilon _{\mathrm {cold}} ( \rho )\) is the finite-temperature part of the specific internal energy with \(\varepsilon _{\mathrm {cold}} ( \rho )\) being the specific internal energy given by a zero-temperature equation of state. Indeed, the ideal-gas equation of state, \(P = ( \varGamma - 1 ) \rho \varepsilon \), which reduces to a polytrope with the adiabatic index \(\varGamma \) for the isentropic fluid and/or at zero temperature assumed in computations of quasiequilibrium, is occasionally adopted in dynamical simulations as a qualitative model of neutron-star matter. A parameter \(\varGamma _{\mathrm {th}}\) represents the strength of the thermal effect, and its appropriate value may be estimated by calibration with simulations performed adopting genuinely finite-temperature equations of state, which are usually given by numerical tables (Bauswein et al. 2010; Figura et al. 2020). It should be cautioned that, however, the constant value of \(\varGamma _{\mathrm {th}}\) is not faithful to nuclear-theory-based calculations (Constantinou et al. 2015) even though uncertain thermal effects at supranuclear density may not be relevant to black hole–neutron star binary coalescences (Carbone and Schwenk 2019). In addition, the use of zero-temperature equations of state is not fully justified after tidal disruption even if the temperature is not increased. This is because, although zero-temperature equations of state are derived as a function of a single variable, e.g., rest-mass density, assuming the \(\beta \)-equilibrium, the rapid decompression of the disrupted material preserves the composition on a dynamical time scale and violates the \(\beta \)-equilibrium condition (Foucart et al. 2017).

To investigate the entire merger and postmerger phases in a self-consistent manner, it is necessary to adopt temperature- and composition-dependent equations of state with an appropriate scheme for neutrino transport (see Appendix B.2.1). These equations of state are usually given in a tabulated form as e.g.,

$$\begin{aligned} P&= P ( \rho , T , Y_{\mathrm {e}} ) , \end{aligned}$$
(37)
$$\begin{aligned} \varepsilon&= \varepsilon ( \rho , T , Y_{\mathrm {e}} ) , \end{aligned}$$
(38)

where T and \(Y_{\mathrm {e}}\) are the temperature and the electron fraction, respectively (see, e.g., Lattimer and Swesty 1991; Shen et al. 1998; Hempel et al. 2012; Steiner et al. 2013; Banik et al. 2014, see also Raithel et al. 2019 for a detailed analytic approach to augment zero-temperature equations of state in a similar manner to Eq. (36)). Because these variables are tightly related to neutrino emission and absorption, neutrino transport is an essential ingredient for accurately determining thermal properties of material in the postmerger phase. Multidimensional neutrino-radiation-hydrodynamics simulations in full general relativity have been developed in the stellar-core collapse (Sekiguchi 2010; Sekiguchi and Shibata 2011) and are later applied to binary neutron stars (Sekiguchi et al. 2011a, b) as well as black hole–neutron star binaries (Deaton et al. 2013; Foucart et al. 2014; Kyutoku et al. 2018).

Current parameter space surveyed

As reviewed in Sect. 2.1, models of black hole–neutron star binaries are characterized by various parameters. In this section, we focus only on the models in which neutron stars are initially in the irrotational state, which is presumably realistic for the majority of compact object binaries as we discussed in Sect. 1.2 (see Foucart et al. 2019a; Ruiz et al. 2020 for studies on spinning neutron stars). Then, the properties of binaries are characterized by the mass of the black hole, \(M_{\mathrm {BH}}\), the spin parameter of the black hole, \(\chi \), its orientation, \(\iota \), and the mass of the neutron star, \(M_{\mathrm {NS}}\). Furthermore, hypothetical equations of state for neutron-star matter should also be regarded as a free parameter (or function) characterizing the models. Properties of an equation of state may be usefully represented by the radius of the neutron star, \(R_{\mathrm {NS}}\), particularly when we focus on tidal disruption.

To date, numerical-relativity simulations of black hole-neutron star binaries have been performed focusing mainly on neutron stars with typical masses in our Galaxy of \(M_{\mathrm {NS}} \approx 1.2\)\(1.5\,M_\odot \) (Tauris et al. 2017; Farrow et al. 2019). Accordingly, the results are reviewed below focusing on the models with these typical values. Because the mass of the neutron star does not vary much among the models, it is useful in many occasions to characterize a binary model by quantities directly related to the criterion for mass shedding to occur outside the innermost stable circular orbit, Eq. (9), namely the mass ratio Q and the compactness \({\mathscr {C}}\) instead of \(M_{\mathrm {BH}}\) and \(R_{\mathrm {NS}}\), respectively. This parametrization is also sufficient for qualitative but scale-free polytropic equations of state (see Sect. 2.3). It will be worthwhile in the future to simulate mergers of black hole–neutron star binaries with \(M_{\mathrm {NS}} \sim 2\,M_\odot \) employing nuclear-theory-based equations of state, particularly in light of possible detections of such neutron stars with gravitational waves (Abbott et al. 2020a, 2021b), although tidal disruption is unlikely to be common due to the large compactness (see also Sect. 4.2.1).

The range of the mass ratio covered by numerical-relativity simulations has been enlarged to \(1 \le Q \lesssim 8.3\) in the last decade. The small value in this range is adopted to clarify the difference between black hole–neutron star binaries and binary neutron stars in the feature of the mergers, and we will discuss this topic in Sect. 4.2.2. The large values of Q are directly related to realistic black hole–neutron star binaries, taking into account the fact that the observed stellar-mass black holes typically have \(M_{\mathrm {BH}} \gtrsim 5\)\(7\,M_\odot \) (Özel et al. 2010; Kreidberg et al. 2012; Abbott et al. 2019a, 2021a, see also Thompson et al. 2019 for a low-mass black-hole candidate with \(\sim 3.3^{+2.8}_{-0.7} \,M_\odot \)).

A wide range of black-hole spins, both in terms of the magnitude and the orientation, have been adopted in simulations of black hole–neutron star binaries. Most of the recent simulations have focused on the prograde spin, because it is required for tidal disruption by massive black holes with \(M_{\mathrm {BH}} \gtrsim 5\,M_\odot \) or \(Q \gtrsim 4\). Notably, the largest value of the spin parameter simulated is increased to \(\chi = 0.97\) (Lovelace et al. 2013). Although this is only the case for a single system with \(Q=3\), the capability of simulating nearly-extremal black holes is important for future investigations of tidal disruption in high mass-ratio systems. Inclined spins of the black holes are also handled in many simulations. Essentially all the orientations of the spin have already been handled, although covering the parameter space becomes computationally demanding simply because of the increased degree of freedom.

After the early days of adopting qualitative ideal-gas (polytropic at zero temperature) equations of state, many simulations have been performed with nuclear-theory-based equations of state. In particular, temperature- and composition-dependent equations of state are routinely adopted in neutrino-radiation-hydrodynamics numerical-relativity simulations. Figure 11 shows the mass-to-radius relations of neutron stars for various equations of state which are popular in numerical-relativity simulations, along with constraints on the neutron-star properties derived by observations of Galactic massive pulsars (Demorest et al. 2010; Antoniadis et al. 2013; Arzoumanian et al. 2018; Cromartie et al. 2020; Fonseca et al. 2021), J0030+0451 by NICER (Miller et al. 2019b), and GW170817 by the LIGO-Virgo collaboration (Abbott et al. 2018). Table 3 summarizes characteristic quantities of neutron stars modeled by these equations of state. Because the maximum mass of the neutron star is widely accepted to exceed \(\sim 2\,M_\odot \) from the observations of massive pulsars, recent numerical simulations seldom employ soft equations of state which are incompatible with these measurements.

Fig. 11
figure 11

Mass-to-radius relation of cold, spherical neutron stars for various equations of state. Unstable configurations with small radii are not shown in this plot. We also display the measured mass (68.3% credibility) of a pulsar in J1614-2230 (cyan band: Demorest et al. 2010; Arzoumanian et al. 2018), that in J0348+0432 (green band: Antoniadis et al. 2013), that in J0740+6620 (magenta band: Cromartie et al. 2020; Fonseca et al. 2021), posterior samples obtained by analysis of J0030+0451 (magenta dot: Miller et al. 2019b, thinned out from provided samples), and those by analysis of GW170817 (green dot: Abbott et al. 2018)

Table 3 List of representative equations of state adopted in simulations of black hole–neutron star binary coalescences and characteristic quantities of neutron stars

Last but not least, various magnetohydrodynamics simulations have been performed. It should be cautioned that the initial strength and geometry of magnetic fields need to be chosen somewhat arbitrarily in current simulations (see Sect. 3.5.3). This limitation might not affect the numerical evolution of the remnant accretion disk if the grid resolution is sufficiently high. This is because the magnetic fields inside the accretion disk are expected to be amplified by the magnetorotational instability and a turbulent state is expected to be developed (see, e.g., Balbus and Hawley 1998 for reviews). Consequently, the relaxed quasisteady state should not depend on the initial conditions. However, the grid resolution is usually not high enough for guaranteeing numerical convergence due to the limited computational resources. Thus, results obtained by current magnetohydrodynamics simulations should be carefully interpreted. We also caution that the global configuration of the magnetic field in the final state could be impacted by the initial choice of a large-scale poloidal field.

Merger process

We begin with the review of the merger process focusing on tidal disruption, subsequent disk formation, and dynamical mass ejection (or absence thereof). In particular, this Sect. 3.3 focuses on the dynamics until \(\sim {10}\, {\hbox {ms}}\) after the onset of merger. Effects of the magnetic field (Chawla et al. 2010; Etienne et al. 2012a, c; Kiuchi et al. 2015b) and/or neutrino transport (Deaton et al. 2013; Foucart et al. 2014; Kyutoku et al. 2018) play a significant role in the dynamical evolution of the system only after the disrupted material winds around the black hole and collides itself to form a circularized disk. Thus, properties of the neutron star are characterized only by the mass and zero-temperature equations of state during the processes discussed here.

The orbital separation of a black hole–neutron star binary decreases due to dissipation of the energy and the angular momentum via gravitational radiation reaction, and eventually two objects merge. As we discussed in Sect. 1, the final fate of black hole–neutron star binary coalescences is classified into two categories (we will later refine this dichotomy). One is the case in which the neutron star is swallowed by the black hole without experiencing tidal disruption. The other is the case in which the neutron star is tidally disrupted outside the innermost stable circular orbit of the black hole. As we described in Sect. 1.3.1, which of these two possibilities is realized is determined primarily by competition between the orbital separation at which the tidal disruption occurs and the radius of the innermost stable circular orbit. Figures 12 and 13 display the snapshots of the rest-mass density and the region inside the apparent horizon on the equatorial plane at selected time slices for typical examples of these two categories (Kyutoku et al. 2015).

Fig. 12
figure 12

Evolution of the rest-mass density profile and the location of the apparent horizon on the equatorial plane for a binary with \(M_{\mathrm {BH}} = 4.05\,M_\odot \), \(\chi = 0\), \(M_{\mathrm {NS}} = 1.35M_{\odot }\), and \(R_{\mathrm {NS}} = {11.1}\, {\hbox {km}}\) (\(Q=3\), \({\mathscr {C}}=0.180\)) modeled by a piecewise-polytropic approximation of the APR4 equation of state (Akmal et al. 1998). The black filled circles denote the regions inside the apparent horizon of the black hole. The color map of each figure shows \(\log _{10} ( \rho [\hbox {gcm}^{-3} ])\). This figure is generated from data of Kyutoku et al. (2015)

Figure 12 illustrates the case in which the neutron star is not tidally disrupted before it is swallowed by the black hole. This system is characterized by \(M_{\mathrm {BH}} = 4.05\,M_\odot \), \(\chi = 0\), \(M_{\mathrm {NS}} = 1.35\,M_\odot \), and \(R_{\mathrm {NS}} = {11.1}\, {\hbox {km}}\) (\(Q=3\), \({\mathscr {C}} = 0.180\)) modeled by a piecewise-polytropic approximation of the APR4 equation of state (Akmal et al. 1998). Because the neutron star is tidally deformed significantly only after it comes very close to the black hole, mass shedding sets in for an orbit too close to induce subsequent disruption outside the innermost stable circular orbit. This is consistent with the expectation from Fig. 10 presented in Sect. 2.4. Accordingly, the masses of the remnant disk and the dynamical ejecta are negligible, say, \(\ll 0.01\,M_\odot \). At the same time, most of the neutron-star material falls into the black hole approximately simultaneously through a narrow region of the horizon. This coherent infall efficiently excites quasinormal-mode oscillations of the remnant black hole. We discuss gravitational waves later in Sect. 3.6. Overall, the behavior of the system in this category universally resembles that of binary-black-hole coalescences with the same masses and spins of components, because the finite-size effect of the neutron star does not play a role (Foucart et al. 2013a). This seems to be the case for all the black hole–neutron star binaries and their candidates reported as of 2021 (Abbott et al. 2021a, b).

Fig. 13
figure 13

Same as Fig. 12 but for a binary with \(M_{\mathrm {BH}} = 4.05M_{\odot }\), \(\chi = 0.75\), \(M_{\mathrm {NS}} = 1.35\,M_\odot \), and \(R_{\mathrm {NS}} = {11.1}\, {\hbox {km}}\) (\(Q=3\), \({\mathscr {C}}=0.180\)) modeled by a piecewise-polytropic approximation of the APR4 equation of state (Akmal et al. 1998). This figure is generated from data of Kyutoku et al. (2015)

Figure 13 illustrates the case in which the neutron star is disrupted before the binary reaches the innermost stable circular orbit. This system is characterized by \(M_{\mathrm {BH}} = 4.05\,M_\odot \), \(\chi = 0.75\), \(M_{\mathrm {NS}} = 1.35\,M_\odot \), and \(R_{\mathrm {NS}} = {11.1}\, {\hbox {km}}\) (\(Q=3\), \({\mathscr {C}} = 0.180\)) modeled by a piecewise-polytropic approximation of the APR4 equation of state (Akmal et al. 1998). In this case, mass shedding from an inner cusp of the deformed neutron star sets in at an orbital separation much larger than that of the innermost stable circular orbit. After a substantial amount of material is removed from the inner cusp, the neutron star is tidally disrupted outside the innermost stable circular orbit. It should be emphasized that tidal disruption does not occur immediately after the onset of mass shedding but occurs for an orbital separation smaller than that for the onset of mass shedding as illustrated by Fig. 13. Thus, conditions such as Eq. (34) are not a sufficient condition but a necessary condition for tidal disruption.

Once the neutron star is disrupted, the material spreads around the black hole and forms a one-armed spiral structure, so-called tidal tail. As a result of the angular momentum transport from the inner to the outer parts of the tidal tail, a large amount of material in the outer part avoids being swallowed immediately by the black hole. Because the tidal tail is in differential rotation, it eventually winds around the black hole and collides with itself (the right middle panel of Fig. 13). This results in circularization and thus formation of an approximately axisymmetric disk surrounding the remnant black hole. The disk material can no longer be treated as zero temperature because of shock heating, and longterm simulations for the disk require appropriate implementations of finite-temperature effects. Still, the disk does not become completely axisymmetric in the typical rotational period of \(\sim {5}\, {\hbox {ms}} (m_0 / 10\,M_\odot )\) (see also Sect. 3.4.1). A one-armed spiral structure with a small amplitude persists for a long time and gradually transports the angular momentum outward. Hence, mass accretion by the black hole continues even if viscous or magnetohydrodynamical processes do not set in. Because the accretion time scale is much longer than the rotational period, the disk is in a quasisteady state on a time scale of \(\gg {10}\, {\hbox {ms}}\). This evolution process agrees qualitatively with that found for longterm evolution of black hole–accretion disk systems (Hawley 1991; Korobkin et al. 2011; Kiuchi et al. 2011; Wessel et al. 2021). In reality, longterm evolution of the disk will be driven by magnetically-induced turbulent viscosity (Balbus and Hawley 1991). We will defer discussions about this stage to Sect. 3.5.

The outermost part of the tidal tail obtains energy sufficient to become unbound from the remnant black hole. Figure 14 visualizes the process of dynamical mass ejection on the phase space of the specific energy and the specific angular momentum for a system with \(M_{\mathrm {BH}} = 3\,M_\odot \), \(\chi = 0\), \(M_{\mathrm {NS}} = 1.35\,M_\odot \), and \(R_{\mathrm {NS}} = {12.3}\, {\hbox {km}}\) (\(Q \approx 2.2\), \({\mathscr {C}}=0.162\)) modeled by a piecewise polytrope called H (Hayashi et al. 2021). Here, unboundedness is identified by the criterion \(-u_t > 1\), which is suitable for dynamical mass ejection as far as the spacetime is approximated as stationary, because the shock heating, and hence, the contribution of the internal energy, does not play a role. If the internal energy contributes significantly, a reasonable criterion may be defined based on \(-hu_t\) taking the offset associated with the composition into account (see, e.g., Fujibayashi et al. 2020a). First, the outer part acquires the angular momentum and also the energy via the tidal torque (from the left top to the right top panels). Next, it gains exclusively the energy via work done by impulsive outward radial force, which is likely to be associated with the infall of the major part of the neutron star to the black hole (from the right top to the left bottom panels). If the energy of a fluid element exceeds the gravitational binding energy, the element escapes from the system as the dynamical ejecta.

Fig. 14
figure 14

Distribution of material on the phase space of specific energy \({\tilde{E}}\) and specific angular momentum normalized by the mass of the remnant black hole \({\hat{J}}\) for a binary with \(M_{\mathrm {BH}} = 3\,M_\odot \), \(\chi = 0\), \(M_{\mathrm {NS}} = 1.35\,M_\odot \), and \(R_{\mathrm {NS}} = {12.3}\, {\hbox {km}}\) (\(Q \approx 2.2\), \({\mathscr {C}}=0.162\)) modeled by a piecewise polytrope called H (Read et al. 2009b). The left top panel shows the distribution in the final stage of the inspiral. The right top panel shows the distribution at the onset of merger. The distribution is broadened and spans a wide range of \({\tilde{E}}\) and \({\hat{J}}\) due to the angular momentum transport. The left bottom panel shows the state after the infall of the material with low \({\hat{J}}\). Only the material with the angular momentum exceeding that for the innermost stable circular orbit, \({\hat{J}}_{\mathrm {ISCO}}\), remains outside the black hole. This material has gained exclusively the energy during the coalescence of the black hole and the major part of the neutron star. The right bottom panel shows the stage in which the remnant disk establishes a quasisteady state. The purple dashed curves denote the relation for stable circular orbits (Bardeen et al. 1972), and the material along this curve is the remnant disk. The cyan dashed curves denote the relation for material with a fixed periastron distance, and the material along this curve consists of the dynamical ejecta and fallback material. Image reproduced with permission from Hayashi et al. (2021), copyright by APS

Material behind the dynamical ejecta also acquires some energy but remains bound to the remnant black hole, and thus it eventually falls back onto the disk. The disk and fallback components may approximately be distinguished by temperature higher and lower than 0.1–1 MeV, respectively, as shown in Fig. 15 generated by Brege et al. (2018), because the shock interaction sets in when the fallback material hits the outer edge of the disk. Although the longterm fallback dynamics cannot be fully tracked in current simulations of black hole–neutron star binary coalescences, estimated fallback rates of the mass are found to coincide with the well-known \(t^{-5/3}\) law for tidal disruption events (Chawla et al. 2010; Kyutoku et al. 2015; Brege et al. 2018, see also Rosswog 2007 for early Newtonian work).

Fig. 15
figure 15

Temperature profile and the location of the apparent horizon (black filled circle) at 7  ms after the onset of merger for a binary with \(M_{\mathrm {BH}} = 7\,M_\odot \), \(\chi = 0.9\), \(M_{\mathrm {NS}} = 1.2\,M_\odot \), and \(R_{\mathrm {NS}} = {13.5}\, {\hbox {km}}\) (\(Q \approx 5.8\), \({\mathscr {C}}=0.130\)) modeled by the FSU2.1 equation of state (Todd-Rutel and Piekarewicz 2005; Shen et al. 2011). The left panel is magnification of the white box in the right panel. The white, red, and black curves denote \(10^{11}\), \(10^{10}\), and \({10^{9}} \hbox {gcm}^{-3}\), respectively. Image reproduced with permission from Brege et al. (2018), copyright by APS

The left bottom panel of Fig. 14 implies that the fallback material and the dynamical ejecta may be considered to be launched from an approximately common periastron. Taking the \(t^{-5/3}\) fallback behavior into account, the process depicted here might seem similar to tidal disruption of stars by supermassive black holes in slightly unbound, hyperbolic encounters (Rees 1988; Phinney 1989). However, it should be remarked that gravitational-wave-driven mergers of compact object binaries occur in a strongly bound, quasicircular orbit. Therefore, it is not trivial a priori that even a finite amount of material could be ejected by tidal disruption in black hole–neutron star binary coalescences.

The process of tidal disruption described in Fig. 13 is qualitatively common for systems with a large neutron-star radius, a small black-hole mass, and/or a high black-hole spin. However, quantitative details depend on the parameters of the binary. The orientation of the black-hole spin also introduces qualitative differences in the merger dynamics and morphology of the remnant. In the following, we review the dependence on these parameters.

Dependence on the equation of state

As found from the analysis of Sects. 1.3 and 2, the merger process depends on the compactness of the neutron star, which is determined by the equation of state. Systematic studies performed employing a variety of piecewise polytropes clearly show that neutron stars with smaller compactnesses are tidally disrupted more easily (Kyutoku et al. 2010, 2011a, 2015). This tendency also holds for nuclear-theory-based tabulated equations of state (Kyutoku et al. 2018; Brege et al. 2018).

Even if the compactness and the mass are identical, the density profiles generally differ among neutron stars modeled by different equations of state. If the density profile is more centrally condensed, the neutron star is less subject to tidal disruption as discussed in Sect. 1.3.2. This tendency is demonstrated by a study employing two-piecewise polytropes with different adiabatic indices for the core region (Kyutoku et al. 2010). Specifically, if the adiabatic index for the core region is smaller, the neutron star with a given compactness is more centrally condensed and less subject to tidal disruption.

Dependence on the mass ratio

As indicated by the analysis of Sects. 1.3 and 2, the possibility of tidal disruption increases as the mass ratio decreases. For example, the amount of the mass remaining outside the black hole is likely to be larger for the lower mass ratio. This dependence is particularly important for nonspinning black holes, because significant tidal disruption occurs only for low-mass black holes such as those in the putative mass gap (Shibata et al. 2009, 2012). Stated differently, for a plausibly realistic mass ratio of \(Q \gtrsim 4\), the neutron star can be tidally disrupted only if the black hole has a high prograde spin, as we discuss in Sect. 3.3.3.

After the discovery of binary neutron stars by gravitational waves, very-low-mass black hole–neutron star coalescences acquire renewed interest (Foucart et al. 2019a, b; Hayashi et al. 2021; Most et al. 2021a, see Sect. 1.4.4 for definition of “very low mass”). The primary reason for this is that they could be potential mimickers of binary-neutron-star coalescences, rendering astrophysical interpretation of gravitational-wave sources ambiguous (Hinderer et al. 2019; Kyutoku et al. 2020). Distinguishing very-low-mass black hole–neutron star binaries and binary neutron stars would be invaluable for gaining knowledge about the maximum mass of neutron stars, the mass gap between black holes and neutron stars (see, e.g., Kreidberg et al. 2012), and the formation mechanism of these compact objects, i.e., stellar core collapse and supernova explosions.

Recent numerical simulations have shown that susceptibility to tidal disruption is not reflected monotonically in the remnant material for very-low-mass-ratio systems (Foucart et al. 2019b; Hayashi et al. 2021; Most et al. 2021a). Rather, the mass of the remnant disk saturates to a value of \(\sim {\mathscr {O}}({0.1}) \,M_\odot \) for the case in which the black hole is nonspinning (see also Brege et al. 2018 and Appendix of Hayashi et al. 2021). Quantitatively, the saturated value of the disk mass depends on the equation of state. Moreover, the mass of the dynamical ejecta, which increases with decreasing Q down to a moderately large values of \(Q = Q_{\mathrm {peak}} \sim 3\), begins to decrease as the mass ratio decreases for \(Q < Q_{\mathrm {peak}}\) (Hayashi et al. 2021). The precise value of \(Q_{\mathrm {peak}}\) again depends on the equation of state. We will discuss quantitative dependence of the remnant disk and the dynamical ejecta on the mass ratio later in this section.

We note that, while the physical reason of the behavior at very low-mass ratios described above is not fully understood yet, it may not be unexpected taking the fact that an extremely low-mass black hole with \(M_{\mathrm {BH}} \ll M_{\mathrm {NS}} (Q \ll 1)\) cannot make the neutron star unbound because of the tiny contribution of such a minute black hole to the dynamics of the entire system. Related simulations have been performed in the context of consumption of a neutron star by an endoparasitic black hole at the center (East and Lehner 2019; Richards et al. 2021).

Dependence on the black-hole spin (I) aligned spin

The spin of the black hole quantitatively modifies the orbital evolution in the late inspiral phase and the merger dynamics. First, we focus on the cases in which the black-hole spin is (anti-)aligned with respect to the orbital angular momentum of the binary. Figure 16 generated by Etienne et al. (2009) shows the trajectories of the black hole and the neutron star for systems characterized by \(Q=3\), \({\mathscr {C}}=0.145\) modeled by a \(\varGamma = 2\) polytrope, and two different values of the spin parameter \(\chi = 0\) (left) and \(\chi = 0.75\) (right). Because a polytropic equation of state is adopted, the mass of the binary components can be scaled arbitrarily as discussed in Sect. 2.3. Both systems have the same values of initial orbital angular velocity normalized by the total mass, \(G m_0 \varOmega / c^3 \approx 0.033\). The nonspinning (\(\chi = 0\)) and spinning (\(\chi = 0.75\)) systems merge after \(\sim 4\) and 6 orbits, respectively. The difference in the number of orbits is ascribed mainly to the spin-orbit interaction. Specifically, this interaction serves as repulsive force for a prograde spin of the black hole, \(\chi > 0\), and vice versa (see, e.g., Kidder et al. 1993; Kidder 1995 for two-body equations of motion in the post-Newtonian approximation). The repulsion for \(\chi > 0\) counteracts the gravitational pull between the binary components and reduces the orbital angular velocity to maintain a circular orbit for a given orbital separation (or a given circumferential radius of the orbit). Because the gravitational-wave luminosity is as sensitive to the orbital angular velocity as \(\propto \varOmega ^{10/3}\), the approaching velocity associated with the radiation reaction is also decreased. This effect increases the lifetime of the binary. In addition, the spin-orbit repulsion decreases the radius of the innermost stable orbit and strengthens gravitational binding there (Bardeen et al. 1972). This further helps to increase the lifetime of a progradely-spinning black hole–neutron star binary, because it needs to emit a larger amount of energy to reach the innermost stable circular orbit than that for a nonspinning black hole. These effects increase the number of inspiral orbits.

Fig. 16
figure 16

Coordinate trajectory of the black hole (black solid curve) and the neutron star (blue dashed curve) on the orbital plane for binaries with \(Q=3\), \({\mathscr {C}}=0.145\) modeled by a \(\varGamma = 2\) polytrope, and \(\chi = 0\) (left) and \(\chi = 0.75\) (right). Image reproduced with permission from Etienne et al. (2009), copyright by APS

The higher the prograde spin of the black hole, the neutron star is disrupted more easily, and thus, the disk formation and the mass ejection are more pronounced. This is clearly shown by comparing Figs. 12 and 13, between which the only difference is the spin of the black hole. Quantitatively, while the mass of the disk for \(\chi = 0\) is less than \({10^{-3}} \,M_\odot \), it increases to \(0.19 \,M_\odot \) for \(\chi = 0.75\) in these examples. The mass of the dynamical ejecta also increases as the black-hole spin increases, specifically from \(\ll {10^{-3}} \,M_\odot \) to \(0.01\,M_\odot \) in these examples (Kyutoku et al. 2015). These increases are ascribed primarily to the small radius of the innermost stable circular orbit with the prograde spin (Bardeen et al. 1972). As an extreme, it has been shown that about more than a half of the neutron-star material remains outside the remnant black hole right after the onset of merger for a binary with \(Q=3\), \({\mathscr {C}}=0.144\) modeled by a \(\varGamma = 2\) polytrope, and \(\chi = 0.97\), which is the largest value of the spin parameter simulated for black hole–neutron star binaries to date (Lovelace et al. 2013). In light of the astrophysically plausible range of Q and \({\mathscr {C}}\) (e.g., \(Q \gtrsim 4\) and \({\mathscr {C}} \gtrsim 0.16\)), it is remarkable that the prograde spin enables tidal disruption to occur for a binary which results in the plunge if the black hole is nonspinning. By contrast, if the spin of the black hole is retrograde, the neutron star is swallowed by the black hole without tidal disruption even if \(Q<3\) for a wide range of equations of state. We defer further quantitative discussions to Sect. 3.4.2.

If tidal disruption occurs in a binary with a spinning black hole and a realistic mass ratio of \(Q \gtrsim 4\), the elongated neutron star can be swallowed by the black hole through a narrow region of its large surface (Kyutoku et al. 2011a). This feature is advantageous for exciting nonaxisymmetric, fundamental quasinormal modes of the remnant black hole as we discuss in Sect. 3.6. This does not occur for nonspinning black holes, because the tidal disruption is possible only for a binary with a low mass ratio of \(Q \lesssim (3{\mathscr {C}})^{-3/2}\) [cf., Eq. (9) and Fig. 10] and thus for a black hole with a small surface. For such a case, the tidally-disrupted material is swallowed through a wide region of the black-hole surface, and the quasinormal-mode excitation is suppressed. These differences are reflected in both gravitational waveforms and spectra as predicted by a black-hole perturbation study (Saijo and Nakamura 2000, 2001).

Figure 17 illustrates the case described above, i.e., the tidally-elongated neutron star is swallowed through a narrow region of the black-hole surface (Kyutoku et al. 2011a). This system is characterized by \(M_{\mathrm {BH}} = 4.05\,M_\odot \), \(M_{\mathrm {NS}} = 1.35\,M_\odot \), \(R_{\mathrm {NS}} = {11.6}\, {\hbox {km}}\) (\(Q=3\), \({\mathscr {C}} = 0.172\)) modeled by a piecewise polytrope called HB, and a moderately high and prograde spin of \(\chi = 0.5\). Tidal disruption occurs at an orbit outside but close to the innermost stable circular orbit. The dense part of disrupted material does not have a sufficient time for winding around the black hole before the infall. Thus, it falls into the black hole in a significantly nonaxisymmetric manner and excites quasinormal-mode oscillations. This behavior is frequently found for a binary with a high mass ratio and a high black-hole spin. Conversely, for a retrograde spin, tidal disruption becomes insignificant even for a small value of \(Q=2\)–3 (Kyutoku et al. 2011a). An alternative interpretation of this finding is that the orientation of the black-hole spin plays an important role. We will discuss this viewpoint for a general inclination angle in Sect. 3.3.4.

Fig. 17
figure 17

Same as Fig. 12 but for a binary with \(M_{\mathrm {BH}} = 4.05\,M_\odot \), \(\chi = 0.5\), \(M_{\mathrm {NS}} = 1.35\,M_\odot \), and \(R_{\mathrm {NS}} = {11.6}\, {\hbox {km}}\) (\(Q=3\), \({\mathscr {C}}=0.172\)) modeled by a piecewise polytrope called HB (Read et al. 2009b). Image reproduced with permission from Kyutoku et al. (2011a), copyright by APS

To summarize, numerical-relativity simulations have revealed that the merger process may be classified into three types according to the mass and the spin of black holes for a given equation of state (see also Pannarale et al. 2013, 2015a for relevant classifications):

  1. 1.

    The neutron star is tidally disrupted at an orbit far from the innermost stable circular orbit. This occurs for the cases in which the black-hole mass is small and/or the black-hole spin is prograde and sufficiently high.

  2. 2.

    The neutron star is tidally disrupted at an orbit close to the innermost stable circular orbit. This occurs for the cases in which the black-hole mass is not small and the black-hole spin is prograde and high.

  3. 3.

    The neutron star is not tidally disrupted. This occurs for the cases in which the black-hole mass is not small and/or the black-hole spin is retrograde or prograde but not high. An approximate criterion for tidal disruption is found in Eq. (9).

These three types are displayed schematically in Fig. 18. The differences of merger processes for these types, particularly that between 1 and 2, are imprinted in gravitational waveforms and spectra described in Sect. 3.6.

Fig. 18
figure 18

Schematic picture for three types of merger processes. The filled circle, the solid red circle, and the black dashed circle denote the black hole, the innermost stable circular orbit, and the radius at which tidal disruption occurs, respectively. The deformed ellipsoid denotes the neutron star. The left, middle, and right panels correspond to the types 1, 2, and 3, described in the body text, respectively. Image adapted from Kyutoku et al. (2011a), copyright by APS

Dependence on the black-hole spin (II) inclined spin

The inclination angle has a qualitative impact on the inspiral and merger dynamics (Foucart et al. 2011, 2013b; Kawaguchi et al. 2015; Foucart et al. 2017, 2021). Figure 19 compares typical orbital evolution of black hole–neutron star binaries for which the spin of the black hole is absent or (anti-)aligned with respect to the orbital angular momentum of the binary (left: \(\chi = 0\)) and inclined (right: \(\chi = 0.75\) and \(\iota \approx {90}^{\circ }\)). The reflection symmetry about the orbital plane is lost in the presence of spin misalignment. Because the orbital angular velocity vector is inclined with respect to the total angular momentum of the system, the vector normal to the orbital plane precesses approximately around the total angular momentum during the inspiral phase (Apostolatos et al. 1994; Kidder 1995; Racine 2008).

Fig. 19
figure 19

Evolution of the coordinate separation between the black hole and the neutron star for binaries with \(M_{\mathrm {BH}} = 4.05\,M_\odot \), \(M_{\mathrm {NS}} = 1.35\,M_\odot \), and \(R_{\mathrm {NS}} = {11.1}\, {\hbox {km}}\) (\(Q=3\), \({\mathscr {C}}=0.180\)) modeled by a piecewise-polytropic approximation of the APR4 equation of state (Akmal et al. 1998). The spin of the black hole is zero for the left panel and \(\chi = 0.75\) with the inclination angle \(\iota \approx {90}^{\circ }\) for the right panel. The z-axis is taken to be the direction of the total angular momentum at the initial instant. This figure is generated from data of Kyutoku et al. (2021)

The spin misalignment also reduces the degree of tidal disruption, as well as the masses of the remnant disk and the dynamical ejecta, for the same magnitude of the spin. This is because the spin-orbit coupling is proportional to, in the post-Newtonian terminology, the inner product \(\mathbf{S} {\varvec{\cdot }}\mathbf{L} \) of the spin angular momentum \(\mathbf{S} \) and the orbital angular momentum \(\mathbf{L} \). The effect of the black-hole spin on the radius of the innermost stable circular orbit, or an innermost stable spherical orbit (Hughes 2001; Buonanno et al. 2006; Fragile et al. 2007; Stone et al. 2013), is also determined primarily by this inner product. Thus, the spin-orbit repulsion for a given magnitude of the spin becomes weak as the inclination angle increases. For systems with the black-hole spin being confined in the orbital plane, \(\mathbf{S} {\varvec{\cdot }}\mathbf{L} \approx 0\), the spin of the black hole is likely to play only a minor role in tidal disruption of the neutron star irrespective of its magnitude, while systematic surveys have not yet been performed.

Orbital precession caused by the spin misalignment introduces qualitative differences also in the morphology of the disrupted material (Foucart et al. 2013b; Kawaguchi et al. 2015). Because the misalignment breaks the reflection symmetry, the remnant also exhibits a reflection-asymmetric structure. Figure 20 generated by Foucart et al. (2013b) shows three-dimensional plots of the rest-mass density and the region inside the apparent horizon. The system on the left column is characterized by \(Q=7\), \({\mathscr {C}}=0.144\) modeled by a \(\varGamma = 2\) polytrope, and \(\chi = 0.9\) aligned with the orbital angular momentum of the binary. The merger dynamics and the morphology of the remnant are essentially the same as those described in Fig. 13. The system on the right column of Fig. 20 has the same parameters as those on the left column, except that the black-hole spin has an inclination angle of \(\iota = {40}^{\circ }\) with respect to the orbital angular momentum. The tidal tail of this system inherits precessing motion of the inspiral phase. Accordingly, it does not form a circularized disk immediately after single orbital revolution. Instead, the tidal tail eventually collides with itself from various directions and forms a thick torus. The inclination angle between the angular momentum of the remnant torus and the spin angular momentum of the remnant black hole becomes smaller than the inclination angle during the inspiral phase, \(\iota \), because a substantial fraction of the orbital angular momentum is brought into the black hole by the infalling neutron-star material (Foucart et al. 2011; Kawaguchi et al. 2015). This effect is more significant for lower mass-ratio systems, for which the initial spin angular momentum of the black hole accounts for a smaller fraction of the total angular momentum.

Fig. 20
figure 20

Three-dimensional plot of the rest-mass density and the location of the apparent horizon (denoted by the filled black region) for binaries with \(Q=7\), \(\chi = 0.9\), and \({\mathscr {C}}=0.144\) (\(R_{\mathrm {NS}} = {14.4}\, {\hbox {km}}\) if we suppose a \(1.4\,M_\odot \) neutron star) modeled by a \(\varGamma = 2\) polytrope. The angles between the spin angular momentum of the black hole and the orbital angular momentum of the binary are \({0}^{\circ }\) and \({40}^{\circ }\) for the left and right panels, respectively. The top and bottom panels show snapshots at different times. Image reproduced with permission from Foucart et al. (2013b), copyright by APS

In the long run, a tilted disk will evolve in a manner different from an aligned one via the Lense-Thirring precession (Bardeen and Petterson 1975; Papaloizou and Pringle 1983), magnetically-induced turbulent viscosity (Fragile et al. 2007) and/or magnetic coupling with the remnant black hole (McKinney et al. 2013). Longterm evolution of a remnant torus of precessing black hole–neutron star binaries with realistic microphysics is a subject for future studies in numerical relativity (but see Mewes et al. 2016 for pure hydrodynamics).

Remnant

In this Sect. 3.4, we present quantitative details of the remnant black hole, disk, fallback material, and dynamical ejecta derived by merger simulations. Unless explicitly stated, we discuss the cases in which the spin of the black hole is aligned with respect to the orbital angular momentum and the system possesses reflection symmetry about the orbital plane. We will make it explicit when we consider the effects of the spin misalignment. The longterm evolution of the remnant disk and associated outflows are discussed separately in Sect. 3.5.

Black hole

The mass and the spin of the black hole change during merger, because it swallows the material of the neutron star. The mass of the remnant black hole \(M_{\mathrm {BH,f}}\) is approximately estimated by (Shibata and Uryū 2007)

$$\begin{aligned} M_{\mathrm {BH,f}} \approx M_{\mathrm {BH}} + M_{\mathrm {NS}} - M_{r>{r_{\mathrm {AH}}}} - E_{\mathrm {GW}} , \end{aligned}$$
(39)

where \(M_{r>{r_{\mathrm {AH}}}}\) denotes the mass of the material remaining outside the black hole, which is composed of the remnant disk, the fallback material, and the dynamical ejecta, and \(E_{\mathrm {GW}}\) denotes the energy carried away by gravitational radiation (see Duez et al. 2008; Shibata et al. 2009 for other methods of estimation). Because a large fraction of the neutron-star material falls into the black hole for most cases and also \(E_{\mathrm {GW}}\) is much smaller than the total rest-mass energy of the system, \(M_{\mathrm {BH,f}}\) is larger than \(0.9 m_0\) except for nearly-extremal spins (Lovelace et al. 2013). Because both \(M_{r>{r_{\mathrm {AH}}}}\) and \(E_{\mathrm {GW}}\) decrease as the spin of the black hole decreases, \(M_{\mathrm {BH,f}}\) becomes close to \(m_0\) for small values of \(\chi \) and/or large inclination angles. Specifically, the difference between \(M_{\mathrm {BH,f}}\) and \(m_0\) is only a few percent for anti-aligned, retrograde spins of \(\chi = -0.5\) (stated differently, \(\chi = 0.5\) and \(\iota = {180}^{\circ }\); Kyutoku et al. 2011a).

The spin parameter of the remnant black hole \(\chi _{\mathrm {f}}\) is determined primarily by the mass ratio of the binary, Q, and the initial spin parameter of the black hole, \(\chi \) (Kyutoku et al. 2011a). Figure 21 shows the spin parameter of the remnant black hole as a function of the initial spin parameter \(\chi \) for various mass ratios obtained by pure hydrodynamics simulations performed by several groups with various numerical implementations. Some particular combinations of Q and \(\chi \) are simulated by several groups independently, and results are overplotted. Taking inherent variation associated with the neutron-star mass, the neutron-star equation of state, and the methods for evaluating \(\chi _{\mathrm {f}}\) (Duez et al. 2008; Shibata et al. 2009) into account, all the results are consistent among independent groups. This figure shows that the dependence on the initial spin parameter is more pronounced for higher mass ratios. The reason for this is that, as we describe below, the spin angular momentum becomes dominant and the orbital angular momentum gives a minor contribution in such systems. Meanwhile, the remnant spin parameter is found to depend only weakly on the equation of state (not shown in this figure), and this fact indicates that the material remaining outside the remnant black hole takes only a minor fraction of the mass and the angular momentum for the systems considered. It is also found that the remnant spin parameter is larger than that for binary black holes with the same values of Q and \(\chi \) for the case in which tidal disruption occurs. For example, the coalescence of equal-mass, nonspinning binary black holes is known to form a Kerr black hole with \(\chi _{\mathrm {f}} \approx 0.686\) (Scheel et al. 2009), which is significantly smaller than \(\chi _{\mathrm {f}} \approx 0.84\) for black hole–neutron star binaries (Etienne et al. 2009; Foucart et al. 2019b). This difference stems from the fact that black hole–neutron star binaries which result in tidal disruption do not experience orbits as close as those in binary black holes. Accordingly, black hole–neutron star binaries do not emit gravitational waves as strongly as binary black holes do.

Fig. 21
figure 21

Spin parameter of the remnant black hole, \(\chi _{\mathrm {f}}\), for a range of binary parameters as a function of the initial spin parameter, \(\chi \). The solid lines denote the fitting for the results of \(Q=2\) and 3 (Kyutoku et al. 2011a). This figure is generated from data of Etienne et al. (2009); Kyutoku et al. (2010); Foucart et al. (2011); Kyutoku et al. (2011a); Foucart et al. (2012); Kyutoku et al. (2015). If the neutron-star mass and/or equation of state are varied in a single paper, the results are averaged

Qualitative dependence of \(\chi _{\mathrm {f}}\) on Q and \(\chi \) can be understood by the following analysis. The total angular momentum of two point particles in a circular orbit with the orbital angular velocity \(\varOmega \) is given in Newtonian gravity by

$$\begin{aligned} J_{\mathrm {orb}} = \frac{G^{2/3} M_{\mathrm {BH}} M_{\mathrm {NS}}}{(\varOmega m_0)^{1/3}} . \end{aligned}$$
(40)

The spin parameter of the system, which is also denoted by \(\chi _{\mathrm {f}}\) here, may be given approximately by

$$\begin{aligned} \chi _{\mathrm {f}}&= \frac{c J_{\mathrm {orb}} / G + \chi M_{\mathrm {BH}}^2}{m_0^2} \nonumber \\&= \frac{\left( G m_0 \varOmega / c^3 \right) ^{-1/3} Q + \chi Q^2}{(Q+1)^2} , \end{aligned}$$
(41)

where we assumed that the spin of the black hole is aligned with the orbital angular momentum. Because the orbital angular velocity at the onset of merger or at tidal disruption is given by \(G m_0 \varOmega / c^3 \sim 0.05\)–0.1 for a wide range of binary parameters, \(( G m_0 \varOmega / c^3 )^{-1/3}\) takes a narrow range of 2.2–2.7. Thus, as far as we may neglect the mass and the angular momentum of the remnant material and the energy carried away by gravitational radiation, \(\chi _{\mathrm {f}}\) gives the spin parameter of the remnant black hole. This expression depends primarily on the mass ratio and the initial spin parameter of the black hole, and furthermore, explains the pronounced dependence on the initial spin parameter for a high mass-ratio system.

Numerical simulations suggest that the remnant black hole is not overspun by the infall of the neutron star, respecting the cosmic censorship conjecture (Penrose 1969, 2002). Rather, fitting formulae derived in Kyutoku et al. (2011a) predict that the spin parameter should decrease during merger for nearly-extremal spins as shown at the right edge of Fig. 21. Simulations of a system with \(Q=3\) and \(\chi = 0.97\) illustrate that the spin parameter indeed decreases below 0.97 (Lovelace et al. 2013), consistently with the prediction of the fitting formulae. Detailed phenomenological models of both the mass and the spin of the remnant black hole are provided by Pannarale (2013, 2014).

The magnitude of the spin parameter of the remnant black hole decreases as the inclination angle increases for a given value of the magnitude of the spin parameter of the initial black hole (Foucart et al. 2011; Kawaguchi et al. 2015; Foucart et al. 2017). This is because the magnitude of the spin angular momentum, which is a vectorial quantity, does not increase as sizably as the mass of the remnant black hole in the presence of the inclination. The magnitude of the spin parameter can even decrease from the initial value for a large inclination angle (Kawaguchi et al. 2015; Foucart et al. 2017) in the same manner as the cases with anti-aligned spins shown in Fig. 21. The direction of the spin of the remnant black hole is approximately aligned with the total angular momentum of the system right before merger, as most of the orbital angular momentum is swallowed by the black hole.

Accretion disk, or material remaining outside the black hole

We separate discussions about the remnant disk into two parts. Here in Sect. 3.4.2, we describe dependence of the disk mass, or the mass of the remnant (see below), on binary parameters. Thermodynamic variables such as the rest-mass density of the remnant disk will be discussed later in Sect. 3.4.3.

Regarding the disk mass, early systematic surveys have rather focused on the total mass of the material remaining outside the black hole, \(M_{r>r_{\mathrm {AH}}}\), without distinguishing bound and unbound components. This quantity can be derived more accurately than the disk or bound mass, which suffers from a subtle task of determining the boundary between bound and unbound components. Furthermore, \(M_{r>r_{\mathrm {AH}}}\) is found to show clearer correlations with the neutron-star compactness than the disk or bound mass is. Thus, we base our discussions on \(M_{r>r_{\mathrm {AH}}}\).

To date, dependence of \(M_{{\mathrm{r}}>{\mathrm{r}}_{\mathrm{AH}}}\) on the spin parameter has been studied most extensively for a qualitative \(\varGamma = 2\) polytrope by several independent groups. Figure 22 plots the fractional baryonic mass of the material remaining outside the black hole obtained by three independent groups as a function of the spin parameter of the black hole for systems with \(Q=3\) and \({\mathscr {C}} \approx 0.145\) modeled by the \(\varGamma = 2\) polytrope. Care must be taken for the fact that these quantities are measured at different times in each simulation, and particularly the result for \(\chi = 0.75\) is inferred as late as \(\approx {25}\, {\hbox {ms}}\) after the onset of merger for a hypothetical value of \(M_{\mathrm {NS}} = 1.35\,M_\odot \) (Etienne et al. 2009, Fig. 13). Taking this limitation into account, the results agree approximately for \(\chi = 0\) (Etienne et al. 2009; Foucart et al. 2011; Shibata et al. 2012). Figure 22 also shows that \(M_{r>r_{\mathrm {AH}}}\) increases as the value of the spin parameter increases for \(Q=3\) and this equation of state. Specifically, the values of \(M_{r>r_{\mathrm {AH}}} / M_{\mathrm {B}}\) are approximately proportional to \(\exp ( b \chi )\) with \(b \approx 2.5\)–3 for a range displayed in Fig. 22. This enables us to reconfirm that the degree of tidal disruption depends strongly on the spin of the black hole.

Fig. 22
figure 22

Summary of the mass of the material remaining outside the black hole, \(M_{r>r_{\mathrm {AH}}}\), as a function of the spin parameter of the black hole for binaries with \(Q=3\) and \({\mathscr {C}} \approx 0.145\) modeled by a \(\varGamma = 2\) polytrope computed by three independent groups. The vertical axis shows the fraction of \(M_{r>r_{\mathrm {AH}}}\) to the baryon rest mass of the neutron star, \(M_{\mathrm {B}}\). Note that these values are measured at different times in each simulation (see also Foucart 2012). This figure is generated from data of Shibata et al. (2012); Etienne et al. (2009); Foucart et al. (2011); Lovelace et al. (2013)

Figure 23 plots \(M_{r>r_{\mathrm {AH}}}\) as a function of the spin parameter of the black hole for systems with \(M_{\mathrm {NS}} = 1.35\,M_\odot \) and \(R_{\mathrm {NS}} = {11.6}\, {\hbox {km}}\) (\({\mathscr {C}} = 0.172\)) modeled by a piecewise polytrope called HB, which approximates soft nuclear-theory-based equations of state. This figure again illustrates the importance of the black-hole spin depicted in Fig. 22. Quantitatively, \(M_{r>r_{\mathrm {AH}}}\) becomes as large as \(\approx 0.1\,M_\odot \) for \(\chi = 0.75\) even if the compactness is realistically large with this soft equation of state and the mass ratio Q is as high as 5. Various subsequent studies have further shown that large values of the spin parameter allow such significant tidal disruption to occur even for \(Q \approx 7\), which corresponds to typical masses of Galactic black holes, \(\approx 10\,M_\odot \) (Foucart et al. 2013b, 2014; Kyutoku et al. 2015). Conversely, it is highly unlikely that the neutron star with \({\mathscr {C}} \gtrsim 0.17\) is disrupted by a black hole with \(M_{\mathrm {BH}} \gtrsim 10\,M_\odot \) to leave material of \(\gtrsim 0.1\,M_\odot \) unless the spin parameter is as high as \(\chi \gtrsim 0.9\).

Fig. 23
figure 23

Mass of the material remaining outside the black hole at \({10}\, {\hbox {ms}}\) after the onset of merger as a function of the spin parameter of the black hole for a variety of the mass ratio. The mass and the radius of the neutron star are fixed, respectively, to \(M_{\mathrm {NS}} = 1.35\,M_\odot \) and \(R_{\mathrm {NS}} = {11.6}\, {\hbox {km}}\) (\({\mathscr {C}} = 0.172\)) modeled by a piecewise polytrope called HB (Read et al. 2009b). This figure is generated from data of Kyutoku et al. (2011a)

As the inclination angle of the black-hole spin increases, the degree of tidal disruption and thus the mass of the remnant disk decrease (Foucart et al. 2011, 2013b; Kawaguchi et al. 2015). Figure 24 shows contours for the mass of the bound material only (not \(M_{r>r_{\mathrm {AH}}}\)). The masses of the black hole and the neutron star are fixed to be \(M_{\mathrm {BH}} = 6.75\,M_\odot \) and \(M_{\mathrm {NS}} = 1.35\,M_\odot \), respectively, and the magnitude of the black-hole spin is fixed to be \(\chi = 0.75\). This figure shows that the mass of the bound material decreases as the inclination angle increases. A similar trend holds for the unbound material and thus for the total mass remaining outside the black hole. Quantitatively, the mass of the bound material decreases from \(0.1\,M_\odot \) (magenta) to \(0.01\,M_\odot \) (black) with the increase of the inclination angle by only \(\approx {20}^{\circ }\)\({30}^{\circ }\) for a given value of the compactness at \({\mathscr {C}} \lesssim 0.17\). It has been pointed out that the mass of the material remaining outside the black hole with inclined spins is approximately reproduced by a model with an aligned black-hole spin whose magnitude derives the same radius of the innermost stable (circular or spherical) orbits as that of the original configuration (Foucart et al. 2013b; Stone et al. 2013) or, more simply, by a model with an aligned spin whose dimensionless magnitude is \(\chi \cos \iota \) (Kawaguchi et al. 2015). The effect of the spin orientation has not been explored systematically over the parameter space, and this is a subject for future study. Figure 24 also shows that the mass of the bound material decreases as the compactness of the neutron star increases. We discuss the dependence of \(M_{r>r_{\mathrm {AH}}}\) on the neutron-star compactness below, going back to the aligned-spin systems.

Fig. 24
figure 24

Contour for the mass of the bound material (denoted by \(M_{\mathrm {disk}}\) in this figure) at \({10}\, {\hbox {ms}}\) after the onset of merger on the compactness-inclination angle (denoted by \(i_{\mathrm {tilt}}\) in this figure) plane for binaries with \(M_{\mathrm {BH}} = 6.75\,M_\odot \), \(\chi = 0.75\), and \(M_{\mathrm {NS}} = 1.35\,M_\odot \) (\(Q=5\)). The mass of the unbound material is excluded. Image reproduced with permission from Kawaguchi et al. (2015), copyright by APS

Figure 25 plots \(M_{r>r_{\mathrm {AH}}}\) as a function of the compactness for \(Q=2\) (left) and \(Q=3\) (right). The mass of the neutron star is fixed to be \(M_{\mathrm {NS}} = 1.35\,M_\odot \), and the compactness is varied by adopting a one-parameter family of piecewise polytropes (see Kyutoku et al. 2011a for the details). The values of the spin parameter are also varied systematically. This figure shows that \(M_{r>r_{\mathrm {AH}}}\) decreases approximately linearly as the compactness increases until the value decreases to \(\lesssim 0.01\,M_\odot \). This trend holds irrespective of the values of Q or \(\chi \). Figure 25 also shows that \(M_{r>r_{\mathrm {AH}}}\) increases steeply as the spin parameter of the black hole increases as already seen in Figs. 22 and 23. Quantitatively, \(M_{r>r_{\mathrm {AH}}} \gtrsim 0.1\,M_\odot \) is achieved for a wide range of compactness, \({\mathscr {C}} \lesssim 0.18\), if the black-hole spin is as high as 0.5 for \(Q=2\) and 0.75 for \(Q=3\).

Fig. 25
figure 25

Mass of the material remaining outside the black hole at \({10}\, {\hbox {ms}}\) after the onset of merger for binaries with \(M_{\mathrm {NS}} = 1.35\,M_\odot \) as a function of the compactness, which is varied by adopting a one-parameter family of piecewise polytropes. The spin parameter of the black hole is also varied. The left and right panels show the results of systems with \(Q=2\) and 3, respectively. Image adapted from Kyutoku et al. (2011a), copyright by APS

Although the primary effect of the equation of state is captured by the compactness of the neutron star, the density profile also affects the susceptibility to tidal disruption. A comparison performed using a two-parameter family of piecewise polytropes shows that the mass of the material remaining outside the black hole decreases by more than a factor of 2 for a centrally-condensed profile of the neutron star even if the compactness is approximately the same (Kyutoku et al. 2010, 2011b). This is because the central condensation tends to suppress the degree of tidal deformation and thus tidal disruption is appreciably delayed from the onset of mass shedding. We note that it has been pointed out that the correlation of \(M_{r>r_{\mathrm {AH}}}\) with the tidal deformability, \(\varLambda \), is not stronger than that with the compactness (Foucart 2012; Foucart et al. 2018). Thus, neither the compactness nor the tidal deformability is fully appropriate to determine the amount of material remaining outside the black hole.

Figure 26 shows the dependence of \(M_{r>r_{\mathrm {AH}}}\) on the neutron-star compactness from another perspective for fixed values of \(\chi = 0.75\) (left) and \(\chi = 0.5\) (right). The mass of the neutron star is fixed to be \(M_{\mathrm {NS}} = 1.35\,M_\odot \). Again, it is found that \(M_{r>r_{\mathrm {AH}}}\) increases approximately linearly as the compactness decreases irrespective of Q or \(\chi \). One notable feature that becomes clearly visible in this plot, although presaged in previous figures, is that the dependence of \(M_{r>r_{\mathrm {AH}}}\) on the mass ratio, Q, becomes weak or even inverted from naive expectations at the small compactness of \({\mathscr {C}} \lesssim 0.15\). Analytic estimation in Sect. 1.3 suggests that the degree of tidal disruption and thus \(M_{r>r_{\mathrm {AH}}}\) is likely to decrease as the black hole becomes massive, i.e., as Q becomes high. This indeed holds for systems with a large neutron-star compactness, \({\mathscr {C}} \gtrsim 0.15\), but does not for \({\mathscr {C}} \lesssim 0.15\). This fact suggests that the dependence on the mass ratio is worth investigating in detail.

Fig. 26
figure 26

Same as Fig. 25 but for \(\chi = 0.75\) (left) and \(\chi = 0.5\) (right). The mass ratio is varied in each plot. Image adapted from Kyutoku et al. (2011a), copyright by APS

The dependence of \(M_{r>r_{\mathrm {AH}}}\) on the mass ratio, Q, has recently been found to become very weak for very-low-mass and nonspinning black holes (Hayashi et al. 2021). Figure 27 shows \(M_{r>r_{\mathrm {AH}}}\) as a function of Q for nonspinning black hole–neutron star binaries with various equations of state. The mass of the neutron star is fixed to be \(M_{\mathrm {NS}} = 1.35\,M_\odot \). It is clearly shown that the value of \(M_{r>r_{\mathrm {AH}}}\) levels off at 0.05–\(0.1\,M_\odot \) for \(Q \lesssim 2\)–3, where their precise values depend on the equation of state (and presumably the black-hole spin). We note that, as we discuss later in Sect. 3.4.5, \(M_{r>r_{\mathrm {AH}}}\) is approximately identical to the mass of the remnant disk at this low-Q saturation regime. In fact, such disappearance of the dependence on binary parameters may be observed in various regions of the parameter space for which the degree of tidal disruption increases (Brege et al. 2018; Hayashi et al. 2021) as is also suggested by Fig. 26. By what and how the saturation values of \(M_{r>r_{\mathrm {AH}}}\) are determined have not been understood yet and may be counted as one of the unsolved problems about black hole–neutron star binaries.

Fig. 27
figure 27

Mass of the material remaining outside the black hole at \({12}\, {\hbox {ms}}\) after the onset of merger as a function of the mass ratio for binaries with \(\chi = 0\) and \(M_{\mathrm {NS}} = 1.35\,M_\odot \). Equations of state are varied by adopting a one-parameter family of piecewise polytropes called HB, H, and 1.25H (Read et al. 2009b; Lackey et al. 2012). Symbols with different sizes show the result obtained with different grid resolutions and are not visible for \(M_{r>r_{\mathrm {AH}}} \gtrsim 0.01\,M_\odot \) on the scale of this figure. Image reproduced with permission from Hayashi et al. (2021), copyright by APS

Fitting formulae for the mass of the material remaining outside the black hole are provided by Foucart (2012); Foucart et al. (2018). These fitting formulae are especially useful for deriving an approximate criterion for tidal disruption. Figure 28 displays the contour above which more than \(1\%\) of the baryon rest mass of the neutron star is left outside the apparent horizon at \({10}\, {\hbox {ms}}\) after the onset of merger for a given value of the neutron-star compactness (denoted by C in this figure) on the Q-\(\chi \) plane adopting a formula of Foucart et al. (2018). This formula is claimed to be accurate within \(\sim 15\%\) for \(1 \le Q \le 7\) and \(-0.5 \le \chi \le 0.9\) for the case in which less than 30% of the baryon rest mass remains outside the apparent horizon. Because the equation of state is uncertain and the mass of the neutron star is different among realistic binaries, we draw contours for various values of the compactness, \({\mathscr {C}}\), which is the only finite-size effect of the neutron star taken into account in the formula adopted here (see Foucart et al. 2018 for another formula based on tidal deformability). This figure clearly shows that high prograde spins enable tidal disruption to occur outside the innermost stable circular orbit of massive black holes for a wide range of the neutron-star compactness. It should be cautioned that, however, all the previous simulations of black hole–neutron star binaries have not systematically varied masses of the neutron stars except for a scale-free, qualitative \(\varGamma =2\) polytrope. Thus, predictions for \(M_{\mathrm {NS}} \lesssim 1.2\,M_\odot \) or \(M_{\mathrm {NS}} \gtrsim 1.5\,M_\odot \) need to be taken with particular care.

Fig. 28
figure 28

Criterion for tidal disruption predicted by the fitting formula of Foucart et al. (2018) for various compactnesses of neutron stars denoted by C. Here, the criterion is defined by the condition that more than 1% of the baryon rest mass of the neutron star is left outside the apparent horizon at \({10}\, {\hbox {ms}}\) after the onset of merger. The baryon rest mass increases toward the left top region of this Q-\(\chi \) plane for a given value of the compactness

Thermodynamic properties of the disk

Continuing the discussions in Sect. 3.4.2, we summarize thermodynamic properties of the remnant disk. For a given value of the mass, the structure and time evolution of the remnant disk depend primarily on the total mass of the system, \(m_0\). This is because the length scale of the system after merger is proportional to the mass of the remnant black hole, which agrees approximately with the total mass as we discussed in Sect. 3.4.1. If we focus on systems with a fixed value of \(M_{\mathrm {NS}}\), the total mass of the system is controlled by the mass ratio as \(m_0 = (1+Q) M_{\mathrm {NS}}\).

Figure 29 displays the profiles of the rest-mass density of the remnant disk on the equatorial plane for systems with \(\chi = 0.75\), \(M_{\mathrm {NS}} = 1.35\,M_\odot \), \(R_{\mathrm {NS}} = {11.6}\, {\hbox {km}}\) (\({\mathscr {C}} = 0.172\)) modeled by a piecewise polytrope called HB but with different values of \(M_{\mathrm {BH}}\) (Kyutoku et al. 2011a). This figure shows that the rest-mass density in the inner region is systematically higher for a smaller value of the total mass. Quantitatively, the maximum rest-mass density is \(\approx {10^{13}} \hbox {gcm}^{-3}\) for \(m_0 = 4.05\,M_\odot \) (\(Q=2\)) and \(\approx {10^{11}} \hbox {gcm}^{-3}\) for \(m_0 = 8.1\,M_\odot \) (\(Q=5\)). This difference cannot be ascribed to the difference in the mass of the material, because \(M_{r>r_{\mathrm {AH}}}\) varies only by a factor of \(\approx 2\) among the systems presented in this figure. Instead, reflecting the fact that the typical length scale such as the radius of the innermost stable circular orbit is smaller for a smaller value of \(m_0\), the remnant disk is spatially more concentrated and the rest-mass density is increased. Conversely, low-density regions with \(\rho \lesssim {10^{10}} \hbox {gcm}^{-3}\) extend to a more distant region for larger values of \(m_0\). Thus, the density gradient becomes shallow as the total mass increases.

Fig. 29
figure 29

Profile of the rest-mass density and the location of the apparent horizon on the equatorial plane at \({10}\, {\hbox {ms}}\) after the onset of merger for binaries with \(\chi = 0.75\), \(M_{\mathrm {NS}} = 1.35\,M_\odot \), and \(R_{\mathrm {NS}} = {11.6}\, {\hbox {km}}\) (\({\mathscr {C}}=0.172\)) modeled by a piecewise polytrope called HB (Read et al.