BH-NS binaries have not been observed yet even in our galaxy in contrast to NS-NS binaries [205, 131].
However, many of statistical studies based on the stellar evolution synthesis suggest that the coalescence
will occur by 1 – 10% as frequently as that of NS-NS binaries in our galaxy and hence in the normal
spiral galaxies [144, 160, 223, 98, 97, 20, 21, 148] (every 10^{6} – 10^{7} years). In addition,
coalescence in elliptic galaxies could contribute to the total coalescence rate of the universe by a
significant fraction [147]. This implies that coalescence is likely to occur frequently in the Hubble
volume, and therefore, the evolution process and the final fate of BH-NS binaries deserve a
detailed theoretical study. In particular, the following two facts have recently enhanced the
motivation for the study of BH-NS binaries: First, BH-NS binaries in close orbits are among
the most promising sources for the large laser-interferometric gravitational-wave detectors,
such as LIGO [126, 2, 1], VIRGO [222, 4, 3], LCGT [120], and Einstein Telescope [91, 92]:
The frequency and amplitude of gravitational waves near the last orbit are estimated to give

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

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

where is the circumferential radius of the NS at . is a function of and an EOS-dependent parameter, which is larger than unity and denotes a degree of tidal deformation of the NS, i.e., the semi-major axis is assumed to be elongated as . The left-hand side denotes the tidal force by the BH at the surface of the NS and the right-hand side is the self-gravitational force of the NS at the inner edge of its surface.We emphasize here that Equation (5) is the necessary condition for the onset of mass shedding, strictly speaking. Tidal disruption occurs after substantial mass is stripped from the surface of the NS, during the decrease of the orbital separation due to gravitational-wave emission. Thus, tidal disruption should occur for a smaller orbital separation (larger orbital angular velocity) than that derived from Equation (5). We also note that the NS radius is assumed to depend weakly on the NS mass. If the radius quickly increases with mass loss, tidal disruption may occur soon after the onset of mass shedding.

Assuming that the binary is in a circular orbit with the Keplerian angular velocity, Equation (5) may be written in terms of the angular velocity as

where denotes the mass ratio defined by . The latest general-relativistic numerical study for quasi-equilibrium states of BH-NS binaries derives a more quantitative result [209, 210] (see Section 2), where the value of the constant, , is for a system composed of a NS of irrotational velocity field and a non-spinning BH. The small value of indicates that mass shedding is enhanced by significant tidal deformation (i.e., ) and/or possibly by general relativistic gravity of the NS. Equation (7) indicates that the frequency of gravitational waves at mass shedding is Thus, tidal disruption is likely to occur for a high frequency (0.7) kHz for a NS of and (15) km, irrespective of the BH mass.Up to now, we implicitly assume that orbits with arbitrary orbital separations may be possible for the binary system. However, BH-NS binaries always have the innermost stable circular orbit (ISCO) determined by the general relativistic effect, and hence, we should impose the condition that mass shedding (and tidal disruption) has to occur before the binary orbit reaches the ISCO in this analysis. According to PN analysis [25], a non-dimensional orbital compactness parameter, , at the ISCO, is for for a system composed of a non-spinning BH and a NS. (The tidal-deformation effect reduces this value by 10 – 20% [209, 210]). In the presence of the ISCO, the condition for the onset of mass shedding is written by

where is the angular velocity at the ISCO, and for a system composed of non-spinning BH (for a spinning BH, this could be much larger than 0.1; see Section 1.2). Equation (9) is rewritten to give where denotes the compactness of the NS, which will be, according to nuclear physics theories for high-density matter, in the range between and [115, 116] for the typical NS of mass . This estimate suggests that tidal disruption by a non-spinning BH could occur for a binary of a relatively low mass ratio of . Equation (10) also shows that the conditions for the onset of mass shedding and for tidal disruption depend strongly on the compactness of the NS.In the above simple estimate, the tidal-deformation effect of the NS to the orbital motion is not taken into account. As a result of the tidal deformation, the gravitational force between two stars is modified, so are the orbital evolution and the criterion for tidal disruption. Lai, Rasio, and Shapiro [111, 113, 112] thoroughly investigated the effects associated with tidal deformation in the Newtonian framework with the ellipsoidal approximation. They found that the two-body attractive force is strengthen by the effect of the tidal deformation and, by this, the orbital separation of the ISCO is increased (see [165, 166, 114, 190] for related issues) and that gravitational waveforms in the late inspiral phase are modified (see [66, 68, 67, 72] for related issues). Uryū and Eriguchi confirmed the fact that the tidal force modifies the orbital motion of BH-NS binaries by numerically computing equilibrium states of a binary composed of a point mass and a fluid star. The essence is that in addition to the Newtonian potential, which has the form , the correction term of the form appears when a NS is tidally deformed. The magnitude of this correction increases steeply with the decrease of the orbital separation, and modifies the location of the ISCO. Also, the tidal effect accelerates the orbital evolution, because the orbital velocity (and the centrifugal force) has to be increased to maintain circular orbits in an enhanced attractive force, and then, the emissivity of gravitational waves is enhanced, leading to a shorter inspiral time.

Living Rev. Relativity 14, (2011), 6
http://www.livingreviews.org/lrr-2011-6 |
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