In the framework of Newtonian gravity, a lot of effort was devoted to qualitatively determining the degree of the tidal deformation, its effects on the orbital motion, and the criterion of mass shedding [71, 111, 114, 213, 191, 217, 225, 94] as reviewed in Section 1.1. Although those approximate analyses are valuable for understanding the qualitative nature of coalescing close binaries, they cannot be appropriate for a quantitatively strict understanding, because coalescing BH-NS binaries in close orbits are in a highly relativistic state. A fully general-relativistic study is obviously required.
Motivated by this fact, numerical computations of quasi-equilibrium states in general relativity have been performed in the past 5 years by several groups, after early attempts, which were done in an approximate formulation of extreme mass ratios [19, 207] or in a preliminary formulation . The first results in general relativity were published in 2006 by three groups: Taniguchi and collaborators , Grandclément [82, 83], and Shibata and Uryū [202, 203]. All these works solved the Hamiltonian and momentum constraint equations, and some components of Einstein’s equation (see Section 2 in detail). The first two groups employed the excision formulation (in which the region inside the BH apparent horizon is excised from the computational domain) and the last one employed the moving-puncture formulation. However, those early-stage results were unsatisfactory for accurately studying quasi-equilibrium sequences. Taniguchi and collaborators treated only mildly-relativistic NS and the numerical computation was not very accurate. The numerical code by Grandclément included some mistakes and the results were not correct. Shibata and Uryū constructed BH-NS binaries only including the corotating motion for the internal velocity field of the NS. However, in subsequent work, this situation was soon improved. Taniguchi and collaborators succeeded in accurately computing BH-NS binaries in quasi-equilibrium  and investigated their nature in detail ; Grandclément corrected his numerical code , and derived results as accurate as those of Taniguchi et al.; Kyutoku et al. succeeded in constructing BH-NS binaries with the irrotational velocity field for the NS [197, 106] developing a formulation originally proposed by Shibata and Uryū [202, 203]. All these groups computed sequences of quasi-equilibrium states and studied the nature of BH-NS binaries in close circular orbits (see Section 2).
Except for the early work of Shibata and Uryū [202, 203], all the computations have been done based on the spectral methods library, LORENE , because this enables high-precision computation, e.g., [27, 28, 84, 85]. (Note also that Grandclément and Taniguchi are two of the main developers of the LORENE library).
The Cornell–Caltech group also developed a numerical code, based on a spectral method for the precise computation of BH-NS binaries in quasi-equilibrium states , extending their original code to BH-BH binaries [158, 47, 40]. Up to now, this group has published quasi-equilibrium sequences only for the equal-mass case, which is not very realistic for BH-NS binaries, although their results for a variety of BH-NS binaries have been used as initial conditions for numerical-relativity simulations [58, 57, 74].
There are also two additional works on BH-BH/BH-NS binaries in quasi-equilibrium [216, 8]. However, these works primarily discuss a numerical method for computing BH binaries in quasi-equilibrium, and do not shown any computational results for BH-NS binaries. Hence, we do not review them in this article.
Living Rev. Relativity 14, (2011), 6
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